Joint Discrete–Continuous Flow Matching for Open-Vocabulary Inverse Design of Multilayer Optical Coatings Zhiyi Li1,2 , Yuheng Jin1,2 , Yidan Huang1,2 , Nan Chen1,2 , Hongyan Fu1,2 , Yikun Bu1,2* 1
arXiv:2607.08392v1 [physics.optics] 9 Jul 2026
School of Electronic Science and Engineering (National Model Microelectronics College), Xiamen University, Fujian, 361005, China. 2 Fujian Key Laboratory of Ultrafast Laser Technology and Applications, Xiamen University, Xiamen, 361005, China.
*Corresponding author(s). E-mail(s): [email protected]; Abstract Amortized neural inverse design typically remains closed-world: component choices are fixed vocabulary tokens, coordinate grids are frozen at training time, and continuous variables are discretized into sequence tokens. Multilayer optical coatings are an industrially important instance, coupling material sequence, layer thickness and wavelength-dependent response. We present IrisFlow, a querybased, open-vocabulary flow-matching framework instantiated in coatings: the target reflectance/transmittance spectrum, wavelength grid, candidate-material optical constants and layer count are supplied at query time. Candidate materials enter as wavelength-aware optical tokens rather than learned identities; material sequences are sampled by discrete flow matching over the query’s candidate bank, thicknesses by continuous flow matching without discretization. A single 136M-parameter model designs 2–100-layer stacks. Across a 224-task benchmark it reconstructs in-distribution targets faithfully and retains same-order accuracy on a 15-material held-out bank without retraining; it reconstructs bands up to 1100 nm beyond its training envelope, designs against analytic application specifications and outperforms an autoregressive baseline on that baseline’s material library. With optical constants calibrated to our deposition process, IrisFlow designs four color-displaying coolers, fabricated by ion-assisted evaporation: the three chromatic devices reach a CIEDE2000 color error of 3.1–5.2 while retaining 93–95% solar near-infrared reflectance, demonstrating open-vocabulary design carried through to fabricated coatings.
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Keywords: query-based inverse design, open-vocabulary models, flow matching, joint discrete–continuous flow matching, multilayer optical coatings, photonics
1 Introduction Amortized inverse design is usually framed as learning a fast map from a desired response to a structure, but in deployment the limiting object is often the interface: which coordinates, components and constraints are fixed into the model, and which can be supplied when a new design is requested. Many physical design tasks are coupled discrete–continuous problems. A designer must choose components from a local bank and tune continuous parameters around those choices; the component bank, coordinate support and fabrication envelope themselves change from one request to the next. If these objects are baked into a neural model as labels, fixed grids or discrete bins, inference becomes fast only inside the closed world seen at training time. Multilayer optical coatings make this interface problem concrete. They are foundational components in imaging systems, displays, photovoltaics, laser cavities, thermal emitters, reconfigurable photonics and chemical sensing, yet their design rarely reduces to a single scalar objective. A useful coating is described by a spectral response over one or more wavelength intervals, a material availability list, a fabrication envelope for layer thicknesses, a permitted layer-count range and application-specific trade-offs among reflectance, transmittance and absorption. The forward problem admits an efficient deterministic solution: the transfer-matrix method (TMM) computes spectra exactly for arbitrary stacks [1]. The inverse problem is much harder. The design space mixes discrete material choices, continuous thicknesses and variable layer counts in a highly non-convex landscape; the same target spectrum can be realized by many physically distinct stacks; and small perturbations to either the spectral target or the available material bank can move the optimum to an entirely different region of the search space. Classical tools such as the needle method [2], gradient-based refinement [3], genetic algorithms (GA) [4], particle-swarm optimization (PSO) [5] and differential evolution (DE) [6] remain valuable because they optimize the TMM simulator directly and respect physical constraints by construction. However, they amortize no experience across tasks (each new target is a fresh optimization run), and they scale poorly with large material banks: local searches (the needle method, gradient refinement) become trapped in nearby minima, population-based metaheuristics pay for broader exploration with thousands of forward simulations per design, and the returned design depends heavily on the initial seed, the search budget and operator expertise. Neural inverse-design methods reduce inference time by learning the mapping from spectra to structures [7–10]. Early convolutional and tandem networks [11, 12], mixture-density network (MDN) [13], conditional variational autoencoder (cVAE) [14] and conditional invertible neural network (cINN) [15] families all address the one-to-many nature of the inverse map. More recent sequence models, including OptoGPT-style autoregressive Transformers [16] and masked diffusion language models [17, 18], naturally handle
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variable-length designs by tokenizing material and thickness sequences, and deep reinforcement learning likewise assembles multilayer stacks layer by layer from a fixed material menu [19, 20]. For coatings, three closed-world assumptions are built into the model architecture and surface as practical barriers when a design is taken from a paper into a fabrication facility. First, a material is represented by a learned categorical token or one-hot identity (ID), which makes the model closed-vocabulary by construction. Thin-film design is often driven by a small set of materials available in a particular facility, a newly measured or computationally screened dispersion curve, or a per-project material bank that changes between jobs. Even when a candidate’s nominal refractive index n(λ) and extinction coefficient k (λ) are tabulated in a handbook, the actual n, k of a deposited film is not uniquely determined by the material name: it depends on deposition technique, rate, temperature, stoichiometry and post-treatment, so each chamber produces its own dispersion curve. A model that only learns material IDs has no way to ingest the measured n, k of a film as deposited in a specific chamber. Second, the wavelength grid is fixed at training time, so the model is wavelength-rigid: it cannot directly handle user-specified sub-bands, stitched non-contiguous passbands or measurements taken on a different grid. Spectral content outside the user’s region of interest still consumes model capacity and can pull the optimum away from the intended working range. Third, the modern sequence and diffusion architectures that have made variable layer counts tractable gain this flexibility by tokenizing thickness alongside materials. The resulting discretization inflates the vocabulary, slows sampling and forces the model to commit to a quantization step that may straddle the spectral optimum rather than land on it. Variable-length design thus comes at the cost of continuous control over thickness. Factored decoders and in-decoder thickness regression reduce these costs [21, 22], but the material vocabulary is still fixed, so the closed world remains. IrisFlow is designed around a query interface rather than a fixed output vocabulary (Fig. 1b–d). In the coating instantiation, the user supplies four inputs at inference time: a target R(λ), T (λ) spectrum, an aligned wavelength grid, a candidate bank of material n(λ), k (λ) curves and a requested layer count. The model returns a material sequence drawn from the candidate bank and a continuous thickness vector. Instead of referencing materials through a fixed vocabulary, IrisFlow injects every candidate material directly into the denoiser as a wavelength-aware optical token. Each candidate’s n(λ), k (λ) curve is paired with the query wavelength array and tokenized into a common token space with the target spectrum, so wavelength alignment is preserved end-to-end and the model reasons about how each candidate behaves on the user’s grid. The denoiser is conditioned on a target-spectrum token together with this candidate memory, and the material head scores each layer against the local candidate bank. Because the coating instantiation mixes continuous thickness variables with discrete material variables, IrisFlow couples two generative processes under a shared flow time and a shared denoising backbone. Thicknesses are modeled by continuous flow matching [23, 24] with a velocity (v ) target, which keeps thickness in its native continuous space and allows arbitrarily fine placement near a spectral optimum. Material
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Fig. 1 The IrisFlow query-interface pattern for coupled discrete–continuous inverse design. a, Conventional neural inverse design is closed-vocabulary (materials are fixed ID tokens) and wavelength-rigid, so a newly measured material cannot enter without retraining. IrisFlow instead (b) takes a target R(λ), T (λ) spectrum on user-specified query bands, a local candidate bank of material n, k curves on an aligned wavelength grid, and a requested layer count; (c) encodes the target and every candidate into a common, wavelength-aware token representation and jointly denoises a material sequence and continuous thicknesses, with each layer scored against the local candidate bank rather than a global class table; and (d) returns diverse TMM-verified stacks. Introducing a new material amounts to supplying its n, k curve on the query grid, with no new class index and no retraining.
stacks are modeled by discrete flow matching (DFM) on a continuous-time Markov chain (CTMC) [25–27] over the local candidate bank, with a uniform transition kernel that injects no a priori similarity bias between candidates. The model instead learns a physically meaningful material representation directly from the supplied n, k curves, in which candidates organize by optical family with no family labels supplied (Methods; Appendix §Q). Together, these design choices turn multilayer coatings into a stress test of three interface principles. First, component choices are open-vocabulary: candidate materials are supplied as optical functions. We validate this on a held-out material bank spanning four chemical families; IrisFlow generates designs whose out-of-distribution (OOD) spectrum root-mean-square error (RMSE) remains on the same order as the matchedcell in-distribution reference. Second, coordinates are query-conditioned. Targets and candidates share the user’s per-query wavelength grid, so arbitrary sub-bands, stitched non-contiguous passbands and wavelength windows restricted to a user-defined region of interest are handled directly: spectral content outside that region is simply absent from the constraint. Third, continuous design variables remain continuous while being generated jointly with discrete choices. Continuous flow matching for thickness, paired with rotary position embedding (RoPE) over layer indices, lets the model emit any number of layers up to Lmax while keeping thickness on a continuous axis. These three properties relax the closed-world constraints that limit the deployment of neural inverse-design systems in fabrication workflows.
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Beyond these three axes, the same query interface yields two further capabilities with no change to the model: any subset of the per-layer material and thickness state can be pinned while the rest is completed (constrained and partial-structure generation), and oblique-incidence and polarized targets are handled by a proxy reparameterization of the candidate bank followed by exact angled-TMM validation. Finally, we close the loop to fabricated hardware: four color-displaying coolers, designed against process-corrected optical constants and built by ion-assisted evaporation. Real fabrication imposes a constraint the benchmark does not: the best-scoring design is not always the most practical to deposit. The diversity of IrisFlow’s TMM-verified outputs supplies a fabrication-ready alternative directly.
2 Results The Results stress-test the query objects that define the IrisFlow interface in multilayer optical coatings: layer count, wavelength support, candidate bank, target spectrum, incidence angle and polarization, and, finally, process-calibrated optical constants. Throughout, this is a single-model test: every result below uses the same Lmax = 100 checkpoint, with no per-task retraining, re-tokenization or vocabulary change, and every generated design is verified by exact TMM re-simulation. We separate these claims with four evaluation suites: an in-distribution layer×wavelength grid (Tier 1), the same grid rebuilt from held-out materials (Tier 2), idealized analytic application targets (Tier 3) and wavelength queries beyond the training envelope (Tier 4). Together they span 224 tasks and 17,152 reported designs, scored by the combined R, T spectral RMSE defined in Methods and aggregated as described in Appendix §G; sampling budgets, cell construction and suite definitions are given in Appendix §F. Beyond these simulation suites, we add four targeted practicaluse tests: a head-to-head comparison with OptoGPT, partial-stack queries that clamp known structure or thicknesses, training-free oblique-incidence and polarized design, and a process-corrected fabrication loop for color-displaying coolers.
2.1 One model spans 2–100 layers and the full spectral range A single IrisFlow model covers the entire 2–100-layer, full-spectrum design space, without an ensemble specialized per layer count or spectral band. Across the in-distribution grid it reconstructs targets to a median combined R, T RMSE of 4.6 × 10−2 (percell range 9.5 × 10−3 to 1.1 × 10−1 , full-bank cells). Figure 2a maps this fidelity over layer count and spectral band, while Fig. 2d, e show representative two-channel reconstructions; the full grid and gallery are in Appendix §H (Table H1, Fig. H2). Two physically expected trends emerge, both mild enough that the one model stays accurate throughout (marginal curves in Fig. H1, Appendix §H). Fidelity degrades only gracefully with layer count, from 1.2 × 10−2 at 2–5 layers to 9.3 × 10−2 at 81–100 layers (pooled over bands); this is the signature of an increasingly degenerate inverse problem at high L, yet the error stays below 10−1 even at the deepest stacks (Appendix Table H1). It varies even less with spectral position: best in the NIR (3.8 × 10−2 ) and extended-NIR (4.4 × 10−2 ), weakest in the short-wave UV–VIS band (5.7 × 10−2 ) where steeper dispersion and the onset of material absorption near the UV edge make 5
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Fig. 2 Benchmark fidelity, reconstruction gallery and OptoGPT comparison examples. a,b, Per-cell median combined R, T spectral RMSE across 7 layer-count bins × 6 spectral bands for (a) Tier 1 (in-distribution materials) and (b) Tier 2 (OOD material bank); cell text is RMSE ×102 and the heatmaps share one color scale. Band abbreviations on the x axis denote UV–VIS (ultraviolet–visible, 380–550 nm), VIS (visible, 400–700 nm), VIS–NIR (visible–near-infrared, 500– 900 nm), NIR (near-infrared, 800–1100 nm), eNIR (extended near-infrared, 1000–1400 nm) and DUAL (dual visible+near-infrared). c, Matched-cell OOD/in-distribution error ratio, plotting Tier 2 against Tier 1 median RMSE; color encodes layer bin, marker shape the candidate-bank mode, and guides mark parity, 2× and 3×; the ratio is a matched-cell distribution summary, not a same-target retention factor (§G). d–k, Reconstruction gallery, two horizontally grouped examples per tier: Tier 1 (d, e), Tier 2 OOD bank (f, g), Tier 3 applications (h, i) and Tier 4 wavelength extrapolation (j, k). Target R, T are solid and IrisFlow re-simulation dashed (R red, T blue); gray marks no-target regions or wavelengths beyond the 1400 nm training edge. The main-text gallery chooses several random cases; full galleries are in Appendices §H–J. l–o, Compact OptoGPT head-to-head examples: target black, IrisFlow 20-block green and OptoGPT saved output red; solid/dashed curves denote R/T . Target specifications taken from the OptoGPT paper are marked “(OptoGPT)” in the panel titles (m– o); panels l and o are both near-infrared high-reflection targets, of which l is the single case where OptoGPT leads. Per-panel RMSE values are the shared dense-grid scores of the OptoGPT head-tohead protocol (Appendix §K, Table K1).
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targets harder, with the dual-band case (4.6 × 10−2 ) handled on par with the single bands. The interface is likewise indifferent to bank size: restricting the candidate bank to only the materials a target needs (plus three distractors), versus offering the full 15-material bank, leaves fidelity practically unchanged (4.61 × 10−2 vs. 4.63 × 10−2 median RMSE).
2.2 New materials enter at query time without retraining The defining test of the query interface is open-vocabulary material generalization: a material the model has never seen should be usable the moment its optical curve is supplied. We test this on a 15-material bank held out from training (five metals, four high-index oxides, four transparent conductors and two polymers; Appendix Table F1) that rebuilds the in-distribution grid cell-for-cell (Tier 2), each held-out material entering at inference only through its n(λ), k (λ) curve on the query grid, with no material-ID embedding and no retraining (construction in Appendix §F.1; exact optical constants in §F.6). The model already carries the structure this requires: probing its n, k encoder shows candidates clustering by optical family with no family labels supplied, and the 15 held-out curves landing in the corresponding trainingmaterial neighborhoods rather than forming new classes (full diagnostic in Appendix §Q, Fig. Q1a). The cost of going open-vocabulary is small. On the held-out bank IrisFlow holds a median combined RMSE of 7.2 × 10−2 against 4.6 × 10−2 on the matched-cell indistribution reference; cell by cell, the OOD/in-distribution ratio has a median of 1.7× (interquartile range 1.3–2.3×), with 90% of cells within 3×. This is a matched-cell distribution summary, not the error from a same-target material swap (Fig. 2c). Even that modest ratio is concentrated where it matters least: the largest ratios fall only where the in-distribution baseline is already vanishingly small (the shallow 2–5-layer cells, in-distribution RMSE ∼ 10−2 ), and the absolute OOD RMSE never exceeds 0.18 on any cell. Error does not track how far a held-out curve sits from the training vocabulary. The bank ranges from near-duplicates of training curves (HfO2 and poly(methyl methacrylate) (PMMA)) to genuinely novel dispersion (Al’s nearest training neighbor is the very different Ag), yet per-material within-cell error shows no detectable relationship with curve-space novelty (Spearman ρ ≈ 0 over the 15 materials); it instead follows the same optical-role pattern as in distribution, with transparent high-index materials hardest both in distribution (TiO2 ) and OOD (indium tin oxide, ITO) and the most novel curves (Al, Nb) among the easiest (Appendix §F.7; representative OOD spectra, including the deep metal/oxide stacks the bank produces, in Fig. 2f, g and Appendix §H, Fig. H3).
2.3 IrisFlow reaches idealized application spectra across nine target families Tier 3 changes the inverse-design question. The other three suites build every target from a known TMM stack, so their spectra are reachable by construction, whether in-distribution (Tier 1), from a held-out material bank (Tier 2) or at out-of-range wavelengths (Tier 4). Tier 3 instead gives idealized analytic application specifications
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whose exact attainability under the allowed materials, thickness window and layer budgets is not assumed. Across 46 such cases spanning nine application families, IrisFlow reaches a median per-case best combined RMSE of 3.9 × 10−2 and reproduces the qualitative behavior of every family in the one-case-per-family showcase (representative cases Fig. 2h, i; full showcase Appendix §I, Fig. I1). The cases cover antireflection (AR), Gaussian and flat-top bandpass (BP and BP-FLAT), notch/edge filter (NF), mirror (MR), solar-selective emitter (SE), structural color (SC), extreme/stress (EX) and generic spectral shapes (S-*). Each is specified as a full two-channel R, T target (Appendix §F.2). Against a retrieval proxy for the oracle (the closest of all 114M training spectra to each target), IrisFlow wins on 39 of the 47 audited target curves (AR-5 contributes two variants) and roughly halves the median spectral distance to the specification, improving on the best design its training corpus offers rather than memorizing it (Appendix §F.8, Fig. F4). Per-family and per-case spectral distances to these specifications are summarized in Table 1, Fig. I2 and Appendix Table I1. The family ordering tracks the optical difficulty of the specifications. Smooth and broad-featured targets are matched best: AR (7.2 × 10−3 RMSE), generic spectral shapes (2.5 × 10−2 ) and notch/edge targets (3.2 × 10−2 ). Families dominated by narrow, high-contrast transmission features are hardest: Gaussian and flat-top bandpass targets sit at 8.2 × 10−2 –8.6 × 10−2 median best RMSE, where a small wavelength offset in a sharp passband produces a large spectral penalty. Tier 3 thus complements the reachable-target reconstruction tests of Tiers 1, 2 and 4. Table 1 Application-family spectral-fidelity scorecard. Tier 3 targets are analytic application specifications. The RMSE column reports the median over each family’s cases of the per-case best combined RMSE (§I).
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broad low-R bands 0.0072 narrow high-T peaks 0.0859 high-T plateaus 0.0818 stop/pass transitions 0.0318 high-R mirrors / splitters 0.0685 solar-selective spectra 0.0456 color-derived spectra 0.0491 stress specifications 0.0306 generic analytic shapes 0.0254
2.4 Wavelength bands outside the training envelope are handled as queries Because the wavelength grid is itself part of the query, the same model generalizes beyond the wavelengths it was trained on. Trained only on [380, 1400] nm, IrisFlow is asked to design for bands reaching out to 2500 nm: pure out-of-range NIR/short-waveinfrared (SWIR) bands (including a dual OOD band), bands straddling the 1400 nm training edge, and multi-octave windows pairing an in-range with an out-of-range band
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across the full 380–2500 nm span (10 cases, 20 targets each; Tier 4). The targets are built from trained materials whose optical constants natively span almost the whole 380–2500 nm window, so what is genuinely OOD here is the wavelength position seen by the model’s λ-encoder, not the materials, and neither the model nor the vocabulary changes (interpolation and band-edge conventions in Methods and Appendix §J). Fidelity barely moves across the training edge: pure-OOD and edge-straddling bands reconstruct at essentially in-distribution quality (median combined RMSE ∼ 3 × 10−3 to 1.5 × 10−2 ; e.g. 3.6 × 10−3 for the 2000–2500 nm deep-SWIR band, a factor 1.8× beyond the edge), and only the multi-octave windows that must satisfy an in-range and an out-of-range band at once are harder, at 3.3–6.5 × 10−2 (representative crossings of the 1400 nm boundary in Fig. 2j, k; full four-regime set and per-case numbers in Appendix §J, Fig. J1, Table J1).
2.5 IrisFlow outperforms an autoregressive sequence baseline IrisFlow also outperforms a published autoregressive baseline on its own terms. Against OptoGPT [16], an autoregressive Transformer that emits stacks as material/thickness tokens, IrisFlow attains lower combined R, T RMSE on an 11-case shared subset, five of them taken directly from the OptoGPT paper and its Supplementary Information (SI) rather than chosen to favor IrisFlow (all methods scored on a shared dense 400–1100 nm grid, best-of-N with N = 500 draws; full protocol Appendix §K). The comparison doubles as an open-vocabulary stress test: IrisFlow receives a 15-material bank drawn from OptoGPT’s own library, supplied purely as n, k curves, none of which are IrisFlow training inputs, so it runs entirely on OOD materials. The methods are not parameter-matched, so Table 2 reports model sizes alongside the scores: the 136M 20-block IrisFlow takes the best median and mean RMSE and wins 10 − 1 pairwise, while the closer-sized 12-block model (86M, 1.5× OptoGPT’s ≈58M) still wins 8 − 3. In the one case OptoGPT leads (a near-infrared high-reflection target, Fig. 2l), it edges ahead on mean in-band reflectance but carries visibly larger spectral ripple (representative spectra spanning both outcomes Fig. 2l–o; per-case spectra Appendix §K).
Table 2 Head-to-head comparison with OptoGPT on the shared 11-case subset. “Case wins” counts all-method wins across the three rows; the pairwise records quoted in the text (10−1, 8−3) are tabulated per case in Appendix §K.
Method IrisFlow 12-block IrisFlow 20-block OptoGPT saved output
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2.6 Partially specified stacks support structure completion and thickness inference In practice, a coating engineer rarely starts from a blank slate: part of the stack is often fixed by the process, leaving open which materials fill the remaining slots, or which thicknesses to deposit for an already-decided structure. Because IrisFlow samples an explicit per-layer state, such constraints are imposed by clamping the known coordinates at every reverse step, and every draw honors them exactly (Methods, Sampling and inference). Figure 3(a,b) shows partial-structure completion on the eight-layer query of Appendix §E.1, with an Ag back layer and the SiO2 spacers pinned and the three interior material slots and all eight thicknesses left free. The model fills the free slots with the absorber the device physics calls for (the amorphous phase-change chalcogenide GSST-a in 54–100% of the lowest-error draws), and the best completed design reaches a combined RMSE of 2.1 × 10−2 , better than the best of 2,500 unconstrained draws on the same target (2.5 × 10−2 ). Pinning the full material sequence instead reduces the query to thickness inference, whose returned per-layer candidates are multimodal across interference orders: a set of TMM-verified thickness initializations for downstream refinement rather than one point estimate. Both modes, with full diagnostics, are in Appendix §O (Fig. O1).
2.7 Oblique incidence and polarization are reached without retraining IrisFlow is trained only at normal incidence, yet oblique-incidence and polarized design follow from a physics-based reparameterization of the candidate bank that leaves the model untouched. At a fixed angle and polarization, each candidate’s optical constants are given a closed-form analytic tilt (exact for s-polarization; for p-polarization, a close admittance proxy with a single thickness rescale), so the existing n, k interface already accepts the oblique problem (Appendix §P). The tilted bank only steers generation: every candidate is re-simulated and ranked by the exact angled TMM on the real material system at the requested angle and polarization, so the reported fidelity is exact even where the tilt is approximate. Figure 3(c–j) follows two fixed feature-rich multilayers, one per polarization, across the full angular range. For each we compute its true polarized R, T at θ0 = 0, 20, 40 and 60◦ and let the model re-design against the angle-tilted candidate bank at every angle; the returned stacks track the angle-shifted reflectance and transmittance from normal incidence through 60◦ , with the per-design combined RMSE (exact angledTMM re-simulation on the real materials) annotated in each panel. The 0◦ panels are the ordinary normal-incidence model under an identity tilt. Across the targets we examined, the oblique fidelity stays within the model’s normal-incidence operating range out to 60◦ for both polarizations, with the full per-target sweep and the small residual error of the p-polarization proxy quantified in Appendix §P. Incidence angle and polarization thus become design-time query coordinates through a forward-model transform; no new training distribution is needed.
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Fig. 3 Query-interface capabilities beyond a target spectrum. Constrained query: partialstructure completion (a,b) on the eight-layer query of Appendix §E.1. a, The query template (Ag back layer and SiO2 spacers pinned; interior material slots and all thicknesses free; hatched “?” marks free slots) beside the lowest-RMSE completed stack drawn to scale; the model fills the free slots with the GSST-a absorber the device physics calls for. b, TMM re-simulation of the selected design against the target R, T ; the query leaves 780–900 nm unspecified, so this no-target band is shaded gray and the target line broken there, while the design spectra stay continuous as real responses. Known coordinates are clamped during sampling, with no retraining or guidance (Methods); the fixed-structure thickness-inference mode, per-slot material marginals and full diagnostics are in Appendix §O (Fig. O1). Oblique incidence and polarization (c–j), without retraining: two fixed feature-rich multilayers (c–f, s-polarization; g–j, p-polarization), each evaluated across θ0 = 0, 20, 40 and 60◦ . At every angle IrisFlow re-designs a stack against the angle-tilted candidate bank; target R, T are solid and the returned design dashed (R red, T blue). Panel titles give the incidence angle and the combined R, T RMSE from exact angled-TMM re-simulation on the real material system; the 0◦ panels are the ordinary normal-incidence model (identity reparameterization). The full pertarget sweep and the p-polarization proxy-error analysis are in Appendix §P.
2.8 Real-world deployment: from calibrated optical constants to fabricated coolers Nominal optical constants rarely match a film as deposited. Because IrisFlow reads candidate materials as n, k curves rather than learned IDs, we calibrate the visible constants of our deposited GSST-a film (markedly lower-index and more absorbing than the handbook values; |∆n| up to 0.31, ∆k up to +0.68) and design against the
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process-corrected bank with no retraining. We exercise this end-to-end on four colordisplaying coolers (black and the subtractive primaries cyan, magenta and yellow), which must reflect the solar near-infrared for passive cooling [28, 29] while absorbing a visible band to display a color; this coupled two-band trade-off has kept most radiative coolers white or silver, and demonstrated colored coolers trade cooling power for the displayed color [30–32]. From each sampled pool we select by a cooling-aware rule: the design with the highest AM1.5G solar-weighted near-infrared reflectance (over 780– 1400 nm) whose color stays within the accepted tolerance ∆E00 ≤ 3.0. Selecting by color alone would instead sacrifice cooling: modestly for the chromatic targets, but severely for the near-black target, whose color-best design is a near-perfect absorber (Appendix §N.3). For yellow, the rule’s cooling optimum carries a thick titanium layer that is impractical to deposit, so we instead fabricate a structurally simpler six-layer in-tolerance design from the same pool, trading ∼0.02 in solar near-infrared reflectance for deposition practicality (Appendix §N.2). The four stacks chosen for deposition (6– 8 layers, Fig. 4) render all four colors within tolerance (∆E00 = 2.6–2.9) and reflect 0.91–0.98 of the solar near-infrared for the three chromatic coolers, the near-black design being the limiting case that cools less well by construction (target construction, candidate bank and selection rule in Appendix §N). The four stacks were fabricated by ion-assisted evaporation on BK7 and measured over 400–2500 nm (Methods); the opaque coatings are characterized by reflectance alone. The three chromatic coolers reproduce their target colors to ∆E00 = 3.1 (cyan), 3.6 (yellow) and 5.2 (magenta) under D65 (near the ∆E00 ≤ 3 tolerance for cyan and yellow, above it for magenta) while retaining a solar-weighted near-infrared reflectance of 0.93–0.95 across the full 780–2500 nm band, extending the cooling evidence well beyond the 1400 nm design grid. These target-referenced errors mix a selection term, fixed before any deposition, with a smaller fabrication-transfer term (Appendix Tables N1 and N2). The cooling-aware rule deliberately trades color margin for near-infrared reflectance: the fabricated designs sit at the tolerance boundary (∆E00 = 2.6–2.9) although the same pools reach ∆E00 = 1.3–2.1 when ranked by color alone, so no headroom remains for deposition error. Scored against its own selected design rather than the analytic target, each measured coating lands at ∆E00 = 0.8 (cyan), 2.1 (yellow) and 4.3 (magenta), and every as-fabricated spectrum tracks its design prediction more closely than the analytic target. The near-black sample is the anticipated limiting case in which blackness and broadband reflectivity are physically anti-correlated: a very dark green (∆E00 = 21.4) reflecting 0.78 of the near-infrared. Its color error reflects the hypersensitivity of near-black colorimetry rather than spectral infidelity; its measured reflectance tracks the design prediction the most closely of the four samples (Appendix §N.4). The experiment closes the loop from chambercalibrated optical constants, through open-vocabulary continuous-thickness design, to measured devices.
3 Discussion IrisFlow is best read as an interface result for amortized inverse design. It replaces a closed-world neural designer, in which components, coordinates and continuous degrees
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Fig. 4 Color-displaying coolers: process-corrected designs and as-fabricated coatings. a–d, Per-color panels for the yellow, magenta, cyan and black coolers, respectively. Each panel shows the reflectance spectra above a summary strip with the thickness-scaled stack (substrate side at left), layer count and total coating thickness, target/design/measured D65 swatches and the main color/cooling metrics; fabricated recipes are in Appendix Table N3 and the structure-selection context in Appendix §N and Fig. N3. The spectra show the design target reflectance (pale gray), the selected design’s BK7-substrate prediction (blue dashed) and the as-fabricated measured reflectance (warm orange). The wavelength axis is compressed beyond 780 nm (dotted divider) to expand the colordefining visible band and condense the flat solar near-infrared cooling band (shaded), which the coatings reflect for passive cooling. The black target leaves 780–900 nm unspecified, so that band is shaded gray and the target line broken; the predicted and measured spectra stay continuous. Strip annotations give the design and measured CIEDE2000 color error against the target and the measured AM1.5G solar-weighted near-infrared reflectance R̄NIR (780–2500 nm). Designs are selected from each compliant pool by the cooling-aware rule: highest solar-NIR reflectance within the accepted color tolerance ∆E00 ≤ 3.0 (yellow is fabricated as a simpler 6-layer in-tolerance stack). GSST-a denotes the piecewise process-corrected curve of Appendix §N. All designs satisfy the Ag–GSST-a adjacency rule. The coatings are opaque (T ≈ 0), so reflectance fully characterizes the response; per-target values are in Appendix Tables N1 and N2. e, Photograph of the fabricated coupons (columns left to right: yellow, magenta, cyan and black; three witness disks and one square per color) on BK7, deposited by ion-assisted evaporation. f, CIE 1931 chromaticity of the reproduced colors: for each chromatic cooler the target (star), selected design (square) and as-fabricated measured (circle) colors under D65, over the filled spectral locus with the sRGB gamut (dashed) and the D65 white point (+). The chromatic coolers sit near the gamut boundary with the design and measured points clustered on the target; the black cooler is achromatic (luminance L∗ ≈ 0, chromaticity undefined) and is characterized in d only.
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of freedom are baked into the training representation, with a query interface: candidate components enter as functions on the user’s coordinate grid, continuous variables remain continuous, and discrete and continuous states are denoised jointly. In multilayer optical coatings, this means materials enter as n(λ), k (λ) curves on the user’s wavelength grid, thickness remains continuous, and a single checkpoint serves every requested layer count from 2 to 100. The benchmark supports each interface claim separately (in-distribution and OOD materials, wavelength extrapolation, idealized application families and a head-to-head baseline), so the three closed-world assumptions of earlier neural designers (fixed material vocabularies, frozen wavelength grids and tokenized thicknesses) are not the price of amortized inverse design. Appendix §M condenses this into a rubric-scored interface-capability map (Fig. M1): classical simulator-in-the-loop optimizers cover the flexible half of the chart and earlier neural families the amortized half, while IrisFlow covers both. Several limitations bound the present claims. First, incidence angle and polarization enter through a forward-model reparameterization rather than as native query inputs: the tilt is exact for s-polarization but only approximate for p-polarization, where an admittance proxy with a thickness rescale leaves a residual (Appendix §P.3). Exact angled-TMM scoring keeps the reported numbers exact, but p-polarized generation is steered less precisely, and a fine-tune of the normal-incidence checkpoint making angle, polarization and substrate native inputs would close the gap. Second, the hardest families are narrow, high-contrast passbands, where a small wavelength offset incurs a large penalty and fidelity is lowest (Results, Tier 3; Appendix §I); there IrisFlow is best used as a structural initializer for local refinement, the deployment mode discussed below. Third, IrisFlow returns a design and its exact nominal spectrum but does not itself report that spectrum’s sensitivity to fabrication error; quantifying it is a cheap post-hoc screen (perturbing the TMM forward model over the chosen stack, since every sample is exactly TMM-verifiable), and folding an errorbudget penalty into generation or best-of-N selection would let the model surface robust designs natively. The fabricated coolers quantify the cost of omitting such a penalty: selected at the color-tolerance boundary with no allowance left for deposition error (Results), even cyan, with a fabrication-transfer error below one ∆E00 unit, lands past the tolerance; re-ranking the chromatic candidate pools under a tighter color cap would have provided that allowance at a cost of at most 0.03 in solar nearinfrared reflectance (Appendix Table N1). The cooler deployment also rests on a target choice we did not optimize: each color admits a family of spectra that render it, and we fixed one representative per color without searching that family, so the reported coolers are a conservative instance rather than a searched optimum (Appendix §N.3). Finally, fidelity degrades gradually with stack depth (Results, Tier 1; Fig. H1); part of this is the intrinsically more degenerate inverse problem at high L, but we cannot yet separate it from a simply under-resourced deep-stack regime (too little and too narrow deep-stack training data, or insufficient backbone capacity), a question the scaling evidence below addresses directly. On that question the evidence favors the under-resourced regime over an intrinsic limit. The capacity comparison of Table 2 already indicates the model is not at the ceiling of its own recipe, and three scaling axes are complementary. The first is a larger,
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more materially diverse long-stack corpus: the deep-stack slice is the smallest and narrowest of the curriculum, its median long stack using two distinct materials against seven in the short-stack regime (Appendix §B, Table B1), which may bias the model toward the periodic high-/low-index (HL)m motif. The second is a larger backbone: on the same data, objective and interface, the deeper 20-block model already beats the 12-block on both median and mean combined RMSE and wins 9−2 pairwise (Appendix §K). The third is a longer training schedule: long-stack error was still falling across the late fine-tuning checkpoints (Appendix Fig. C3). These limitations inform the deployment mode; they do not weigh against the interface result. Because every IrisFlow sample is well-formed by construction and cheap to verify exactly, the model composes naturally with the existing design stack: as a generator of diverse, physically valid starting points for needle or gradient refinement; as a completion engine for partially specified stacks (Appendix §O); as a design-space probe over a facility’s actual material bank, including dispersion curves measured in its own chambers; and as the amortized inner loop of larger photonic design problems whose figures of merit are computed downstream of R, T . The process-corrected-GSST cooler workflow of the final Results section realizes the first instance: a chambercalibrated n, k curve enters the candidate bank in the same way as a handbook curve, and the resulting designs were carried through to fabricated devices. This interface framing also clarifies how far the claim should be extrapolated. The structure IrisFlow addresses is broader than coatings: amortized inverse design over a non-separably coupled discrete–continuous space, where the optimal continuous parameters depend on the discrete choices and vice versa, so a design cannot be reached by choosing components and then sizing them. The same coupled structure recurs across generative design, from protein sequence–structure co-design [33] to the joint generation of composition and geometry in inorganic materials [34]. The model meets this coupling directly (a single joint flow samples material and thickness together, without decomposing, relaxing or enumerating the mixed-integer search), and the partial-specification experiments (Appendix §O), where fixing some materials reshapes both the remaining material and the thickness predictions, indicate that the two are genuinely entangled. The same query interface therefore applies, in principle, to any inverse problem of the same joint discrete–continuous form, in which components are selected from a bank and each carries continuous parameters that must be co-designed with the selection: acoustic and radio-frequency multilayer stacks, transmission-line and filter synthesis, and composite-laminate layup are direct analogs, in each of which a per-facility component bank would enter, as here, through feature tokens. Three boundaries delimit the claim: these targets are untested here; like the present work, training and exact verification presuppose a fast forward model for the domain; and three-dimensional design spaces, where geometric symmetry favors equivariant backbones, would retain the coupled discrete–continuous flow but not the plain-sequence architecture. In this sense IrisFlow is a methodology as much as a coating-specific tool: a demonstrated instance of an open-vocabulary, coupled discrete–continuous inverse-design interface.
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4 Methods 4.1 Problem definition and query interface A multilayer thin-film design is defined by a material sequence m = (m1 , . . . , mL ) and a thickness vector d = (d1 , . . . , dL ), where each material mℓ is selected from a candidate bank supplied at query time and each thickness dℓ is a continuous value in nanometers. IrisFlow receives four query objects: (i) a target optical response y (λ) = [R(λ), T (λ)], (ii) a wavelength grid λ = {λs }Ss=1 defining the spectral support of the query, (iii) a candidate material bank C = {ci (λ)}C i=1 , where ci (λ) = [ni (λ), ki (λ)], and (iv) a requested layer count L. The objective is to generate one or more stacks whose TMM response [1] matches the target response over the specified wavelength points. Unless stated otherwise, all experiments assume normal incidence, a vacuum (n = 1) incidence medium and a fused-silica substrate; the design-to-fabrication experiment uses BK7 substrates, for which the substrate correction to the selected designs is quantified in Appendix §N.2. The reported model supports requested layer counts up to Lmax = 100 and predicts continuous thicknesses within the configured fabrication window dmin = 5 nm and dmax = 300 nm.
4.2 Optical inputs and wavelength handling The target response and candidate bank enter the model as discretized arrays on the query grid: an S × 2 array of (R, T ) values and a C × S × 2 tensor of (n, k ) curves. This grid is inverse-wavelength-uniform (the coordinate in which thin-film interference is naturally resolved) and is itself part of the query rather than a fixed model index, so both target and candidates are resampled onto it before tokenization. The grid length is fixed by the curve tokenizer at S = 128 points (Appendix Table C1); a query chooses the spectral support and placement of these points, not their number. Training grids are drawn from the training data distribution, but at inference IrisFlow accepts any user-specified wavelength support, including unseen ones. Candidate n, k values are interpolated within their tabulated support and held at the nearest endpoint outside it (no extrapolated dispersion); for reliable design beyond the tabulated support, the candidate curve should be extended from measurement or a separate validated source first.
4.3 Candidate material banks and curve tokenization IrisFlow has no global material-ID output layer. Each query defines its own local candidate bank, and the model scores every layer against it: a material enters through its optical curve n(λ), k (λ) rather than a learned class index. We set the maximum number of candidate materials to 15, matching the 15 materials in the training data. A wavelength-aware curve-token encoder maps any two-channel wavelength function c(λ) to a token. The target R(λ), T (λ) curve and the candidate n(λ), k (λ) curves are each encoded by an instance of this encoder (a spectrum branch and a candidate branch) into a common token space. Wavelength enters both encoder branches as a per-grid-point signed inverse-wavelength feature, 2 (0.30 µm)/λ − 1 (Appendix 16
Table C1). For the target spectrum, the encoder emits one conditioning token. For the candidate bank, it emits a memory of C candidate tokens. These candidate tokens are used by the denoiser through candidate cross-attention and by the material head as the local scoring basis.
4.4 Joint discrete–continuous denoising model The architecture and the learned material-token geometry are summarized in Fig. 5. a
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Fig. 5 IrisFlow architecture and learned material-token geometry. a, Query inputs: target R, T spectra, a query-local candidate bank of material n, k curves and a requested layer count. b, Wavelength-aware optical-curve tokenizers encode the target spectrum and every candidate material curve into one spectrum token and a memory of candidate n, k tokens. c, The joint denoising core updates coupled material and thickness layer states using FiLM conditioning by flow time, RoPE self-attention over layers, a target-spectrum bias and candidate cross-attention over the local material memory. Self-attention over these fused per-layer tokens couples both the layers and the two modalities. d, Separate output heads predict continuous thickness velocities and discrete material scores against the query’s candidate bank, yielding a multilayer stack. e, Training couples continuous flow matching for thicknesses with uniform-CTMC discrete flow matching for materials. f, Principalcomponent projection of learned candidate n, k tokens for the 15 training materials (circles; h4 denotes LaTiO3 ) and 15 held-out OOD materials (triangles), colored by material family for display only. The family structure is not supplied to the encoder; the full labeled representation and cross-attention diagnostic is shown in Appendix §Q (Fig. Q1).
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IrisFlow jointly denoises a continuous thickness state and a discrete material-stack state under a shared flow time. The noisy state at time t ∈ [0, 1] is (xt , mt ) where xt is the normalized noisy continuous thickness variable and mt is the noisy material state over the local candidate bank. The model is conditioned on the query tuple q = (y, λ, C, L). The inverse-design target is the conditional joint distribution p(x0 , m0 | q ) over clean thicknesses and materials. It does not factor across modalities or layers (the TMM response couples every layer’s material, thickness and neighbors), so IrisFlow denoises both coordinates with a single backbone, conditioning every prediction on the full current joint state (xt , mt ). Appendix §A gives a formal account showing that the forward corruption, the training objective below and the reverse sampler are mutually consistent: the loss learns exactly the denoising statistics the sampler consumes. The denoiser maps the full layer sequence (xt , mt ) and the query conditioning q to two outputs: a thickness velocity prediction
vθ (xt , mt , q, t) and clean-material posterior distributions over the local candidate bank,
pθ,ℓ (· | xt , mt , q, t). The continuous and discrete states are embedded into a shared layer-token representation. Thickness states are embedded with sinusoidal thickness features, while material states are embedded by gathering the corresponding candidate n, k token from the local candidate memory. The denoising core is a stack of Transformer blocks [35] with query-key normalization [36], FiLM modulation by flow time [37], a learnable gated bias derived from the target-spectrum token and candidate cross-attention over the local candidate memory. Self-attention over the fused per-layer tokens resolves the coupling identified above: each layer is updated conditioned on the full stack, and because thickness and material share one token, one layer’s thickness can shift another layer’s material posterior and vice versa; this is the cross-layer, cross-modality structure that the factorized reverse kernels (Appendix §A) would otherwise drop. RoPE [38] over layer indices supplies the ordering signal and lets the same core span the staged expansion to 100-layer stacks, with separate thickness and material heads on top.
4.5 Continuous flow matching for thicknesses The thickness branch treats layer thickness as a continuous variable rather than a discretized token. Let x0 denote the clean thickness vector after normalization to the model scale, and let ϵ ∼ N (0, I ) denote Gaussian noise. Physical thicknesses are affinely mapped from [dmin , dmax ] to the model interval [−1, 1] before applying the
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flow-matching path. For flow time t ∈ [0, 1], we use the linear flow-matching path [23, 24] xt = (1 − t)x0 + tϵ with target velocity
v ⋆ = ϵ − x0 . The thickness head predicts vθ (xt , mt , q, t). The thickness loss is the mean squared velocity error pooled over active layers. Here 1ℓ equals 1 for active layers and 0 for padded positions, and Lseq denotes the padded sequence length used in the batch:
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E(x0 ,m0 ,q),t,ϵ,mt
hP
Lseq 2 ℓ=1 1ℓ ∥vθ (xt , mt , q, t)ℓ − (ϵℓ − x0,ℓ )∥2
E(x0 ,m0 ,q),t,ϵ,mt
hP
Lseq ℓ=1 1ℓ
i
i .
The batch estimate divides the summed per-layer errors by the batch’s total activelayer count, so every active layer contributes equally. At its population optimum over the conditioning c = (xt , mt , q, t), this regression recovers the conditional mean velocity vθ (xt , mt , q, t) = E[ϵ−x0 | xt , mt , q, t], which is exactly the marginal flow-matching velocity integrated by the reverse ODE (Appendix §A). During sampling, the predicted velocity field is integrated from noise to data on a decreasing time grid, dxt = vθ (xt , mt , q, t). dt
4.6 Uniform CTMC DFM for material stacks The material branch follows a CTMC-based DFM formulation [25–27] over the local candidate bank. Instead of directly predicting an explicit probability-velocity or transition-rate tensor, IrisFlow uses the denoiser parameterization of DFM [26]: the material head predicts clean-material posterior distributions that, together with the CTMC transition kernels, define the reverse categorical updates. For a query with C valid candidate materials, the clean material sequence is m0 = (m0,1 , . . . , m0,L ), with m0,ℓ ∈ {1, . . . , C}. For C > 1, we construct the CTMC generator from a uniform off-diagonal transition kernel [39] over valid candidate states,
( Kij =
1 C−1 ,
0,
i ̸= j, i, j ∈ {1, . . . , C} i=j
and the CTMC rate matrix
Q = K − I. For the degenerate case C = 1, the material state is deterministic and no discrete corruption is applied. For variable-size candidate banks, padded candidates are masked before constructing Q, and C denotes the number of valid candidates. The transition matrix at time t is Pt = exp(τt Q) 19
where τt is determined by the discrete noise schedule. We use a log-linear schedule t ᾱt = αend with αend = 10−4 , and define τt = − log ᾱt . This anchors the process at the clean endpoint (P0 = I) and drives the forward marginal to within 10−4 of the uniform distribution over the valid local candidate bank at t = 1; ᾱt parameterizes the CTMC time rather than the literal per-state stay probability. Because the kernel is uniform, Pt has a closed form under which each layer stays with probability Pt [i, i] and otherwise jumps to any other valid candidate uniformly (Appendix §A.1). For each layer, the noisy material state is sampled as
mt,ℓ ∼ Categorical(Pt [m0,ℓ , :]). We train this denoiser parameterization, which predicts the clean-material posterior over the local candidate bank, with a CTMC-reweighted cross-entropy over active layers, normalized like Lth by the total (here, weighted) layer count,
Lst = −
E(x0 ,m0 ,q),t,ϵ,mt
hP
i
Lseq ℓ=1 1ℓ wℓ log pθ,ℓ (m0,ℓ | xt , mt , q, t)
E(x0 ,m0 ,q),t,ϵ,mt
hP
Lseq ℓ=1 1ℓ wℓ
i
where wℓ = λsoft + (1 − λsoft )(1 − Pt [m0,ℓ , m0,ℓ ]) is the CTMC-soft reweighting factor built from the stay probability, with λsoft = 0.1 in the reported model. Under the uniform kernel wℓ depends on t and C but not on the class label (within one draw it is constant across active layers), so under the total-weight normalization above it reweights draws, down-weighting nearly clean (small-t) ones, rather than layers. Being label-independent, it leaves the objective a proper scoring rule: at its population optimum the material head recovers the per-layer clean-material posterior pθ,ℓ (i | xt , mt , q, t) = Pr(m0,ℓ = i | xt , mt , q, t), conditioned on the full joint state (Appendix §A). The final joint objective combines continuous thickness flow matching and CTMCbased DFM, L = Lth + λst Lst with λst = 0.4 in the reported configuration, fixed at the value where the balancing coefficient of a task-uncertainty-weighting pilot [40] stabilized. The uniform kernel injects no a priori material-similarity bias: the optical role of a material is set by its neighbors, thickness and target spectrum, and the denoiser learns this context-dependent substitutability from data (Appendix §Q).
4.7 Sampling and inference At inference time, IrisFlow receives the four query objects q = (y, λ, C, L). Sampling is performed only on the first L active layers, and padded positions are masked throughout. The continuous thickness state is initialized from Gaussian noise, and the material state is initialized uniformly at random over the local candidate bank. Sampling follows a shared decreasing time schedule for the thickness and material states; the reported evaluations use the 15-step adaptive-power schedule described in Appendix §D. At each step, the denoiser predicts a thickness velocity field and clean-material 20
posterior probabilities conditioned on the full current state and query. Thicknesses are updated by integrating the flow-matching ordinary differential equation (ODE) with a 15-step Euler integrator. Material states are updated with an ancestral categorical reverse-CTMC sampler: the denoised clean-material posterior is combined with the CTMC transition matrices between adjacent time steps to form a posterior over the previous material state, from which the next reverse sample is drawn. Concretely, for a reverse step from grid time t to the next time s < t, the per-layer reverse posterior marginalizes the predicted clean-material posterior against the CTMC posterior bridge,
p̃θ,ℓ (ms,ℓ = j | xt , mt , q, t) =
C X Ps [k, j ] Pt|s [j, mt,ℓ ]
Pt [k, mt,ℓ ]
k=1
pθ,ℓ (k | xt , mt , q, t)
where
Pt|s = exp (τt − τs )Q and the material vectorQis advanced in one reverse step from the product-form L categorical kernel, ms ∼ ℓ=1 Categorical p̃θ,ℓ (· | xt , mt , q, t) over active layers. This reverse parameterization consumes only the conditional mean velocity and the per-layer clean-material posteriors, exactly the two population optima the heads are trained to recover. Although each reverse step factors into a deterministic thickness kernel and a product-form, mean-field material kernel, both are read from one shared denoising state conditioned on the full (xt , mt ), so the coordinates stay coupled; Appendix §A formalizes this consistency and states the remaining approximation gaps. We visualize this reverse trajectory for one representative run in Appendix §E.1 (Figs. E1–E3). After the final step, thicknesses are mapped back to nanometers and constrained to the configured fabrication window, and material indices are decoded through the local candidate bank, so every sampled design is well-formed by construction: each layer selects a valid candidate, every thickness lies inside the window, and the stack admits a TMM evaluation. Because the initial states are stochastic, repeated sampling from the same query can produce multiple valid designs; the evaluation protocol below specifies how independent samples are budgeted and selected for best-of-N reporting. Because sampling operates on an explicit per-layer joint state, any subset of these coordinates can additionally be supplied as part of the query and held fixed during generation, with no retraining, fine-tuning or guidance machinery: user-specified materials and thicknesses are clamped at every reverse step (both in the state passed to the denoiser and after each update), so the sampler evolves only the unspecified coordinates while conditioning on the clamped layers through self-attention, and every draw honors the constraint exactly rather than through a soft penalty. This yields a fixedstructure mode, in which the complete material sequence is specified and sampling reduces to thickness inference, and a partial-structure mode, in which an arbitrary subset of layer materials (and optionally thicknesses) is specified and the rest is completed jointly. Both modes reuse the same checkpoint and best-of-N protocol; worked examples are reported in Appendix §O. 21
4.8 Layer-count expansion training IrisFlow is trained with a staged layer-count expansion strategy. The architecture is kept fixed across stages (the same curve-token encoders, joint denoising core, heads and uniform CTMC process throughout), and only the maximum active layer count, layer masks and training distribution are expanded. Training directly on the full L ≤ 100 distribution is harder because the inverse-design space becomes increasingly degenerate at high L, where many distinct stacks produce similar spectra; the curriculum first establishes the input–output mapping in the comparatively well-constrained shortstack regime and then preserves it by replay while gradually exposing the model to the degenerate long-stack regime. In Stage 1, the model is pretrained on a short-to-medium-stack corpus with Lmax = 20, containing approximately 110 million TMM samples. In Stage 2, the Stage1 checkpoint is fine-tuned with the maximum active layer count expanded to 60, on samples spanning the full range L ≤ 60 rather than only newly introduced long stacks; this replay preserves short-stack performance while exposing the model to longer optical paths: tracked across fine-tuning checkpoints, short-stack error stays flat while long-stack error falls (Appendix Fig. C3). In Stage 3, the same procedure reaches the final reported Lmax = 100, again mixing shorter and longer stacks across L ≤ 100. Final evaluations report performance over the full requested layer-count range up to 100 layers.
4.9 Training data The training data are TMM-simulated multilayer stacks paired with target R, T spectra, material sequences, continuous thickness vectors, wavelength grids and candidate material banks, all generated within the configured thickness window by the same TMM implementation used for evaluation. The corpus includes single-band and multiband targets sampled over the available optical range. Each sample carries its own wavelength grid and band metadata, with points spaced at equal intervals in 1/λ within each band, and candidate n, k curves are interpolated to the sample’s grid before tokenization. This data interface matches the inference interface: every field is supplied per query. For material-generalization experiments, the training material pool and the held-out OOD set are kept disjoint; OOD materials are introduced at inference time through their n, k curves and are never assigned learned material-ID embeddings. The model is trained with Adam with decoupled weight decay (AdamW) [41], bfloat16 (bf16) mixed precision, gradient clipping and a linear-warmup, cosine-decay learningrate (LR) schedule. Stage-specific sample counts, layer-count ranges, LRs, update counts and hardware are reported in Appendix Tables C1–C2; the full data-generation procedure, per-corpus statistics and training-material optical constants in Appendix §B; the 15-material training vocabulary and the disjoint OOD bank in Appendix §F.5; and the sampler settings (timesteps and best-of-N ) in Appendix §D.
22
4.10 Evaluation Every generated design is verified by re-simulation: we run the same TMM solver used to construct the targets and score the design only on its query wavelength support. Fidelity is reported as the combined R,T error RMSERT , with the per-channel RMSER and RMSET recorded alongside; the metric itself and the selection and aggregation conventions are defined in Appendix §G. Because sampling is stochastic and inverse design is one-to-many, each target is queried under a fixed best-of-N budget: we draw N independent designs, re-simulate each, and report the one with the lowest RMSERT (oracle ranking). We use N = 100 for the layer×band grids (Tiers 1 and 2) and N = 500 for the application and wavelength suites (Tiers 3 and 4), with 15 reverse-time steps throughout (Appendix §D). We report two baseline comparisons, each with its full protocol in a dedicated appendix: a head-to-head against the autoregressive model OptoGPT, with the perconfiguration best-of-N budget matched at N = 500 on a shared scoring grid and target set, and OptoGPT’s materials supplied to IrisFlow as OOD n, k curves rather than learned vocabulary tokens (Appendix §K); and a comparison against four classical simulator-in-the-loop optimizers (needle, GA, PSO and random search) sharing the target, candidate pool, constraints and TMM solver, compared by simulator-evaluation count (Appendix §L).
4.11 Fabrication The color-displaying coolers (Fig. 4) exercise this design-to-device loop end to end on chamber-remeasured optical constants. We obtain these constants from single-layer witness films, so the design is built on the films as deposited rather than on handbook values: each material is grown alone on glass by ion-assisted evaporation, and its n, k are retrieved by fitting the reflectance and transmittance measured by spectrophotometry across the visible and near-infrared. We substitute only the process-corrected visible curve GSST − ameas−vis into the candidate bank (the other films match the remeasurement and keep their nominal n, k ) and design on a 400–1400 nm grid under the same material-adjacency and thickness constraints, selecting from the best-of-N pool by the cooling-aware rule (Results); the target construction, candidate bank and selection rule are given in Appendix §N. Each selected sequence is converted directly into a deposition recipe and grown on BK7 substrates by ion-assisted evaporation, with silver deposited by resistive evaporation and all other layers by electron-beam evaporation. Because silver and GSST-a oxidize readily, all four coatings are deposited oxygen-free at low substrate temperature, so these two films and their neighbors never see reactive oxygen during growth; chamber preparation and source conditioning are detailed in Appendix §N.4. Layer thicknesses are controlled during growth by quartz-crystal monitoring calibrated per material. The coatings are opaque (T ≈ 0), so reflectance fully characterizes the response; we measure it over 400–2500 nm. Each coating is scored against both the analytic target and the TMM prediction of its selected stack by the D65 CIEDE2000 color difference [42] and the AM1.5G solar-weighted near-infrared reflectance R̄NIR [43] (Appendix §N, Table N2).
23
Supplementary information. The Supplementary Information (SI) of this article is contained in Appendices A–Q below. Acknowledgments. This work was supported by the National Natural Science Foundation of China (62375231); Fujian Province Science and Technology Planning Project of China [2022H6015]; Xiamen University President’s Fund Project of China [20720242021].
Declarations Author contributions Z.L. conceived and designed the study, developed the IrisFlow framework, implemented the code, performed the simulations and benchmark experiments, and analyzed the data. Y.J. fabricated the multilayer thin-film samples. Y.B. proposed the initial research direction. N.C., H.F. and Y.B. supervised the project. All authors discussed the results. Z.L. wrote the manuscript with input from all authors. Y.H. reviewed and edited the manuscript.
Competing interests The authors declare no competing interests.
Data availability The data underlying this work (including the effective training-material opticalconstant library, the Tier 2 OOD optical-constant library and the benchmark case registry referenced in the Supplementary Information) are available from the corresponding author upon reasonable request.
Code availability The code underlying this work is available from the corresponding author upon reasonable request.
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Appendix / SI The SI appendices are organized as follows. §A gives the formal account, deferred to in the Methods, of how the forward process, the joint training objective and the reversetime sampler fit together. §§B–E then document the model behind every reported number: the training-data generation pipeline and corpus statistics (§B), the architecture and multi-stage training configuration (§C), the sampler settings (timestep schedule and best-of-N protocol; §D) and the sampler’s reverse-time dynamics and solution diversity (§E). §§F–G define the evaluation: benchmark construction and the material vocabularies (§F) and the exact metric definitions (§G). §§H–J tabulate the full results underlying the Results section: the Tier 1/2 layer×band grid (§H), the Tier 3 application scorecard (§I) and the Tier 4 wavelength-extrapolation suite (§J). §§K–O report the comparisons and case studies: the OptoGPT head-to-head (§K), the classical-optimizer comparison (§L), the interface-capability rubric (§M), the process-specific GSST-a optical constants (§N) and constrained generation (§O), and §P extends the interface to oblique incidence and polarization without retraining. Finally, §Q examines the learned representations: curve tokenization and candidate cross-attention. All numbers correspond to the single Lmax = 100 checkpoint and the best-of-N oracle-ranked protocol described in Methods.
Appendix A
Joint discrete–continuous flow matching: forward process, training objective and sampler
This appendix states the consistency relation among the forward corruption, the joint objective and the reverse-time sampler: at the population optimum, the loss recovers the statistics consumed by the sampler. The claim is not that the finite network 29
reaches that optimum, that any draw is a global TMM optimum, or that the productform sampler is the exact joint reverse process; the controlled approximation gaps are listed in §A.5. Throughout, q = (y, λ, C, L) is the query, c = (xt , mt , q, t) is the t conditioning state, C = |C|, and the forward kernels are Pt = exp(τt Q) with ᾱt = αend , τt = − log ᾱt and αend = 10−4 . A.1 The forward corruption is a valid joint kernel. The clean design z0 = (x0 , m0 ) ∈ [−1, 1]L × {1, . . . , C}L is drawn from the coupled inverse-design target p(x0 , m0 | q ). Only the noising factorizes: qt (zt | z0 , q ) = qtx (xt | x0 ) qtm (mt | m0 , q ), with xt = (1−t)x0 +tϵ, ϵ ∼ N (0, I ), and mt,ℓ ∼ Categorical(Pt [m0,ℓ , :]). The thickness path is Gaussian for t > 0 and a Dirac measure at t = 0; the material path is valid because Q = K − I is a CTMC generator and Pt is row-stochastic. C I− Because the kernel is uniform, the generator is a scaled projector, Q = − C−1 1 ⊤ C with 1 ∈ R the all-ones vector, so its matrix exponential has the closed form C 11
1 ⊤ 1 ⊤ −τt C/(C−1) Pt = exp(τt Q) = 11 + e I − 11 , C C −τt C/(C−1) and under which each layer stays with probability Pt [i, i] = C1 + C−1 C e otherwise jumps to any other valid candidate with equal probability Pt [i, j ] = C1 1 − e−τt C/(C−1) , i ̸= j . A.2 The two heads are trained to Bayes-optimal denoising statistics. Let v ⋆ = ϵ − x0 . For fixed c, the thickness risk E[∥f (c) − v⋆ ∥22 | c] is minimized by vθ⋆ (c) = E[v⋆ | c], a joint denoising quantity because c includes mt . For the material head, the CTMC-soft reweight is label-independent under the −τt C/(C−1) uniform kernel: Pt [i, i] = C1 + C−1 is the same for every clean state i, so C e wℓ is a function of the conditioning state alone (t and C ), not of the class. Under the total-weight normalization of Methods, the weight therefore only rescales how much each conditioning state contributes to the risk (shifting weight toward strongly corrupted draws), while at every fixed c the loss over predictions remains an ordinary proper scoring rule, so the population optimum is p⋆θ,ℓ (i | c) = Pr(m0,ℓ = i | c). Thus the additive objective has population optimum vθ = E[v⋆ | c] and pθ,ℓ = Pr(m0,ℓ = · | c). The material head estimates per-layer posterior marginals, not a sequence-level posterior over m0 . A.3 The sampler needs only these statistics. The clean posterior pt (z0 | zt , q ) ∝ pdata (x0 , m0 | q ) qtx qtm does not factor because the data prior is coupled by TMM. Representing a full categorical over clean material sequences would require C L logits per query and time (15100 ≈ 4 × 10117 states for C = 15, L = 100). IrisFlow instead uses only the two statistics identified above. For thickness, the probability-flow update integrates the conditional-mean velocity E[v⋆ | c] [23, 24], with the conditioning on the joint state c (including mt ) as in multimodal flow matching [27]; the full density pt (x0 | c) is not needed.
30
For materials, the one-layer chain m0,ℓ → ms,ℓ → mt,ℓ yields the CTMC posterior bridge Pr(ms,ℓ = j | mt,ℓ = i, c) =
C X
Pr(m0,ℓ = k | c)
k=1
Ps [k, j ] Pt|s [j, i] , Pt [k, i]
which depends on the clean material only through the marginal posterior returned by the material head. IrisFlow multiplies these single-layer factors into a product-form material kernel rather than representing the exact sequence-level reverse bridge, Pr(ms | mt , c) =
X
Pr(m0 = k | c)
k∈{1,...,C}L
L Y Ps [kℓ , ms,ℓ ] Pt|s [ms,ℓ , mt,ℓ ] ℓ=1
Pt [kℓ , mt,ℓ ]
,
which in general Q equals the product form only when the clean posterior factorizes, Pr(m0 | c) = ℓ Pr(m0,ℓ | c). The product kernel is therefore the mean-field discrete reverse approximation. Thus the reverse sampler consumes η ⋆ (c) = E[v⋆ | c], {Pr(m0,ℓ = i | c)}ℓ,i , exactly the pair recovered at the population optimum of the joint objective. A.4 Validity and coupling of the reverse process. For t > 0, each reverse x m x step Kθ,t→s = Kθ,t→s KQ is a Dirac update at θ,t→s is a valid Markov kernel: K m xt + (s − t)v̂, and K = ℓ Categorical(p̃θ,ℓ ) is a product of normalized categoricals because Chapman–Kolmogorov gives Ps Pt|s = Pt . The coordinate factorization is not independence: both factors are read from the same hidden state H = Fθ (xt , mt , q, t), so each modality can affect the other’s update through the shared denoiser. A.5 Approximation gaps. Four gaps separate this construction from an exact reverse process. (i) Discrete prior endpoint. The continuous prior N (0, I ) is the exact t = 1 thickness marginal, but the discrete t = 1 marginal P1 [i, ·] is only within C−1 C/(C−1) ≤ αend = 10−4 of the uniform prior used to initialize the sampler; C αend the mismatch vanishes as αend → 0. (ii) Mean-field discrete step. The product-form kernel does not represent the cross-layer correlation of the exact sequence-level bridge Pr(ms | mt , c). IrisFlow keeps this approximation tractable by re-evaluating every perlayer factor from the full current joint state at every reverse step, so joint structure can still enter through self-attention in the shared denoiser even though each categorical transition factorizes (§A.4). (iii) Terminal step. At the last step the bridge reduces to the identity and the returned materials are a factorized draw or decode from the per-layer posteriors, with no successor step to re-condition them as in (ii). The crosslayer error of a factorized step is controlled by the probability mass it transports, which the reverse schedule drives to zero as t → 0: by the terminal step the per-layer posteriors have largely concentrated and the sampled stack has all but settled, so the final draw perturbs only a vanishing fraction of layers and introduces negligible crosslayer incoherence. Refining the reverse discretization suppresses this residual within the sampler itself: as the step count grows, the terminal posteriors sharpen and the fraction of layers still changing near t = 0 falls monotonically. Exact TMM scoring in best-of-N (Methods) therefore ranks the N candidate stacks rather than repairing
31
coupling left unresolved by the sampler. (iv) Finite model and selection. The optima above are over measurable functions; a finite trained network need not attain them, and best-of-N (Methods) is an oracle-ranked evaluation protocol, not a global-optimality guarantee.
Appendix B
Training data: generation and statistics
IrisFlow is trained on three TMM corpora, one per stage of the staged layer-count curriculum (Methods, Layer-Count Expansion Training). All three use the 15-material training vocabulary (Table F1), the 380–1400 nm wavelength envelope on a 128-point per-sample grid, and the [5, 300] nm thickness window. B.1 Generation. Two generators populate complementary regimes.
• Forward sampling builds the short/standard regime with no optimization: for each sample it draws a layer count, a material sequence constrained to three physical classes (dielectrics: TiO2 , SiO2 , Ta2 O5 , h4, Al2 O3 ; metals: Ag, Cr, Ti; absorbers / phase-change media: Si, gst-a/b, gsst-a/b, ss-a/b) and per-layer continuous thicknesses inside per-material windows; it then samples a random spectral support (≈70% of samples a single band of width 120–700 nm, otherwise a double band with each segment ≤ 350 nm and an intervening gap) and computes R(λ), T (λ) by an incremental TMM. Here gst, gsst and ss denote the phase-change chalcogenides GST (germanium antimony telluride), the selenium-substituted variant GSST and Sb2 S3 (antimony trisulfide), with the -a/-b suffixes marking their amorphous and crystalline states. This generator produces all of Stage 1 (˜110M samples, 2–20 layers) and the short-stack portion of Stages 2–3. • Optimization-refined long stacks populate the deep 21–100-layer regime with spectrally meaningful designs rather than the near-opaque stacks that random deep sampling would yield: each long sample is refined by Powell optimization [44] toward one of six spectral-target families (r peak, t peak, t multiband, r stopband, edge filter, absorber) from expert or random material initializations. A per-sample candidate bank and a cross-layer-range shard mix produce the final training format for Stages 2 and 3. Stage 1 is forward-only; Stage 2 mixes 37% forwardshort with 63% Powell-long; Stage 3 mixes 24% / 76% (Table B1, Fig. B1). Forward-short = directly sampled stacks (≤20 layers); Powell-long = optimizationrefined long stacks (21–100 layers) across 6 spectral-target families (r peak, t peak, t multiband, r stopband, edge filter, absorber). B.2 Distribution. Figure B1 visualizes the three corpora. The curriculum does not merely grow the layer count (median 11 → 29 → 47, max 20 → 60 → 100); it shifts the kind of design. Stage 1, built by unconstrained forward sampling, is materially diverse (median 7 distinct materials per stack) and optically opaque (mean absorption 0.46, with 14% metal and 28% absorber / phase-change layers), because random multimaterial stacks mostly absorb. Stages 2 and 3, dominated by the Powell-optimized interference designs, move sharply toward low-loss dielectric filters: dielectric usage rises to 93% then 96%, mean absorption falls to 0.21 then 0.17, mean transmission rises from 0.09 to ≈ 0.32, and the median stack uses 4 then 2 distinct materials,
32
Table B1 Training-data summary. Statistics estimated from sampled shards (≈0.15–0.2M designs per corpus); corpus sizes are the full shard totals.
Property
20l (Stage 1)
60l (Stage 2)
100l (Stage 3)
approx. corpus size ˜110M ˜1.6M ˜2.5M layer range 2–11–20 2–29–60 2–47–100 (min–median–max) wavelength coverage 380–1400 380–1400 380–1400 (nm) single-band fraction 70% 89% 93% median band width 317 327 328 (nm) mean R / T / A 0.46 / 0.09 / 0.46 0.47 / 0.32 / 0.21 0.51 / 0.32 / 0.17 in-band R std 0.151 0.203 0.223 (structure) dielectric / metal / 58% / 14% / 28% 93% / 3% / 4% 96% / 2% / 2% absorber–phasechange unique materials/stack 7 4 2 (median) composition 100% forward-short 37% fwd / 63% 24% fwd / 76% Powell-long Powell-long
(a) Layer-count distribution
(b) Material usage 20l (2–20) 60l (2–60) 100l (2–100)
0.07
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Ti
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TiO2
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Ta2O5
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ss-a
h4
SiO2
gst-b
0.0000 gst-a
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Cr
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Ag
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Al2O3
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density
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(c) Target band width 20l (2–20) 60l (2–60) 100l (2–100)
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% of all layers
density
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(f) 100l composition (forward-short + Powell-long families)
(e) Spectral structure mean R mean T mean A
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t_
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Fig. B1 Training-data overview. Sampled-shard statistics of the three curriculum corpora (20l / 60l / 100l, Table B1): a, layer-count distribution; b, per-layer material usage; c, single-band target width; d, per-sample mean R/T/A of the 100l corpus; e, spectral structure (in-band reflectance standard deviation); f, 100l composition across the forward-short generator and the six Powell-long target families.
33
matching periodic two-material interference coatings. In parallel the targets become more spectrally structured (in-band reflectance std 0.151 → 0.203 → 0.223) and more single-band (70% → 89% → 93%), while the wavelength envelope (380–1400 nm) and median band width (≈ 320 nm) stay fixed. This matches the curriculum rationale in Methods: establish the input–output mapping on the comparatively well-posed shortstack regime, then progressively expose the model to the structured, more degenerate long-stack regime that the application benchmark ultimately probes. B.3 Training-material optical constants. Candidate optical constants are direct model inputs, so the exact effective curves form part of the reported model and training-data definition. Table B2 records the native tabulated support of every training material together with the range of the effective curve supplied to the model. Figure B2 plots all 15 effective curves, separated into dielectrics, metals and absorbers / phase-change materials so that their different scales remain readable. The exact effective training curves are archived as a checksummed machinereadable library: all 15 materials on the 1-nm 380–1400 nm grid (15 × 1021 × 2 (n, k ) values); the library is available from the corresponding author on request (Data availability). The native table supports in Table B2 document the support of the laboratory-measured records; the effective training arrays are archived directly and are not reconstructed from those files. When an effective curve is queried outside its tabulated support, its nearest endpoint value is used; no new dispersion trend is fitted or extrapolated. Training and the reported in-range evaluation use only these 380–1400 nm effective curves. Dielectrics
Metals
(a)
Al2O3 h4 SiO2 Ta2O5 TiO2
2.6 2.4
3.5
Absorbers / phase-change
(b)
(c) 6
3.0 5
n
n 2.0
Ag Cr Ti
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4
1.5
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3 gsst-a gsst-b gst-a gst-b
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2.5 2.2
Metals 10
(d)
Si ss-a ss-b
Absorbers / phase-change
(e)
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(f)
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9 3.0
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k
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0.000 400
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800 1000 Wavelength (nm)
1200
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400
600
800 1000 Wavelength (nm)
1200
1400
0.5 0.0 400
600
800 1000 Wavelength (nm)
1200
1400
Fig. B2 Effective training-material optical constants. Refractive index n (top) and extinction coefficient k (bottom) for every material in the fixed 15-material training library. Curves are grouped by material family but plotted from the exact effective arrays supplied to the model.
34
Table B2 Effective optical constants used for training. The native tables are laboratory-measured material records retained in local material-table files (§F.5). Codes: gst = GST (germanium antimony telluride), gsst = Se-substituted GSST, ss = Sb2 S3 ; the -a/-b suffixes denote the amorphous and crystalline states.
Material Family
Native tabulated support (nm)
Ag
Metals
400.0–2500.0 210
380–1400
Al2 O3
Dielectrics
422.9–980.5
9
380–1400
Cr
Metals
354.2–1239.8 51
380–1400
gsst-a
Absorbers / phase-change Absorbers / phase-change Absorbers / phase-change Absorbers / phase-change Dielectrics
400.0–1600.0 121
380–1400
400.0–1600.0 121
380–1400
350.0–2500.0 216
380–1400
350.0–2500.0 216
380–1400
400.0–800.0
37
380–1400
gsst-b gst-a gst-b
Native points
Effective Effective n Effective k training range range support (nm)
h4 (LaTiO3 ) Si Absorbers / phase-change SiO2 Dielectrics
430.0–2497.3 152
380–1400
420.0–2457.0 87
380–1400
ss-a
401.0–1600.0 1200
380–1400
401.0–1600.0 1200
380–1400
Ta2 O5
Absorbers / phase-change Absorbers / phase-change Dielectrics
400.0–2500.0 214
380–1400
Ti
Metals
380.0–1100.0 64
380–1400
TiO2
Dielectrics
412.6–3000.0 24
380–1400
ss-b
Appendix C
0.0516– 0.2084 1.5992– 1.6384 1.9175– 3.6807 3.3180– 4.0750 2.6049– 5.3196 2.6177– 4.4526 1.8615– 6.4925 1.9667– 2.0574 3.3483– 4.8812 1.4372– 1.4504 2.7506– 3.6672 3.3321– 4.2028 2.0528– 2.2207 1.5000– 3.3000 2.2392– 2.7094
1.7209– 9.7945 0.0000– 0.0002 2.7364– 3.7844 0.0000– 2.0526 0.5354– 3.1509 0.0756– 2.1882 1.3338– 3.5459 0.0016– 0.0120 0.0027– 1.2609 0.0000– 0.0000 0.0003– 1.5681 0.0048– 2.0705 0.0000– 0.0038 2.0000– 3.5000 0.0000– 0.0024
Model architecture and training configuration
The reported model is the single Lmax = 100 checkpoint (136.3M parameters, Stage-3 epoch 79). Its architecture is fixed across all three curriculum stages; only the maximum active layer count and the training corpus change. Tables C1–C2 give the full architecture and training configuration. Common training settings (all stages). Fused AdamW (weight decay 0.01), bf16 mixed precision with TensorFloat-32 (TF32) matrix multiplications, torch.compile (max-autotune), an exponential moving average (EMA) of weights (decay 0.999), and a shared warmup–cosine LambdaLR schedule. The main LR warms from the per-stage base LR to max lr over pct start=0.03 and then cosine-decays to the explicit min lr reported in Table C2. The same LR multiplier is applied to 35
Table C1 Architecture. The reported 20-block settings are identical for Stages 1–3; the 12-block checkpoint is included as a parameter-count reference.
Component
Setting
Total parameters (20-block reported model) ≈ 136.3 M Total parameters (12-block reference model) ≈ 85.8 M Denoising backbone 20 Transformer blocks (the smaller one has 12 Transformer blocks), dmodel = 512, 8 heads, Swish-gated linear unit (SwiGLU) feed-forward network (FFN) [45] (×4.0), dropout 0.1 Normalization / position query–key normalization; RoPE over layer index (base 1000) Target-spectrum encoder curve-token encoder (flat mode): multilayer perceptron (MLP; hidden 1024, depth 2) over the flattened 128-point two-channel curve plus per-point wavelength features → one conditioning token; 2 channels (R, T ) Candidate n, k encoder same curve-token architecture as the spectrum branch; one token per candidate curve, forming the candidate memory used for denoiser cross-attention, material-state embedding and material-head scoring Wavelength features signed inverse-wavelength channel 2 (0.30 µm)/λ − 1 per grid point, shared by both tokenizer branches Thickness head continuous flow matching, velocity (v) target, window [5, 300] nm, linear interpolant path xt = (1 − t)x0 + tϵ Material head DFM on a uniform-kernel CTMC over the local candidate bank; log-linear schedule (ᾱend = 10−4 ); CTMC-soft reweighting λsoft = 0.1 Flow-matching timesteps 1000 Joint loss L = Lth + 0.4 Lst Material vocabulary 15 (curve-injected; no ID embedding)
all optimizer parameter groups. Gradient clipping is applied by parameter partition: the main parameters are clipped at 1.0, while the curve-token spectrum encoder and the n, k -curve/candidate encoder parameters are grouped together and clipped at 5.0. Global batch size is 1024 and the seed is 42. The candidate optical constants are interpolated online from a fixed 15-material n, k library. Stage 1 was trained on one NVIDIA RTX PRO 6000D GPU; Stages 2 and 3 used one NVIDIA RTX PRO 6000D and two NVIDIA L20 GPUs jointly through NVIDIA Collective Communications Library (NCCL) distributed data parallel (DDP). Final training losses are not directly comparable across stages because the corpora differ (§B); the joint loss combines the thickness flow-matching mean-squared error (MSE) and the CTMC-reweighted material cross-entropy with the λst = 0.4 weight above. The model, optimizer and EMA states are stored per checkpoint; inference uses the Stage-3 epoch-79 EMA weights, by which point the held-out spectrum error has saturated (Fig. C2). Fig. C3 repeats the Stage-2 and Stage-3 checkpoint sweeps
36
Table C2 Per-stage training. The architecture is shared; each stage expands the maximum active layer count and fine-tunes from the previous stage (Stage 1 is trained from scratch). Stage 1 comprises two sub-stages, reported as one column (“+” separates the sub-stages): a 5-epoch warmup–cosine run from scratch followed by a 2-epoch warmup–cosine warm restart at reduced LR.
Stage 1 corpus (§B, Table B1) max active layers initialized from
20l 20 scratch (sub-stage 2 warm-restarts sub-stage 1) epochs 7 (5 + 2) steps / epoch 107,422 total optimizer steps 751,954 (537,110 + 214,844) samples seen ≈ 770 M GPUs (NCCL DDP) 1 global batch (per-rank) 1024 base / max / min LR
spectrum-encoder, n, k-encoder LR wall-clock final training loss
1.2e-5 / 1.2e-4 / 2e-5, then 2e-5 / 4e-5 / 5e-6 1.2e-5
Stage 2
Stage 3 (reported)
60l 60 Stage 1
100l 100 Stage 2 (epoch 79)
80 ≈ 1,553 124,240
80 ≈ 2,432 194,560
≈ 127 M ≈ 199 M 3 3 1024 1024 (256/512/256) (256/512/256) 1e-5 / 5e-5 / 1e-6 1e-5 / 5e-5 / 1e-6 1e-6
1e-6
≈ 32 h (23 + 9, 1 GPU) ≈ 7.5 h (3 GPU) ≈ 18 h (3 GPU) 1.12 0.71 0.61
with the evaluation set split by depth, separating what each fine-tune is learning (long stacks) from what its replay fraction must preserve (short stacks).
Appendix D
Sampler hyperparameters: timesteps and best-of-N
The two inference knobs (the number of sampler timesteps and the best-of-N pool size) are characterized by the two studies below. The timesteps study (§D.1) sweeps the full candidate distribution of three representative queries (an 8-layer, a 40-layer and an 80-layer design) across 28 step budgets. The best-of-N study (§D.2) runs on one fixed held-out validation subset (1000 targets, seed 42, all layer counts), not the curated Tier 1/2/3 benchmark; its absolute RMSE sits well above the curated tier medians (a mixed-depth mean with no minimum-structure floor, dominated by a few hard deep stacks), so both studies are read only as relative trends for hyperparameter selection. D.1 Timesteps. We swept the sampler step budget on three representative queries spanning the supported depth range: an 8-layer edge filter, a 40-layer all-dielectric high-contrast target and an 80-layer all-dielectric high-contrast target. Each query used the same 28 step budgets (10–20 in unit steps, 30–100 in tens, 200–1000 in hundreds), 2500 stochastic candidates per budget (five seeds × 500 samples), the reported checkpoint’s EMA weights and the adaptive-power schedule of §D.3; every
37
Stage 1 20 layers
1.6
Stage 2 60 layers
Stage 3 100 layers
total training loss
1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 2.25
thickness loss stack loss
2.00 component loss
1.75 1.50 1.25 1.00 0.75 0.50 0.2
0.4
0.6 cumulative training step
0.8
1.0 1e6
Fig. C1 Multi-stage progressive training loss. Joint training loss across the three curriculum stages of Table C2, laid end-to-end on a shared cumulative-step axis (Stage 1: 20 layers from scratch, including its 2-epoch warm-restart sub-stage, whose start is marked by a dotted line; Stage 2: 60 layers; Stage 3: 100 layers, each warm-started from the previous checkpoint). Top: the total loss L = Lth + 0.4 Lst (raw in gray, per-stage EMA in color). Bottom: the thickness (Lth ) and stack (Lst ) joint-loss components. Each stage begins near the previous stage’s loss level and then improves; the level shifts at the two stage boundaries (dashed) because the corpus and maximum layer count change (§B), so absolute levels are not directly comparable across stages.
candidate was re-simulated and scored by combined R, T RMSE before any best-of-N selection. The selection-relevant view fixes total sequential model evaluations, B = steps ×N , and estimates the expected best-of-N candidate by exact order statistics from the empirical candidate distributions. Across budgets from 750 to 6000 model evaluations, the optimum remains in a low-step band on all three queries: 10–12 steps for the 8layer case, 10–13 for 40 layers and 10–16 for 80 layers, with the benchmark’s 15-step setting inside the statistically tied band for the deeper query and close to it for the shorter ones (Fig. D1). The reason is visible in the per-draw distribution (Table D1): fewer steps degrade the median but leave the low-error p10 tail nearly flat, so at fixed compute extra draws are usually more valuable than extra refinement of each draw. The per-draw (equal-N ) view is tabulated in Table D1. Per-draw quality is nonmonotone: on the eight-layer query the median improves to a shallow optimum around 30 steps and then slowly degrades. The benchmark’s 15-step setting is therefore not the per-draw optimum for every query, but it sits close to the tied low-step region while preserving the equal-compute selection advantage.
38
combined R/T spectral RMSE
0.14
Stage 1 20 layers
Stage 2 60 layers
Stage 3 100 layers
0.12 0.10 0.08 0.06 0.04 mean median
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R component T component
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Fig. C2 Held-out spectrum error across training checkpoints. The deliverable counterpart of Fig. C1: the training loss is a flow-matching/cross-entropy surrogate, so its minimum need not coincide with the best designs. Here every saved checkpoint (EMA weights, every 6,000 optimizer steps) instead samples designs that are re-simulated with TMM against a fixed 1,000-target heldout test set matched to the stage’s depth range (Stage 1: 2–20 layers, Stage 2: 2–60, Stage 3: 2–100; best-of-20 draws per target, 15 sampler timesteps, seed 42), laid on the same cumulative-step axis as Fig. C1 (stage boundaries dashed, Stage 1’s warm-restart sub-stage dotted). Top: mean (solid) and median (dashed) best-of-20 combined R, T spectral RMSE. Bottom: the R and T components. Within each stage the spectrum error of sampled designs falls and then saturates (Stage 1: 0.112 → 0.053 mean RMSE across the two sub-stages; Stage 2: 0.118 → 0.099; Stage 3: 0.115 → 0.110), with checkpoint-to-checkpoint differences beyond ≈ 130 k Stage-3 steps within sampling noise, supporting the epoch-79 checkpoint used everywhere else in the paper. Because each stage is scored on one fixed evaluation set, the within-stage trajectory directly tracks design fidelity; as in Fig. C1, absolute levels are not comparable across stages, since the evaluation set follows the curriculum and deeper stacks are intrinsically harder (§B). (Best-of-20 trials use batched RNG streams in Stages 1–2 and per-trial seeds in Stage 3; this changes only the random-number handling, not the within-stage trend.)
We therefore use one cross-tier setting, 15 steps with N = 100 or 500 depending on the benchmark suite. It is slightly above the equal-compute optimum but statistically tied with the low-step band at the Tier 1/2 operating budget, and it remains close to the per-draw optima across depths. D.2 Best-of-N . Holding the sampler at the benchmark’s 15-step budget (plain evenly spaced steps; §D.3 shows the deployed adaptive-power schedule is better or equal at this step count on every query, so the trend read here is, if anything, conservative), we draw a pool of stochastic samples per target and report the lowest re-simulated combined RMSE (oracle-ranked). Accuracy improves monotonically with N with strongly diminishing returns (Table D2): most of the gain is in place by N ≈ 32–64, and growing the pool from the benchmark’s N = 100 all the way to 39
Stage 2 (60-layer fine-tune)
Stage 3 (100-layer fine-tune)
combined R/T spectral RMSE
0.175 0.150 0.125
long stacks (21–60 layers) short stacks (2–20 layers) mean median
0.100 0.075
long stacks (21–100 layers) short stacks (2–20 layers) mean median
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Fig. C3 Short-stack retention vs. long-stack learning during the staged fine-tunes. The checkpoint sweep of Fig. C2 with the evaluation set split by depth instead of following the curriculum: every Stage-2 (left) and Stage-3 (right) checkpoint (EMA weights, every 6,000 optimizer steps) is scored on a fixed 1,000-target short-stack set (2–20 layers, the Stage-1 evaluation set of Fig. C2; blue) and a fixed 1,000-target long-stack set (Stage 2: 21–60 layers, green; Stage 3: 21–100 layers, red), with best-of-20 draws per target, 15 sampler timesteps and seed 42; mean (solid) and median (dashed) combined R, T spectral RMSE. In both stages the long-stack error on the newly introduced depth range falls steadily as fine-tuning progresses (Stage 2: 0.176 → 0.132 mean RMSE; Stage 3: 0.150 → 0.138), while the short-stack error of the replayed regime stays essentially flat (Stage 2: 0.056 → 0.060; Stage 3: 0.059 → 0.068), a drift of less than 0.01 RMSE around a level roughly half the longstack error. The replayed short stacks in the Stage-2/3 corpora (§B) therefore prevent catastrophic forgetting of the short-stack mapping while each fine-tune learns the longer, more degenerate regime. As in Fig. C2, the depth ranges differ in intrinsic difficulty, so short- and long-stack levels are not directly comparable.
N = 1000 trims the mean combined RMSE only a further ˜16% (the median ˜21%). The single deterministic greedy draw (N = 1) is a different sampling mode and is much worse than even best-of-2 stochastic sampling, so the gain cannot come from selection alone: the stochasticity itself is what places a good design within reach. Sampling cost scales linearly with N , so the operating N is a budget choice along this curve; the one-to-many spread that makes best-of-N effective is characterized separately in the solution-diversity analysis of §E.2. D.3 Adaptive stack-time schedule and matched-compute ablation. The reported benchmarks use an adaptive-power stack-time schedule (p = 2, 15 steps) rather than evenly spaced time nodes. At each update, the exponent e = 1 + (p − 1)u is set by the current normalized entropy u of the material posterior, so uncertain early states take larger steps and resolved late states take smaller ones. Figure D2 shows one realized schedule; Figure D3 and Table D3 compare it with a uniform 15-step schedule at matched compute. The ablation uses the three queries of §D.1, 5000 seed-paired candidates per query and schedule, and the same TMM re-scoring for every candidate. Adaptive-power is better or equal at every pool size on all three queries: it substantially improves the 8and 40-layer per-draw distributions, remains neutral-to-positive on the 80-layer case, and improves the expected best-of-100 by 17% and 16% on the short and medium queries while staying within ∼ 3% on the long query (Table D3). It also concentrates
40
A. Tied step band — 8-layer query
B. 40-layer query
C. 80-layer query
B = 6000
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per-budget optimum benchmark (15 steps)
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0.08
B = 750 NFE B = 1500 NFE B = 3000 NFE B = 6000 NFE
0.06 10
1
10
2
steps per candidate
10
3
expected best-of-(B/steps) RMSE
0.12
expected best-of-(B/steps) RMSE
expected best-of-(B/steps) RMSE
0.16 0.14
0.30
0.25
0.20
0.15
B = 750 NFE B = 1500 NFE B = 3000 NFE B = 6000 NFE
0.10
0.05 10
1
10
2
10
3
steps per candidate
Fig. D1 Equal-compute timesteps selection across stack depths. The step budget is chosen at fixed total model evaluations B = steps × N ; the expected best-of-N candidate (N = B/steps) is estimated by exact order statistics on the 2500 empirical candidates per setting. Columns: 8-, 40- and 80-layer queries. (A–C) Step settings statistically tied with each budget’s optimum (inside its 95% bootstrap CI); stars: per-budget optima; diamond: the Tier 1/2 benchmark operating point (15 steps × N = 100) on its B = 1500 row; dotted line: the 15-step setting. (D–F) Expected best-of-(B/steps) combined R, T RMSE versus steps at fixed total budget (B = 750–6000 model evaluations; shaded bands: 95% bootstrap CIs; stars: per-budget optima; dashed line: 15 steps). The equal-compute optimum stays in a low-step band on all three queries (10–12 steps 8-layer, 10–13 40-layer, 10–16 80-layer, each widening at larger budgets) and the 15-step setting sits inside the 80-layer band. The per-draw mechanism (the median degrades as steps shrink while the best-of-N p10 tail stays nearly flat) is read from Table D1.
material-sequence sampling without hurting best-of-N quality, so the uniform schedule is not preferable at the deployed step budget.
Appendix E
Sampler dynamics and solution diversity
E.1 Reverse-trajectory dynamics of the discrete-flow material head. The material head is a uniform-CTMC discrete-flow-matching sampler (Methods; §C): at every reverse-time step it emits a categorical clean-material posterior P (x0 ) over the candidate material at each layer, which is combined with the CTMC bridge to form the reverse-step sampling distribution for the material IDs. To make the qualitative behavior of this process visible we saved the full reverse trajectory of one representative joint run: an eight-layer design against a combined R, T target, drawn from a five-material pool {SiO2 , Ag, Ti, GSST-a, h4}, where GSST-a denotes the processcorrected curve of §N, as in the §D.1 sweep. The run drew 2500 stochastic candidates; the trajectory shown is that of the design selected by angular MSE (§G.2) rather than by RMSERT , so the displayed stack (combined R, T RMSE 0.028) is not the single lowest-RMSE draw of the same run, which reaches 2.5 × 10−2 (§O). The selected stack 41
Table D1 Sampler-timesteps distribution sweep (per-draw / equal-N view). One representative eight-layer query (setup above), 2500 seed-paired stochastic candidates per step budget, the reported checkpoint’s EMA weights, adaptive-power stack-time schedule; statistics are over the per-candidate combined R, T RMSE with no best-of-N selection. “Better vs 1000” is the fraction of seed-paired candidates scoring better than their 1000-step counterparts. Per-draw median quality peaks at 30 steps (†) and degrades slowly at larger budgets; the equal-compute selection derived from these distributions is shown in Figure D1. steps
mean
median
p10
p90
better vs 1000
10 11 12 13 14 15 16 17 18 19 20 30 † 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000
0.1754 0.1722 0.1692 0.1664 0.1643 0.1652 0.1653 0.1625 0.1597 0.1605 0.1599 0.1546 0.1545 0.1551 0.1545 0.1565 0.1589 0.1584 0.1606 0.1675 0.1687 0.1705 0.1710 0.1693 0.1704 0.1708 0.1706 0.1690
0.1628 0.1627 0.1595 0.1576 0.1536 0.1550 0.1568 0.1542 0.1502 0.1535 0.1513 0.1448 0.1452 0.1482 0.1457 0.1476 0.1485 0.1476 0.1519 0.1552 0.1594 0.1598 0.1603 0.1597 0.1592 0.1602 0.1610 0.1565
0.0893 0.0866 0.0851 0.0848 0.0856 0.0871 0.0860 0.0859 0.0845 0.0849 0.0845 0.0821 0.0800 0.0813 0.0838 0.0842 0.0847 0.0855 0.0845 0.0858 0.0878 0.0899 0.0897 0.0900 0.0884 0.0911 0.0898 0.0890
0.2740 0.2700 0.2667 0.2600 0.2580 0.2540 0.2534 0.2468 0.2453 0.2470 0.2451 0.2416 0.2422 0.2381 0.2377 0.2418 0.2479 0.2449 0.2451 0.2625 0.2651 0.2655 0.2660 0.2643 0.2626 0.2651 0.2612 0.2662
48.7% 48.9% 50.2% 51.7% 52.4% 52.2% 51.8% 54.0% 53.7% 52.4% 55.2% 57.2% 56.6% 56.3% 57.7% 55.8% 54.8% 54.2% 52.8% 51.9% 50.5% 50.1% 49.2% 50.1% 50.2% 48.9% 47.9% —
is Ag/SiO2 /GSST-a/h4/GSST-a/SiO2 /GSST-a/SiO2 (73/36/21/45/43/36/7/75 nm). Figures E1–E3 visualize how the posterior over materials, and the underlying CTMC jump kernel, evolve along the reverse path. This is an illustrative single run, not a benchmark statistic. Three features are robust across layers. (i) Resolution is progressive and backloaded. The mean normalized posterior entropy H (P (x0 ))/ log V starts at 1 (uniform prior), drops to ≈ 0.65 after the first update, holds a long metastable plateau through the middle of the trajectory, and only collapses to ≈ 0.31 over roughly the final third; correspondingly the mean posterior mass on the eventually selected material rises from ≈ 0.2 to ≈ 0.77, most of it in the last few recorded steps (Fig. E1b,c). (ii) Commitment timing is layer-dependent. The bottom layer locks onto Ag almost immediately (nearzero entropy from the first recorded step), whereas interior layers remain genuinely in contention between competing candidates (e.g. SiO2 versus GSST-a, or Ti) until late, 42
Table D2 Best-of-N sweep. Fixed held-out validation subset (1000 targets, see the §D preamble), 15 evenly spaced steps (the benchmark’s step budget), oracle-ranked by re-simulated combined RMSE. N = 1 is a single deterministic greedy draw (different sampling mode); N ≥ 2 are stochastic draws selected from one shared 1000-sample pool, so the rows are nested prefixes and the curve is monotone. N
mean RMSE RT
median RMSE RT
p90 RMSE RT
sampling
1 2 4 8 16 32 64 100 128 256 512 768 1000
0.5884 0.1835 0.1627 0.1476 0.1352 0.1253 0.1180 0.1132 0.1107 0.1049 0.0997 0.0968 0.0951
0.6253 0.1614 0.1414 0.1204 0.1048 0.0926 0.0846 0.0801 0.0765 0.0724 0.0663 0.0646 0.0630
0.8276 0.3499 0.3188 0.2990 0.2892 0.2744 0.2676 0.2615 0.2595 0.2454 0.2341 0.2315 0.2259
deterministic stochastic stochastic stochastic stochastic stochastic stochastic stochastic stochastic stochastic stochastic stochastic stochastic
Table D3 Schedule comparison summary. Per query and schedule: median and p10 of the per-candidate combined R, T RMSE, the expected best-of-100 and best-of-500 by exact order statistics, the pool minimum (best-of-5000), the fraction of seed-paired candidates where adaptive-power scores better than uniform, and the number of unique material sequences among the 5000 draws.
query
schedule
median p10
8L edge filter 8L edge filter 40L dielectric 40L dielectric 80L dielectric 80L dielectric
adaptivepower uniform
0.1558 0.0872 0.0450
0.0329
0.2248 0.1155 0.0541
adaptivepower uniform adaptivepower uniform
E[best-of- E[best-of- best-of100] 500] 5000
paired wins
unique seq.
0.0227
73.6%
1856
0.0370
0.0225
—
3260
0.1607 0.1168 0.0791
0.0655
0.0540
81.5%
12
0.2117 0.1424 0.0945
0.0786
0.0678
—
4084
0.3147 0.1855 0.0876
0.0551
0.0321
56.3%
104
0.3208 0.1907 0.0899
0.0572
0.0320
—
465
with several posterior crossovers (Figs. E1, E2). (iii) The jump kernel concentrates as t → 0. The CTMC transition kernel K (t) = exp(τ (t) Q) moves from near-uniform mixing toward a concentrated stay-kernel over the schedule (Fig. E3), so the per-step reverse-transition entropy collapses in the final steps. This entropy-resolution profile is precisely the per-position uncertainty signal that the optional adaptive-power schedule (§D.3) reads out online. E.2 Solution diversity: the one-to-many structure behind best-of-N . Best-of-N selection helps only if independent draws land in different basins of the inverse map. We test this on the four color-displaying coolers of the main text (Fig. 4): 43
(a) Reverse-time nodes 1.0
zero-uncertainty width tk /mk
0.12
power-2 grid (u ≡ 1) adaptive nodes tk
0.8
full-uncertainty width adaptive Δtk
0.10 step width Δtk
reverse time t
(b) Per-step widths
uniform grid (u ≡ 0)
0.6
0.4
0.08 0.06 0.04
0.2 0.02 0.0
0.00 2
4
6 8 10 update index k
12
14
0
2
6 8 update index k
10
12
14
(d) Step width vs. uncertainty
14
2.0
0.8
1.8
0.6
1.6
0.4
1.4
0.2
1.2
0.0
1.0 0
2
4
6 8 update index k
10
12
0.10
step width Δtk
1.0
power exponent ek = 1 + (p − 1) uk
uncertainty uk (normalized entropy of P(x0 ))
(c) Posterior uncertainty and exponent
4
12 10
0.08
8 0.06
6
update index k
0
4
0.04
2 0.02
14
0 0.3
0.4 0.5 uncertainty uk
0.6
Fig. D2 Adaptive-power stack-time schedule. A representative adaptive-power run (15 fixedbudget updates, p = 2). Each update applies tk+1 = tk ((mk − 1)/mk )ek with mk the remaining update budget and ek = 1 + (p − 1) uk set by the current uncertainty uk . a, The realized reverse-time nodes tk between the rule’s two analytic envelopes: the uniform grid recovered at zero uncertainty (ek ≡ 1) and the power-p grid at full uncertainty (ek ≡ p). b, The per-step width ∆tk against the same zero- and full-uncertainty reference widths; high early entropy drives larger early steps. c, The per-step uncertainty uk (normalized entropy of P (x0 )); the right axis reads out the induced exponent ek . d, Step width versus uncertainty, colored by update index.
for each exact-color target we take the eight lowest-RMSE compliant candidates of its 12,500-sample pool (§N) and re-simulate every stack with the TMM solver. Because the pool mixes requested layer counts 4–8, these near-best sets vary in depth as well as material order and thickness. We report, per target (Table E1): material disagreement ⟨∆mat⟩ (mean pairwise fraction of differing layer positions over the union length, a missing layer counting as a mismatch), thickness RMS ⟨∆d⟩ (over the common prefix) and spectral spread (mean pairwise RMS difference of the re-simulated R, T spectra). The cooler design space is strongly multimodal. All four targets realize five to eight distinct material sequences among their eight near-best designs, with material disagreement 0.35–0.62 and thickness differences of 20–71 nm. The near-black target commits hardest (five sequences, disagreement 0.35), magenta is the most multimodal (eight sequences) and yellow shows the largest material disagreement (0.62). Yet the spectral spread stays small (0.013–0.057): these structurally distinct stacks (differing in materials, thicknesses and layer count) are spectrally near-equivalent; this is the one-to-many degeneracy that makes stochastic sampling with best-of-N selection the
44
(B) Per-draw RMSE distributions 1.0
adaptive-power uniform short (8L, edge filter) medium (40L, dielectric) long (80L, dielectric)
0.30 0.25
ECDF
0.20
0.8
0.15 benchmark N = 100
0.10
0.6 0.4 0.2
0.05 0.0 10
0
1.0
10
1
2
10 pool size N
10
3
0.1
(C) Seed-paired win rate (adaptive better)
0.8 fraction of paired draws
0.0
0.6
0.4
0.2
10
medium (L=40)
0.7
adaptive-power uniform
4
4084
3260 1856
10
3
465
10
10
0.0 short (L=8)
0.2 0.3 0.4 0.5 0.6 per-candidate combined R, T RMSE
(D) Material-sequence concentration unique material sequences / 5000 draws
expected best-of-N combined R, T RMSE
(A) Equal-step best-of-N (15 steps)
104
2
12
1
long (L=80)
short (L=8)
medium (L=40)
long (L=80)
Fig. D3 Adaptive-power versus uniform stack-time schedule. Three queries (8/40/80 layers), both schedules at the benchmark’s 15-step budget, 5000 seed-paired candidates per (query, schedule) cell, the reported checkpoint’s EMA weights. (A) Expected best-of-N combined R, T RMSE by exact order statistics on the empirical candidate distributions (solid: adaptive-power; dashed: uniform; dotted line: the benchmark pool N = 100); adaptive-power is better or equal at every pool size on all three queries. (B) Per-draw RMSE ECDFs: the per-draw distribution improves wholesale on the 8- and 40-layer queries and marginally on the 80-layer query. (C) Fraction of seed-paired candidates scoring better under adaptive-power (Wilson 95% intervals; dotted line: parity). (D) Unique material sequences among the 5000 draws (log scale): the adaptive schedule concentrates material-sequence sampling without degrading best-of-N quality.
right inference mode here (§D.2). Figure E4 overlays the near-best re-simulations and stack diagrams per target. Table E1 Solution diversity of the cooler designs (eight lowest-RMSE compliant candidates per target).
Target
k near-best RMSE range #distinct ⟨∆mat⟩ ⟨∆d⟩ nm spectral spread layers
black cyan magenta yellow
8 8 8 8
0.0258–0.0269 0.0526–0.0742 0.0396–0.0539 0.0434–0.0510
5 7 8 6
45
0.35 0.54 0.51 0.62
20 71 53 67
0.013 0.057 0.030 0.041
4–7 7–8 6–8 6–8
(a) sampled material id 7
4
6
3
layer
5 4
2
3 2
1
1
0
0 15
13
10
8
6
4
1
-1
(b) posterior entropy H(P(x0 ))/log V
1.0
7 6
0.8
layer
5
0.6
4 3
0.4
2
0.2
1 0
0.0 15
13
10
8
6
4
1
-1
(c) posterior mass on final material
1.0
7 6
0.8
layer
5
0.6
4 3
0.4
2
0.2
1 0
0.0 15
13
10
8
6
4
1
-1
recorded reverse step
Fig. E1 Discrete-flow resolution dynamics. Reverse trajectory of the example eight-layer run, one row per layer, columns = recorded reverse steps (early at left). a, The discrete sampled material ID per step. b, The normalized posterior entropy H(P (x0 ))/ log V (bright early/uncertain, dark late/resolved; layer mean 1 →≈ 0.31); the bottom layer (Ag) is dark from the first step (immediate commitment) while interior layers stay bright longer. c, The posterior mass P (x0 ) on the finally selected material, rising toward 1 (layer mean ≈ 0.2 → 0.77), showing the initial-drop / metastableplateau / late-resolution shape of discrete commitment. layer 1
layer 2
layer 3
layer 5
1.0
0.8
0.6 P(x0 )
SiO2 Ti h4 gsst-am Ag
0.4
gsst-am Ti Ag h4 SiO2
h4 SiO2 Ti gsst-am Ag
SiO2 h4 gsst-am Ti Ag
0.2
0.0 15
SiO2
11
7
3
-1
15
11
7
3
-1
15
11
7
3
-1
15
11
7
reverse step
reverse step
reverse step
reverse step
h4
h4
Ti
h4
Ti
gsst-am
Ti
h4
SiO2
SiO2
3
-1
gsst-am
Fig. E2 Per-layer material competition. For the four layers with the highest mean posterior entropy: top, the posterior probability P (x0 ) of each layer’s five most-probable materials across reverse steps, with the finally selected material dashed and late crossovers (e.g. SiO2 overtaking Ti) visible; bottom, the same posteriors projected onto the 2-simplex of each layer’s three leading materials (marker color encodes reverse-step index), migrating from the barycenter (uncertain) toward a vertex (committed).
46
K(t) at step 14, τ=9.21
K(t) at step 7, τ=4.91
K(t) at step 0, τ=0.614
kernel diagnostics (log-linear)
1.0 SiO2
1.0 0.8
Ag
SiO2
1.0 0.8
Ag
0.6 Ti
0.0
h4
0.6
0.2 0.0
h4
0.7 mean K(t)[i, i]
0.6
0.4 gsst-am
0.9
0.8 0.8
Ag
Ti
0.4 0.2
1.0
0.6 Ti
gsst-am
SiO2
0.4 gsst-am
mean row entropy / log V
0.5
0.2 0.4 0.0
h4
0.3
h4
gsst-am
Ti
Ag
SiO2
h4
Ti
gsst-am
Ag
SiO2
h4
Ti
gsst-am
Ag
SiO2
0.2 0
5
10
forward schedule step
Fig. E3 CTMC transition kernel. The exact discrete-flow kernel K(t) = exp(τ (t) Q) at an early, middle and late schedule step for the five-material bank, where Q is the uniform CTMC rate matrix and τ (t) = − log ᾱ(t): early t gives near-uniform rows (strong mixing), late t a concentrated staykernel. Also shown are the mean stay probability K(t)[i, i] and the mean row entropy across the forward schedule, which quantify the mixing→freezing transition that drives the resolution profile in Fig. E1.
Appendix F
Benchmark construction
F.1 Layer×wavelength grid (Tier 1 and Tier 2). Both tiers share a crossed grid of 7 layer-count bins × 6 spectral bands × 2 material-availability modes = 84 cells per tier, evaluated on 100 held-out targets per cell (Fig. F1a). The layer bins are 2–5, 6– 10, 11–20, 21–40, 41–60, 61–80, 81–100 layers. The band positions are five single bands (UV–VIS, 380–550 nm; VIS, 400–700 nm; VIS–NIR, 500–900 nm; NIR, 800–1100 nm; extended-NIR, 1000–1400 nm) plus one in-distribution dual band (VIS 450–700 nm + NIR 850–1150 nm). The band bins are coarse position anchors, not fixed bands: at target-synthesis time each sample either uses the bin’s exact extent or a random sub-band (≥ 60 nm wide) drawn inside every base segment, so spectral width varies within each bin rather than being confounded one-to-one with position across bins. A dedicated band RNG, decoupled from the material/thickness stream, makes corresponding Tier 1 and Tier 2 cells draw identical bands. The cells are therefore matched in layer, wavelength support and availability mode, but their target stacks and spectra are independently realized from different material banks. The OOD/in-distribution ratio is therefore a matched-cell distribution comparison, not a target-matched degradation estimate. Targets are accepted only if their in-band spectral structure clears a minimum-variation floor (to avoid degenerate near-flat showcase targets); otherwise the underlying stack is redrawn. The two material-availability modes are FULL (all 15 bank materials offered as candidates) and NEEDED (only the materials a target uses, plus three distractors). F.2 Idealized target spectra (Tier 3). Tier 3 is a benchmark of idealized analytic target spectra of unknown attainability: each of the 46 cases prescribes a desired R, T shape, and IrisFlow is scored purely on how closely a fixed-budget design reproduces it. Each case is built from a shape primitive (flat, Gaussian peak/valley, step, linear gradient, multi-Gaussian, soft rectangle, or piecewise-segmented); all 46 target curves are plotted, with their nearest training neighbors, in Fig. F4 (§F.8). Every case is defined as a full two-channel R(λ), T (λ) target; a case whose specification prescribes a single channel is completed by the lossless complement (the other channel set to one minus the prescribed one, i.e. zero absorption), so both channels are always
47
0.026
0.026
0.026
SiO2 54
LaTiO3 54
0.026
0.027
0.027
0.027
0.027
LaTiO3 56
LaTiO3 51
1.0 SiO2 65
black reflectance
0.8
LaTiO3 41
SiO2 53
0.6
SiO2 62
SiO2 77
LaTiO3 32 GSST-a 33
SiO2 73 SiO2 85
0.4
SiO2 81
SiO2 43
SiO2 44
GSST-a 40 GSST-a 40
GSST-a 32
GSST-a 42
GSST-a 41
LaTiO3 62
SiO2 57
SiO2 58
SiO2 63
SiO2 88
Ag 78
0.2
Ag 73
Ag 75
Ag 73
Ag 76
0.071
0.072
Ag 68
Ag 68
Ag 67
0.0 400
600
800
1000
1200
1400 0.053
0.063
SiO2 114
SiO2 119
0.067
1.0
LaTiO3 67 SiO2 162
0.8
cyan reflectance
LaTiO3 79
SiO2 170
SiO2 47
0.6
Ag 30
0.4
SiO2 159
Ag 30
LaTiO3 108
Ag 73
Ag 71
Ag 69
0.074
SiO2 105
LaTiO3 112 Ag 30
SiO2 178
LaTiO3 123
0.074
SiO2 120
SiO2 166 SiO2 179
Ag 34
SiO2 168
SiO2 184
Ag 33
SiO2 174
SiO2 276
Ti 56
SiO2 168
LaTiO3 108 LaTiO3 86
Ti 33
SiO2 95
LaTiO3 66
Ti 47
Ag 52
SiO2 48
Ag 64
Ag 71
0.050
0.050
Ti 47
0.2
0.073 SiO2 91
Ag 64
0.0 400
600
800
1000
1200
1400 0.040
0.043
0.045
SiO2 125
LaTiO3 78
SiO2 127
0.051
0.053
0.054
SiO2 130
SiO2 123
1.0
0.8
Ag 33
Ag 37
LaTiO3 79
SiO2 127
Ag 35
SiO2 122 Ag 33
magenta reflectance
Ag 34 SiO2 151
0.6
SiO2 137
Ti 41 Ti 43
GSST-a 36
0.4
LaTiO3 176
Ag 67
SiO2 141
SiO2 139
Ag 40
Ag 35
Ag 33
SiO2 142
SiO2 156
SiO2 145
GSST-a 41
Ti 31 LaTiO3 98
0.2
Ag 33
SiO2 137
GSST-a 31 LaTiO3 175
Ag 55
Ti 30
Ti 46
Ag 50
Ag 71
0.048
0.049
LaTiO3 177
Ag 39
SiO2 83
SiO2 102
Ag 71
Ti 50
0.0 400
600
800
1000
1200
1400 0.043
0.045
0.047
0.049
0.050
0.051
1.0 SiO2 201
SiO2 91
LaTiO3 143 SiO2 96
0.8
yellow reflectance
SiO2 100 SiO2 104
0.6
SiO2 254
0.4
Ag 34 LaTiO3 60
Ag 36 Ag 62
SiO2 94
SiO2 93
SiO2 86
Ag 37
Ag 35 Ag 35
SiO2 105
SiO2 109
GSST-a 38
SiO2 107
SiO2 252
GSST-a 41
SiO2 91
SiO2 84
Ti 31
Ti 31
SiO2 61
Ag 68
Ag 62
Ag 78
GSST-a 41
SiO2 145
0.2
Ag 38
Ti 42 Ag 68
Ag 53
Ag 62
LaTiO3 91 Ag 56
SiO2 96
Ti 30 GSST-a 48
0.0 400
600
800
1000
1200
1400
wavelength (nm)
Fig. E4 Solution diversity of the cooler designs. Per target (rows): left, the re-simulated reflectance of the eight lowest-RMSE compliant candidates (thin lines) overlaid on the target (black; for the black target the unspecified 780–900 nm no-target band is shaded gray and its line broken); right, the corresponding stack diagrams (substrate side at the bottom; per-stack combined RMSE above each diagram). Structurally distinct stacks (different material sequences, layer counts and interference orders) realize near-identical colored-cooler responses; this one-to-many degeneracy is what best-of-N selection exploits (§D.2).
scored. Each case defines one analytic target (AR-5 additionally defines a second variant of its specification). Like the best-of-N pool, the requested layer count is part of the evaluation budget rather than of the case definition: each target is queried over a per-case layer sweep (e.g. AR-1 at L = 2, 5, 8), and every (target, L) query is scored as its own best-of-N evaluation. The per-case specifications, sweep layer counts and seeds are recorded in the case registry as the reproducibility anchor. Because the targets are idealized analytic constructions, no Tier 3 case is a training row by construction; the nearest-training-sample audit of §F.8 (Fig. F4) quantifies this directly against every spectrum the model saw in training. F.3 Wavelength-OOD bands (Tier 4). The training wavelength domain is strictly [380, 1400] nm (verified from the training shards), and Tier 4 probes whether 48
(a) Layer × wavelength grids (Tiers 1–2)
(b) Application families (Tier 3)
(c) Wavelength extrapolation (Tier 4) training envelope 380–1400 nm
Tier 2 — held-out OOD 15-material bank Tier 1 — trained 15-material bank
81–100
Antireflection
Bandpass (Gaussian)
Bandpass (flat-top)
5 cases
5 cases
2 cases
Notch / edge
Mirror
Solar
4 cases
5 cases
4 cases
Structural color
Extreme / stress
Spectral shape
8 cases
5 cases
8 cases
WO-NIR-NAR WO-NIR-MID WO-NIR-WIDE
61–80
identical targets, cell-for-cell paired
41–60 21–40
WO-SWIR-DEEP WO-OOD-DUAL WO-EDGE
11–20
WO-NEAR-EDGE 6–10
IR
AL
WO-WIDEST
U
eN
WO-MIX-FOOD
D
IR
IR N
S– N
S VI
WO-MIX-VN
VI
U
V– VI
S
2–5
7 layer bins × 6 bands × 2 modes (FULL / NEEDED) 100 targets per cell, best-of-100
46 analytic cases, layer-sweep queries, best-of-500 full two-channel R, T targets; family device metrics
pure OOD
380
700
edge
1000
mixed
1400
1800
2200
2500
wavelength (nm)
Fig. F1 Benchmark structure. a, The matched-cell Tier 1 / Tier 2 layer×band grids: 7 layercount bins × 6 spectral bands × 2 material-availability modes, 100 targets per cell; corresponding cells use identical band draws from the shared band RNG, while each tier’s target stacks and spectra are independently realized from its own bank. b, The nine Tier 3 application families and their case counts (46 cases). c, The ten Tier 4 wavelength-extrapolation cases drawn as band intervals against the 380–1400 nm training envelope, out to 2500 nm, colored by regime (pure OOD / edge / mixed).
IrisFlow’s λ-encoder extrapolates beyond it. The suite reaches 2500 nm; in the encoder’s signed inverse-wavelength coordinate (Table C1) this extends ≈ 16% beyond the trained interval, or 1.8× in wavelength. Its 10 cases place the query support partly or wholly above 1400 nm in three regimes: pure OOD (the whole band exceeds 1400 nm, including a 2000–2500 nm deep-SWIR band and a dual OOD band), edge (the band straddles the 1400 nm boundary) and mixed (an in-range band stitched to an OOD band, up to the full 380–2500 nm span). Each case uses 20 TMM-synthesized targets, the trained-15 vocabulary and 10–15-layer stacks, scored under the same bestof-500 protocol as Tier 3 (§F.4). To isolate wavelength position as the OOD factor, target synthesis is restricted to the wide-7 subset of trained materials whose measured tables span nearly the full 380–2500 nm window (Table B2), keeping their n, k essentially in-distribution; only narrow band-edge margins use the standard nearestendpoint hold, and no dispersion trend is fitted. The full trained-15 vocabulary is offered as the candidate bank and every design is re-simulated with the same library, so scoring is self-consistent. F.4 Sampling and scoring. At inference IrisFlow receives the four query objects (y, λ, C, L) and draws N stochastic samples per target (N = 100 for the Tier 1/2 grid cells and N = 500 for the Tier 3 application and Tier 4 wavelength suites), all with 15 reverse-time steps under the adaptive-power stack-time schedule (schedule power p = 2, §D.3). Every design is re-simulated with the same TMM solver used to generate the targets, then selected and aggregated by the best-of-N oracle protocol of §G.2. F.5 Material vocabularies. Tier 1 uses the 15 materials in the training vocabulary; Tier 2 uses a 15-material held-out bank (Table F1), none of which appear in the training vocabulary. OOD materials enter the model only through their n(λ), k (λ) curves on the query grid. The Tier 2 bank spans four chemical families: metals, high-index oxides, transparent conductors and polymers. The exact effective training curves (§B, Table B2; archived as a checksummed machine-readable library) are laboratory-measured records from our material files,
49
Table F1 Training vocabulary and OOD bank.
Set
Size Materials
Training vocabulary (Tier 1) 15
OOD bank (Tier 2)
15
Ag, Cr, Ti (metals); Al2 O3 , SiO2 , Ta2 O5 , TiO2 , h4 (LaTiO3 ) (oxides); Si (semiconductor); gst-a, gst-b, gsst-a, gsst-b, ss-a, ss-b (phase-change chalcogenides: GST, Se-substituted GSST and Sb2 S3 ; -a/-b = amorphous/crystalline) Au, W, Nb, Pt, Al (metals); CeO2 , Y2 O3 , HfO2 , ZrO2 (high-index oxides); ZnO, fluorine-doped tin oxide (FTO), ITO, aluminum-doped zinc oxide (AZO) (transparent conductors); polyethylene terephthalate (PET), PMMA (polymers)
supplied to the model as the effective training arrays. They should therefore be interpreted as the process-specific measurements used for this training corpus, not as universal material constants: deposition conditions and later calibration can change the n(λ), k (λ) assigned to the same nominal material. The process-corrected GSST-a workflow in §N illustrates this point, where the nominal training gsst-a curve was replaced over 400–800 nm by a remeasured segment for the fabrication process used in the cooler experiment. F.6 Tier 2 OOD-bank optical constants. The exact optical constants supplied to Tier 2 form part of the benchmark definition because the model receives candidate n(λ), k (λ) curves directly. Table F2 records the source and source support of each curve; Figure F2 plots the exact effective 380–1400 nm arrays used by the benchmark. Records obtained from refractiveindex.info cite both the database [46] and the original measurement or dispersion-model paper identified by that record. Target synthesis for the Tier 1/2 grids restricts metal layers to a 5–50 nm thickness window. All effective curves contain 1021 points on the 1-nm 380–1400 nm grid; when the source record does not cover that full range, the nearest endpoint value is used outside source support. The exact Tier 2 curves are archived in machine-readable form alongside the training library and are available on request (Data availability). The Au, W, Nb, HfO2 and Al curves are optical constants measured in our laboratory. They are primary experimental data supplied by the authors. The retained local measured tables and archived effective arrays identify the exact curves used by the benchmark. F.7 Curve-space novelty of the OOD bank and per-material error. The main text reports that within-cell OOD error does not track a held-out curve’s distance from the training vocabulary, but instead follows the same optical-role pattern seen in distribution. This subsection substantiates that reading with two material-resolved diagnostics, both defined exactly in §G.4: the curve novelty dstd (draw ), the nearestneighbor distance from each held-out n(λ), k (λ) curve to the training vocabulary in per-wavelength z-scored (raw) (n, k ) units, and the per-material error score βm , the expected within-cell log2 RMSE excess of a hypothetical pure-m stack (0 = cell-typical; +1 = double the cell median).
50
51
Transparent conductors High-index oxides High-index oxides Metals
ZnO
Transparent conductors High-index oxides Polymers
FTO
Transparent conductors Metals Transparent conductors High-index oxides
ITO
Al
AZO
ZrO2
PMMA
HfO2
Metals
Metals
Nb
Pt
Polymers
PET
W
Y2 O3
Synowicki, refractiveindex.info
Measured in our laboratory Treharne, refractiveindex.info
Minenkov-glass, refractiveindex.info
Measured in our laboratory Werner, refractiveindex.info von Rottkay & Rubin (1996) Measured in our laboratory Zhang-Mitsubishi, refractiveindex.info
Nigara, refractiveindex.info Measured in our laboratory Zhang, refractiveindex.info
Measured in our laboratory Aguilar, refractiveindex.info Vangelista et al. (2017)
Metals
Au
CeO2
Source
Material Family
Source support (nm)
This work local measured table 413.3– 2480.0 [47] main/ZnO/nk/ 300.0– Aguilar.yml 3200.0 [48] local table 260.0– 1000.0 [49] main/Y2O3/nk/ 250.0– Nigara.yml 9600.0 This work local measured table 406.5– 2480.0 [50] organic/(C10H8O4)n 400.0– - polyethylene 19900.0 terephthalate/nk/ Zhang.yml This work local measured table 413.3– 2755.0 [51] main/Pt/nk/ 17.6– Werner.yml 2480.0 [52] local table 310.0– 2500.0 This work local measured table 410.0– 2500.0 [50] organic/(C5H8O2)n 400.0– - poly(methyl 19900.0 methacrylate)/nk/ Zhang-Mitsubishi.yml [53] other/mixed 191.0– crystals/ 1690.0 In2O3-SnO2/nk/ Minenkov-glass.yml This work local measured table 400.0– 2480.0 [54] other/doped 300.0– crystals/Al-ZnO/ 900.0 nk/Treharne.yml [55] main/ZrO2/nk/ 130.0– Synowicki.yml 33000.0
Reference Source record
Table F2 Effective optical constants used by the fixed Tier 2 OOD-15 bank.
380–1400
380–1400
380–1400
380–1400
380–1400
380–1400
380–1400
380–1400
380–1400
380–1400
380–1400
380–1400
380–1400
380–1400
380–1400
Effective Tier 2 support (nm)
2.1078– 2.2260
0.4900– 2.7998 1.6183– 2.0592
0.2400– 2.0807
1.3500– 2.9300 0.4611– 1.4757 1.2149– 2.1059 1.9540– 2.0566 1.4770– 1.5082
0.1600– 1.6360 1.7361– 1.9074 2.2026– 2.6836 1.8944– 1.9933 3.0000– 3.8498 1.5433– 1.6103
0.0000– 0.0000
4.8600– 14.2444 0.0009– 0.0143
0.0071– 0.9859
2.6800– 7.3416 2.9108– 14.6106 0.0095– 0.2092 0.0060– 0.0060 0.0000– 0.0000
1.7962– 8.8268 0.0175– 0.1544 0.0000– 0.0792 0.0000– 0.0000 2.4200– 4.4500 0.0000– 0.0000
Effective n Effective k range range
Metals
High-index oxides
2.6
3.0
2.5 Au W Nb Pt Al
2.0 1.5 1.0 0.5 0.0
1.58
1.50
1.56
1.25
2.2
1.00
2.1
0.75
2.0
0.50
1.9
0.25
1.52 ZnO FTO ITO AZO
1.50 1.48
Transparent conductors
(f)
1.0
CeO2 Y2O3 HfO2 ZrO2
0.07
0.05
8
0.04
6
1.2
(h)
1.0
0.6
0.8
ZnO FTO ITO AZO
0.4
0.03
Polymers
1e−5
(g)
0.8
0.06
10
1.54
k
0.08
k
k
12
1.75
High-index oxides
Au W Nb Pt Al
PET PMMA
1.60
k
(e)
(d)
2.00
2.3
Metals 14
Polymers
(c)
CeO2 Y2O3 HfO2 ZrO2
2.4
n
n
2.5
Transparent conductors
(b)
n
2.7
(a)
3.5
n
4.0
PET PMMA
0.6 0.4
0.02 4
0.2
0.2
0.01
2
0.00 400
600
800
1000
Wavelength (nm)
1200
1400
0.0 400
600
800
1000
1200
1400
Wavelength (nm)
0.0 400
600
800
1000
Wavelength (nm)
1200
1400
400
600
800
1000
1200
1400
Wavelength (nm)
Fig. F2 Effective Tier 2 OOD-bank optical constants. Refractive index n (top) and extinction coefficient k (bottom) for every material in the fixed OOD-15 bank. Curves are grouped into metals, high-index oxides, transparent conductors and polymers, and are plotted from the exact rows supplied to Tier 2.
Because the same regression is applied to Tier 1, its β values fix the in-distribution reference spread against which the OOD values are read, and we use β in preference to the raw per-material median RMSE (also tabulated, but less discriminating because deep stacks mix many bank materials). Table F3 and Figure F3 bear this reading out. Novelty and error are decoupled: across the 15 OOD materials the Spearman rank correlation between dstd and β is near zero (−0.01; −0.06 for draw and +0.19 against the raw per-material median RMSE), even though the bank ranges widely in novelty: HfO2 , ZrO2 , PMMA and PET are near-duplicates of training curves (dstd ≤ 0.04), whereas Al, Pt and Nb are the most distant held-out curves (dstd = 1.81, 1.23 and 1.14). What error structure there is mirrors the in-distribution pattern by optical role: the largest positive β values belong to transparent conductors (ITO +1.06, FTO +0.46), the same transparent high-index role that is hardest in distribution (TiO2 ), while the most novel metals are among the easiest (Nb −1.49, W −1.17, Al −1.07). Every Tier 2 β but one lies inside the Tier 1 reference spread (−2.46 to +1.03); the exception, ITO (+1.06), essentially coincides with the in-distribution maximum (TiO2 , +1.03). β should be read as a target-composition association, not an isolated per-curve skill estimate: the negative metal coefficients partly reflect that metal-rich stacks tend to be smoother and easier in this grid, and because Tier 1 and Tier 2 use the identical regression, the meaningful comparison is the material-family pattern rather than any single coefficient. On that comparison the conclusion is clear: curve-space novelty does not predict OOD difficulty; optical role does. F.8 Tier 3 target novelty: nearest-training-sample audit. Tier 3 targets are analytic constructions, but the long-stack training corpora include Powell-refined shapes that can resemble application targets. We therefore scan all 114.1M spectra seen during training and, for each of the 47 distinct Tier 3 target curves (46 cases, with AR-5 retaining two variants), record the nearest training spectrum under the paper’s combined R, T RMSE on the target grid. Training spectra are interpolated only over their own band support, never across inter-band gaps; 41 of 47 targets have a 52
Table F3 Curve-space novelty vs per-material Tier 2 error. Sorted by descending novelty; “# targets” counts the Tier 2 targets (of 8400) whose ground-truth stack uses the material.
OOD material Family
Nearest training curve
dstd (draw )
β (log2 excess)
Median RMSE # targets (targets using it)
Al
metal
Ag
-1.07
0.0846
5938
Pt
metal
Ag
+0.21
0.0850
5920
Nb
metal
Ti
-1.49
0.0819
6192
Au
metal
Ag
+0.14
0.0833
6250
W
metal
Cr
-1.17
0.0831
6054
ITO
SiO2
+1.06
0.0853
6198
+0.46
0.0841
6338
+0.40
0.0835
6416
-0.04
0.0846
6232
-0.08
0.0841
6238
+0.11
0.0840
6348
PET
transparent conductor transparent conductor transparent conductor transparent conductor high-index oxide high-index oxide polymer
Al2 O3
+0.31
0.0843
6232
PMMA
polymer
SiO2
-0.18
0.0833
6402
ZrO2
high-index oxide high-index oxide
Ta2 O5
1.81 (3.64) 1.23 (3.28) 1.14 (2.16) 0.59 (0.78) 0.58 (0.82) 0.46 (0.65) 0.22 (0.26) 0.14 (0.15) 0.13 (0.16) 0.07 (0.09) 0.06 (0.07) 0.04 (0.05) 0.04 (0.04) 0.03 (0.04) 0.02 (0.02)
-0.13
0.0827
6386
-0.29
0.0838
6208
FTO AZO ZnO Y2 O3 CeO2
HfO2
Al2 O3 Al2 O3 Al2 O3 h4 TiO2
h4
full-coverage neighbor (≥ 99.5%), while the six widest targets can reach only 91–95% coverage because they exceed the training band-width cap. Figure F4 reports the nearest neighbor for the most conservative target variant in each case. The median nearest-neighbor distance is 8.1 × 10−2 , above the model’s median best-of-sweep error on the same targets (3.7 × 10−2 ; 3.9 × 10−2 after collapsing AR-5’s two variants in the §I scorecard). Only saturated mirror/absorber targets fall within 10−2 of a training spectrum, which is expected because the training generator includes mirror and absorber families. As an oracle retrieval baseline, the best single training spectrum strictly outperforms the generated design on only 5 of 47 targets, with 3 further effective ties (the remaining 39 going to IrisFlow); those 5 genuine losses mostly reflect a depth asymmetry, with the nearest retrieved training stack deeper than the case’s allowed layer sweep. The audit therefore argues against memorization as the source of Tier 3 performance: IrisFlow improves on the best design the training corpus itself offers, generalizing beyond retrieval.
53
Curve novelty vs. OOD error (Spearman ρ = -0.01)
→ Ag → Ag
Pt
→ Ti
Nb Au
→ Ag
W
→ Cr → SiO2
ITO → Al2O3
FTO AZO
→ Al2O3
ZnO
→ Al2O3 → h4
Y2O3
→ TiO2
CeO2 PET
→ Al2O3
PMMA
→ SiO2
ZrO2
→ Ta2O5
HfO2
→ h4
0.0
metal oxide TCO polymer
per-material error score β (log2 within-cell RMSE excess)
Curve novelty of each held-out material Al
0.5 1.0 1.5 2.0 curve novelty dstd (z-scored n, k distance to nearest training curve)
ITO
1.0 0.5 0.0 −0.5
PET
ZrO2
in-distribution per-material spread (Tier 1: -2.46..+1.03)
AZO FTO Pt
Au
CeO2 ZnO Y2O3 PMMA HfO2
−1.0
Al
W Nb
−1.5 −2.0 −2.5 0.00
0.25
0.50
0.75 1.00 curve novelty dstd
1.25
1.50
1.75
Fig. F3 Curve-space novelty vs per-material OOD error. Left: distance from each held-out curve to its nearest training-vocabulary curve (z-scored (n, k) RMS over the shared 380–1400 nm grid), annotated with the nearest neighbor; colors mark the four chemical families. Right: novelty against the per-material error score β; the shaded band is the Tier 1 (in-distribution) per-material spread from the identical regression.
Appendix G
Metric definitions
This section gives the exact definitions behind every reported number: the headline fidelity metric and selection/aggregation protocol (§§G.1–G.2) and the secondary spectral metrics (§§G.3–G.4). G.1 Combined spectral RMSE. Every reported design is re-simulated with the TMM solver on its query wavelength support {λs }Ss=1 (band-limited and multi-band targets are queried and scored only on their in-band grid points). With re-simulated spectra R̂, T̂ andqtarget R∗ , T ∗ , PS 1 ∗ 2 ∗ 2 RMSERT = 2S s=1 (R̂s − Rs ) + (T̂s − Ts ) , i.e. the two per-channel mean squared errors are averaged before the root; equivalently, the RMSE over the 2S concatenated reflectance and transmittance samples, with all wavelength points weighted uniformly. The per-channel values RMSER = q P 1 ∗ 2 s (R̂s − Rs ) (and likewise RMSET ) are recorded alongside. The combined metS ric is well-defined for every case: Tier 1/2/4 targets are TMM-realized stacks specifying both channels, and Tier 3 analytic targets are defined on both channels via the lossless complement (§F.2). G.2 Best-of-N selection and aggregation. Each target is queried with N independent stochastic draws (N = 100 for the Tier 1/2 grid, N = 500 for Tiers 3/4; §F.4). Every draw is re-simulated and the reported design is the one with the lowest RMSERT : the ranking score is the reported metric itself (oracle ranking). For Tier 1/2 grid cells and Tier 4 cases, the table statistic is the median of the selected designs’ RMSERT over the targets in that cell or case. For the Tier 3 full scorecard (§I), each target variant is instead first reduced to its best layer-sweep result, i.e. the lowest selected-design RMSERT over the queried layer counts; the case value is that targetbest RMSE, except for AR-5 where the two target-best RMSEs are summarized by their median. Other pooled values quoted in the text follow the statistic named in 54
AR-1
AR-2
AR-3
AR-4
AR-5
BP-1
1.0
1.0
1.0
1.0
1.0
1.0
0.8
0.8
0.8
0.8
0.8
0.8
0.6 0.4
0.6
dNN =0.034 (cov 1.00, L=2) model 0.001, L=2
0.4
0.6
dNN =0.047 (cov 1.00, L=3) model 0.004, L=5
0.4
0.6
dNN =0.186 (cov 1.00, L=3) model 0.010, L=2
0.4
0.6
dNN =0.045 (cov 1.00, L=2) model 0.007, L=5
0.4
0.2
0.2
0.2
0.2
0.2
0.0
0.0
0.0
0.0
0.0
520
525
530
535
540
545
400
500
BP-2 1.0
0.4
0.6 0.4
0.0 800
900
600
0.6 0.4
700
800
600
700
0.4
500
600
600
700
0.6 0.4
0.4
0.4
700
800
0.6 0.4
900
0.6 0.4
0.2
0.2
0.2
0.0
0.0
0.0
1000
1200
400
600
800
1.0
0.4
0.6 0.4
600
800
1000
1200
1400
0.4
600
700
0.4
700
0.4
600
700
0.4
0.6 0.4
700
0.6 0.4
0.2
0.2
0.2
0.2
0.0
0.0
0.0
0.0
0.0
500
600
700
500
550
S-STEP 1.0
0.4
650
700
0.6 0.4
0.2 0.0
0.6 0.4
700
800
900
700
800
900
500
600
700
400
500
800
900
600
dNN =0.092 (cov 1.00, L=18) model 0.058, L=5
0.4 0.2 0.0
500
600
700
400
500
700
800
900
650
700
600
700
EX-3 1.0 0.8 0.6
dNN =0.208 (cov 1.00, L=10) model 0.102, L=50
dNN =0.056 (cov 1.00, L=77) model 0.031, L=65
0.4 0.2 0.0
600
800
1000
600
620
640
660
S-VALLEY 1.0 0.8 0.6
dNN =0.072 (cov 1.00, L=26) model 0.046, L=30
dNN =0.077 (cov 1.00, L=4) model 0.049, L=10
0.4 0.2 0.0
500
600
700
800
400
500
600
700
800
900
1.0 0.8 0.6
dNN =0.167 (cov 1.00, L=64) model 0.167, L=9
0.4
dNN =0.034 (cov 1.00, L=4) model 0.020, L=11
0.2
0.0 400
0.6
S-ND
0.2
0.0 600
600
0.8
dNN =0.060 (cov 1.00, L=2) model 0.016, L=5
0.2
500
500
1.0
0.8
dNN =0.133 (cov 1.00, L=20) model 0.031, L=38
400
400
S-MULTI
1.0
0.8 0.6
600
S-GRAD
1000
0.8
0.8
dNN =0.047 (cov 1.00, L=3) model 0.009, L=5
800
SC-5
1.0
0.8
0.4
600
S-PEAK
1.0
0.6
400
dNN =0.096 (cov 1.00, L=17) model 0.137, L=15
S-FLAT-L
dNN =0.012 (cov 1.00, L=5) model 0.008, L=20
1000 1.0
400
0.2
400
800
0.0 600
1090
0.0 600
0.2
500
1080
0.2
0.8
dNN =0.047 (cov 1.00, L=3) model 0.014, L=2
1070
dNN =0.196 (cov 1.00, L=19) model 0.047, L=40
0.4
1.0
400
0.8
0.6
EX-2
0.0 600
1060
0.8
dNN =0.198 (cov 1.00, L=3) model 0.030, L=6
400
0.2
500
1.0
0.6
700
0.8
0.4
1050
SE-3
0.0 500
1040 1.0
EX-1
0.6
700
0.2
400
dNN =0.021 (cov 1.00, L=2) model 0.012, L=5
600
0.8
0.4
950
0.0
400
0.6
900
dNN =0.000 (cov 1.00, L=17) model 0.002, L=20
0.4
SC-4
dNN =0.055 (cov 1.00, L=20) model 0.022, L=10
S-FLAT-H
dNN =0.001 (cov 1.00, L=23) model 0.001, L=22
1400
1.0
400
0.8 0.6
700
0.0 600
1.0
dNN =0.182 (cov 1.00, L=16) model 0.125, L=30
600
0.2
500
1200
0.0 500
0.8 0.6
1000
0.2
EX-5
0.8
0.4
0.6
1.0
SC-8
dNN =0.109 (cov 1.00, L=3) model 0.082, L=10
EX-4 1.0
0.6
1.0
400
800
0.8
dNN =0.110 (cov 1.00, L=2) model 0.024, L=10
400
0.0 700
600
850
0.2
500
SC-3
0.0 500
0.2
600
400
0.2
400
0.6
0.0
0.4
0.4
0.8
dNN =0.094 (cov 1.00, L=16) model 0.068, L=8
0.2
0.6
0.6
1.0
0.8
700 1.0
SC-7
1.0
500
600
0.8
dNN =0.114 (cov 1.00, L=15) model 0.040, L=10
SC-6
400
500
1.0
0.0 400
400
SC-2
0.2
0.0
0.4
1400
0.8
dNN =0.427 (cov 0.91, L=20) model 0.132, L=25
0.2
0.6
1200
1.0
0.8 0.6
1000
SC-1
800
0.8
dNN =0.007 (cov 1.00, L=83) model 0.007, L=15
0.8
0.0 800
750
MR-2
1.0
0.2
SE-4
700 1.0
400
0.0 600
600
SE-2
dNN =0.482 (cov 0.91, L=67) model 0.045, L=10
900
0.0 500
0.0 800
800
dNN =0.013 (cov 1.00, L=73) model 0.045, L=30
0.4
0.2
700
700
0.2
400
0.2
400
0.6
0.8
dNN =0.072 (cov 1.00, L=33) model 0.037, L=40
0.8
0.4
600
MR-1
1.0
0.6
500
0.8
1.0
600
400
BP-FLAT-2
dNN =0.044 (cov 1.00, L=89) model 0.119, L=60
SE-1
dNN =0.080 (cov 1.00, L=18) model 0.068, L=30
1400
0.0 600
500
0.8 0.6
0.4
0.0
1.0
dNN =0.446 (cov 0.91, L=20) model 0.164, L=20
0.6
0.2
1000
1200
1.0
NF-4
0.6
1000
0.2
MR-5
0.8
800
0.8
dNN =0.064 (cov 1.00, L=89) model 0.075, L=15
0.8
800
600
1.0
500
dNN =0.098 (cov 1.00, L=16) model 0.013, L=35
600
0.0 400
BP-FLAT-1
1.0
800
1.0
dNN =0.411 (cov 0.95, L=5) model 0.106, L=20
700
0.0 500
1000
0.0 400
0.6
900
0.2
MR-4
0.8
0.4
0.2
400
MR-3 1.0
0.6
0.8
dNN =0.079 (cov 1.00, L=49) model 0.037, L=40
800
0.8
dNN =0.181 (cov 1.00, L=15) model 0.127, L=8
1.0
0.0 500
700
NF-3
0.2
0.0
600
dNN =0.108 (cov 1.00, L=8) model 0.086, L=8
0.4 0.2
BP-5
0.0 500
0.8
0.2
0.4
0.4
1.0
dNN =0.081 (cov 1.00, L=34) model 0.027, L=40
500 1.0
NF-2
0.8
1000
0.2
400
1.0
0.6
0.6
0.0
400
800
0.8
dNN =0.244 (cov 1.00, L=16) model 0.116, L=34
NF-1
0.4
600
1.0
0.2
700
400
BP-4
0.8
dNN =0.047 (cov 1.00, L=99) model 0.075, L=8
0.2
0.6
700
1.0
0.8 0.6
600
BP-3
0.6
dNN =0.440 (cov 0.91, L=23) model 0.015, L=2
0.0 400
500
600
700
800
900
450
wavelength (nm)
500
550
600
target R(λ) target T(λ)
nearest training R(λ) nearest training T(λ)
Fig. F4 Tier 3 nearest-training-sample gallery. For each of the 46 application cases: the case’s target curve closest to the training corpus (solid: target R, T ) overlaid with the realized spectrum of its nearest training sample among all 114M Stage 1–3 training spectra (dashed). Each panel reports the nearest-neighbor distance dNN (combined R, T RMSE on the target grid) with that neighbor’s grid coverage and layer count, and the model’s lowest best-of-N RMSE across the case’s layer sweep, with the layer count L that achieves it. Only the saturated specifications (the perfect mirrors MR-1/MR2 and the near-perfect absorber EX-5) find close neighbors; these are shapes the corpus contains by construction.
55
their corresponding table or caption. One illustrative run (the §E.1 reverse-trajectory example) instead ranks its p draws by the angular MSE, thepvariance-stabilizing score p p PS 1 aMSE = 2S s=1 (arcsin R̂s − arcsin Rs∗ )2 + (arcsin T̂s − arcsin Ts∗ )2 (the angular mse option of the inference script), whose arcsin-square-root transform upweights deviations near the R, T = 0 and 1 boundaries. All reported benchmark statistics use the RMSERT oracle above; angular MSE changes only which single draw is displayed in §E.1, not any reported number. G.3 Matched-cell OOD/in-distribution error ratio. For each of the 84 matched (mode, layer, band) grid cells, ρ = medT2 /medT1 , the ratio of the Tier 2 (held-out bank) to Tier 1 (training bank) per-cell median RMSERT , with corresponding cells sharing band-support draws, layer bins and availability modes (§F.1), but not target stacks or target spectra. The Results section summarizes ρ by its median and interquartile range over cells and the fraction of cells with ρ ≤ 3; the ratio of the two overall (pooled) medians is quoted alongside. Thus ρ describes relative error across matched cell-level target distributions and is not interpreted as a causal material-OOD degradation or same-target retention factor. Because ρ is inflated on cells whose indistribution baseline is already ∼ 10−2 , the absolute OOD RMSE is always reported next to it. G.4 Per-material error score β and curve novelty dstd . Defined here and analyzed in §F.7 (Fig. F3, Table F3). Each grid target’s selected-design RMSE is converted to a within-cell excess e = log2 (RMSE/cell median) (which removes the layer×band difficulty of §H) and P regressed by least P squares on the target’s groundtruth composition fractions, e ≈ m fm βm with m fm = 1 (no intercept), over a tier’s 8400 grid targets; βm reads as the expected log2 excess of a hypothetical pure-m stack (0 = cell-typical, +1 = double the cell median). Curve novelty dstd (draw ) is the nearest-neighbor RMS distance from a held-out curve to the training vocabulary in per-wavelength z-scored (raw) (n, k ) space, the z-scoring taken over the union of the 15 training and 15 OOD curves so that the larger absolute scale of metallic k does not dominate. The solution-diversity statistics (number of distinct material sequences, mean pairwise material disagreement ⟨∆mat⟩, mean pairwise RMS thickness difference ⟨∆d⟩) are defined where they are used, in §E.2.
Appendix H
Full Tier 1 / Tier 2 grid
The pooled-by-layer and pooled-by-band fidelity medians are collected in Table H1 (which duplicates the marginal information in the main-text heatmaps, Fig. 2a, b), together with the marginal curves of Fig. H1; both are aggregated from the full per-cell grid that Table H2 reports for every Tier 1 and Tier 2 grid cell (all 168 layer×band×availability-mode cells). The full per-tier reconstruction galleries (Tier 1 in-distribution, Fig. H2; Tier 2 OOD material bank, Fig. H3) provide the complete case sets from which the main-text overview figure (Fig. 2d–g) draws two representative cases each.
56
Table H1 Tier 1 / Tier 2 spectral fidelity (median combined RMSE over the FULL availability-mode cells; NEEDED-mode cells in the full grid below, Table H2; independently realized frozen target sets under matched cell definitions, Appendix §§F.1, F.6). Pooled values are medians of per-cell medians; because fidelity is (near-)monotone in layer count, most by-band entries coincide with the middle (21–40-layer) bin, a property of the median.
By layer count (pooled over bands) 2–5 6–10 11–20 21–40 41–60 61–80 81–100 By target band (pooled over layers) UV–VIS 380–550 VIS 400–700 VIS–NIR 500–900 NIR 800–1100 eNIR 1000–1400 DUAL VIS+NIR
Tier 1 (in-distribution)
Tier 2 (OOD)
Tier 2 / Tier 1
0.0117 0.0281 0.0332 0.0468 0.0689 0.0767 0.0927
0.0520 0.0582 0.0662 0.0702 0.0875 0.1112 0.1248
4.44 2.07 2.00 1.50 1.27 1.45 1.35
0.0567 0.0503 0.0480 0.0382 0.0436 0.0455
0.0555 0.0836 0.0674 0.0583 0.0758 0.0746
0.98 1.66 1.40 1.52 1.74 1.64
Fidelity vs. layer count
Fidelity vs. band, in-distribution vs. OOD median spectrum RMSE (pooled over layers)
NIR eNIR DUAL
median spectrum RMSE
10
UV–VIS VIS VIS–NIR
−1
10
−2
2–5
6–10
11–20 21–40 41–60 requested layer count
61–80
8 × 10
7 × 10
6 × 10
5 × 10
4 × 10
81–100
in-distribution (Tier 1) OOD (Tier 2)
−2
−2
−2
−2
−2
UV–VIS
VIS
VIS–NIR NIR target band
eNIR
DUAL
Fig. H1 Marginal fidelity trends. Median combined spectral RMSE pooled over bands vs. layercount bin (left) and pooled over layers vs. spectral band (right), for Tier 1 and Tier 2.
Table H2: Full Tier 1 / Tier 2 layer×band grid (all 168 cells; per-cell median combined R, T RMSE). Any worst-point statistic is omitted; fidelity is reported through the best-of-N median alone. Case IDs abbreviate the NEEDED material-availability mode of §F.1 as NEED. Case
Tier
Mode
Layers
Band
median RMSE
T1-FULL-L02 05-BUV T1-FULL-L02 05-BVIS T1-FULL-L02 05-BVISNIR T1-FULL-L02 05-BNIR T1-FULL-L02 05-BENIR T1-FULL-L02 05-BDUAL T1-FULL-L06 10-BUV
T1 T1 T1 T1 T1 T1 T1
FULL FULL FULL FULL FULL FULL FULL
L02 05 L02 05 L02 05 L02 05 L02 05 L02 05 L06 10
BUV BVIS BVISNIR BNIR BENIR BDUAL BUV
0.0208 0.0121 0.0105 0.0113 0.0290 0.0095 0.0382
57
Case
Tier
Mode
Layers
Band
median RMSE
T1-FULL-L06 10-BVIS T1-FULL-L06 10-BVISNIR T1-FULL-L06 10-BNIR T1-FULL-L06 10-BENIR T1-FULL-L06 10-BDUAL T1-FULL-L11 20-BUV T1-FULL-L11 20-BVIS T1-FULL-L11 20-BVISNIR T1-FULL-L11 20-BNIR T1-FULL-L11 20-BENIR T1-FULL-L11 20-BDUAL T1-FULL-L21 40-BUV T1-FULL-L21 40-BVIS T1-FULL-L21 40-BVISNIR T1-FULL-L21 40-BNIR T1-FULL-L21 40-BENIR T1-FULL-L21 40-BDUAL T1-FULL-L41 60-BUV T1-FULL-L41 60-BVIS T1-FULL-L41 60-BVISNIR T1-FULL-L41 60-BNIR T1-FULL-L41 60-BENIR T1-FULL-L41 60-BDUAL T1-FULL-L61 80-BUV T1-FULL-L61 80-BVIS T1-FULL-L61 80-BVISNIR T1-FULL-L61 80-BNIR T1-FULL-L61 80-BENIR T1-FULL-L61 80-BDUAL T1-FULL-L81 100-BUV T1-FULL-L81 100-BVIS T1-FULL-L81 100-BVISNIR T1-FULL-L81 100-BNIR T1-FULL-L81 100-BENIR T1-FULL-L81 100-BDUAL T1-NEED-L02 05-BUV T1-NEED-L02 05-BVIS T1-NEED-L02 05-BVISNIR T1-NEED-L02 05-BNIR T1-NEED-L02 05-BENIR T1-NEED-L02 05-BDUAL T1-NEED-L06 10-BUV T1-NEED-L06 10-BVIS T1-NEED-L06 10-BVISNIR T1-NEED-L06 10-BNIR T1-NEED-L06 10-BENIR T1-NEED-L06 10-BDUAL T1-NEED-L11 20-BUV T1-NEED-L11 20-BVIS T1-NEED-L11 20-BVISNIR T1-NEED-L11 20-BNIR T1-NEED-L11 20-BENIR T1-NEED-L11 20-BDUAL T1-NEED-L21 40-BUV T1-NEED-L21 40-BVIS T1-NEED-L21 40-BVISNIR T1-NEED-L21 40-BNIR T1-NEED-L21 40-BENIR
T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1
FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED
L06 10 L06 10 L06 10 L06 10 L06 10 L11 20 L11 20 L11 20 L11 20 L11 20 L11 20 L21 40 L21 40 L21 40 L21 40 L21 40 L21 40 L41 60 L41 60 L41 60 L41 60 L41 60 L41 60 L61 80 L61 80 L61 80 L61 80 L61 80 L61 80 L81 100 L81 100 L81 100 L81 100 L81 100 L81 100 L02 05 L02 05 L02 05 L02 05 L02 05 L02 05 L06 10 L06 10 L06 10 L06 10 L06 10 L06 10 L11 20 L11 20 L11 20 L11 20 L11 20 L11 20 L21 40 L21 40 L21 40 L21 40 L21 40
BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR
0.0279 0.0267 0.0209 0.0297 0.0283 0.0380 0.0340 0.0305 0.0291 0.0382 0.0323 0.0567 0.0503 0.0480 0.0382 0.0436 0.0455 0.0725 0.0775 0.0743 0.0472 0.0493 0.0652 0.0764 0.0877 0.0842 0.0524 0.0665 0.0769 0.0933 0.1027 0.0921 0.0575 0.0583 0.1073 0.0169 0.0102 0.0092 0.0092 0.0300 0.0072 0.0372 0.0269 0.0238 0.0192 0.0288 0.0276 0.0400 0.0313 0.0304 0.0291 0.0352 0.0331 0.0568 0.0500 0.0473 0.0364 0.0411
58
Case
Tier
Mode
Layers
Band
median RMSE
T1-NEED-L21 40-BDUAL T1-NEED-L41 60-BUV T1-NEED-L41 60-BVIS T1-NEED-L41 60-BVISNIR T1-NEED-L41 60-BNIR T1-NEED-L41 60-BENIR T1-NEED-L41 60-BDUAL T1-NEED-L61 80-BUV T1-NEED-L61 80-BVIS T1-NEED-L61 80-BVISNIR T1-NEED-L61 80-BNIR T1-NEED-L61 80-BENIR T1-NEED-L61 80-BDUAL T1-NEED-L81 100-BUV T1-NEED-L81 100-BVIS T1-NEED-L81 100-BVISNIR T1-NEED-L81 100-BNIR T1-NEED-L81 100-BENIR T1-NEED-L81 100-BDUAL T2-FULL-L02 05-BUV T2-FULL-L02 05-BVIS T2-FULL-L02 05-BVISNIR T2-FULL-L02 05-BNIR T2-FULL-L02 05-BENIR T2-FULL-L02 05-BDUAL T2-FULL-L06 10-BUV T2-FULL-L06 10-BVIS T2-FULL-L06 10-BVISNIR T2-FULL-L06 10-BNIR T2-FULL-L06 10-BENIR T2-FULL-L06 10-BDUAL T2-FULL-L11 20-BUV T2-FULL-L11 20-BVIS T2-FULL-L11 20-BVISNIR T2-FULL-L11 20-BNIR T2-FULL-L11 20-BENIR T2-FULL-L11 20-BDUAL T2-FULL-L21 40-BUV T2-FULL-L21 40-BVIS T2-FULL-L21 40-BVISNIR T2-FULL-L21 40-BNIR T2-FULL-L21 40-BENIR T2-FULL-L21 40-BDUAL T2-FULL-L41 60-BUV T2-FULL-L41 60-BVIS T2-FULL-L41 60-BVISNIR T2-FULL-L41 60-BNIR T2-FULL-L41 60-BENIR T2-FULL-L41 60-BDUAL T2-FULL-L61 80-BUV T2-FULL-L61 80-BVIS T2-FULL-L61 80-BVISNIR T2-FULL-L61 80-BNIR T2-FULL-L61 80-BENIR T2-FULL-L61 80-BDUAL T2-FULL-L81 100-BUV T2-FULL-L81 100-BVIS T2-FULL-L81 100-BVISNIR
T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T1 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2
NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL FULL
L21 40 L41 60 L41 60 L41 60 L41 60 L41 60 L41 60 L61 80 L61 80 L61 80 L61 80 L61 80 L61 80 L81 100 L81 100 L81 100 L81 100 L81 100 L81 100 L02 05 L02 05 L02 05 L02 05 L02 05 L02 05 L06 10 L06 10 L06 10 L06 10 L06 10 L06 10 L11 20 L11 20 L11 20 L11 20 L11 20 L11 20 L21 40 L21 40 L21 40 L21 40 L21 40 L21 40 L41 60 L41 60 L41 60 L41 60 L41 60 L41 60 L61 80 L61 80 L61 80 L61 80 L61 80 L61 80 L81 100 L81 100 L81 100
BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR
0.0448 0.0801 0.0781 0.0776 0.0520 0.0475 0.0666 0.0812 0.0852 0.0808 0.0531 0.0725 0.0857 0.0939 0.1047 0.0964 0.0611 0.0620 0.1098 0.0510 0.0576 0.0455 0.0353 0.0577 0.0530 0.0544 0.0715 0.0627 0.0551 0.0556 0.0609 0.0547 0.0721 0.0629 0.0561 0.0758 0.0696 0.0555 0.0836 0.0674 0.0583 0.0730 0.0746 0.0652 0.1087 0.0866 0.0675 0.1108 0.0885 0.0788 0.1241 0.0984 0.0856 0.1633 0.1319 0.0996 0.1316 0.1180
59
Case
Tier
Mode
Layers
Band
median RMSE
T2-FULL-L81 100-BNIR T2-FULL-L81 100-BENIR T2-FULL-L81 100-BDUAL T2-NEED-L02 05-BUV T2-NEED-L02 05-BVIS T2-NEED-L02 05-BVISNIR T2-NEED-L02 05-BNIR T2-NEED-L02 05-BENIR T2-NEED-L02 05-BDUAL T2-NEED-L06 10-BUV T2-NEED-L06 10-BVIS T2-NEED-L06 10-BVISNIR T2-NEED-L06 10-BNIR T2-NEED-L06 10-BENIR T2-NEED-L06 10-BDUAL T2-NEED-L11 20-BUV T2-NEED-L11 20-BVIS T2-NEED-L11 20-BVISNIR T2-NEED-L11 20-BNIR T2-NEED-L11 20-BENIR T2-NEED-L11 20-BDUAL T2-NEED-L21 40-BUV T2-NEED-L21 40-BVIS T2-NEED-L21 40-BVISNIR T2-NEED-L21 40-BNIR T2-NEED-L21 40-BENIR T2-NEED-L21 40-BDUAL T2-NEED-L41 60-BUV T2-NEED-L41 60-BVIS T2-NEED-L41 60-BVISNIR T2-NEED-L41 60-BNIR T2-NEED-L41 60-BENIR T2-NEED-L41 60-BDUAL T2-NEED-L61 80-BUV T2-NEED-L61 80-BVIS T2-NEED-L61 80-BVISNIR T2-NEED-L61 80-BNIR T2-NEED-L61 80-BENIR T2-NEED-L61 80-BDUAL T2-NEED-L81 100-BUV T2-NEED-L81 100-BVIS T2-NEED-L81 100-BVISNIR T2-NEED-L81 100-BNIR T2-NEED-L81 100-BENIR T2-NEED-L81 100-BDUAL
T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2 T2
FULL FULL FULL NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED NEED
L81 100 L81 100 L81 100 L02 05 L02 05 L02 05 L02 05 L02 05 L02 05 L06 10 L06 10 L06 10 L06 10 L06 10 L06 10 L11 20 L11 20 L11 20 L11 20 L11 20 L11 20 L21 40 L21 40 L21 40 L21 40 L21 40 L21 40 L41 60 L41 60 L41 60 L41 60 L41 60 L41 60 L61 80 L61 80 L61 80 L61 80 L61 80 L61 80 L81 100 L81 100 L81 100 L81 100 L81 100 L81 100
BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL BUV BVIS BVISNIR BNIR BENIR BDUAL
0.0950 0.1680 0.1386 0.0505 0.0524 0.0469 0.0400 0.0802 0.0563 0.0565 0.0609 0.0523 0.0561 0.0567 0.0649 0.0529 0.0710 0.0593 0.0561 0.0747 0.0663 0.0600 0.0798 0.0648 0.0557 0.0800 0.0805 0.0632 0.1042 0.0870 0.0605 0.1157 0.0892 0.0823 0.1216 0.0976 0.0802 0.1542 0.1203 0.0973 0.1345 0.1150 0.1016 0.1744 0.1414
Appendix I
Full Tier 3 application scorecard
The full one-case-per-family application showcase (Fig. I1) and the per-family scorecard (Fig. I2) accompany the per-case fidelity table below; the main-text overview figure (Fig. 2h, i) shows two representative cases drawn from these families.
60
Tier 1 reconstruction showcase — in-distribution materials, across layer count and band 02–05 layers · VIS–NIR 500–900 L=4, RMSE=0.019
02–05 layers · DUAL VIS+NIR L=5, RMSE=0.039
06–10 layers · VIS 400–700 L=7, RMSE=0.031
11–20 layers · DUAL VIS+NIR L=14, RMSE=0.087
1.0
1.0
1.0
1.0
0.8
0.8
0.8
0.8
R target R pred T target
T pred no target band
0.6
0.6 R target R pred
T target T pred
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0.0
0.0
0.0
500
600 700 800 wavelength (nm)
900
600
21–40 layers · NIR 800–1100 L=23, RMSE=0.053
800 wavelength (nm)
1000
560 580 wavelength (nm)
600
61–80 layers · DUAL VIS+NIR L=61, RMSE=0.013
1.0
1.0
1.0
0.8
0.8
0.8
0.8
T target T pred
R target R pred T target
R, T
R target R pred
T pred no target band
0.6 R, T
0.6
R, T
R, T
0.6 T target T pred
R target R pred
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0.0
0.0 800
900 1000 wavelength (nm)
1100
0.0 800
900 1000 wavelength (nm)
1100
800 1000 wavelength (nm)
81–100 layers · VIS–NIR 500–900 L=83, RMSE=0.014
1.0
R target R pred
T pred no target band
0.0 540
41–60 layers · NIR 800–1100 L=49, RMSE=0.013
0.6
R target R pred T target
R, T
T target T pred
R, T
R target R pred
R, T
0.6
R, T
0.6
T target T pred
0.0 600
800 1000 wavelength (nm)
620
640 660 680 wavelength (nm)
700
Fig. H2 In-distribution reconstructions (Tier 1). Representative best-of-N designs (markers) vs. target R(λ), T (λ) (solid) spanning 2–100 layers and all band positions; gray regions carry no target band. Table I1: Full Tier 3 per-case scorecard (all 46 cases; best combined R, T RMSE after selecting the lowest-RMSE layer-sweep result for each target; AR-5 reports the median of its two target-best RMSEs). Case
best RMSE
AR-1 AR-2 AR-3 AR-4 AR-5 BP-1 BP-2 BP-3 BP-4 BP-5 BP-FLAT-1 BP-FLAT-2 NF-1 NF-2 NF-3 NF-4 MR-1 MR-2 MR-3 MR-4 MR-5 SE-1 SE-2 SE-3 SE-4 SC-1 SC-2 SC-3 SC-4
0.0015 0.0038 0.0100 0.0072 0.0209 0.0859 0.0752 0.1155 0.1275 0.0749 0.1189 0.0447 0.0271 0.0365 0.0129 0.0370 0.0068 0.0022 0.1055 0.1636 0.0685 0.0446 0.0297 0.0466 0.1320 0.0403 0.0237 0.0225 0.1365
61
Appendix J
Case
best RMSE
SC-5 SC-6 SC-7 SC-8 EX-1 EX-2 EX-3 EX-4 EX-5 S-FLAT-H S-FLAT-L S-PEAK S-VALLEY S-STEP S-GRAD S-MULTI S-ND
0.0578 0.0679 0.0818 0.0119 0.0137 0.1016 0.0306 0.1250 0.0013 0.0084 0.0088 0.0464 0.0494 0.0307 0.0159 0.1668 0.0201
Tier 4: wavelength-encoding extrapolation (out-of-range bands)
Target construction for this suite (the out-of-range bands and three OOD regimes, the wide-7 synthesis subset, and the endpoint-hold convention that keeps the optical constants in-distribution so that wavelength position is the factor under test) is described in §F.3. Table J1 reports the per-case combined R, T RMSE. Table J1 Tier 4 wavelength-OOD fidelity. Per-target median combined R, T RMSE (median RMSE); mean RMSE is the case mean (outlier-sensitive). Case
band(s) nm
type
median RMSE
mean RMSE
WO-NIR-NAR WO-NIR-MID WO-NIR-WIDE WO-SWIR-DEEP WO-OOD-DUAL WO-EDGE WO-NEAR-EDGE WO-MIX-VN WO-MIX-FOOD WO-WIDEST
1450–1550 1500–1800 1400–2000 2000–2500 1500–1700 + 2200–2500 1300–1600 1200–1700 450–700 + 1700–2000 380–1400 + 1500–1900 380–2500
pure OOD pure OOD pure OOD pure OOD pure OOD edge edge mixed mixed mixed
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The pure-OOD and edge-straddling bands are reconstructed at essentially indistribution fidelity (median RMSE ∼ 3 × 10−3 to 1.5 × 10−2 ), including the deepest 2000–2500 nm SWIR band; the mixed multi-octave windows, which must satisfy an in-range and an OOD band at once over a coarsely sampled span, are harder but reconstruct at ∼ 3–7 × 10−2 median RMSE. Figure J1 shows representative reconstructions for four cases spanning the regimes (for each, the most spectrally structured target in the best-reconstructed quartile); the main-text overview (Fig. 2j, k) draws two.
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Tier 2 reconstruction showcase — OOD materials introduced at inference via n,k curves only 02–05 layers · VIS–NIR 500–900 L=4, RMSE=0.080
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Fig. H3 Out-of-distribution reconstructions (Tier 2). Representative designs built from the 15-material held-out bank across the layer range; the model recovers the target response from the candidate n, k curves alone.
Appendix K
OptoGPT head-to-head comparison
This section reports the full protocol and per-case results of the OptoGPT [16] headto-head comparison (Results, Table 2), covering the eleven shared cases. Both methods are scored as best-of-N stochastic samplers with N = 500 draws per sampling configuration: each OptoGPT saved output is the best of 500 autoregressive samples for its case, while IrisFlow draws 500 samples at each requested layer count from 2 to 20 and reports the best across the sweep. The two interfaces differ in stack-depth control. IrisFlow takes the layer count as a query input, so each requested depth is its own configuration (a practical fabrication lever, where depth is itself a cost and yield constraint), whereas OptoGPT emits its own depth per sample and cannot be pinned to a requested one. The total draw budgets therefore differ (500 for OptoGPT against 19 × 500 = 9,500 for IrisFlow), but the per-configuration budget that bestof-N selection actually operates on is matched at N = 500; the 2–20 sweep exercises that controllable depth axis rather than giving IrisFlow extra sampling. Both methods output stacks, and they are compared on a common dense TMM evaluation grid, decoupled from either model’s input grid: 7001 wavelength-uniform points over 400– 1100 nm (0.1 nm spacing). Each predicted stack is rendered to a spectrum on this grid and scored against the case target interpolated onto the same 7001 points: IrisFlow’s 63
Application case studies: best-of-N design vs. analytic target (one case per family) AR-3 solar AR (R low, 400–1100) (RMSE=0.010, L=2)
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Fig. I1 Application showcase (Tier 3). One representative case per application family: best-ofN design vs. the analytic target R, T specification. Tier 3 per-family spectral fidelity (median of per-case best combined RMSE) 0.086
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Fig. I2 Application scorecard (Tier 3). Per-family spectral fidelity (median of the per-case best combined RMSE) across the nine application families.
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Tier 4 wavelength-extrapolation showcase — reconstructions on bands beyond the 380–1400 nm training envelope WO-NEAR-EDGE · edge L=14, RMSE=0.028
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by re-simulation through the local Abeles TMM path (OptoGPT’s material library and the inference layer order), OptoGPT’s from the high-resolution spectra saved with its predicted stacks. The two models are conditioned through different native input grids: OptoGPT on its fixed 71-point wavelength-uniform grid (10 nm spacing), and IrisFlow on a 128-point inverse-wavelength-uniform grid (uniform in 1/λ) resampled from that same target. Neither input grid enters the scoring (both stacks are evaluated on the dense grid above), so the resampling affects only what each model is shown, not how the two are compared. The IrisFlow query interface accepts at most 15 candidate materials per query, so the IrisFlow runs use one fixed 15-slot candidate bank constructed from OptoGPT’s material library: the 12 distinct materials the saved OptoGPT structures use on this subset (Ag, Al2 O3 , AlN, HfO2 , MgF2 , MgO, Si, SiO2 , Ta2 O5 , TiO2 , ZnO and ZnSe), plus three additional materials (TiN, Si3 N4 and ZnS) drawn at random from the
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remainder of OptoGPT’s material pool to fill the bank. Thus this table is a samebenchmark, OptoGPT-aligned material-bank transfer comparison; it is not a strict identical-candidate-set comparison (OptoGPT decodes over its full learned vocabulary, IrisFlow over this 15-curve bank). Because IrisFlow is curve-conditioned, this entire bank is presented through its open-vocabulary n, k interface with no retraining; none of the supplied curves are IrisFlow training inputs (even materials whose names coincide with IrisFlow’s training vocabulary are supplied here with OptoGPT-library optical constants, which can differ substantially), so the head-to-head is also an OOD material test. Five of the eleven targets are drawn directly from the showcase examples in the OptoGPT paper and its SI. Model sizes are reported alongside the scores (Results, Table 2) because the methods are not parameter-matched: OptoGPT is a 6-block decoder-only Transformer with embedding dimension 1024, 8 attention heads and a 901-token output vocabulary (900 material–thickness tokens plus an “EOS” token), totaling an author-reported ≈58M parameters; the IrisFlow 12- and 20-block models carry 85.8M and 136.3M parameters respectively (§C). The 12-block IrisFlow, at 1.5× OptoGPT’s capacity, is the closer size match and still wins the pairwise comparison 8 − 3. The capacity gap also runs in the opposite direction on the task definition: OptoGPT’s parameters serve one fixed material library and spectral grid, whereas the IrisFlow budget covers the open-vocabulary curve encoder and a 100-layer range. Case wins are all-method wins among the three scored folders (per-case results in Table K1). Pairwise, IrisFlow 20-block beats OptoGPT saved output 10 − 1, IrisFlow 12-block beats OptoGPT saved output 8 − 3, and 20-block beats 12-block 9 − 2. These are wins by combined R, T RMSE only. On high reflection NIR, OptoGPT’s saved spectrum has lower R, T RMSE because its mean reflectance in the 800–1100 nm highreflection band is closer to unity, but its R-channel ripple is much larger (the standard deviation of the point-to-point R difference on the shared dense grid over that band is 5.6 × 10−3 , versus 1.7 × 10−4 for 12-block and 1.6 × 10−4 for 20-block). The per-case combined R, T RMSE is bar-charted in Fig. K1, and all eleven cases are shown in the Fig. K2 grid. IrisFlow 20-block
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Table K1 Per-case results.
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OptoGPT Winner saved output R, T RMSE
AR
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bandstop 700-800 extend
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1.181e-01
1.427e-01
bandstop dual optogpt
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2.732e-01
2.864e-01
high reflection 600-1100
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1.509e-01
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2.231e-01
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Appendix L
IrisFlow 20-block IrisFlow 20-block IrisFlow 20-block IrisFlow 20-block IrisFlow 12-block IrisFlow 20-block IrisFlow 12-block OptoGPT saved output IrisFlow 20-block IrisFlow 20-block IrisFlow 20-block
Classical-optimizer comparison on an SC target
This section compares IrisFlow against four classical simulator-in-the-loop optimizers on the radiative-structure-color black (RSC-BLACK) target: the needle method [2] (insertion plus local refinement), GA [4] (population 100 × 10,000 generations), PSO [56] (swarm 100 × 10,000 iterations, constriction form [57]) and uniform random search (106 trials). All methods share the identical target R, T on the same 400–1400 nm 128-point grid, the same five-material candidate pool (Ag, SiO2 , Ti, the process-corrected GSST-a curve of §N, h4; identical optical constants supplied to every method), the same [5, 300] nm thickness window, an 8-layer cap, and the same TMM engine. The baselines optimize the combined R, T MSE directly; all entries below report the combined R, T RMSE of the best design found. All runs enforce the standard fabrication-practice adjacency constraint that Ag must never be deposited in contact with GSST-a: silver is a fast diffuser in chalcogenide films and reacts with the chalcogens to form silver chalcogenides, degrading both the metal layer and the phase-change film [58]. The classical optimizers enforce the constraint natively in their feasibility check, rejecting violating stacks before any simulation. The comparison axis is TMM evaluations, not wall-clock: each baseline’s count is its optimizer-reported total, and each IrisFlow stochastic candidate costs exactly one
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TMM evaluation (the re-simulation that scores it; model forward passes are excluded from this axis by construction). IrisFlow (the reported 20-block checkpoint, 15 sampling steps) drew 12,500 candidates (5 seeds × 500 samples at each requested layer count 4–8) with the adjacency constraint applied by screening: a violating stack is rejected by inspection before any simulation, so screening is free on the TMM axis. 12,350 of the 12,500 generated candidates (98.8%) comply, and the expected best-ofN curve in Figure L1 is computed over this compliant pool by exact order statistics, so intermediate budgets are read directly off the same run. Each classical optimizer’s recorded best-so-far trajectory is drawn as a convergence curve on the same axis. Table L1 RSC-BLACK comparison (best combined R, T RMSE).
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IrisFlow’s best compliant design (0.0252) outperforms every classical optimizer (Table L1). The strongest baselines, GA (0.0292) and PSO (0.0297), need ∼ 5.5 × 105 and ∼ 8.9 × 105 TMM evaluations to plateau, whereas IrisFlow’s expected best-of500 (0.0299) matches their final fidelity at roughly three orders of magnitude fewer evaluations, and its expected best-of-N curve lies below every baseline’s recorded bestso-far trajectory at every budget (Fig. L1). The greedy needle method is trapped by the adjacency constraint and stalls at 0.0936: once the growing stack approaches an Ag/GSST-a contact, the locally best insertions are rejected and growth stops early. Sequential local construction is brittle against feasibility rules of exactly this kind, which the rejection-screened generative pool absorbs at negligible cost. Random search reaches only 0.0354 after 106 evaluations. IrisFlow requires no per-target optimization run: the same checkpoint, sampler and interface used everywhere else in this paper. This is a single-target comparison; the systematic multi-target version remains future work.
Appendix M
Interface-capability comparison across method families
This section condenses the paper’s interface claims into one qualitative comparison. It is deliberately rubric-scored rather than measured: each cell of Table M1 is a statement about what a method family’s published interface admits, scored 0 (absent), 1 (partial) or 2 (full) against the axis definitions below, and Figure M1 renders Table M1 as a radar, adding no information beyond it. Quantitative fidelity is intentionally excluded 69
RSC-BLACK: design fidelity vs simulation budget
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from the radar: mixing a measured error axis into a qualitative rubric would lend it false precision, and those comparisons live in Tables 1–2 and Appendices §H, §K and §L. Four method families are compared. IrisFlow is scored on the capabilities demonstrated in this paper, with each cell pointing at the section that evidences it. OptoGPT is scored as the concrete representative of closed-world autoregressive sequence models, from its published description [16]: a 6-block decoder-only Transformer of ≈58M parameters with 900 learned material–thickness tokens plus an “EOS” token, thickness tokenized at 10 nm steps over 10–500 nm, variable stack depth up to 20 layers, a fixed 400–1100 nm spectral grid, and decode-time constraint imposition by probability resampling (masking and renormalizing the token distribution); it is also the §K head-to-head baseline (model sizes are tabulated there). Direct-inverse networks (tandem, MDN, cVAE and cINN designs [11, 13–15]) are scored as a class on their shared architecture: a fixed-length spectrum vector in, a fixed structure template out. Classical simulator-in-the-loop optimizers (needle, GA, PSO, DE [2, 4–6]) are scored as a class: because their merit function calls the TMM simulator directly, they score full on the first five interface axes, partial on solution diversity (one run converges to one design) and fail only on amortization: each target is a fresh optimization run. The intended reading of Figure M1 is therefore not that IrisFlow dominates, but that
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the flexible half of the chart (historically classical territory) and the amortized half (historically neural territory) are covered by a single model. Axis definitions (each scored 0/1/2): 1. Open material vocabulary: a new candidate material enters as data (an n, k curve or simulator table) at query/run time, with no retraining or vocabulary change. 2. Wavelength-grid flexibility: targets may be specified on arbitrary user grids, including sub-bands and stitched non-contiguous bands, without retraining. 3. Continuous thickness: layer thickness is a continuous output coordinate, not snapped to a discretization grid by the method itself. 4. Variable layer count (one model): a single model or procedure serves variable stack depths natively. 5. Constrained / partial queries: an arbitrary subset of layer materials and/or thicknesses can be pinned exactly while the method completes the rest, with the free coordinates conditioned on the pinned ones. 6. Solution diversity: one query yields multiple structurally distinct candidate designs. 7. Amortized inference: designs are produced by forward inference without a pertarget optimization run. Table M1 Interface-capability rubric (0 = absent, 1 = partial, 2 = full).
Axis Open material vocabulary Wavelength-grid flexibility Continuous thickness Variable layer count (one model) Constrained / partial queries Solution diversity Amortized inference
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The non-obvious cells deserve their justifications spelled out. IrisFlow’s row is evidenced by Tier 2 and §N (open vocabulary), Tier 1 dual band and Tier 4 (wavelength), Methods (continuous flow-matching thickness), the 2–100-layer Results (variable depth), §O (constrained queries) and §E.2 (diversity). OptoGPT, constrained queries = 1: decode-time probability resampling genuinely imposes constraints (e.g. restricting the material set or thickness ranges), and a token can be forced when its position is reached; but conditioning is causal (layers emitted before a pinned position are not informed by it), whereas clamping in a bidirectional denoiser conditions every free coordinate on every pinned one (§O), and the constraints live on the discrete token grid. OptoGPT, continuous thickness = 0: the model emits 10 nm thickness tokens; published finer-grained results use post-hoc local fine-tuning, which is external refinement rather than model output (any method composes with local refinement). Direct-inverse nets, continuous thickness = 2: regression heads emit continuous thicknesses for their fixed template, an honest strength of the class. “Direct-inverse nets, 71
diversity = 1”: MDN/cVAE/cINN variants sample multiple solutions while tandemstyle regression is deterministic, so the class is partial. Classical, diversity = 1: one run converges to one design (population methods end with a collapsed population); diversity requires restarts. Classical, variable layer count = 2: needle insertion/deletion changes stack depth within a single run. IrisFlow (this work)
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Fig. M1 Interface-capability radar. Rubric scores of Table M1 (0 = center, 1 = partial, 2 = full), one panel per method family: IrisFlow, OptoGPT (autoregressive sequence model), directinverse networks (tandem / MDN / cVAE / cINN, as a class) and classical simulator-in-the-loop optimizers (needle / GA / PSO / DE, as a class). The dashed outline in the three baseline panels is IrisFlow’s polygon, repeated for reference. Axes are grouped: the three closed-world axes (vocabulary, wavelength, thickness), the two structure axes (depth, constrained queries), then sampling (diversity, amortization).
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Appendix N
Process-specific optical constants and fabrication reporting
N.1 Witness-film calibration and GSST-a visible-range correction. In the fabrication loop, all candidate materials were checked by process witness films: each material was deposited as a single layer under the coating process, and its n, k were fitted from measured reflectance and transmittance. Only the amorphous GSST entry required a library replacement, because its visible-range measurement deviated from the nominal gsst-a table, whereas the other remeasured materials matched their nominal data within the accuracy needed for the cooler designs. The corrected material is therefore reported in the paper as GSST − ameas−vis . This name distinguishes the piecewise process-corrected amorphous GSST curve from nominal gsst-a and records that the replacement is the measured visible-range segment. The general GSST material background is cited from prior work [59]; this citation is not used as the provenance of the new visible-range measurement, of the other process witness-film checks, or of the nominal gsst-a table. The measurement procedure is described in the Methods (Fabrication). The supplied curve is piecewise: it replaces the inaccurate 400–800 nm portion of the original training curve with the new process measurement and retains the original 810–1600 nm portion unchanged because the longer-wavelength data already matched our measurement. The retained portion is therefore validated source data, not a fitted or extrapolated continuation of the visible measurement. The resulting hybrid table has a discontinuity between 800 and 810 nm, which is retained in the source data and should be considered when using queries that cross this boundary. The reported cooler queries span 400–1400 nm and therefore exercise both parts of the hybrid curve. Figure N1 isolates the replacement segment and compares it against the original training values. Quantitatively, over the corrected 400–800 nm interval the measured-minus-nominal index change has a mean of −0.100 (RMSE 0.210, maximum |∆n| = 0.314), and the extinction coefficient is higher throughout (mean ∆k = +0.501, RMSE 0.532, maximum 0.681): the deposited film is lower-index and markedly more absorbing than the nominal table, changing both the phase accumulation and the absorption predicted for every GSST-containing layer. N.2 Selected cooler designs and cooling-aware selection. The four colordisplaying cooler designs of the main text (Fig. 4) are each selected from a 12,500candidate pool (500 stochastic draws at each requested layer count 4–8, five seeds) over the five-material process-corrected n, k bank {SiO2 , Ag, Ti, GSST − ameas−vis , LaTiO3 (h4)}, screened for the Ag–GSST-a adjacency rule (§L); 96–99% of each pool complies. Selection follows the cooling-aware rule of §N.3: among designs within ∆E00 ≤ 3.0 of the target color under D65, choose the one with the highest AM1.5G-weighted near-infrared reflectance. By construction of the exact-color targets, the D65 target color equals the sRGB-secondary reference (IEC standard 61966-2-1 [60, 61]). For black, cyan and magenta this best-cooler design is the one fabricated; for yellow the cooling optimum (§N.3, Table N1) is a 7-layer stack carrying a thick titanium layer, so a structurally simpler 6-layer in-tolerance design (∆E00 = 2.9, R̄NIR = 0.910, a single thin Ti layer) was fabricated instead, trading ∼0.02 in solar-NIR reflectance for
73
Refractive index
Extinction coefficient
4.2 2.00 4.0
1.75 1.50 k
n
3.8 1.25 1.00
3.6
0.75 3.4
Nominal GSST-a Measured GSST-a (GSST−ameas−vis )
0.50 0.25
3.2
0.7 0.6
0.1
Δk = kmeas − knom
Δn = nmeas − nnom
0.2
0.0 −0.1 −0.2
0.5 0.4 0.3 0.2 0.1
−0.3
0.0 400
500
600 Wavelength (nm)
700
800
400
500
600 Wavelength (nm)
700
800
Fig. N1 Visible-range process correction for GSST-a optical constants. Original trainingcurve values vs. the remeasured refractive index n and extinction coefficient k over the 400–800 nm interval replaced for the cooler designs. The original 810–1600 nm values are retained unchanged because they were already accurate.
electron-beam deposition practicality. The exact-color targets are analytic reflectance profiles constructed to meet color coordinates, not guaranteed multilayer spectra, so residual errors include target realizability as well as model or fabrication error. The main-text designs use the cooling-aware ranking analyzed in §N.3; their one-to-many near-best pools are summarized spectrally in §E.2 and by rendered color in Fig. N3. Re-simulating the selected stacks on BK7 changes predicted reflectance by at most 10−3 , which is applied as the first-order substrate correction in Fig. 4. The exact substrate-to-air material sequences and nominal deposition thicknesses are reported in Table N3. N.3 Balancing solar-infrared cooling against color, and the diversity of admissible designs. A colored cooler must render a visible color while reflecting solar near-infrared light. We therefore score each compliant candidate by color error ∆E00 under D65 and AM1.5G-weighted near-infrared reflectance R̄NIR over 780–1400 nm, then select the highest-R̄NIR design within the accepted color tolerance ∆E00 ≤ 3.0. This rule differs from minimizing spectral RMSE: a design can be a better cooler precisely by reflecting more near-infrared light than the analytic target. Figure N2 and Table N1 show the resulting Pareto structure. The near-black target is the limiting case: the color-best design is a near-perfect absorber with only 15% solarNIR reflectance, whereas the best acceptably black cooler reaches 74.5% and higher cooling would visibly gray the surface. The chromatic targets are more favorable: cyan 74
and yellow trade a small acceptable color shift for higher cooling, while magenta is already near its cooling ceiling. Table N1 Cooling–color balance and admissible-design diversity of the four cooler targets. R̄NIR is the AM1.5G solar-weighted reflectance over 780–1400 nm (cooling figure of merit; higher is better). Color-best = lowest-∆E00 compliant design; recommended = the main-text design, the best cooler (highest R̄NIR ) within the accepted color tolerance ∆E00 ≤ 3.0; “max R̄NIR ” is the highest cooling attainable at all (reached only at the large ∆E00 in parentheses). The last three columns describe the five lowest-∆E00 compliant designs with distinct material sequences (Fig. N3): layer-count range, mean pairwise material disagreement ⟨∆mat⟩ (defined as in §E.2), and the spread of their cooling figure of merit.
Target
color-best recommended max R̄NIR div. R̄NIR tgt R̄NIR ∆E00 / R̄NIR ∆E00 / R̄NIR (∆E00 ) layers ⟨∆mat⟩ range
black cyan magenta yellow
0.804 0.984 0.990 0.990
1.3 / 0.152 1.7 / 0.947 2.1 / 0.976 2.0 / 0.903
2.9 / 0.745 2.6 / 0.974 2.7 / 0.976 2.2 / 0.930
0.982 (97) 0.991 (26) 0.991 (43) 0.991 (30)
4–8 7–8 6–8 4–8
0.53 0.66 0.43 0.84
0.13–0.79 0.95–0.97 0.98–0.98 0.87–0.93
Within the color tolerance the inverse map remains one-to-many. The five lowest∆E00 distinct-sequence designs for each target span 4–8 layers, material disagreement 0.43–0.84 and broad cooling ranges (for example 13–79% solar-NIR reflectance for black), so secondary criteria such as cooling, deposition cost and layer count can be chosen after the color constraint is met (Fig. N3, Table N1). This design-side degeneracy has an upstream, input-side counterpart that we do not exploit. A color fixes only the integrated tristimulus coordinates, so the visible reflectance shape that renders a given color is itself under-determined; our construction resolves it with a fixed analytic recipe (a smoothness-regularized fit to the exact color coordinates over a chosen visible/near-infrared partition), which selects one representative from a whole family of admissible target spectra, all rendering the same color (cf. the realizability caveat of §N.2). We did not search this family for the member best matched to the model’s reach (lowest reconstruction RMSE) or to cooling (highest R̄NIR ). The reported coolers are therefore one admissible instance: jointly optimizing the visible target shape and the visible/near-infrared partition within the color tolerance is an unexploited lever that can only improve the best attainable color-constrained cooler, a direction we leave to future work. N.4 As-fabricated reflectance and measured color fidelity. The four designs selected for fabrication in §N.2 were deposited by ion-assisted evaporation on BK7 substrates and their reflectance measured over 400–2500 nm (Methods); the coatings are opaque (T ≈ 0), so reflectance fully characterizes the response. The chamber was pumped to a base pressure of ≈3 × 10−3 Pa; during pump-down the LaTiO3 (h4), SiO2 and Ti sources were pre-melted to stabilize their deposition rates, and the substrate was then ion-beam cleaned for 3 min to remove surface dust and improve adhesion before growth began. Figure 4 overlays each measured spectrum on the target and the selected design’s BK7 prediction together with the rendered target, design and measured colors, and Table N2 reports the measured color error and solar-weighted 75
black
color
target
color
target
4
4
0
2
0 0.94 R̄
0.96 NIR
0.98
1.00
0 0.96
(solar-NIR reflectance, cooling)
0.97 0.98 0.99 1.00 R̄ NIR (solar-NIR reflectance, cooling)
1.01
0.90
0.92 0.94 0.96 0.98 R̄ NIR (solar-NIR reflectance, cooling)
1.0
1.0
1.0
1.0
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
color-best: ΔE00 1.3, R̄ NIR 0.152 recommended: ΔE 2.9, R̄ 0.745 00
color-best: ΔE00 1.7, R̄ NIR 0.947 recommended: ΔE 2.6, R̄ 0.974
NIR
00
0.2
AM1.5G (scaled) target color-best recommended
0.0 400
600
800
1000
wavelength (nm)
1200
1400
00
AM1.5G (scaled) target color-best recommended
0.0 400
600
800
1000
1200
color-best: ΔE00 2.0, R̄ NIR 0.903 recommended: ΔE 2.2, R̄ 0.930
NIR
00
0.2
AM1.5G (scaled) target color-best recommended
0.0
1400
wavelength (nm)
1.00
0.4 color-best: ΔE00 2.1, R̄ NIR 0.976 recommended: ΔE 2.7, R̄ 0.976
NIR
0.2
cooler
accepted color (ΔE00 ≤ 3)
2
1.0
color
4 accepted color (ΔE00 ≤ 3)
2
0.4 0.6 0.8 R̄ NIR (solar-NIR reflectance, cooling)
target
cooler
6
accepted color (ΔE00 ≤ 3)
0
color
6
accepted color (ΔE00 ≤ 3)
0.2
8
cooler
6
4
yellow
8
cooler
6
2
reflectance
magenta
8 target
ΔE00 vs target (color)
cyan
Pareto front color-best recommended
8
400
600
800
1000
wavelength (nm)
1200
1400
NIR
0.2
AM1.5G (scaled) target color-best recommended
0.0 400
600
800
1000
1200
1400
wavelength (nm)
Fig. N2 Balancing solar-infrared cooling against color over the fixed cooler pool. Per target. Top: CIEDE2000 color error (against the sRGB-secondary reference under D65) versus the AM1.5G solar-weighted near-infrared reflectance R̄NIR (780–1400 nm; the cooling figure of merit, higher is better) for every compliant candidate (gray; the color axis is clipped to the informative range), with the non-dominated Pareto front (black), the color-best (lowest ∆E00 , diamond) and recommended (best cooler within the accepted color tolerance, filled plus) selections. The green shaded band and dotted line mark the accepted-color region ∆E00 ≤ 3.0 (with a fainter line at the goodmatch level ∆E00 = 2); the blue dashed vertical line marks a high-performance cooler (R̄NIR = 0.95) and the gray dashed line the target’s own R̄NIR ; rendered swatches show the target color, the colorbest and the recommended cooler. Bottom: reflectance of the selections overlaid on the target, with the near-infrared cooling band (≥ 780 nm) shaded and the AM1.5G solar spectrum drawn faintly (scaled); cooling is the height of the reflectance over the shaded band. For black, the unspecified 780–900 nm band is shaded gray and the target line broken (selection spectra continuous). The color– cooling tension varies by target (Table N1): the chromatic recommended designs buy cooling for a small, still-acceptable color change or sit near the ceiling, whereas the near-black color-best is a nearperfect absorber that cannot be made a good cooler within the acceptable-color band.
near-infrared reflectance per target. Beyond the headline color and cooling figures given in Results, the as-fabricated spectra track the model’s design prediction more closely than the idealized analytic target for every sample (reflectance RMSE 0.03–0.09 against the design versus 0.07–0.14 against the target), and the measured colors land within ∆E00 = 0.8–4.3 of the chromatic designs, so the predicted stack is the operative reference for the deposited film. The residual target–design offset is design-side rather than fabrication error, and it separates by metric: in color it is the selection margin spent deliberately by the cooling-aware rule of §N.3 (the fabricated designs sit at the tolerance boundary, ∆E00 = 2.6–2.9, although the same pools reach ∆E00 = 1.3– 2.1 when ranked by color alone; Table N1), while in the spectral residual it is the realizability term of §N.2, the analytic targets not being guaranteed multilayer spectra. The near-black sample shows the same separation: its measured reflectance tracks the design most closely of the four samples (0.034 RMSE), but near L∗ ≈ 0 the cuberoot lightness response of CIE colorimetry magnifies small absolute visible residuals into large color distances, so the target-referenced ∆E00 = 21.4 (a very dark green) overstates the spectral discrepancy rather than indicating a failed deposition.
76
ΔE 1.3 R̄ NIR 0.15
ΔE 2.9 R̄ NIR 0.75
ΔE 3.4 R̄ NIR 0.13
SiO2 82
SiO2 87
SiO2 75
ΔE 4.9 R̄ NIR 0.53
ΔE 5.1 R̄ NIR 0.79
black 0
target
SiO2 44 GSST-a 36
GSST-a 36
Ag 70
Ag 66
Ag 72
−1
b
SiO2 43 GSST-a 38
SiO2 65 *
SiO2 70 SiO2 34 GSST-a 34
−2
LaTiO3 97 Ag 78
−3
Ag 60
target
−4 −3
−2
−1
a ΔE 1.7 R̄ 0.95 NIR
ΔE 2.3 R̄ 0.97
ΔE 2.3 R̄ 0.97
ΔE 2.6 R̄ 0.97
ΔE 2.8 R̄ 0.97
SiO2 160
SiO2 162
SiO2 162
SiO2 158
SiO2 178
Ag 38 LaTiO3 116
NIR
NIR
NIR
NIR
0
1
−44
−42
*
−12
cyan
−13
LaTiO3 137 GSST-a 35
target
−14 *
SiO2 168
b
SiO2 105
SiO2 155
LaTiO3 86
LaTiO3 180
LaTiO3 108
SiO2 127
Ag 64
Ag 51
Ag 69
Ag 73
Ag 36 Ag 51
ΔE 2.1 R̄ 0.98
ΔE 2.7 R̄ 0.98
ΔE 3.1 R̄ 0.98
ΔE 3.3 R̄ 0.98
ΔE 3.5 R̄ 0.98
SiO2 125
SiO2 125
SiO2 127
SiO2 129
Ag 35 SiO2 137
LaTiO3 85
−15 −16 −17
−52
−50
−48
−46
a NIR
NIR
NIR
NIR
NIR
magenta
target
−52
LaTiO3 176
LaTiO3 244
Ag 67
SiO2 139
Ti 41 Ti 43
Ti 46
LaTiO3 175
Ag 61
Ag 71
ΔE 2.0 R̄ 0.90
ΔE 2.2 R̄ 0.93
ΔE 2.3 R̄ 0.87
LaTiO3 136
LaTiO3 133
−54 −56
*
SiO2 141 GSST-a 39
b
SiO2 151 GSST-a 36
SiO2 127
Ti 35 LaTiO3 237
−58
Ag 55
Ag 60
−60
ΔE 2.9 R̄ 0.91
ΔE 3.3 R̄ 0.88
LaTiO3 142
SiO2 214
85
90
a NIR
NIR
NIR
NIR
95
*
NIR
yellow
100.0
SiO2 95
LaTiO3 93
Ag 40
Ag 49 SiO2 79 Ag 47
SiO2 99 Ti 49
SiO2 105 LaTiO3 107
SiO2 133
Ag 78
Ag 53
97.5 95.0 *
SiO2 107
SiO2 160 Ag 63 LaTiO3 93 GSST-a 45
b
SiO2 220
target
*
92.5 90.0 87.5 85.0 −21
−20
−19
a
−18
−17
*
Fig. N3 Diversity of admissible color solutions and their cooling. Per target (rows): the target color (left); the five lowest-∆E00 compliant designs with distinct material sequences (thicknessscaled stacks, substrate side at the bottom, each design’s rendered color drawn as a band above its stack and labeled with its ∆E00 and solar-NIR cooling R̄NIR ); and the CIE a∗ b∗ chromaticities of those designs against the target (×). Structurally distinct stacks render a tightly clustered family of colors around each target but span a wide range of cooling, so cooling rather than color discriminates among admissible designs. For the near-neutral black target the residual color spread is mainly in lightness L∗ and is understated by the a∗ b∗ projection.
Table N2 As-fabricated measurements of the four color coolers. Measured (M) reflectance of the deposited coatings against the target (T) color and the selected design (D) of Fig. 4. ∆E00 is the CIEDE2000 color difference under D65; R̄NIR is the AM1.5G solar-weighted reflectance over the indicated band (nm; the cooling figure of merit, higher is better). RMSE is over the 400–1400 nm design grid (reflectance only; the coatings are opaque, T = 0). The design R̄NIR (780–1400 nm) reproduces the recommended column of Table N1, except for yellow, whose fabricated stack is the structurally simpler 6-layer in-tolerance design of §N.2.
Target
∆E00 M–T
∆E00 M–D
R̄NIR meas (780–1400 / 780–2500)
R̄NIR design (780–1400)
RMSE(R) M–T / M–D
black cyan magenta yellow
21.4 3.1 5.2 3.6
21.5 0.8 4.3 2.1
0.72 / 0.78 0.94 / 0.93 0.95 / 0.95 0.92 / 0.93
0.75 0.97 0.98 0.91
0.070 / 0.034 0.136 / 0.056 0.092 / 0.090 0.113 / 0.083
77
Table N3 Final deposited stack recipes for the four color-displaying coolers. Layer order runs from the BK7 substrate side to the air side, matching the left-to-right stack strips in Fig. 4. Parentheses give the nominal design/deposition thickness in nm; totals are rounded to 0.1 nm. GSST-a denotes the process-corrected amorphous GSST curve of §N.1, and LaTiO3 is the material denoted h4 internally.
Target
Layers Total (nm) Ordered stack: material (thickness in nm)
yellow
6
496.6
magenta
8
675.6
cyan
7
529.1
black
6
272.7
Appendix O
Ag (52.5) / SiO2 (133.4) / Ti (33.7) / SiO2 (104.8) / Ag (29.8) / LaTiO3 (142.4) Ag (61.5) / LaTiO3 (244.4) / GSST-a (38.8) / Ti (15.4) / SiO2 (141.3) / Ag (31.9) / SiO2 (124.6) / Ag (17.9) Ag (70.8) / LaTiO3 (76.7) / GSST-a (35.0) / LaTiO3 (131.1) / Ag (32.2) / SiO2 (162.9) / Ag (20.3) Ag (66.3) / LaTiO3 (30.6) / GSST-a (36.1) / SiO2 (43.9) / GSST-a (9.3) / SiO2 (86.5)
Constrained generation: partial-structure completion and fixed-structure thickness inference
This section documents the two constrained query modes of the sampler on a worked example. Both are imposed by exact clamping at every reverse step (Methods, Sampling and inference), so all N draws honor the constraint without retraining, fine-tuning or guidance. At the interface level both modes are inference-time query arguments: the user supplies the material sequence with unknown slots marked free, optionally pinning per-layer thicknesses. Both modes are exercised on the example query of §E.1: eight layers, the combined R, T absorb-below/reflect-above edge target (R = 0 below 780 nm, R = 0.99 at 900– 1400 nm, T = 0 throughout) on the 400–1400 nm 128-point grid, over the five-material candidate bank {Ag, SiO2 , GSST-a, h4, Ti}, with GSST-a again the process-corrected curve of §N. Sampling uses the same reported checkpoint (EMA weights), joint sampler and adaptive-power stack-time schedule (15 steps, p = 2; §D.3) as the benchmark tiers, with N = 500 stochastic draws per query; every draw is re-simulated with the TMM solver and ranked by combined R, T RMSE. The unconstrained reference on this target is the §E.1 trajectory run itself, which uses the identical sampler configuration at N = 2,500: its lowest combined-RMSE draw reaches 2.5 × 10−2 . O.1 Partial-structure completion. The template fixes Ag as the bottom layer and SiO2 as the four spacer layers, like Ag/SiO2 /?/SiO2 /?/SiO2 /?/SiO2 , while leaving the three interior material slots and all eight thicknesses free (Fig. O1a). The completion statistics are strongly peaked on the physically appropriate candidate: among the 50 lowest-RMSE draws, the free slots are filled with the absorber GSST-a in 54%, 82% and 100% of draws respectively (Fig. O1b), and 10% of all 500 draws select GSSTa at all three slots simultaneously. The best completed design (GSST-a at all three slots, combined RMSE 2.1 × 10−2 ; Fig. O1c) recovers the same metal-backed absorber motif that unconstrained generation finds on this target, at better fidelity than the N = 2,500 unconstrained best and a fifth of its sampling budget. 78
O.2 Fixed-structure thickness inference. Pinning the full eight-layer material sequence Ag/SiO2 /GSST-a/SiO2 /GSST-a/SiO2 /GSST-a/SiO2 reduces sampling to thickness inference over the given structure (Fig. O1d). The best of 500 thickness sets reaches a combined RMSE of 2.0 × 10−2 , and 19% of all draws fall below 5 × 10−2 (versus 6% in the partial mode, whose draws also vary the materials); the three constrained modes are summarized in Table O1. The returned thickness candidates are visibly multimodal (Fig. O1e): the second and fourth (SiO2 cavity) layers split into two distinct interference orders (≈20–45 nm and ≈180–260 nm), with the selected design drawn from the thick-order family; these structurally distinct thickness sets realize near-identical responses and are the continuous analog of the discrete solution diversity of §E.2. This is the intended deployment mode for spec-grade refinement workflows: for a frozen structure the constrained sampler proposes diverse thickness initializations that respect the fabrication window, each costing one TMM evaluation to verify, which a local optimizer (needle, gradient) can then polish.
Table O1 Constrained-generation summary on the §E.1 example target (all rows: same checkpoint, sampler and schedule; designs ranked by combined R, T RMSE).
Query mode
constrained coordinates
N best RMSE median RMSE draws < 0.05
Unconstrained none 2,500 (§E.1) PartialAg + 4 SiO2 slots pinned; 3 500 structure material slots + 8 completion thicknesses free Fixed-structure all 8 materials pinned; 8 500 thickness thicknesses free inference
Appendix P
0.025
0.156
1.2%
0.021
0.225
5.6%
0.020
0.110
19.4%
Training-free oblique-incidence and polarized design
The reported model is trained and queried at normal incidence (Methods). Obliqueincidence and polarized design are obtained without retraining by a forward-model reparameterization: the oblique problem is mapped to a normal-incidence proxy that the existing n, k candidate interface already accepts, candidates are generated against the proxy, and every design is validated and ranked by the exact angled TMM on the real material system. This appendix states the transform, its exactness for spolarization and the residual error for p-polarization, the experimental protocol, and the per-case reconstructions (Figs. P1 and P2) behind the oblique-incidence result reported in Results (Fig. 3). P.1 Effective-index reparameterization. For an isotropic, non-magnetic stack at external angle θ0 in an incident medium of index ninc , tangential-wavevector
79
b
partial-structure completion
SiO2 73 nm
?
GSST-a 10 nm
SiO2 ?
fraction of 50 lowest-RMSE draws
1.0 SiO2
SiO2 39 nm
IrisFlow
GSST-a 28 nm
SiO2
SiO2 93 nm
?
GSST-a 14 nm
SiO2
SiO2 23 nm
Ag
Ag 71 nm
substrate
substrate
query template
best design (to scale)
c
materials chosen at the masked slots Ag SiO2
light
h4 Ti
GSST-a
0.8
★
0.8 0.6 0.6
★
0.0
e
light
SiO2 72 nm GSST-a 11 nm
SiO2
SiO2 24 nm
IrisFlow
GSST-a
GSST-a 26 nm
query template
best design (to scale)
1400
0.8
0.6
L4 SiO2
R target R design
T target T design
0.4
L3 GSST-a
Ag 72 nm substrate
1200
1.0
selected design
L5 GSST-a
SiO2 253 nm
Ag substrate
800 1000 wavelength (nm)
best of 500: combined RMSE 0.020
R, T
SiO2 222 nm
600
f
thickness candidates per layer
L6 SiO2
GSST-a 44 nm
SiO2
SiO2
400
layer 7 = ?
L8 SiO2
SiO2
GSST-a
layer 5 = ?
L7 GSST-a
GSST-a
T target T design
0.2
0.2
layer 3 = ?
fixed-structure thickness inference
R target R design
0.4 0.4
0.0
d
best of 500: combined RMSE 0.021
1.0
★
R, T
a
0.2
L2 SiO2 L1 Ag
0.0 0
50
100 150 200 layer thickness (nm)
250
300
400
600
800 1000 wavelength (nm)
1200
1400
Fig. O1 Constrained generation on the §E.1 example query. a,d, The query template (left column: pinned layers colored and labeled, unspecified slots hatched ?; all thicknesses free) and the lowest-RMSE completed stack (right column, layer heights to scale), for partial-structure completion (a) and fixed-structure thickness inference (d). b, Material marginals at the three free slots over the 50 lowest-RMSE of 500 draws (⋆ = the selected design’s choice). e, Per-layer thickness candidates among the 50 lowest-RMSE draws (diamonds = selected design); the SiO2 cavity layers L2/L4 are bimodal across interference orders. c,f, TMM re-simulation of the selected design against the target R, T ; the unspecified 780–900 nm band is shaded gray and the target line broken (design continuous).
conservation gives each layer a normal-wavevector factor
Qj (λ) =
q
Nj (λ)2 − (ninc sin θ0 )2 ,
Nj = nj − ikj
(passive branch Im Q ≤ 0), a layer phase δj = k0 dj Qj with k0 = 2π/λ, and a polarization-dependent admittance Yjs = Qj , Yjp = Nj2 /Qj . A normal-incidence proxy uses one index for both phase and admittance (Q = Y = N ). For s-polarization Qj and Yjs coincide, so the single tilted index
Ñj = Qj =
q Nj2 − (ninc sin θ0 )2
reproduces the s-polarized layer matrix exactly. For p-polarization Qj ̸= Yjp unless θ0 = 0, so no single ordinary index matches both; we use the admittance proxy Ñjp = Yjp = Nj2 /Qj , which matches the p-polarized interface admittance at every waveproxy length, and restore the layer phase by a single real thickness rescale dreal j = aj dj P ∗ p P 2 with the broadband least-squares factor aj = Re λ Qj Yj λ |Qj | . The tilted indices are supplied as an ordinary candidate n, k library, preserving material order
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so generated tokens map back to the real materials by identity; only the conditioning changes and the model itself is unmodified. The proxy library only steers generation. Each candidate design is re-simulated and ranked by the exact angled TMM on the real materials at the true angle and polarization (real ambient and substrate, T from the true Poynting flux), so the reported RMSE is the genuine oblique fidelity of the design. A layer-only tilt does not tilt the semi-infinite ambient and substrate admittances; this boundary term therefore affects only how well the proxy steers sampling, not the scored numbers, and is reabsorbed by the true-angle ranking and any subsequent thickness refinement. P.2 Targets and protocol. For each polarization we sample six feature-rich multilayer targets (10–16 layers) from a transmissive vocabulary (loss-free dielectrics plus thickness-capped Si) so that both R and T carry angle-dependent structure, on pertarget narrow k -uniform sub-bands drawn from the training band distribution (§B). Each target is one fixed physical stack; its true polarized R(λ), T (λ) is computed at θ0 = 0, 20, 40, 60◦ with the reference TMM. For every (target, angle) pair we run the reported Lmax = 100 checkpoint under the best-of-N protocol of §D with the angle/polarization-tilted library, and report the best design’s combined R, T RMSE under the exact angled TMM. Figures P1 and P2 show every (target, angle) reconstruction, with the per-case RMSE annotated in each panel; the 0◦ column serves as the internal baseline. Six targets per polarization are too few to support aggregate statistics, so we report the cases individually: qualitatively, the oblique reconstructions stay about as faithful as their normal-incidence counterparts out to 60◦ for both polarizations, the sole exception being the wide-band target 4, matched poorly at every angle including normal incidence (target difficulty, not an angular penalty). P.3 Residual error of the p-polarization proxy. The admittance proxy Ñ p = N 2 /Q is exact per wavelength; the sole approximation is the collapse of the complex, wavelength-dependent phase factor cos2 θ(λ) = Q2 /N 2 to the single real thickness scale aj , which vanishes for narrow bands, non-dispersive layers, or normal incidence. We quantify it directly: for 300 random 6–24-layer stacks over a 525–1075 nm visible– near-infrared band, we compare the proxy spectrum (admittance swap plus the leastsquares thickness scale) against the exact angled p-polarized TMM, itself validated to ≤ 2 × 10−12 RMSE against an independent characteristic-matrix implementation. With the boundary media tilted, the irreducible cost of the replacement is combinedR, T RMSE ≈ 0.0013, 0.0053 and 0.0075 at 20, 45 and 60◦ (Table P1); this is below the model’s own in-distribution RMSE floor (∼ 0.02–0.08) and scales as sin2 θ0 and with the band-relative dispersion of the layer indices. The larger error of a naive layeronly swap (≈ 0.06 at 45◦ ) is the boundary term shared with s-polarization, removed by the true-angle scoring described above.
Appendix Q
Learned representations: curve tokenization and candidate cross-attention
IrisFlow conditions on optics through two curve-token branches of the same form (§C, Fig. 5): a target-spectrum branch that reads the R, T query and a candidate branch 81
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Fig. P1 Oblique s-polarized design: every case. All six feature-rich targets (rows) re-designed at θ0 = 0, 20, 40 and 60◦ (columns); the 0◦ column is the ordinary normal-incidence model (identity reparameterization). Target R, T are solid and the IrisFlow design dashed (R red, T blue); gray marks no-target bands; each panel title gives the incidence angle and the combined R, T RMSE from exact angled-TMM re-simulation on the real material system. Target 4 is a wide-band case the model matches poorly at every angle, normal incidence included.
that reads every n, k curve in the local material bank. Both embed a 128-point twochannel curve, together with the same signed inverse-wavelength feature, into a single dmodel = 512 token, so the target and the candidate library are described in a common token space; the two roles meet only inside the denoiser, where the spectrum token is injected as an additive conditioning bias and the candidate tokens form the memory that the layer states cross-attend to; the same memory serves every layer slot and every denoising step. Because this conditioning representation is reused unchanged, probing
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it exposes how a stack is assembled. Figure 5f gives the compact main-text view of the candidate-token geometry; Fig. Q1 projects the learned tokens of the reported Stage3 model (§C) and reads out the candidate cross-attention for the four Gaussian-CMY radiative structure-color targets (black and the subtractive primaries cyan, magenta and yellow). The per-curve representations are physically structured. The candidate branch arranges the 15 training materials by family even though it is never given a family label: the metals (Ag, Cr, Ti), the dielectric spacers (SiO2 , Al2 O3 , TiO2 , Ta2 O5 and h4), the semiconductor Si and the phase-change/absorbing chalcogenides (GST/GSST
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Table P1 Residual error of the p-polarization proxy. Combined R, T RMSE between the proxy spectrum and the exact angled p-polarized TMM, averaged over 300 random 6–24-layer stacks on a 525–1075 nm band. “p-pol proxy” tilts the ambient/substrate admittances (the irreducible cost of the N 2 /Q replacement with a single real thickness rescale); “naive layer-only swap” leaves the boundary media untilted. The proxy cost is below the model’s own RMSE floor; the larger boundary term of the naive swap is reabsorbed by the true-angle scoring used in design.
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and the Sb2 S3 states ss-a/ss-b) form separate groups (Fig. Q1a). The same projection places the 15 held-out OOD materials (triangles) in the matching family regions: the OOD metals (Au, W, Nb, Pt, Al) land among the trained metals and the highindex oxides (CeO2 , Y2 O3 , HfO2 , ZrO2 ) among the dielectric spacers. A candidate absent from training thus enters the existing geometry from its optical constants alone; this is the property the open-vocabulary benchmark relies on. A nonlinear t-SNE projection of the same candidate tokens gives the same qualitative picture (Fig. Q1b). The “common token space” of Fig. 5 denotes one coordinate system and tokenizer interface for wavelength-aware optical curves, not a claim that target-spectrum tokens and candidate-material tokens must occupy coincident regions. The cross-attention is what reconciles them. For every color target it concentrates on the building blocks of a thin-film color filter (the low- and high-index dielectric spacers SiO2 , Ta2 O5 , Al2 O3 , h4 and TiO2 , together with Ag) and places little weight on the phase-change materials (Fig. Q1c). Projecting the candidate keys and the spectrum-induced query of that cross-attention together gives a qualitative view of the same attention readout; the gray links mark the three most-attended materials for each target in this two-dimensional projection (Fig. Q1d). The correspondence between a target spectrum and the materials that realize it is therefore learned by the cross-attention, not built into the encoders. This yields a mechanistic reading of the query interface (Fig. 5): a design is composed by repeatedly routing the evolving per-layer states, through a shared comparison space, to a query-local material memory whose tokens are organized by optical family; a candidate that was absent from training enters as one more curve token in the same geometry, and material choice is mediated by an attention-weighted comparison to the query-local material memory rather than a lookup in a global closed vocabulary. The projections summarize one checkpoint on representative targets in two dimensions and are intended as a qualitative, interpretability-level view rather than a quantitative claim.
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a Candidate n, k tokens: trained + held-out (OOD)
b Candidate n, k tokens: t-SNE Metal Oxide / dielectric Semiconductor Phase-change Transparent conductor Polymer trained held-out (OOD)
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Fig. Q1 Learned conditioning representations and candidate cross-attention. Probes of the reported Stage-3 model (§C) on the four Gaussian-CMY radiative structure-color targets. a, Principal-component projection of the candidate n, k tokens for the 15 training materials (circles) and the 15 held-out OOD materials (triangles), colored by material family (assigned for display only); the held-out materials fall into the same family regions as the training set. b, t-SNE projection of the same candidate n, k tokens, shown as a nonlinear view of the material-token geometry. c, Candidate cross-attention weight from the layer states to each material (mean over heads, layer slots and denoising blocks), per target; the bright columns mark the most-attended candidate materials. d, The candidate keys (circles, by family) and the spectrum-induced queries (stars) of the candidate cross-attention projected together; gray lines join each target to its three most-attended materials, without implying a nearest-neighbor rule in the two-dimensional projection. Panels a and d are twodimensional principal-component projections, with the variance explained shown on each axis; panel b is t-SNE. The cross-attention panels c,d are scored over each Gaussian-CMY target’s trained 15material candidate bank.
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