Entropy Across the Bridge: Conditional–Marginal Discretization for Flow and Schrödinger Samplers
Bruno Trentini1,2∗
Dejan Stancevic3
arXiv:2605.16126v1 [cs.LG] 15 May 2026
Alexander Tong4 1
Michael M. Bronstein2,4
Luca Ambrogioni3∗
NVIDIA Corporation, Santa Clara, CA, USA
2
University of Oxford, Dept. of Computer Science, Oxford, UK 3 Donders Institute for Brain, Cognition, and Behaviour, Radboud University, Nijmegen, NL 4 AITHYRA, Research Institute for Biomedical AI, Vienna, AT
Abstract For a fixed flow-based generative model under a small inference budget, sample quality can depend strongly on where the sampler spends its few function evaluations. Flow matching and Schrödinger bridges define probability paths, yet their inference grids are usually heuristic or inherited from one-endpoint diffusion. We derive a conditional–marginal entropy-rate objective for bridge-aware discretization, separating endpoint-conditioned bridge geometry from marginal flow evolution, and use it to build a training-free entropic inference-time scheduler from first principles. For Gaussian Brownian bridges this rate is closed-form and U-shaped, motivating boundary-heavy nonuniform grids. On trained twodimensional bridge/flow models, the estimated profile recovers the predicted shape and improves 10-step ODE-Heun MMD over linear by 18.1%, with a paired 22.7% SDE-Heun improvement in the same low-NFE sweep. On EDM/CIFAR-10, the entropic time-discretization gives the best tested five-step FID (186.3 ± 4.0 versus 200.5 ± 2.9 for linear and 238.0 ± 5.3 for cosine). On AlphaFlow protein generation, entropic conditional–marginal (cond-marg) scheduling shows advantage in low-NFE regimes on both CAMEO22 and ATLAS benchmarks. These results support entropy-rate scheduling as a practical low-budget allocation signal for high-dimensional bridge and flow samplers.
1
Introduction
Modern generative samplers often define a continuous-time path and then approximate it with a small number of discrete evaluations. Diffusion models, flow matching, stochastic interpolants, and Schrödinger bridges all share this numerical bottleneck [2, 8, 15, 20, 21, 26, 27, 50]. We use NFE2 to denote the number of neural network function evaluations performed during sampling. At high NFE, many grids become similar because integration error is small. In the low-NFE regime, a misplaced step can dominate the behavior of the sampler. This issue is no longer only about image synthesis. The same sampling budget appears in protein design, molecular generation, materials discovery, biological trajectory modeling, and other AI-for-science systems where each extra model call can be expensive. The standard response is to choose a useful time grid. Linear, cosine, sigmoid, power, and log-SNR grids encode different beliefs about where denoising or transport is difficult [20, 21, 25, 36]. Higher∗ Correspondence to: [email protected], [email protected] 2 NFE counts neural network calls during sampling. It is a useful proxy for latency, energy use, and serving cost because these calls dominate inference in many diffusion, flow, and bridge samplers.
Preprint.
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Figure 1: Entropy profiles for EDM and AlphaFlow. EDM is sharply endpoint-concentrated and requires log tempering; AlphaFlow recovers the boundary-heavy cond–marg profile predicted by the Brownian-bridge calculation. Shaded bands mark BCR endpoint windows.
order solvers and predictor-corrector schemes change the update rule [31, 32, 46, 55, 56]. Optimized or learned schedulers search for better timesteps or trajectory parameterizations, including Align Your Steps, optimized-step rules, dynamic-transport schedules, and recent few-step scheduler learning [35, 40, 51, 54]. Align Your Steps optimizes a discrete schedule for diffusion sampling, Lipschitz or transport-based rules use smoothness proxies, and learned scheduler families fit parameterized curves. These methods show that the grid matters, albeit they do not by themselves say which bridge quantity should be measured. Scheduling and distillation also solve different problems: distillation changes the model weights, while the present method changes the inference grid and can therefore be combined with a distilled sampler. The distinction matters because the bridge and flow literature has moved beyond one-endpoint image diffusion. Schrödinger bridges now appear in simulation-free flow and score matching, image translation, energy-based sampling, biological trajectories, discrete spaces, and structured domains [22, 27, 28, 45, 47, 50, 53]. Flow matching has also become central in protein and molecular generation, including AlphaFlow, Proteina, BioEmu, MolFlow, FlowMM, and Wasserstein flow matching [9, 13, 17, 18, 24, 34]. In these settings, a sampler is not just reversing a noising process because it transports between structured endpoints, often under geometric constraints. Recent entropic time work gives a principled schedule for diffusion models by using conditional entropy as a time coordinate [43, 44]. A Schrödinger bridge carries an additional layer: conditional paths indexed by endpoint information and a marginal population flow obtained after those conditions are mixed. A single-field entropy story is therefore too small for the bridge, because the rate must compare how volume changes along the paired conditional path with how volume changes after endpoint mixing. The bridge story needs a two-term object. Let Z denote the bridge condition, such as a data endpoint or endpoint pair, and let Xt denote the state at time t. Under a smooth probability-flow description, we show that d H(Z | Xt ) = EZ,Xt |Z [∇· vt (Xt | Z)] − EXt [∇· v̄t (Xt )] . dt
(1)
The first term measures conditional volume change along paired bridge paths. The second measures marginal volume change after the bridge conditions have been mixed. Their difference is the conditional–marginal (cond–marg) information-rate signal studied here as a bridge-aware scheduling signal. The proposed grid is obtained by normalizing the magnitude of this signal and inverting its cumulative distribution. This paper makes five contributions: (a) an entropy-rate criterion for inference-time discretization in flow and bridge samplers; (b) a cond–marg estimator that implements Equation (1); (c) a Brownianbridge closed form showing why the conditional term is U-shaped and singular near the endpoints; (d) a link between entropy-weighted grids and local ODE error; (e) a staged empirical path from controlled two-dimensional bridge/flow models to high-dimensional samplers. 2
2
Preliminaries
Probability flows. Let p0 be a data distribution on Rd and p1 be a reference distribution. A deterministic probability flow is a density path (pt )t∈[0,1] and vector field v̄t satisfying ∂t pt (x) + ∇· pt (x)v̄t (x) = 0. (2) Sampling integrates the learned field from the reference endpoint toward data on a decreasing time grid 1 = t0 > t1 > · · · > tN = 0. This numerical grid is the object we study. Flow matching. Flow matching often starts from a conditional interpolation. Given a condition Z, define a conditional law pt (x | Z) and a conditional vector field vt (x | Z) satisfying ∂t pt (x | Z) + ∇· pt (x | Z)vt (x | Z) = 0. (3) The marginal field is the posterior average v̄t (x) = E[vt (x | Z) | Xt = x], whenever the conditional and marginal descriptions are compatible. This is the population field targeted by standard flow matching losses [2, 26, 29]. Optimal transport and Schrödinger bridges. Optimal transport asks for a low-cost coupling between two endpoint distributions. A Schrödinger bridge adds stochasticity by selecting the path measure closest in relative entropy to a reference process while matching the same endpoints [23, 41, 42]. In the Brownian reference case, the bridge path measure can be written as a mixture of Brownian bridges weighted by an entropic optimal transport plan: Z P ((Xt )t∈[0,1] ) = Q((Xt )t∈[0,1] | x0 , x1 ) dΠ2σ2 (x0 , x1 ). (4) For this case the natural condition is Z = (X0 , X1 ). The distinction between the conditional field and the marginal field is not cosmetic. It is the mathematical expression of the bridge constraint. Time grids and NFE. Given a nonnegative rate r(t), an entropic grid is constructed through Rt r(s) ds k Q(t) = R 01 , tk = Q−1 . N r(s) ds
(5)
0
The number of intervals, or equivalently the number of model evaluations for a one-step solver, is the NFE budget. The central question is what rate should be used for a bridge. The rate derived below is d r(t) = | dt H(Z | Xt )|, optionally transformed by log(1 + r) when the raw signal is too concentrated near singular endpoints. Expanded derivations for the entropy identities, Brownian bridge field, and ODE allocation rule are given in Sections A, B, B.2 and E. The empirical sections are guided by three questions summarized in Section H. First, does the Brownian-bridge calculation predict a U-shaped conditional rate, and do small learned twodimensional bridge probes show the same boundary-heavy pattern? Second, does the cond–marginal estimator reveal bridge-specific allocation in a high-dimensional model? Third, when the rate is converted into a grid, does it help low-NFE sampling without being confused with endpoint quality or model training effects?
3
Conditional–marginal entropy rate
The bridge constraint asks for more than the entropy of the current marginal state. A conditional bridge path can contract because it is resolving a particular endpoint pair, while the population flow can expand or contract differently after those endpoint pairs are averaged. A schedule based on only one of these effects confounds path geometry with population geometry. The derivation below isolates the difference. It uses the standard regularity needed to differentiate entropy through a continuity equation. Assumption 1. All conditional and marginal densities are smooth and positive on their support, the relevant vector fields are continuously differentiable in x, and boundary terms vanish under integration by parts. The conditional and marginal paths satisfy Equations (2) and (3).
3
Theorem 1 (Conditional–marginal entropy identity). Under Assumption 1, EZ,Xt |Z [∇· vt (Xt | Z)] − EXt [∇· v̄t (Xt )].
d H(Z | Xt ) = dt (6)
Use H(Z | Xt ) = H(Xt | Z) + H(Z) − H(Xt ). Since H(Z) is constant and the continuity equation d d gives dt H(Xt ) = EXt [∇· v̄t (Xt )], with the same calculation conditionally giving dt H(Xt | Z) = EZ,Xt |Z [∇· vt (Xt | Z)], substitution yields Equation (6). Full integration-by-parts details are in Section A. The theorem is useful because it separates endpoint-specific volume change from population volume change. In a bridge, the conditional field may expand or contract because it is resolving endpoint information, while the marginal field may show a different expansion after averaging over endpoints. The conditional–marginal rate is the difference. Proposition 1 (Score form of the same rate). Under the assumptions of Theorem 1, d H(Z | Xt ) = −EZ,Xt |Z [(∇x log pt (Xt | Z) − ∇x log pt (Xt ))⊤ vt (Xt | Z)]. dt
(7)
Proposition 1 is not the estimator used in the high-dimensional experiments, since conditional scores are usually not exposed by pretrained models. It is included because it explains what the divergence contrast measures: the conditional field is weighted by the gap between the conditional score and the marginal score. 3.1
Gaussian bridges
For a Gaussian conditional path, pt (x | z) = N x; µ(z, t), σ 2 (z, t)I , a sample can be written as Xt = µ(z, t) + σ(z, t)ϵ, with ϵ ∼ N (0, I). Differentiating the sample path gives vt (x | z) = x−µ(z,t) ∂t µ(z, t) + ∂t σ(z, t) x−µ(z,t) σ(z,t) . Using the Gaussian score ∇x log pt (x | z) = − σ 2 (z,t) , we obtain the score form vt (x | z) = ∂t µ(z, t) − σ(z, t)∂t σ(z, t)∇x log pt (x | z). (8) p For a Brownian bridge between x0 and x1 , write mt = (1 − t)x0 + tx1 and σ(t) = σ0 t(1 − t). Substitution into Equation (8) gives the deterministic probability-flow field 1 − 2t vt (x | x0 , x1 ) = (x1 − x0 ) + (x − mt ). (9) 2t(1 − t) The noise scale σ0 cancels. This is important because the schedule shape is a property of bridge geometry, not of an arbitrary Brownian scale. Proposition 2 (Closed-form conditional divergence). For the Brownian-bridge probability-flow 1 − 2t . (10) field in Equation (9), ∇· vt (x | x0 , x1 ) = d 2t(1 − t) Proof. The term x1 −x0 is independent of x and has zero divergence. The term mt is also independent of x. Therefore 1 − 2t 1 − 2t 1 − 2t ∇· (x − mt ) = ∇· x = d . (11) 2t(1 − t) 2t(1 − t) 2t(1 − t)
The magnitude of Equation (10) is singular near both endpoints and vanishes at the midpoint. This is the mathematical source of the U-shaped bridge profile. Because the conditional divergence is spatially constant and endpoint-independent in the Brownian case, averaging over an entropic optimal-transport coupling preserves the same conditional expectation. The full conditional–marginal rate further subtracts the marginal divergence term in Equation (6); the conditional closed form supplies a model-independent anchor and the marginal term adapts it to the learned population path. The U-shape is a theorem for the Gaussian Brownian bridge, not for every learned bridge. Outside this class, the identity remains the general object and the profile must be estimated or derived from the model. Other interpolants can yield flat or monotone analytic profiles, as summarized in Section C. 4
3.2
SDE drift and probability flow
For a Brownian bridge, the conditional SDE drift can be written uot (x | x0 , x1 ) = (x1 − x0 ) +
1 − 2t (x − mt ), t(1 − t)
(12)
mt − x . σ02 t(1 − t)
(13)
with conditional score ∇x log pt (x | x0 , x1 ) =
The ODE field in Equation (9) and the SDE drift in Equation (12) are distinct conditional objects. They satisfy uot (x | x0 , x1 ) = 2vt (x | x0 , x1 ) − (x1 − x0 ). (14) The usual probability-flow relation applies to marginal fields after integrating over bridge endpoints. It should not be applied naively to the conditional drift. Section B.2 gives the step-by-step algebra, including the cancellation that explains the factor of two. This distinction is operationally important: using the conditional SDE drift as if it were already the probability-flow field removes the bridge contraction term and makes the conditional profile appear artificially flat.
4
Estimator, grids, and experimental loops
We organize the method into three layers. The inner layer is mathematical: choose the rate that a bridge actually exposes, namely the conditional divergence minus the marginal divergence. The middle layer is computational: estimate that rate without forming a Jacobian. The outer layer is empirical: use the estimated rate to build a grid, then ask whether the resulting allocation explains the observed behavior of high-dimensional samplers. The layers are tied together because the outer evidence is meaningful only if the computational estimator matches the mathematical rate. The cond–marg rate is estimated by evaluating both divergence terms in Equation (6). Exact traces are infeasible in high dimension, so the middle loop uses Hutchinson estimation: ∇· vt (x) = Tr∇x vt (x) = Eu u⊤ ∇x vt (x)u , E[uu⊤ ] = I. (15) Using the same random probe for the conditional and marginal terms reduces the variance of their difference. After averaging over calibration samples and time points, the rate is smoothed, clipped to a small positive floor, transformed by the default regularized map ϕ(r) = log(1 + r), and converted into a grid through Equation (5). We also report the raw transform ϕ(r) = r as an ablation. The raw grid preserves the uncompressed cond–marg signal, whereas the log transform keeps the ordering of high-rate regions while reducing endpoint domination. Computational note. With M calibration times, n calibration states, and m Hutchinson probes, the estimator uses O(M nm) derivative-vector products. A typical calibration with 50 times and four probes uses 200 derivative-vector products, which is then amortized across all samples drawn with the same grid. Once the grid is built, sampling at a fixed NFE has the same number of model evaluations as linear, cosine, sigmoid, or power grids. The high-dimensional evidence uses the same inference budgets as the baselines and treats entropy estimation as a calibration step. Memory and arithmetic-intensity details are in Section G. The output of this construction is a density over time and the grid obtained from its inverse CDF. Figure 2 summarizes the AlphaFlow cond–marg allocation, while Figure 3 compares the regularized node geometry against linear and against the ten-step DTW alignment. Algorithmic summaries are given in Section F. We also define a boundary concentration ratio (BCR) to summarize how much grid mass is placed near the endpoints. Visual inspection of schedule curves can be misleading when two grids look similar but allocate different endpoint density, so BCR gives a quantitative check of boundary concentration. This auxiliary measurement can be useful beyond this work for comparing schedule concentration across models; details and derivation are in Section D. We use BCR only as a grid-geometry diagnostic, not as an endpoint-quality metric.
5
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Figure 2: AlphaFlow cond–marg schedule allocation. The inverse-CDF grid places nodes according to the estimated conditional–marginal rate for 5, 10, and 25 steps; shaded endpoint bands mark the boundary regions used for the schedule-geometry diagnostic. This is grid-allocation evidence, not endpoint-quality evidence. Table 1: Synthetic low-NFE results. Left: learned ODE-Heun MMD averaged across steps, scenarios, σ, and seeds. Right: entropic-vs-linear improvement with bootstrap intervals. Scheduler
MMD ↓ (×10−3 )
Entropic vs (%)
Linear Power-2 Power-3 Log Cosine Sigmoid Entropic
6.028±0.603 6.620±1.070 6.912±1.216 8.164±1.855 5.548±0.152 5.639±0.339 5.537±0.131
8.1 [6.6, 9.7] 16.4 [14.1, 18.6] 19.9 [17.6, 22.3] 32.2 [29.7, 34.6] 0.2 [0.1, 0.3] 1.8 [0.5, 3.1] —
Steps
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SDE-Heun (%)
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18.1 [-4.9, 35.9] 12.3 [-10.8, 31.1] 7.0 [-16.8, 26.7]
22.7 [7.3, 34.9] 13.1 [-3.7, 27.0] 10.7 [-5.2, 24.8]
Proposition 3 (Weighted step-size rule). For an order-p one-step ODE solver with local error R1 R1 constant C(t), the continuous step density minimizing 0 C(t)ρ(t)−p dt subject to 0 ρ(t)dt = 1 is ρ⋆ (t) ∝ C(t)1/(p+1) . (16) The entropy rate is not asserted to equal C(t) exactly. It is used as a tractable proxy for regions where the vector field compresses, expands, or bends sharply. Therefore, the default entropic grid uses the regularized rate log(1 + r). The raw rate is kept as an ablation because it preserves the mathematical signal but can over-concentrate the grid near singular endpoints. This is the same practical lesson seen across diffusion scheduler and fast-solver work: a principled time signal still needs a numerically stable parametrization at low-NFE [5, 20, 25, 31, 44]. The same rate can matter differently for deterministic and stochastic solvers: ODE integration exposes grid-placement error directly, while SDE noise can smooth some placement errors. The full derivation of Proposition 3 is in Section E.
5
Results
Before moving to high-dimensional samplers, we ran a controlled two-dimensional pre-flight experiment. We trained residual vector-field models on four transport geometries: continuous–continuous (C-C), continuous–discrete (C-D), discrete–continuous (D-C), and discrete–discrete (D-D), across bridge scales. We then built grids from the measured entropy-rate profiles. These learned probes recover the boundary-heavy Brownian-bridge pattern, place more nodes near endpoints, and show the largest gains at low NFE before schedules converge. We report both ODE and SDE sweeps, but use the probability-flow ODE for the high-dimensional study because it removes sampling noise and matches the cond–marg estimator. Full transport visuals and ODE/SDE curves are in Section K and the controlled transport setup and entropy maps are summarized in in Figures 4 and 6. We next test the same allocation signal in two high-dimensional settings. EDM/CIFAR-10 is a mismatch case: it is not a bridge model and was not trained with the entropy objective, but it tests whether a tempered cond–marg-inspired grid can help a pretrained image sampler [20]. AlphaFlow operates on protein ensembles and is closer to the bridge setting because its flow-matching sampler exposes the fields needed for the cond–marg estimator, though it was also not trained with this criterion [18]. We use ODE sampling throughout, evaluate CAMEO22 and ATLAS proteins[12, 52], and report the evaluation funnel in Section J. 6
Table 3: EEDM/CIFAR-10 ODE-Heun FID. Values are means with 95% CIs over five seeds; lower is better. Entropic uses cond–marg log1p, with raw entropy shown as an ablation. Scheduler
5 steps
ODE + Linear 200.52 ± 2.91 ODE + Cosine 238.03 ± 5.29 ODE + Sigmoid 355.06 ± 2.20 ODE + Power-2 316.54 ± 2.68 ODE + Entropic (raw) 322.58 ± 4.80 186.26 ± 3.97 ODE + Entropic
10 steps
25 steps
172.78 ± 2.83 176.58 ± 2.49 212.15 ± 8.03 189.13 ± 3.51 290.11 ± 4.94 1å72.10 ± 3.74
170.85 ± 3.50 170.80 ± 3.36 172.68 ± 2.39 171.28 ± 4.06 227.96 ± 1.43 170.78 ± 3.42
AlphaFlow. The cond–marg profile in Figure 1 is strongly U-shaped: the intervals [0, 0.1] and [0.9, 1] each carry about 31.1% of the normalized mass, while the center window [0.4, 0.6] carries about 5.0%. The resulting grid allocation is shown in Figure 2; the log1p node geometry and ten-step DTW alignment are shown in Figure 3. Protein structures lie on a thin constrained manifold, where bond geometry, steric exclusions, chirality, and fold commitment can make both endpoints stiff. This motivates boundary allocation, but it does not by itself prove endpoint-quality improvement. Endpoint predicted local distance difference test (pLDDT) is mixed in the available AlphaFlow sweeps summarized in Table 2: sigmoid and cosine are strongest in many displayed cells, raw entropic scheduling is strongest in the medium and large five-step cells, and raw entropic pLDDT degrades sharply at 10 and 25 steps. pLDDT is an AlphaFold-style per-residue local distance difference test, and we use it here as an endpoint confidence proxy rather than as a measure of schedule geometry [19, 33]. Our method leads pLDDT performance on both medium and largesized proteins (97.62 ± 0.22 and 97.91 ± 0.14) at the low 5-step NFE count, and it is close to the sigmoid leader for small proteins (95.06 ± 1.12 versus 95.26 ± 0.99). Its limitations in this paper are summarized in Section J. Additional AlphaFlow grid and structure-evolution visualizations are kept in the Evaluation Grids appendix, including the 25-step AlphaFlow trajectory diagnostic in Section L; Figure 15 explains how endpoint plDDT should be interpreted so that we keep our emphasis on the schedule geometry. Table 2: AlphaFlow endpoint pLDDT under ODE-Heun. Values are means with 95% CIs; higher is better. Highlighting marks the best scheduler per column and bold marks entropic. Small ≤ 50 aa
Large > 400 aa
Medium 51–400 aa
Scheduler
5
10
25
5
10
25
5
10
25
Linear Cosine Sigmoid Power-2 Entropic
89.97 ± 1.83 92.73 ± 1.44 95.26 ± 0.99 84.81 ± 2.64 95.06 ± 1.12
94.15 ± 0.88 96.08 ± 0.66 96.15 ± 0.80 90.44 ± 1.82 78.79 ± 3.42
95.82 ± 0.51 96.07 ± 0.54 95.64 ± 0.71 94.87 ± 0.78 75.45 ± 3.81
93.10 ± 0.61 95.23 ± 0.47 97.49 ± 0.26 90.92 ± 0.80 97.62 ± 0.22
95.08 ± 0.31 97.57 ± 0.13 98.11 ± 0.08 93.99 ± 0.52 87.85 ± 1.11
96.25 ± 0.12 97.54 ± 0.06 97.55 ± 0.06 95.27 ± 0.22 86.02 ± 1.25
94.27 ± 0.57 95.92 ± 0.45 97.82 ± 0.23 92.54 ± 0.73 97.91 ± 0.14
94.30 ± 0.30 97.09 ± 0.13 97.95 ± 0.08 94.89 ± 0.45 90.26 ± 0.68
94.12 ± 0.12 96.39 ± 0.07 96.67 ± 0.06 93.52 ± 0.20 89.26 ± 0.76
EDM. EDM is a mismatch test because it lies outside the Schrödinger-bridge setting [20]. We use log-tempered cond–marg entropy as the default scheduler and raw entropy as the ablation: raw allocation over-concentrates endpoints and performs poorly, while log(1 + r) keeps the high-rate regions without exhausting the low-NFE budget at singular endpoints. Table 3 reports the full ODE-Heun FID sweep: at five steps entropic achieves 186.26 ± 3.97, compared with 200.52 ± 2.91 for linear, 238.03 ± 5.29 for cosine, and 355.06 ± 2.20 for sigmoid, while 10/25-step schedules are close. Additional trajectory-grid diagnostics are in Section L. The schedule-geometry comparison in Figure 3 shows the same calibration point: log1p stays close to linear but remains endpoint-biased, whereas raw cond–marg allocation requires stronger warping. Cross-domain allocation. Figure 1 puts the EDM and AlphaFlow entropy profiles on the same page, Figure 2 shows the AlphaFlow allocation, and Figure 3 shows the log1p-vs-linear node geometry together with the ten-step DTW alignment. In these diagnostics the bridge-aware quantity is the two-term rate rcm (t) = EZ,Xt |Z [∇· vt (Xt | Z)] − EXt [∇· v̄t (Xt )] . (17) The profiles are not the same, which is the point of measuring the entropy rate. EDM needs tempering because its raw profile is too sharp for stable low-NFE integration. AlphaFlow shows a bridge-like U-shape, but endpoint confidence remains influenced by the training objective and protein-specific geometry. The evidence separates three ideas that are easy to conflate: the entropy profile, the 7
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Figure 3: AlphaFlow schedule-geometry diagnostics. Left: the conditional–marginal log1p grid remains visually close to the linear grid across 5, 10, and 25 steps while keeping endpoint bias quantitatively. Right: dynamic-time-warping paths at 10 steps quantify this distinction; the log1p grid has lower distance to linear than the raw cond–marg grid, while raw cond–marg requires stronger index warping.
numerical grid derived from that profile, and the downstream metric used to judge the generated sample. Additional grid and evolution visualizations for EDM and AlphaFlow are in Section L.
6
Discussion
This work follows a simple sequence of tests. The first obstacle is remaining competitive with, and sometimes improving over, the strong fixed grid baselines. Linear, cosine, sigmoid, and power schedules often work because they encode plausible assumptions about where a trajectory is hard to integrate. The entropy construction must do more than replace one smooth curve with another, because it must measure a path-dependent signal that fixed grids can only approximate. Cosine, sigmoid, and power grids can be good because they approximate common shapes of entropy-rate profiles. The two-dimensional sweep makes this point concrete,although it does not prove that any fixed heuristic is optimal. In those controlled experiments, cosine was often close to the entropic scheduler because its node density can approximate a boundary-heavy curve. However, this was not a separate theory of cosine schedules, and the near-tie was scenario-dependent. The result should be read as shape compatibility between cosine and the measured entropic grid, not as equivalence between the two schedules [20, 25, 36]. When the true profile is monotone, U-shaped, or boundaryheavy, these heuristics can be close. Linear grids can work when the profile is flat. The advantage of the entropy construction is that it measures the shape instead of assuming it. The same 2D runs also explain why the training-loss curves matter. In the flow-matching ideal, reducing the loss should make the learned vector field closer to the target bridge field on the training distribution. When train and validation curves decrease together, the estimated entropy-rate profile is expected to move toward the analytic U-shape. We use this only as a diagnostic for alignment between the learned field and the analytic bridge calculation, not as a convergence proof for all learned bridges. This distinction matters most at low NFE, where a few misplaced steps can determine the outcome.
7
Related work
Diffusion, flow, and bridge models. Diffusion models and score-based samplers established continuous-time generation as a numerical integration problem [15, 20, 21]. Flow matching and stochastic interpolants later simplified training by regressing vector fields along prescribed conditional paths [2, 26, 29, 49]. Schrödinger bridges connect this view to entropy-regularized optimal transport and stochastic control [8, 23, 27, 41, 42, 50]. Recent work expands bridges to energy-based sampling, discrete spaces, structured domains, and branched trajectories [22, 28, 45, 47, 53]. Our contribution is orthogonal to these training objectives: it asks how to place inference steps once a path has been learned. 8
Schedulers and fast samplers. Classical schedules and design-space analyses remain important baselines [20, 21, 25, 36]. DPM-Solver, DPM-Solver++, UniPC, DEIS, and STORK change the numerical update or exploit ODE structure [31, 32, 46, 55, 56]. Align Your Steps, optimized-step methods, dynamic-transport scheduling, and BézierFlow search for better timesteps or scheduler parameterizations [35, 40, 51, 54]. These approaches can be combined with an entropy-derived grid, but they do not identify the bridge-specific conditional–marginal rate. Entropy and information geometry. Entropy-based scheduling for diffusion models was introduced by Stančević et al. [44] and further connected to information dynamics by Stančević and Ambrogioni [43]. Entropy and mutual information also appear widely as learning objectives or regularizers: entropy-regularized optimal transport and maximum-entropy control use entropy to shape couplings or policies [7, 11, 57]; information bottlenecks, InfoMax, InfoGAN, neural mutualinformation estimators, and Mutual Information Machines use mutual information as a compression, representation, or generative-model objective [3, 4, 6, 30, 48]; and protein inverse folding has used diversity-regularized DPO to increase differential entropy in ProteinMPNN log-probability space [37]. This work extends the information view to bridge and flow settings through Equation (6). The main difference is the object being estimated: diffusion entropic time uses a one-endpoint entropy signal, while a bridge exposes a contrast between conditional and marginal divergence; Table 8 summarizes the comparison. Low-step training and scientific domains. Distillation compresses many steps into a trained fewstep model [1, 10, 14, 38]. Entropic discretization is training-free and can be used with pretrained samplers. In scientific domains, flow and bridge methods increasingly model constrained objects such as proteins, molecules, materials, and distributions over distributions [9, 13, 17, 18, 24, 34]. These domains motivate measuring the path geometry directly because heuristic grids may miss endpoint stiffness or manifold constraints. A compact comparison is given in Section I.
8
Limitations and future work
Overall, the results support conditional–marginal entropy rate as a practical low-NFE allocation signal, provided it is regularized into a stable grid and interpreted separately from endpoint-quality metrics. The estimator depends on access to conditional and marginal fields. Some pretrained models expose only a single field or require a surrogate for one of the terms. In such cases the resulting grid should be treated as a diagnostic unless both terms are matched to the theory. Hutchinson estimation introduces variance, and raw boundary singularities can be too sharp for numerical stability. Log tempering is a practical calibration, not a theorem. The closed-form U-shape is also a Gaussian Brownian-bridge result. For non-Gaussian learned bridges, Equation (6) remains the target, but the shape of the rate is an empirical or model-specific question. While the current results are encouraging, optimized entropic scheduling for protein generation deserves a separate research program. Short proteins, long proteins, flexible loops, and disordered regions may have different entropy profiles. Models such as AlphaFlow, BioEmu, Proteina, and Peptron make it possible to study those profiles as biological signals rather than as nuisance quantities [9, 16, 18, 24]. Besides protein generation, the same question could be tested in image generation by training or calibrating EDM- or latent-diffusion models with the entropy criterion, and then sampling them with an entropic grid at inference time [15, 20, 39]. In both domains, the next question is whether training and inference can be aligned around the same cond–marg information rate.
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13
Appendix A
A guided derivation of the conditional–marginal rate
Marginal entropy rate. We assumed in the main text that the density and vector field are smooth enough for integration by parts and that boundary terms vanish. We now walk through the calculation. Consider first the marginal entropy Z H(Xt ) = − pt (x) log pt (x) dx, (18) and differentiate with respect to time: Z Z d ∂t pt (x) H(Xt ) = − ∂t pt (x) log pt (x) dx − pt (x) dx dt pt (x) Z Z d = − ∂t pt (x) log pt (x) dx − pt (x) dx dt Z = − ∂t pt (x) log pt (x) dx. The second term vanishes because pt remains normalized. Next, use the continuity equation, Z d H(Xt ) = ∇· (pt (x)v̄t (x)) log pt (x) dx dt Z = − pt (x)v̄t (x)⊤ ∇x log pt (x) dx Z = − v̄t (x)⊤ ∇x pt (x) dx Z = pt (x)∇· v̄t (x) dx = EXt [∇· v̄t (Xt )] .
(19) (20) (21)
(22) (23) (24) (25) (26)
Conditional entropy rate. The conditional calculation repeats the same story while holding the bridge condition fixed. For each condition Z = z, the conditional density follows its own continuity equation, so d H(Xt | Z = z) = EXt |Z=z [∇· vt (Xt | z)] . (27) dt Averaging over Z gives d H(Xt | Z) = EZ,Xt |Z [∇· vt (Xt | Z)] . dt Now take into consideration the entropy chain rule, H(Z | Xt ) = H(Xt | Z) + H(Z) − H(Xt ),
(28)
(29)
and subtract the marginal entropy rate derived above. We obtain d H(Z | Xt ) = EZ,Xt |Z [∇· vt (Xt | Z)] − EXt [∇· v̄t (Xt )] . dt
(30)
Score representation. The divergence form is the estimator used in this paper, but the score form clarifies what is being measured. The marginal field is Z v̄t (x) = pt (z | x)vt (x | z) dz. (31) Taking divergence, Z ∇· v̄t (x) =
Z pt (z | x)∇· vt (x | z) dz +
14
∇x pt (z | x)⊤ vt (x | z) dz.
(32)
Multiply by pt (x) and integrate over x. The first term becomes the expected conditional divergence. For the second term, Bayes’ rule gives ∇x log pt (z | x) = ∇x log pt (x | z) − ∇x log pt (x).
(33)
EXt [∇· v̄t (Xt )] = EZ,Xt |Z [∇· vt (Xt | Z)] + EZ,Xt |Z (∇ log pt (Xt | Z) − ∇ log pt (Xt ))⊤ vt (Xt | Z) .
(34)
Therefore
Rearranging yields Equation (7). This shows that the rate measures the conditional field against the gap between the conditional score and the marginal score. Gaussian conditional vector field. The Brownian-bridge calculation in the main text is a special case of a Gaussian conditional path. Let ϵ ∼ N (0, I).
Xt = µ(z, t) + σ(z, t)ϵ,
(35)
For fixed noise ϵ, ∂t Xt = ∂t µ(z, t) + ∂t σ(z, t)ϵ.
(36)
Since ϵ = (Xt − µ(z, t))/σ(z, t), vt (Xt | z) = ∂t µ(z, t) + ∂t σ(z, t)
Xt − µ(z, t) . σ(z, t)
(37)
The Gaussian score is ∇Xt log pt (Xt | z) = −
Xt − µ(z, t) . σ 2 (z, t)
(38)
Substitution gives vt (Xt | z) = ∂t µ(z, t) − σ(z, t)∂t σ(z, t)∇Xt log pt (Xt | z).
B
Brownian bridge details
B.1
Probability-flow field
For mt = (1 − t)x0 + tx1 and σ(t) = σ0
p
(39)
t(1 − t),
∂t mt = x1 − x0 .
(40)
The time derivative of the standard deviation is 1 ∂t σ(t) = σ0 p (1 − 2t), 2 t(1 − t)
(41)
so σ(t)∂t σ(t) = σ0
p σ0 (1 − 2t) σ 2 (1 − 2t) t(1 − t) p = 0 . 2 2 t(1 − t)
The conditional score is ∇x log pt (x | x0 , x1 ) = −
x − mt . σ02 t(1 − t)
(42)
(43)
Substituting into the Gaussian field, σ02 (1 − 2t) x − mt vt (x | x0 , x1 ) = x1 − x0 − − 2 2 σ0 t(1 − t) 1 − 2t = x1 − x0 + (x − mt ). 2t(1 − t) 15
(44) (45)
B.2
SDE drift and the factor of two
The Brownian-bridge SDE drift is 1 − 2t (x − mt ). t(1 − t)
(46)
uot (x | x0 , x1 ) = 2vt (x | x0 , x1 ) − (x1 − x0 ).
(47)
uot (x | x0 , x1 ) = x1 − x0 + Comparing with the probability-flow field gives
The factor of two arises from the chain-rule term σ(t)∂t σ(t) in the probability-flow derivation. If one applies the marginal probability-flow transformation to the conditional objects, the score correction cancels the bridge contraction term incorrectly: uot + σ02 (1 − 2t)∇x log pt (x | x0 , x1 ) = x1 − x0 + = x1 − x0 .
1 − 2t 1 − 2t (x − mt ) − (x − mt ) (48) t(1 − t) t(1 − t) (49)
The result differs from Equation (9). The probability-flow relation is valid for marginal fields after endpoint mixing, not for this conditional conversion.
C
Interpolation profiles and bridge specificity
We raised an important scope question: is the U-shape a generic property of every interpolation, or a bridge-specific calculation? The answer is bridge-specific. Different interpolants produce different analytic divergence profiles, and the learned field can deviate from the exact conditional formula. This is why the main text treats the Brownian-bridge profile as an anchor and the conditional–marginal estimator as the general object. Table 4: Analytic divergence profiles for common conditional interpolants. Here d is the ambient dimension. The table illustrates that different path choices lead to different schedule signals. Interpolation
Exact conditional divergence
Qualitative profile
Linear optimal-transport interpolation Variance-preserving diffusion path Cosine diffusion path
0 for the exact linear conditional field
Flat before learning error Monotone boundary growth Monotone boundary growth Symmetric U-shape in magnitude
Brownian bridge
d t/(1 − t2 ) under the standard VP parameterization dπ tan(πt/2)/2 under the cosine noise parameterization d(1 − 2t)/[2t(1 − t)] for the probability-flow field
For Schrödinger bridges represented as mixtures of Brownian bridges, the conditional Brownian divergence in Equation (10) does not depend on the endpoint pair (x0 , x1 ) or on the spatial position x. Therefore 1 − 2t E(X0 ,X1 ),Xt |X0 ,X1 [∇· vt (Xt | X0 , X1 )] = d (50) 2t(1 − t) for the conditional term of the Brownian reference bridge. The marginal term in Equation (6) remains model- and coupling-dependent, which is why the full bridge rate is not reduced to the conditional term alone.
D
Boundary concentration ratio
We introduced boundary concentration ratio after observing that many useful schedules differ mainly in where they place density near t = 0 and t = 1. The entropy rate gives a full curve, but comparisons across many schedules are easier when one also has a scalar diagnostic. The diagnostic should answer a narrow question: how much more schedule mass lies near the two endpoints than would lie there under a uniform grid? 16
Take into consideration a normalized schedule density q(t) on [0, 1] with boundary width ϵ ∈ (0, 1/2), define the endpoint mass Z ϵ Mϵ (q) =
R1 0
q(t)dt = 1. For a
Z 1 q(t) dt +
0
q(t) dt.
(51)
1−ϵ
The same two endpoint intervals have mass 2ϵ under a uniform grid density. We therefore define BCRϵ (q) =
Mϵ (q) . 2ϵ
(52)
A uniform grid has BCRϵ = 1. Values larger than one mean that the schedule assigns more node density to the boundary windows than uniform spacing. Values below one mean that the schedule avoids the endpoints relative to the uniform grid. For a discrete grid, one can estimate q from the interval widths or from the inverse-CDF density used to construct the grid. The density-based version is preferable because it is less sensitive to whether the endpoints themselves are counted as nodes. The assumptions behind BCR are simple: the same ϵ must be used for all schedules, the grid must be monotone, and the diagnostic should be computed on the schedule density rather than on downstream metrics. BCR is useful beyond this paper because it separates a geometric property of the grid from sample quality. A high BCR can indicate endpoint stiffness, bridge contraction, or a raw entropy singularity. It can also warn that tempering may be needed, since over-resolving a singular endpoint can waste low-NFE budget. BCR is therefore a schedule diagnostic and not a main claim. A model can have high boundary concentration and still perform poorly if the estimator is mismatched or if the training objective favors another grid.
E
ODE error derivation
For an order-p one-step method, suppose the local truncation error on a step of width h is bounded by ∥ek+1 ∥ ≤ C(tk )hp+1 + O(hp+2 k k ).
(53)
Let ρ(t) be a continuous density of grid points. Locally h(t) ≈ 1/(N ρ(t)). Ignoring constants independent of ρ, a continuous proxy for total error is Z 1 Z 1 −p E[ρ] = C(t)ρ(t) dt, ρ(t) dt = 1. (54) 0
0
The Lagrangian is Z 1 L[ρ] =
C(t)ρ(t)−p dt + λ
0
Z 1
ρ(t) dt − 1 .
(55)
0
The stationarity condition is −pC(t)ρ(t)−p−1 + λ = 0,
(56)
ρ(t) ∝ C(t)1/(p+1) .
(57)
which gives
Entropy-rate scheduling uses |Ḣ(Z | Xt )| as a measurable proxy for C(t).
F
Algorithms
This appendix states the inference-time pipeline as algorithms. The algorithms do not assume a particular implementation. They separate the calibration stage, where the entropy-rate curve is estimated once, from the sampling stage, where all schedulers are compared at matched NFE. 17
Algorithm 1 Calibration Set and Time Mesh Construction Require: Calibration conditions {zi }ni=1 , time range [ϵ, 1 − ϵ], number of mesh points M Ensure: Calibration states {xij } and paired conditions {zi , sj } 1: Choose a monotone mesh 0 < ϵ = s1 < · · · < sM = 1 − ϵ < 1 2: for i = 1, . . . , n do 3: Draw or retrieve the bridge condition zi 4: for j = 1, . . . , M do 5: Construct the state xij ∼ psj (· | zi ) when the conditional simulator is available 6: Otherwise, use a model trajectory state paired with its condition and time 7: end for 8: end for 9: return Paired calibration set {(xij , zi , sj )}
Algorithm 2 Hutchinson Divergence of a Vector Field ⊤ Require: Vector field ft , state x, time t, probes {uℓ }m ℓ=1 with E[uℓ uℓ ] = I Ensure: Estimate of ∇· ft (x) 1: a ← 0 2: for ℓ = 1, . . . , m do 3: Evaluate the Jacobian-vector product wℓ ← ∇x ft (x)uℓ 4: a ← a + u⊤ ℓ wℓ 5: end for 6: return a/m
Algorithm 3 Conditional–Marginal Entropy-Rate Estimation Require: Calibration set {(xij , zi , sj )}, conditional field vt (· | z), marginal field v̄t , probes per state m Ensure: Estimated rate values {r̂(sj )}M j=1 and uncertainty estimates 1: for j = 1, . . . , M do 2: dj1 , . . . , djn ← empty 3: for i = 1, . . . , n do 4: Draw shared probes {uℓij }m ℓ=1 5: if analytic conditional divergence is available then 6: cij ← ∇· vsj (xij | zi ) from the analytic formula 7: else 8: cij ← Algorithm 2 applied to vsj (· | zi ) at xij 9: end if 10: mij ← Algorithm 2 applied to v̄sj at xij with the same probes 11: dji ← cij − mij 12: end for Pn 13: r̂(sj ) ← n−1 i=1 dji 14: se(s b j ) ← standard error of {dji }ni=1 15: end for 16: return Rate curve r̂ and uncertainty curve se b 18
Algorithm 4 Rate Postprocessing and Grid Construction Require: Mesh {sj }M j=1 , estimated rate r̂, transform ϕ ∈ {r, log(1 + r)}, sampling steps N Ensure: Monotone grid 0 = t0 < t1 < · · · < tN = 1 1: Replace non-finite values by local interpolation and clip r̂ to a small positive floor 2: Smooth the clipped curve only on the calibration mesh, preserving endpoint ordering 3: aj ← ϕ(r̂(sj )) for j = 1, . . . , M 4: Normalize aj into a density qj by numerical quadrature R sj 5: Compute the cumulative curve Q(sj ) ≈ 0 q(u)du 6: Enforce monotonicity of Q and set Q(0) = 0, Q(1) = 1 7: for k = 0, . . . , N do 8: Set tk ← Q−1 (k/N ) by monotone interpolation 9: end for 10: return Grid {tk }N k=0 and density summary q
Algorithm 5 Matched-NFE Evaluation Protocol Require: Model sampler, scheduler family S, NFE budgets N , seeds B, evaluation metrics M Ensure: Matched comparison table across schedulers 1: for scheduler S ∈ S do 2: for budget N ∈ N do 3: Build the grid for S with exactly N intervals 4: for seed b ∈ B do 5: Generate samples with the same model, solver family, and NFE budget 6: Evaluate all metrics in M on the generated samples 7: end for 8: end for 9: end for 10: Aggregate means, standard deviations, and paired deltas relative to selected baselines 11: return Matched-NFE metric table and per-sample records
Algorithm 6 Paired Bootstrap Confidence Intervals Require: Per-unit metric records for scheduler A and baseline B, bootstrap draws R, confidence level 1 − α Ensure: Mean paired difference and percentile interval 1: Match records by evaluation unit, seed, dataset split, and NFE budget 2: for r = 1, . . . , R do 3: Resample matched units with replacement 4: Compute the paired difference ∆r = metric(A) − metric(B) on the resample 5: end for ¯ on the original matched records 6: Report ∆ 7: Report percentiles α/2 and 1 − α/2 of {∆r }R r=1 8: return Paired estimate and confidence interval
Algorithm 7 Boundary-Concentration Diagnostic Require: Schedule density q(t) or grid density estimate, boundary width ϵ ∈ (0, 1/2) Ensure: Boundary concentration ratio BCRϵ Rϵ R1 1: Estimate boundary mass Mϵ = 0 q(t)dt + 1−ϵ q(t)dt 2: Normalize by the uniform-grid boundary mass 2ϵ 3: BCRϵ ← Mϵ /(2ϵ) 4: return BCRϵ
19
G
Computational notes
Table 5: Complexity of grid construction. Sampling cost at fixed NFE is unchanged after the grid has been constructed. Method
Calibration compute
Extra memory
TFLOPS/byte note
Analytic bridge rate Exact divergence
O(M ) scalar evaluations O(M nd) derivative products O(M nm) derivative-vector products None during sampling
O(N ) grid Large Jacobian or repeated products Probe buffers plus activations O(N ) grid
Negligible Usually impractical
Hutchinson rate Reused entropy grid
Kernel-dependent Same NFE as baseline
Here M is the number of calibration times, n the number of calibration states, m the number of Hutchinson probes, d the dimension, and N the number of sampling steps. Hardware-level TFLOPS/byte depends on kernels, precision, and hardware, so the robust comparison is symbolic.
H
Hypotheses and conclusions
We guided the work with three overall questions. The first asks whether the bridge calculation predicts a distinctive information geometry. The second asks whether learned two-dimensional probes preserve that geometry after the vector field is trained. The third asks whether cond–marg estimator and its grid help interpret high-dimensional low-NFE behavior without turning an allocation diagnostic into an endpoint-quality claim.
Table 6: Questions that guided the evidence and the corresponding conclusions. Guiding question
Evidence
Conclusion
What quantity should a bridge schedule measure?
Equation (6) and Equation (10) show that the bridge rate is a conditional–marginal divergence contrast, with a U-shaped Brownian-bridge conditional term. AlphaFlow conditional–marginal profiles show a boundary-heavy U-shape, while EDM gives a different profile and therefore a useful mismatch case.
Use a two-term bridge rate, not a single-field proxy.
Does this geometry appear in high dimension?
Can the measured rate guide low-NFE sampling?
EDM five-step FID improves with log-tempered entropy, and AlphaFlow allocation diagnostics align with the bridge calculation while endpoint pLDDT remains training-objective dependent.
20
The rate is model-dependent and must be measured or derived. Entropy is a grid signal, not a universal endpoint-quality guarantee.
I
Additional method comparison
Table 7: Comparison with related acceleration and scheduling families. Family
Main object
Relation to this work
Heuristic grids
Fixed analytic time maps
Align Your Steps
Optimized discrete schedule
Lipschitz and transport scheduling
Smoothness or transport-cost proxy
Fast ODE solvers Distillation
Numerical update formula New few-step model
Bridge and flow coupling
Training path geometry
Strong baselines, but not path-measured Empirical optimization rather than entropy identity Related to error constants, no conditional–marginal bridge decomposition Complementary to grid choice Requires training, unlike inference-time grid construction Changes the learned path rather than measuring the grid density
Table 8: Closest entropy-based scheduler comparison. The bridge-specific change is the conditional– marginal divergence contrast.
J
Aspect
Entropic diffusion time
Conditional–marginal bridge time
Object Signal
One-endpoint diffusion process Entropy of data given noised state
Estimator
Diffusion loss or single-field formula
Shape Use
Often asymmetric in noise time Inference-time reparameterization
Conditional bridge or flow path Ḣ(Z | Xt ) from conditional minus marginal divergence Matched conditional and marginal divergence estimates Symmetric U-shape for Brownian bridges Inference-time grid for bridge and flow samplers
Protein endpoint confidence metrics
Protein endpoint quality is reported with predicted local distance difference test (plDDT), a perresidue confidence estimate associated with distance agreements [19, 33]. The score is usually reported on a 0–100 scale. Higher values indicate higher local confidence. We average plDDT over generated endpoint structures and use it as an endpoint confidence proxy. This is useful because it is cheap, standardized, and sensitive to local structural plausibility. However, it is not the same object as the cond–marg entropy rate. plDDT does not measure where the sampler placed time steps, and it can depend on the target protein, sequence length, flexible or disordered regions, model calibration, and the evaluator. Hence, pLDDT is interpreted together with diversity and trajectory diagnostics, not as direct proof that one grid is the correct entropy grid. Protein evaluation funnel. CAMEO22 and ATLAS are used here as evaluation sources, not as evidence that the entropy grid was optimized during AlphaFlow training. The Sankey diagram in Figure 11 separates the available benchmark proteins from the subset that passed the native-score testing path. The count table behind the figure reports 48 available ATLAS proteins and 72 available CAMEO proteins; 115 proteins were tested after filtering (45 ATLAS, 70 CAMEO), with give large proteins not tested due to system timeouts. These counts describe evaluation coverage. They should be read separately from endpoint pLDDT, which remains a confidence proxy, and separately from the cond–marg rate, which is the schedule-geomtry signal. 21
1.0
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induced time map τ(u)
analytic Brownian-bridge reference
linear reference τ(u) = u
D→D
0.000
Figure 4: Controlled transport setup and entropy-induced grids. Left: entropy-rate profiles and induced inverse-CDF time grids show the core mechanism: measure where the path carries information, normalize that rate, and place time nodes by the corresponding cumulative distribution. Right: synthetic transport scenarios used as controlled probes, contrasting continuous–continuous, continuous–discrete, discrete–continuous, and discrete–discrete endpoint structure.
K
Toy and synthetic data
The synthetic appendix is included to show what the theory predicts before the high-dimensional stress tests. These plots are not used as the main evidence. They help the reader see why a bridge can ask for a non-uniform grid even when the downstream model is simple. Linear Power-2 (t 2 )
Power-3 (t 3 ) Logarithmic
Cosine Sigmoid
Entropic (Ḣ)
ODE-HEUN
SDE-HEUN
0.011 0.010
MMD
0.009 0.008 0.007 0.006 0.005 25
50
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125
150
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200
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50
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125
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Figure 5: Scenario-level comparison on the toy transports. The figure is included to make the synthetic evidence explicit: the schedule effect depends on the endpoint geometry, and the role of the toy data is to isolate this dependence before moving to EDM and AlphaFlow.
Linear Power-2 (t 2 )
Power-3 (t 3 ) Logarithmic
Cosine Sigmoid
Entropic (Ḣ)
ODE-HEUN
SDE-HEUN
0.011 0.010
MMD
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Figure 6: Initial ODE/SDE probing on synthetic data. The main observation is that low-NFE behavior is more sensitive to grid placement than high-NFE behavior, and deterministic integration exposes this sensitivity more directly than stochastic integration. 22
step 1
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step 6
step 7
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step 10
(a) Linear.
(b) Cond–marg log1p.
Figure 8: EDM ten-step ODE-Heun trajectory grids. The quantitative five-step claim is carried by Table 3; Log1p allocation is visually close to linear while remaining a distinct measured grid.
1.00
EDM time t
0.75 0.50 0.25 0.00
step index
AlphaFlow time t
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linear cosine sigmoid
power2 power3 log
std std log1p std rev
std log1p rev cond-marg cond-marg log1p
Figure 7: Reference scheduler shapes at σ = 0.5. This plot is kept as a reference for how common heuristic grids distribute nodes relative to the bridge-inspired profiles. It is not used to claim that one heuristic is universally best.
L
Evaluation Grids
The following figures are grid and trajectory diagnostics referenced from the main text. 23
t = 0.00
t = 0.25
t = 0.50
t = 0.75
t = 1.00
C→C
2.818
1.691
learned field speed
2.254
C→D
1.127
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0.564
D→D
0.000
Figure 9: Synthetic transport scenarios used as controlled probes. The grid contrasts continuous– continuous, continuous–discrete, discrete–continuous, and discrete–discrete endpoint structure. The point of this panel is to show that the bridge path can face different endpoint constraints even before any high-dimensional neural sampler is introduced..
run41 AlphaFlow CAMEO condmarg Trajectory pLDDT Short
Medium
Trajectory pLDDT
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Figure 10: CAMEO AlphaFlow trajectory pLDDT by dataset size. This panel documents endpointconfidence trends across the trajectory and does not replace the cond–marg schedule diagnostic. 24
ATLAS Small 1 proteins
ATLAS 48 proteins
36 11
CAMEO 72 proteins
ATLAS Medium 36 proteins ATLAS Large 11 proteins
10
CAMEO Small 10 proteins
43
CAMEO Medium 43 proteins
19
CAMEO Large 19 proteins
36 8 10
Tested 115 proteins
43 17
Not tested 5 proteins
ATLAS
CAMEO
Figure 11: AlphaFlow protein-testing flow summary. This figure records the filtering and evaluation path behind the protein evidence.
step 1
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Figure 12: EDM ODE-Heun trajectory grid at 25 steps. The reader should observe that well-placed schedules begin to converge in visual behavior, matching the quantitative convergence in FID. 25
7skj_A
7skj_A
L=275
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7
(a) 2D view.
(b) 3D view.
Figure 13: AlphaFlow cond–marg protein evolution at 5 steps for the same target. The paired 2D and 3D renderings are trajectory diagnostics, not endpoint-quality metrics.
7skj_A
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17.5
(a) 2D view.
(b) 3D view.
Figure 14: AlphaFlow cond–marg protein evolution at 10 steps for the same target. The paired views make the trajectory easier to inspect without changing the endpoint metric interpretation.
7skj_A
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current protein trace
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40
(a) 2D view.
(b) 3D view.
Figure 15: AlphaFlow cond–marg protein evolution at 25 steps for the same target. This higher-NFE view supports trajectory inspection and remains separate from quantitative pLDDT claims.
M
Computational Notes
The inference data for our test runs was collected on a SLURM-managed HPC GPU cluster using NVIDIA H100 80GB HBM3 GPUs . The main inference sweeps ran as exclusive 8-GPU node jobs, typically requesting gpus-per-node=8 with torchrun-nproc_per_node=8, about 80 requested CPU cores per job, 128 CPU cores allocated by SLURM, and roughly 2 TB of node memory. Our runs also included several smaller 1-GPU AlphaFlow precompute jobs with 16 CPUs and about 250 GB allocated memory. Counting allocated GPU wall time, including failed and timed-out retry attempts because they still consumed resources, and excluding CPU-only postprocessing, our runs consumed about 333.67 H100 GPU-hours in total: 183.52 GPU-hours for EDM and 150.15 GPU-hours for AlphaFlow, including experimentation. If the earlier invalid or cancelled launches are included as an all-in accounting number, the estimate rises slightly to about 334.42 H100 GPU-ours. 26
Computational cost. Our method is computationally light relative to both EDM and AlphaFlow because it is a scheduler rather than a new generative backbone. At a fixed number of function evaluations (NFE), it uses the same trained-model calls as the corresponding baseline linear, cosine, sigmoid, or power schedule, and only changes the time grid on which those evaluations are placed. The additional online overhead consists of reading or interpolating a precomputed entropy-rate curve and constructing the schedule, which is O(K) for K sampling steps and negligible compared with a single neural-network evaluation. The offline entropy-curve estimation incurs a one-time cost proportional to the number of time points and probe samples, but this cost is amortized across all subsequent samples and does not change the asymptotic inference complexity. For EDM, inference is dominated by denoising-network evaluations. If B is the batch size, K is the number of model evaluations, and CEDM is the cost of one EDM U-Net evaluation at the target image resolution, then sampling costs approximately O(BKCEDM ), up to constant-factor differences between Euler, Heun, and SDE-corrector implementations. Our entropic schedule has the same complexity at matched NFE; any improvement comes from allocating the same evaluations more effectively, or from reaching a target quality with fewer evaluations. For AlphaFlow, the dominant cost is substantially larger because each denoising or integration step invokes a protein-structure model, here the ESMFold-based AlphaFlow checkpoint. For a protein of length L, each model call scales at least quadratically in L due to attention and pairwise geometric representations, with additional architecture-dependent costs in the structure module. AlphaFlow sampling therefore scales roughly as O(BKCAF (L)), where CAF (L) grows steeply with protein length and memory also increases strongly with L. Our scheduler does not introduce additional AlphaFlow passes; it only changes the temporal allocation of the same 5-, 10-, or 25-step budgets.
27