Equivariant Graph Neural Networks Improve Optical Spectra Prediction for Materials Screening
arXiv:2606.19133v1 [physics.optics] 17 Jun 2026
Kasper Helverskov Petersen 1 François Raymond J Cornet 1 Martin Ovesen 2 Mikkel Jordahn 1 Kristian S. Thygesen 2 Mikkel N. Schmidt 1 prediction of optical spectra. Experimental measurement and ab initio methods such as density functional theory (DFT) (Hohenberg & Kohn, 1964; Kohn & Sham, 1965) and many-body perturbation theory (MBPT) (Onida et al., 2002) are resource-intensive and computationally prohibitive for large-scale screening. Machine learning surrogates trained on first-principles databases offer a compelling alternative, but must be accurate enough to reliably distinguish promising candidates from poor ones, making architectural improvements a key priority.
Abstract Scalable prediction of optical spectra is a critical component of high-throughput materials screening for optoelectronic applications such as solar cells. Existing surrogate models are trained on spectra computed from lower levels of theory or rely on rotation-invariant scalar features, limiting their geometric expressiveness. We explore the use of equivariant graph neural networks for optical spectra prediction, adapting GotenNet to this task and evaluating it on multiple datasets including a recently published collection of 10,533 structures with spectra computed at the level of the random phase approximation (RPA). The proposed model outperforms the current state of the art, with the largest gains in the 0–8 eV range and on predicting the static real permittivity, both of particular relevance for thin-film optics.
Existing approaches (Hung et al., 2024; Ibrahim & Ataca, 2024; Ginter et al., 2025) are trained on limited data at lower levels of theory, such as the independent-particle approximation (IPA) or tight-binding (TB). Meanwhile, Grunert et al. (2026c) recently introduced a dataset of 10,533 optical spectra at the level of the random phase approximation (RPA) using an IPA-pretrained model and achieving state-of-the-art results. Their architecture, OptiMate3B, encodes geometry only through rotation-invariant scalar features. Furthermore, it explicitly constructs a line-graph over edges to capture angular features, which necessitates a Voronoi graph construction to limit connectivity. Equivariant architectures avoid both limitations: they encode directional information by construction, providing a geometric inductive bias that is especially valuable given limited training data, and capture angular correlations implicitly, allowing distance-based graph construction that more naturally reflects the physical interaction ranges of atoms.
1. Introduction AI is increasingly applied across scientific disciplines, and materials discovery, sitting at the confluence of physics, chemistry, and engineering, is a natural target. The combinatorial vastness of chemical and structural space makes AIdriven exploration particularly compelling: generative models propose candidate materials; surrogate models screen them for target properties; and first-principles calculations and experimental characterization evaluate the most promising candidates. Together, these form a pipeline bridging machine learning, computational physics, and materials synthesis and characterization. For discovering improved materials for solar cells, a critical component is the scalable
In this work, we adapt GotenNet (Aykent & Xia, 2025), a state-of-the-art equivariant graph neural network (GNN), to optical spectra prediction, and evaluate it on both the IPA and RPA OptiMate3B datasets. The resulting model, GotenNetOpt , achieves consistent improvements with the most pronounced gains in the 0–8 eV range and on predicting the static real permittivity, both of particular relevance for materials screening in thin-film optics. We further benchmark against GNNOpt (Hung et al., 2024) on their smaller dataset of 944 IPA-computed spectra, demonstrating improvements in a low-data setting.
1
Department of Applied Mathematics and Computer Science, Technical University of Denmark, Kongens Lyngby, Denmark 2 Department of Physics, Technical University of Denmark, Kongens Lyngby, Denmark. Correspondence to: Kasper Helverskov Petersen <[email protected]>. Proceedings of the AI4Physics Workshop at the 43 rd International Conference on Machine Learning (AI4Physics@ICML 2026), Seoul, South Korea. 2026. Copyright 2026 by the author(s).
1
Equivariant Graph Neural Networks Improve Optical Spectra Prediction for Materials Screening
While the real and imaginary parts of ε̄ (or equivalently, χ̄) can be calculated independently of each other, a consequence of causality is that they must satisfy the criterion that they are related to each other by Hilbert transformations (Toll, 1956) Z ∞ 1 Im (χ̄ (ω ′ )) ′ Re (χ̄(ω)) = P dω , (8) π ω′ − ω −∞
2. Background 2.1. The dielectric function The predicted optical spectra consist of the complex dielectric function, which describes the response of a solid to an applied electric field E of angular frequency ω. The dielectric function is actually a second-rank tensor, εij (ω), relating the electric polarization density along the Cartesian direction i induced by an electric field polarized along the direction j. The tensorial nature reflects that the polarization response can differ along different crystallographic directions. Mathematically, this can be expressed as the linear relationship between the components of the electric displacement field D and the components of the applied electric field E X Di (ω) = ε0 εij (ω)Ej (ω), (1)
and 1 Im (χ̄(ω)) = − P π
out = Re[Tr(ε(ω))/3] ∥ Im[Tr(ε(ω))/3],
where the relationship between εij and χij is simply (3)
outRe = Re[Tr(ε(ω))/3],
outIm = Im[Tr(ε(ω))/3], (11) each yielding a 251-dimensional output vector. A brief background section on DFT is provided in Section A.
Typically, determining all 9 tensor elements is unnecessary due to the crystal structure being invariant w.r.t. certain sets of symmetry operations. For example, for materials with cubic symmetry or isotropic structure, the off-diagonal elements of the tensor vanish while the diagonal elements reduce to a scalar, i.e. εij (ω) = ε(ω)δij . More generally, the isotropic part of the dielectric function tensor can be defined as ε̄(ω) = Tr(ε(ω))/3, (4)
2.2. 3D graph representation of crystal structures A crystal is an infinite periodic solid described by a unit cell: a parallelepiped spanned by lattice vectors a1 , a2 , a3 ∈ R3 and a finite set of atoms at positions {xi }, given in Cartesian or fractional coordinates {si } ⊂ [0, 1)3 via xi = P k si,k ak . Under periodic boundary P conditions (PBC), each atom at xi has images at xi + k nk ak for all nk ∈ Z.
which captures the mean linear response across all crystallographic directions. It consists of a real and imaginary part which are related to the refractive index n(ω) and the extinction coefficient κ(ω), describing the amount of light attenuation, through p n(ω) + iκ(ω) = ε̄(ω), (5)
sp κ=
Re(ε̄)2 + Im(ε̄)2 − Re(ε̄) . 2
(10)
which is a 4002-dimensional output vector. Whereas, for the GNNOpt dataset, a separate model is trained and evaluated for the real and imaginary components, with each predicting a single component interpolated onto a uniform grid of 251 points over 0 ≤ ℏω ≤ 50 eV, where ℏ is the reduced Planck constant:
j
and
(9)
For the OptiMate3B dataset, we predict the concatenated real and imaginary parts of ε̄(ω), sampled from 0 to 20 eV in 10 meV steps, yielding
where ε0 is the permittivity of free space and the secondrank tensor is unitless. Alternatively, the second-rank tensor can be formulated as the electric susceptibility, χij , describing the linear relationship between the polarization density P and the applied electric field X Pi (ω) = ε0 χij (ω)Ej (ω), (2)
which can also be written as sp Re(ε̄)2 + Im(ε̄)2 + Re(ε̄) n= , 2
Re (χ̄ (ω ′ )) ′ dω , ω′ − ω −∞
where P denotes that the singularity at ω ′ = ω is resolved as the Cauchy principal value. The equations (8) and (9) are known as the Kramers-Kronig relations.
j
εij (ω) = δij + χij (ω).
Z ∞
We represent crystals as a graph G = (V, E), where nodes V are the unit cell atoms and edges E encode atomic interactions, with the neighborhood of atom i defined as N (i) = {j ∈ V | (i, j) ∈ E}. PBC means the neighborhood search includes periodic images, so multiple edges may exist between the same pair of atoms, each with a distinct edge vector rij . We use a distance-based cutoff of rc = 6.0 Å, and additionally evaluate a Voronoi-based construction, which yields fewer edges and is the method used by OptiMate3B.
(6)
Each atom i is featurized by concatenating one-hot encodings of its periodic table group (18 classes) and period (7
(7) 2
Equivariant Graph Neural Networks Improve Optical Spectra Prediction for Materials Screening
classes), giving zi ∈ R25 . Each edge carries the unit vector r̂ij = rij /∥rij ∥ and the interatomic distance ∥rij ∥.
spherical harmonics in Cartesian coordinates, normalized to unit norm. Each node i similarly carries steerable features (l) (l) (2l+1)×F max {X̃i }L transforming as a rankl=1 , with X̃i ∈ R l irreducible representation of SO(3). Both the edge and node tensor features thus capture increasingly higher-order geometric information: degree l = 1 encodes directional information, while higher degrees encode more complex angular patterns such as planar orientations (l = 2). The edge tensor features are fixed geometric inputs, while the steerable node features are initialized to zero and iteratively updated across message passing layers.
2.3. Equivariant graph neural networks GNN architectures for atomic systems are commonly formalized within the message passing neural network (MPNN) framework (Gilmer et al., 2017), where atom embeddings are iteratively updated by aggregating information from neighbors. At each round t ∈ {1, . . . , T }, messages are computed and aggregated, the embeddings updated, and after T rounds a readout layer produces the final output: M (t+1) (t) (t) mi = Mt hi , hj , (12)
3.1.2. E MBEDDING L AYER Scalar node feature initialization. Scalar node features hi ∈ RF are initialized by aggregating over neighbors j ∈ N (i) using element-wise products of atom, covalent radius, and distance embeddings. Inspired by GNNOpt (Hung et al., 2024), which demonstrated that incorporating covalent radius as an explicit physical descriptor improves prediction of optical spectra, we augment the original GotenNet embedding with covalent radius gating. Each atom’s covalent radius ci ∈ R is expanded into a fixed-width representation via a Gaussian RBF basis ψ(ci ), providing a continuous encoding of atomic size.
j∈N (i) (t+1) hi = Ut
ŷ = R
(t) (t+1) hi , mi
n
(T )
hi
|i∈V
, o
(13) ,
(14)
where Mt , Ut , L and R are the message, update, and readout functions, and is a permutation-invariant aggregation. Early architectures relied on invariant scalar features derived from pairwise distances, angles, and torsions (Gilmer et al., 2017; Schütt et al., 2017; Lu et al., 2019; Klicpera et al., 2020; Liu et al., 2021). Equivariant architectures instead propagate vector or higher-order tensor features that transform consistently under symmetry operations (Brandstetter et al., 2021; Satorras et al., 2021; Schütt et al., 2021; Le et al., 2022; Thölke & Fabritiis, 2022; Batatia et al., 2023; Simeon & de Fabritiis, 2023; Liao & Smidt, 2023; Aykent & Xia, 2025). Formally, a function f is equivariant to the action of a group G if f (g · x) = g · f (x)
∀ g ∈ G.
Neighbor messages are then formed by gating the projected species embedding with the neighbor’s covalent radius embedding: X mi = σ(Wnbr zj ) ◦ Wcbr ψ(cj ) ◦ dij , (16) j∈N (i)
where dij = fcut (∥rij ∥) · Wndp φ(∥rij ∥)
(15)
(17)
projects a exponential RBF expansion φ(∥rij ∥), cosinecutoff weighted to suppress distant neighbors. The source atom embedding is similarly gated by its own covalent radius: ai = σ(Wna zi ) ◦ Wcna ψ(ci ). (18)
By design, this property guarantees orientation-consistent predictions, eliminating the need for the model to learn these geometric symmetries directly from the training data.
This multiplicative gating allows atomic size to modulate the contribution of species identity, encouraging the model to distinguish atoms that share a period or group but differ in effective radius. Node features are then initialized by a two-layer MLP over the concatenation of ai and mi :
3. Method 3.1. GotenNetOpt Architecture Figure 1 outlines the proposed architecture, which builds upon GotenNet (Aykent & Xia, 2025). For clarity, we reformulate the model below and highlight our specific modifications.
hi,init = Wnru σ(LN(Wnrd [ai , mi ])) + bnru ,
(19)
where σ denotes SiLU and LN denotes layer normalization.
3.1.1. T ENSOR F EATURES
Scalar edge feature initialization. The scalar edge features tij ∈ RF are initialized as the element-wise product of the summed initial node features and a linear projection of the RBF-expanded edge distances:
For each edge (i, j), we encode tensor edge features: (l) (0) max r̃ij = {r̃ij }L l=0 , where r̃ij = ∥rij ∥ is the scalar dis(1)
tance, r̃ij = r̂ij is the normalized direction vector, and (l)
r̃ij = Y (l) (r̂ij ) ∈ R2l+1 for l > 1 are degree-l real
tij,init = (hi,init + hj,init ) ◦ (Werp φ(||rij ||) + berp ) (20) 3
Equivariant Graph Neural Networks Improve Optical Spectra Prediction for Materials Screening
Figure 1. Architecture of GotenNetOpt . (a) The model follows the MPNN framework: an embedding layer initializes atomic and bond features from elemental information and 3D coordinates, T message-passing rounds update the representations, and a readout P Player predicts the optical spectra. (b) The message block. (c) Scalar edge feature update. (d) Node feature update. The summation l m is over l = 1, . . . , Lmax and m = 2l + 1. Symbols: · dot product, ◦ element-wise product. Figure adapted from (Aykent & Xia, 2025).
3.1.3. M ESSAGE B LOCK
brs ∈ RS·F and an MLP-transformed neighbor node feature hcj = γs (hj ) ∈ RS·F , both reshaped into H heads so that
The message block updates node features via the GeometryAware Tensor Attention (GATA) module from GotenNet, adopted without modification except for removing the layer normalization on scalar node features.
c,(h)
tij
(h)
kj = Wk hj + bk ,
(22) (23)
(h)
(h)
∈ RFch .
c,(h)
◦hj
∈ RFch . (27)
j∈N (i)
e (l) = ∆X i
M d,(l) (l) t,(l) e (l) , cij ◦ e rij + cij ◦ X j
(30)
j∈N (i)
which are added to the respective features:
where |N (i)| denotes the degree of node i. The value vector vj = γv (hj ) ∈ RS·F is obtained via a two-layer MLP γv : RF → RS·F , with S = 2Lmax + 1. Reshaping into H heads of dimension Fch = S · F/H, the scaled edge attention (SEA) for head h is: (h)
c,(h)
These coefficients define the messages to the scalar and steerable node features: M ∆hi = csij , (29)
Splitting into H heads of dimension Fhead = F/H, the attention score for head h is computed and normalized as: (h) (h) ⊤ (h) a,(h) αij = qi kj ◦ tij , (24) p (h) exp αij |N (i)| (h) αij ← P · √ , (25) (h) F exp α k∈N (i) ik
seaij = αij vj
(h)
Concatenating across all heads and reshaping gives cij ∈ RS×F , which we decompose into S components of dimension F : h d,(l) Lmax t,(l) Lmax i cij = csij , cij , cij l=1 . (28) l=1
(21)
taij = σ(Wre tij + bre ) .
∈ RFch :
cij = seaij +fcut (∥rij ∥)· tij
First, queries, keys, and a projected edge embedding are computed: qi = Wq hi + bq ,
c,(h)
, hj
hi ← hi + ∆hi ,
e (l) ← X e (l) + ∆X e (l) . X i i i
(31)
Note that the high-degree steerable features are initialized t,(l) (l) as X(l) ≡ 0, which makes the cij ◦ X̃j term equal zero in the first message block, resulting in the initial features:
(26)
To compute the per-head coefficient vectors, we additionally introduce a second edge projection tcij = Wrs tij +
(l)
X̃i,init =
M j∈N (j)
4
d,(l)
(cij
(l)
◦ r̃ij )
(32)
Equivariant Graph Neural Networks Improve Optical Spectra Prediction for Materials Screening
3.1.4. U PDATE B LOCK
3.1.5. R EADOUT L AYER
The update block updates the scalar edge features through the Hierarchical Tensor Refinement (HTR) module and the scalar and steerable node features through the Equivariant Feed-Forward (EQFF) network proposed in GotenNet.
For the readout layer, we use the same design as OptiMate3B. It consists of an attention-weighted global pooling to produce a graph-level embedding, followed by an output MLP to predict the spectrum. First, a linear layer with SiLU activation computes a per-node attention vector ai = σ(Wpool hi + bpool ) ∈ RF from the node features, which is then normalised across all nodes in the graph:
Scalar edge feature update. For each degree l = 1, . . . , Lmax , we compute transformed steerable features: (l)
(l)
Qi = X̃i Wvq ,
(l)
(l)
(l)
Kj = X̃j Wvk ,
exp(ai ) ∈ RF , exp(a ) j j∈V
(33)
αi = P
where exp and division are applied elementwise, and the normalisation is taken over all nodes j belonging to the same graph. The graph-level embedding is then the weighted sum: X hG = αi ◦ hi ∈ RF . (41)
(l)
where Wvq is shared across degrees and Wvk is degree(l) (l) specific. We then obtain Q̂i and K̂j by applying vector (l)
(l)
(l)
(l)
rejection of Qi onto r̃ij and of Kj onto −r̃ij , respectively. This ensures the outputs retain only what the network has learned beyond the edge geometry.
i∈V
Second, the graph embedding is passed through a final MLP, with D hidden layers of dimension Fs , each followed by a SiLU activation (replacing the ReLU used in OptiMate3B), and a plain linear output layer.
Next an interaction vector wij is obtained by summing element-wise products across all degrees and components: wij =
L max 2l+1 X X
(l)
(l)
Q̂i,m ◦ K̂j,m ,
(34)
For the OptiMate3B dataset, the MLP produces ŷ = γout (hG ) ∈ R4002 , where the first 2001 elements correspond to the imaginary part and the remaining elements to the real part. For the GNNOpt dataset, the output is instead ŷ = γout (hG ) ∈ R251 , predicting either the real or imaginary part.
l=1 m=1
capturing learned atomic interactions not explained by edge geometry alone. Finally, the residual update to the scalar edge features is: ∆tij = γw (wij ) ◦ γt (tij ),
(35)
4. Results
where γw and γt are two-layer MLPs, giving tij ← tij + ∆tij . Node feature updates.
4.1. OptiMate3B Dataset We follow the same procedure as OptiMate3B, training first on IPA spectra and finetuning on the RPA dataset. Training details and dataset descriptions are provided in Sections B.1 and C.1. Code to reproduce results is available at https://github.com/khelverskovp/GotenNetOpt. We evaluate using three metrics over the frequency range [0, ωmax ]: the similarity coefficient (SC), mean squared error (MSE), and mean absolute error (MAE), defined as:
For each l = 1, . . . , Lmax the (l)
steerable feature go through a linear projection X̂i = (l) X̃i Wvu . Second, we compute a channel-wise norm over all projected features: v uLmax 2l+1 uX X (l) n =t (X̂ )2 + ϵ i
i,m
(40)
(36)
l=1 m=1
Next, the scalar node features and this norm are concatenated and passed through a two-layer MLP, which is split into two parts:
SC[X(ω); Y (ω)] = 1 −
MSE[X(ω); Y (ω)] = [mi,1 , mi,2 ] = γm ((ni , hi ))
(37)
After which, the scalar and high-degree steerable node features are updated as: hi ← m1,i + hi (l) (l) (l) X̃i ← (mi,2 ◦ X̂i ) + X̃i
MAE[X(ω); Y (ω)] =
(38)
R
1
|X(ω) − Y (ω)| dω R , |Y (ω)| dω Z
ωmax
1 ωmax
(42)
|X(ω) − Y (ω)|2 dω, (43)
Z
|X(ω) − Y (ω)| dω. (44)
where X(ω) is the target spectra and Y (ω) is the predicted spectra for Im(ε̄) or Re(ε̄).
(39) 5
Equivariant Graph Neural Networks Improve Optical Spectra Prediction for Materials Screening Table 1. IPA and RPA test set results for the imaginary (Im) and real (Re) parts of the dielectric function, measured by mean absolute error (MAE), mean squared error (MSE), and similarity coefficient (SC). Each cell shows the mean with median in parentheses below. Best results are shown in bold. I M(ε̄IPA )
R E(ε̄IPA )
I M(ε̄RPA )
R E(ε̄RPA )
M ODEL
MAE ↓
MSE ↓
SC ↑
MAE ↓
MSE ↓
SC ↑
MAE ↓
MSE ↓
SC ↑
MAE ↓
MSE ↓
SC ↑
O PTI M ATE 3B
0.265 (0.193)
0.744 (0.108)
0.883 (0.903)
0.276 (0.201)
0.677 (0.108)
0.882 (0.904)
0.195 (0.115)
0.539 (0.035)
0.906 (0.930)
0.199 (0.120)
0.393 (0.035)
0.910 (0.933)
G OTEN N ET O PT
0.238 (0.173)
0.643 (0.089)
0.894 (0.913)
0.247 (0.178)
0.601 (0.089)
0.894 (0.914)
0.164 (0.093)
0.453 (0.024)
0.919 (0.942)
0.168 (0.099)
0.318 (0.024)
0.923 (0.946)
Table 2. Percentage improvement of GotenNetOpt over OptiMate3B in median MAE, MSE, and SC across the test set, reported for the full 0–20 eV range and the low-energy 0–8 eV sub-range.
Table 1 reports mean and median values for each metric on both the IPA and RPA datasets, separated by imaginary and real parts. Additionally, we report results on using Voronoi graph construction and a scaled down version of GotenNetOpt in Section D. Median values are considerably lower than means reflecting a tail of outlier samples. RPA predictions are substantially better than IPA for both models due to the fine-tuning procedure and the greater physical accuracy of RPA spectra. The distribution of each evaluation metric on the RPA test set are shown in Figure 2. GotenNetOpt OptiMate3B
12 10
M ETRIC
∆% (0–20 eV)
∆% (0–8 eV)
I M(ε̄IPA )
MAE ↓ MSE ↓ SC ↑
10.1% 18.0% 1.1%
8.9% 16.0% 1.6%
R E(ε̄IPA )
MAE ↓ MSE ↓ SC ↑
11.0% 17.9% 1.1%
11.4% 18.2% 1.1%
I M(ε̄RPA )
MAE ↓ MSE ↓ SC ↑
18.4% 31.2% 1.2%
19.2% 31.3% 1.4%
R E(ε̄RPA )
MAE ↓ MSE ↓ SC ↑
17.9% 31.6% 1.4%
21.5% 33.2% 1.3%
AVERAGE
MAE/MSE ↓ SC ↑
19.6% 1.2%
20.0% 1.4%
Density
8 6 4 2 0 −0.2
70
0.0
0.2
0.4
0.6
0.8
1.0
Similarity Coefficient (SC) GotenNetOpt OptiMate3B
60
Density
50
Table 3 compares both models on predicting the static real permittivity by evaluating the median MAE and MSE on Re(ε̄(0)) from the predicted spectra for both the IPA and RPA test sets.
40 30 20 10 0 10−4
10−3
10−2
10−1
100
101
102
Mean Squared Error (MSE) 8
Table 3. Test set MAE and MSE (median) on the static real permittivity, and the percentage improvement of GotenNetOpt .
GotenNetOpt OptiMate3B
7
Density
6 5 4
M ODEL
3
10−2
10−1
100
Mean Absolute Error (MAE)
Figure 2. Density histograms of SC (top), MSE (middle), and MAE (bottom) across the RPA test samples for the two models. MSE and MAE are shown on logarithmic x-axes to accommodate their heavy-tailed distributions.
R E(ε̄(0))RPA
MSE ↓
MAE ↓
MSE ↓
O PTI M ATE 3B G OTEN N ET O PT
0.131 0.080
0.017 0.006
0.121 0.074
0.015 0.006
∆%
39.0%
62.8%
38.5%
62.1%
1 0
R E(ε̄(0))IPA MAE ↓
2
Figure 3 shows predicted versus target spectra at the 10th, 40th, 60th and 90th SC percentiles of the RPA test set. For additional spectra plots, Figures 7–10 in Section D.4 show representative spectra for the cases where GotenNetOpt outperforms OptiMate3B, where OptiMate3B outperforms GotenNetOpt , and where both models perform worst and best simultaneously. The imaginary and real parts are predicted with comparable accuracy across percentiles, consistent with the Kramers-Kronig relations.
The difference in distributions is most pronounced for MAE and MSE, particularly the latter, as evidenced by the performance summary in Table 2. We report percentage improvements for both the full spectrum (0–20 eV) and the low-energy sub-spectrum (0–8 eV), which is most relevant for thin-film optics. The relative performance improvements are largest on the RPA dataset and in the 0–8 eV sub-spectrum. 6
Equivariant Graph Neural Networks Improve Optical Spectra Prediction for Materials Screening
Q10% SC=0.858
Im(ε̄RPA ) AgBr
Re(ε̄RPA ) AgBr
4.5
6.0
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4.5
1.5 0.0
Target Predicted
3.0 1.5 0
250
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Q40% SC=0.935
Tl2 HgTe 10
1750
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5
Q60% SC=0.953
1500
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SrHClO
3
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CsTlBr2
4.5 Q90% SC=0.974
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1.5
0.0 0
250
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1000
1250
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1000
Frequency index
Figure 3. Predicted (dashed, red) vs. target (solid, blue) dielectric spectra at the 10th, 40th, 60th, and 90th percentiles of the RPA test set ranked by SC. Each row shows Im(ε̄RPA ) and Re(ε̄RPA ) components for a single structure (chemical formula shown above).
Figure 4 confirms this explicitly, having a strong-per sample correlation between SC on Re(ε̄) and Im(ε̄). 1.0
4.2. GNNOpt Dataset Table 4 presents the performance of GotenNetOpt on the GNNOpt dataset; training details and dataset descriptions are provided in Sections B.2 and C.2. Following Hung et al. (2024), we report MSE and R2 on the test set. 2 R P is evaluatedPon the energy-weighted mean X̄i = k ωk Xi (ωk )/ k ωk of each predicted and target spectrum, 2 PN i=1 X̄i − Ȳi 2 R [X; Y ] = 1 − PN (45) 2 , i=1 X̄i − X
Test sample Kr y=x
0.8
SC Re(ε̄RPA )
0.6
0.4
0.2
where Ȳi is defined analogously and X is the mean of X̄i over the test set. MSE is computed over all K = 251 frequency points, consistent with the definition in Equation (43).
0.0
−0.2
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
SC Im(ε̄RPA )
Figure 4. Kramers-Kronig consistency check. Per-sample SC for Re(ε̄) against SC for Im(ε̄).
Table 4. Test set results on the GNNOpt dataset measured by MSE and coefficient of determination (R2 ) along with percentage improvement compared to GNNOpt.
Five outliers are Kr-only crystals. Kr appears in only 2 training samples with no other group-18 elements (Figure 5), leaving its atom embedding poorly informed; Kr also forms no covalent bonds under ordinary conditions, meaning the covalent radius feature is physically meaningless. These outliers are an extreme case of a broader log-linear relationship between prediction accuracy and per-element training count (Figure 6), consistent with findings reported for OptiMate3B (Grunert et al., 2026c). 7
M ETRIC
GNNO PT
G OTEN N ET O PT
∆%
I M(ε̄)
MSE ↓ R2 ↑
1.765 0.737
1.371 0.864
22.3% 17.2%
R E(ε̄)
MSE ↓ R2 ↑
2.363 0.715
1.907 0.806
19.3% 12.7%
Equivariant Graph Neural Networks Improve Optical Spectra Prediction for Materials Screening
0.4
1
0.5
0.6 0.7 Median SC
H
0.8
0.9
2
He
0.894 3
4
Li
0.917 11
5
Be
B
0.918
0.859
12
Na
0.927 19
13
Mg
Al
0.935
0.927
20
K
21
Ca
Sc
22
23
Ti
24
V
Cr
25
Mn
26
Fe
27
28
Co
Ni
29
Cu
30
Zn
31
Ga
6
C
7
N
8
O
0.885
0.887
0.897
14
15
16
0.917
0.923
0.923
Si 32
Ge
P
33
As
S
34
Se
9
F
0.907 17
Cl
0.927 35
Br
0.937
0.930
0.875
0.869
0.865
0.819
0.861
0.890
0.868
0.881
0.890
0.923
0.939
0.934
0.926
0.919
0.922
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
0.935
0.874
0.890
0.922
0.926
57
Hf
Rb
0.940 55
Cs
Sr
Y
56
0.904
Ba
La
0.902
Zr
Nb
Mo
0.899
72
0.782
73
74
Ta
0.874
W
0.864
0.898
Tc
0.846
Ru
0.874
76
77
0.913
0.862
75
Re
0.633
Rh
0.829
Os
Pd
0.849 78
Ir
Pt
0.868
Ag
0.891 79
Au
0.870
Cd
0.910 80
Hg
0.878
In
0.936 81
Tl
0.889
Sn
0.928 82
Pb
0.893
Sb
0.926 83
Bi
0.896
Te 84
Po
I
85
At
10
Ne 18
Ar 36
Kr
0.380 54
Xe 86
Rn
Figure 5. Median SC for each compound in the IPA test set that contains a given element. Elements that are not present in any compounds are colored white.
informative, and more generally, the most useful atomic features may depend on the chemical subspace being screened. Future work could explore other physical properties as node features, tailored to the target application. The readout architecture is similarly not necessarily optimal, and other designs remain to be explored. More broadly, other equivariant GNN architectures have not yet been tested on this task.
Te
Element Mn Cr
Ru
Median SC – IPA
Mo
Re
6 × 10−1
4 × 10−1
Despite performance improvements, the dominant failure mode is still distributional: prediction accuracy correlates with per-element training frequency. Discarding these outliers, the similarity coefficient metric suggests the model is approaching highly accurate predictions on the current data. Hence, for elements well represented in the training data, the model is viable as a large-scale screening tool, but for structures containing underrepresented elements, predictions should be treated with caution, and expanding the training set would be necessary before using the model on poorly covered chemical spaces.
Kr
3 × 10−1
101
102
103
Training set count (# compounds containing element)
Figure 6. Log-log plot of median SC for all compounds in the test set that contain a given element as a function of how often that element occurs in the training set for the IPA data.
5. Discussion This work suggests that equivariant GNNs can meaningfully improve optical spectra prediction, with the most pronounced gains in the 0–8 eV range and for prediction of the static real permittivity. Gains are also proportionally larger on the RPA dataset than on the IPA dataset, which may reflect that physically accurate spectra encoding richer many-body interactions benefit more from expressive geometric representations
6. Conclusion We have presented GotenNetOpt , an adaptation of the equivariant GNN GotenNet for optical spectra prediction. The results show that using an equivariant architecture leads to consistent performance increases, indicating that geometric equivariance is a beneficial inductive bias for this task. Gains are most pronounced in the low-energy range and in the static real permittivity, quantities of direct relevance for high-throughput screening of thin-film optical materials, and are proportionally larger on the RPA dataset, suggesting that geometric expressiveness matters most when the training data is physically accurate.
The adaptation of GotenNet involved two main changes: incorporating covalent radius embeddings into the scalar node feature initialization, and replacing the readout layer with the OptiMate3B readout design. Covalent radius is a physically motivated feature, though it is not universally 8
Equivariant Graph Neural Networks Improve Optical Spectra Prediction for Materials Screening
0003-6951. doi: 10.1063/5.0305586. URL https: //doi.org/10.1063/5.0305586.
Impact Statement This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Hohenberg, P. and Kohn, W. Inhomogeneous electron gas. Phys. Rev., 136:B864–B871, Nov 1964. doi: 10.1103/PhysRev.136.B864. URL https://link. aps.org/doi/10.1103/PhysRev.136.B864.
References
Hung, N. T., Okabe, R., Chotrattanapituk, A., and Li, M. Ensemble-embedding graph neural network for direct prediction of optical spectra from crystal structure, 2024. URL https://arxiv.org/abs/2406.16654.
Aykent, S. and Xia, T. Gotennet: Rethinking efficient 3d equivariant graph neural networks. In The Thirteenth International Conference on Learning Representations, 2025. URL https://openreview.net/forum? id=5wxCQDtbMo.
Ibrahim, A. and Ataca, C. Prediction of frequencydependent optical spectrum for solid materials: A multioutput & multi-fidelity machine learning approach, 2024. URL https://arxiv.org/abs/2407.07736.
Batatia, I., Kovács, D. P., Simm, G. N. C., Ortner, C., and Csányi, G. Mace: Higher order equivariant message passing neural networks for fast and accurate force fields, 2023. URL https://arxiv.org/abs/ 2206.07697.
Jain, A., Ong, S. P., Hautier, G., Chen, W., Richards, W. D., Dacek, S., Cholia, S., Gunter, D., Skinner, D., Ceder, G., and Persson, K. A. Commentary: The materials project: A materials genome approach to accelerating materials innovation. APL Materials, 1(1):011002, 07 2013. ISSN 2166-532X. doi: 10.1063/1.4812323. URL https://doi.org/10.1063/1.4812323.
Brandstetter, J., Hesselink, R., van der Pol, E., Bekkers, E. J., and Welling, M. Geometric and physical quantities improve E(3) equivariant message passing. CoRR, abs/2110.02905, 2021. URL https://arxiv.org/ abs/2110.02905.
Klicpera, J., Giri, S., Margraf, J. T., and Günnemann, S. Fast and uncertainty-aware directional message passing for non-equilibrium molecules. CoRR, abs/2011.14115, 2020. URL https://arxiv.org/abs/2011. 14115.
Gilmer, J., Schoenholz, S. S., Riley, P. F., Vinyals, O., and Dahl, G. E. Neural message passing for quantum chemistry. CoRR, abs/1704.01212, 2017. URL http://arxiv.org/abs/1704.01212.
Kohn, W. and Sham, L. J. Self-consistent equations including exchange and correlation effects. Physical review, 140(4A):A1133, 1965.
Ginter, C., Choudhary, K., and Mandal, S. Accelerated prediction of dielectric functions in solar cell materials with graph neural networks, 2025. URL https:// arxiv.org/abs/2510.08738.
Le, T., Noé, F., and Clevert, D.-A. Equivariant graph attention networks for molecular property prediction, 2022. URL https://arxiv.org/abs/2202.09891.
Grunert, M., Großmann, M., and Runge, E. Data for evaluating OptiMate3B. 1 2026a. doi: 10.6084/m9.figshare.30257551.v1. URL https: //figshare.com/articles/dataset/Data_ for_evaluating_OptiMate3B/30257551.
Liao, Y.-L. and Smidt, T. Equiformer: Equivariant graph attention transformer for 3d atomistic graphs, 2023. URL https://arxiv.org/abs/2206.11990.
Grunert, M., Großmann, M., and Runge, E. Dielectric Function of Semiconductors and Insulators under the Independent Particle Approximation and the Random Phase Approximation. 1 2026b. doi: 10.6084/m9.figshare.30257689.v1. URL https://figshare.com/articles/ dataset/Dielectric_Function_of_ Semiconductors_and_Insulators_ under_the_Independent_Particle_ Approximation_and_the_Random_Phase_ Approximation/30257689.
Liu, Y., Wang, L., Liu, M., Zhang, X., Oztekin, B., and Ji, S. Spherical message passing for 3d graph networks. CoRR, abs/2102.05013, 2021. URL https://arxiv.org/ abs/2102.05013. Lu, C., Liu, Q., Wang, C., Huang, Z., Lin, P., and He, L. Molecular property prediction: A multilevel quantum interactions modeling perspective, 2019. URL https: //arxiv.org/abs/1906.11081. Onida, G., Reining, L., and Rubio, A. Electronic excitations: density-functional versus many-body green’s-function approaches. Rev. Mod. Phys., 74:601–659, Jun 2002. doi: 10.1103/RevModPhys.74.601. URL https://link. aps.org/doi/10.1103/RevModPhys.74.601.
Grunert, M., Großmann, M., and Runge, E. Optical spectra prediction using three-body information. Applied Physics Letters, 128(5):053302, 02 2026c. ISSN 9
Equivariant Graph Neural Networks Improve Optical Spectra Prediction for Materials Screening
Satorras, V. G., Hoogeboom, E., and Welling, M. E(n) equivariant graph neural networks. CoRR, abs/2102.09844, 2021. URL https://arxiv.org/abs/2102. 09844. Schütt, K. T., Unke, O. T., and Gastegger, M. Equivariant message passing for the prediction of tensorial properties and molecular spectra. CoRR, abs/2102.03150, 2021. URL https://arxiv.org/abs/2102.03150. Schütt, K. T., Kindermans, P.-J., Sauceda, H. E., Chmiela, S., Tkatchenko, A., and Müller, K.-R. Schnet: A continuousfilter convolutional neural network for modeling quantum interactions, 2017. URL https://arxiv.org/ abs/1706.08566. Simeon, G. and de Fabritiis, G. Tensornet: Cartesian tensor representations for efficient learning of molecular potentials, 2023. URL https://arxiv.org/abs/2306. 06482. Thölke, P. and Fabritiis, G. D. Torchmd-net: Equivariant transformers for neural network based molecular potentials. CoRR, abs/2202.02541, 2022. URL https: //arxiv.org/abs/2202.02541. Toll, J. S. Causality and the dispersion relation: logical foundations. Physical review, 104(6):1760, 1956. Yang, R. X., Horton, M. K., Munro, J., and Persson, K. A. High-throughput optical absorption spectra for inorganic semiconductors, 2022. URL https://arxiv.org/ abs/2209.02918.
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Equivariant Graph Neural Networks Improve Optical Spectra Prediction for Materials Screening
A. Density functional theory Optical response tensors of materials are calculated through quantum mechanical perturbation theory which necessitates an accurate description of the electronic ground state of the material to be perturbed first. For many-body systems, that is, systems containing many electrons, the ground state of the system is typically obtained through DFT. Strictly speaking, determining the ground state of a quantum system with N electrons means determining the ground state many-body wave function Ψ(r1 , r2 , . . . , rN ); however, within DFT the ground state can be understood entirely through the ground state electron density ρ(r) which is otherwise considered an observable that can be calculated directly from the many-body wave function. While many-body wave functions describe the ground states of interacting electrons governed by the Schrödinger equation, most modern formulations of DFT use an auxiliary set of single-particle wave functions which obey the Kohn-Sham equations (Kohn & Sham, 1965). These single-particle wave functions describe a fictitious set of non-interacting electrons; however, the Kohn-Sham equations are cleverly constructed such that they (in theory) yield the same electron density as the true many-body wave function. Density functional theory thus discards the staggeringly complicated many-body wave function in favor of the much simpler set of single-particle wave functions as both will construct the true ground state electron particle density ρ. Although DFT is a popular and versatile method for obtaining ground state information, trouble arises when considering excited states of the system as only the ground state can be described entirely by the density, while excited states need the true many-body wave function. This leads to a set of perturbation theory methods tiered by how diligently they account for many-body effects, which are otherwise missing in the non-interacting Kohn-Sham system. The simplest of which is the independent-particle approximation, which simply treats the optical response as a sum of optical responses of the independent-particle system.
B. Training Details B.1. Training on the OptiMate3B Dataset Table 5 shows the hyperparameters used for training on the OptiMate3B dataset. Training proceeds in two stages. In the first stage, the model is trained to predict IPA dielectric spectra. In the second stage, the best IPA checkpoint is used to initialise the model, which is then fine-tuned on RPA dielectric spectra. Both stages use the AdamW optimiser with a learning rate of 1 × 10−5 and weight decay of 0.01. Gradients are clipped to a maximum norm of 10. The learning rate is reduced on plateau by a factor of 0.5, with a minimum learning rate of 10−8 and 10−9 for IPA and RPA training respectively. The plateau scheduler patience is set to 15 epochs. Early stopping is applied with a patience of 50 epochs in both stages. The loss function is mean absolute error (MAE) in both stages. To stabilize early stopping, an exponential moving average (EMA) of the validation loss is maintained with smoothing factor α = 0.9. Validation loss spikes exceeding 2× the current smoothed value are ignored when updating the EMA, preventing transient instabilities from triggering premature learning rate decay or stopping. B.2. Training on the GNNOpt dataset Table 6 shows the hyperparameters used for training on the GNNOpt dataset. We follow the GNNOpt training setup, including their graph construction procedure and the scaling and interpolation of the target spectra. Unlike the OptiMate3B dataset setup, where the real and imaginary parts of the dielectric function are predicted jointly, here a separate model is trained and evaluated for each component. The optical spectra are interpolated onto a uniform grid of 251 points over the energy range 0 ≤ ℏω ≤ 50 eV. Spectra are then normalized by a global scale factor computed as the median of the per-sample maximum values across the training set. The loss function is mean squared error (MSE) on the normalized spectra. Graphs are constructed by placing edges between all atoms within a cutoff radius of rc = 6.0 Å. All experiments use a fixed random seed of 12 (same seed used in the original GNNOpt experiments). Training is performed for 200 epochs with a batch size of 8, and gradients are clipped to a maximum ℓ2 norm of 1.0.
11
Equivariant Graph Neural Networks Improve Optical Spectra Prediction for Materials Screening
Table 5. Hyperparameters for the two-stage training procedure (IPA pre-training and RPA fine-tuning). Shared architecture parameters apply to both stages. (S) are parameter values used for the smaller variant.
Category
Hyperparameter
Value
Optimisation
Learning rate Minimum learning rate Weight decay Gradient clipping Batch size Max epochs
1 × 10−5 (S: 1 × 10−4 ) 1 × 10−8 (RPA: 1 × 10−9 ) 0.01 10 32 1000
Scheduler & Stopping
LR scheduler LR reduction factor LR patience Early stopping patience
ReduceLROnPlateau 0.5 15 epochs 50 epochs
Loss
Loss function Val. loss EMA α Spike filter factor
MAE 0.9 2.0
Data Split
Training fraction Validation fraction Test fraction
80% 10% 10%
Architecture
Feature embedding dim (F ) Message passing rounds (T ) Radial basis functions Cutoff radius Max degree (Lmax ) Attention heads (H) Dropout Readout depth (D) Readout hidden dim (Fs )
1024 (S: 256) 6 64 6.0 Å 2 8 0.1 4 (S: 2) 8192 (S: 256)
12
Equivariant Graph Neural Networks Improve Optical Spectra Prediction for Materials Screening Table 6. Hyperparameters for training on the GNNOpt dataset.
Category
Hyperparameter
Value
Optimisation
Optimizer Learning rate Weight decay Gradient clipping Batch size Epochs
AdamW 6 × 10−4 0.015 1.0 8 200
Scheduler
LR scheduler LR decay (γ)
Exponential 0.98
Loss & Seed
Loss function Random seed
MSE 12
Architecture
Feature embedding dim (F ) Message passing rounds (T ) Radial basis functions Cutoff radius Max degree (Lmax ) Attention heads (H) Dropout Readout depth (D) Readout hidden dim (Fs )
256 3 64 6.0 Å 2 8 0.3 1 256
C. Dataset Descriptions C.1. OptiMate3B Dataset The OptiMate3B datasets consists of an IPA dataset with 23,834 materials and an RPA dataset with 10,533 materials, which we obtain from Grunert et al. (2026a) and Grunert et al. (2026b). The structures were originally drawn from the Alexandria database and filtered according to the following criteria: • no elements beyond Bismuth and no Lanthanides; • a direct or indirect bandgap above 50 meV; • an indirect bandgap below 100 meV or above 10 eV • energy above hull below 150 meV per atom; • symmetry of at least space group 75 (tetragonal or higher); • at most 8 atoms per primitive unit cell. For the RPA dataset, structures were further restricted to at most 4 atoms per unit cell. We adopt the same split procedure as Grunert et al. (2026c), splitting by unique chemical composition in an 80:10:10 ratio yielding the exact same splits as in the original paper of 19,036, 2413, and 2385 materials for training, validation and test sets respectively. The same compositional split is preserved for the RPA dataset, giving 8,414, 1,040, and 1,079 materials.
13
Equivariant Graph Neural Networks Improve Optical Spectra Prediction for Materials Screening
C.2. GNNOpt Dataset The GNNOpt dataset consists of 944 crystal structures obtained from the GNNOpt GitHub repository. The optical spectra were originally computed using IPA by (Yang et al., 2022), with structures sourced from the Materials Project (Jain et al., 2013). The dataset was constructed by filtering for materials satisfying the following criteria: • an energy band gap between 0 and 5.0 eV; • at most 10 atoms per primitive unit cell. We adopt the same train/validation/test split as the original GNNOpt paper, taken directly from the repository, corresponding to an 80%, 10%, 10% division of 733, 97, and 110 materials, respectively.
D. Additional Results and Analysis D.1. Covalent Radius Embeddings Ablation The effect of covalent radius embeddings is modest but generally positive, as shown in Table 7. The clearest gains are in the RPA targets, where MAE and MSE improve consistently (up to 3.13% and 4.00% respectively). For the IPA targets the picture is more mixed; MAE improves slightly but the median MSE is marginally worse (+1.14%), suggesting that covalent radius information is more beneficial for the RPA computed dielectric function. Overall, the average median improvement across MAE and MSE is 1.18% and 0.14% for SC. Table 7. IPA and RPA test set results ablating covalent radius embeddings in GotenNetOpt . Each cell shows the mean with median in parentheses below. Best results are shown in bold. The ∆% row shows the relative change of GotenNetOpt compared to the model without covalent radius embeddings based on the medians. I M(ε̄IPA )
R E(ε̄IPA )
I M(ε̄RPA )
R E(ε̄RPA )
M ODEL
MAE ↓
MSE ↓
SC ↑
MAE ↓
MSE ↓
SC ↑
MAE ↓
MSE ↓
SC ↑
MAE ↓
MSE ↓
SC ↑
W / O COV. RADIUS
0.241 (0.174)
0.664 (0.088)
0.893 (0.912)
0.251 (0.180)
0.623 (0.088)
0.893 (0.913)
0.167 (0.096)
0.465 (0.025)
0.920 (0.940)
0.171 (0.102)
0.332 (0.024)
0.923 (0.945)
G OTEN N ET O PT
0.238 (0.173)
0.643 (0.089)
0.894 (0.913)
0.247 (0.178)
0.601 (0.089)
0.894 (0.914)
0.164 (0.093)
0.453 (0.024)
0.919 (0.942)
0.168 (0.099)
0.318 (0.024)
0.923 (0.946)
∆%
0.57%
+1.14%
0.11%
1.11%
+1.14%
0.11%
3.13%
4.00%
0.21%
2.94%
0.00%
0.11%
D.2. Effect of Model Size on Performance Table 8 reports performance on the OptiMate3B dataset of GotenNetOpt -S, a smaller variant (16M parameters) using readout depth 2, feature dimension 256, and readout hidden dimension 256, compared to GotenNetOpt (474M). Despite its reduced size, GotenNetOpt -S trails GotenNetOpt by only 6.9% in median MAE/MSE and 0.3% in median SC on average, suggesting that performance gains stem primarily from architectural differences rather than increased parameter count. Nevertheless, since inference is run once per crystal in the intended screening use case, the computational overhead of the larger model is negligible, making it the preferred choice over the smaller variant. Table 8. IPA and RPA test set results comparing GotenNetOpt -S (small variant) and GotenNetOpt (main model). Each cell shows the mean with median in parentheses below. Best results are shown in bold. The ∆% row shows the relative improvement of GotenNetOpt compared to GotenNetOpt -S based on the medians. I M(ε̄IPA )
R E(ε̄IPA )
I M(ε̄RPA )
R E(ε̄RPA )
M ODEL
MAE ↓
MSE ↓
SC ↑
MAE ↓
MSE ↓
SC ↑
MAE ↓
MSE ↓
SC ↑
MAE ↓
MSE ↓
SC ↑
G OTEN N ET O PT -S
0.243 (0.178)
0.679 (0.092)
0.893 (0.910)
0.252 (0.183)
0.640 (0.092)
0.893 (0.913)
0.170 (0.102)
0.452 (0.028)
0.918 (0.938)
0.173 (0.105)
0.326 (0.028)
0.921 (0.943)
G OTEN N ET O PT
0.238 (0.173)
0.643 (0.089)
0.894 (0.913)
0.247 (0.178)
0.601 (0.089)
0.894 (0.914)
0.164 (0.093)
0.453 (0.024)
0.919 (0.942)
0.168 (0.099)
0.318 (0.024)
0.923 (0.946)
∆%
2.81%
3.26%
0.33%
2.73%
3.26%
0.11%
8.82%
14.29%
0.43%
5.71%
14.29%
0.32%
14
Equivariant Graph Neural Networks Improve Optical Spectra Prediction for Materials Screening
D.3. Effect of Graph Construction on Performance We compare GotenNetOpt trained with Voronoi-based graph construction against the distance-based cutoff of 6 Å on the OptiMate3B dataset. As shown in Table 9, the cutoff-based method consistently outperforms the Voronoi-based method across all targets and metrics, with an average combined median improvement of 4.3% in MAE and MSE and 0.3% in SC. While the improvements are modest, they are consistent across all four targets and both the IPA and RPA datasets, suggesting that the fixed-radius cutoff provides a slightly better graph structure for this task than the Voronoi approach. Table 9. IPA and RPA test set results comparing GotenNetOpt on Voronoi vs. distance based graph construction. Each cell shows the mean with median in parentheses below. Best results are shown in bold. The ∆% row shows the relative change of the 6 Å cutoff method compared to Voronoi based on the medians. I M(ε̄IPA )
R E(ε̄IPA )
I M(ε̄RPA )
R E(ε̄RPA )
M ODEL
MAE ↓
MSE ↓
SC ↑
MAE ↓
MSE ↓
SC ↑
MAE ↓
MSE ↓
SC ↑
MAE ↓
MSE ↓
SC ↑
VORONOI
0.243 (0.178)
0.678 (0.094)
0.892 (0.910)
0.254 (0.185)
0.635 (0.094)
0.892 (0.911)
0.169 (0.099)
0.497 (0.026)
0.917 (0.939)
0.173 (0.103)
0.352 (0.026)
0.921 (0.944)
6 Å CUTOFF
0.238 (0.173)
0.643 (0.089)
0.894 (0.913)
0.247 (0.178)
0.601 (0.089)
0.894 (0.914)
0.164 (0.093)
0.453 (0.024)
0.919 (0.942)
0.168 (0.099)
0.318 (0.024)
0.923 (0.946)
∆%
2.81%
5.32%
0.33%
3.78%
5.32%
0.33%
6.06%
7.69%
0.32%
3.88%
7.69%
0.21%
D.4. Additional spectra plots Top 5 GotenNet wins as meaured by ∆SC Figure 7 shows the predicted and target dielectric spectra for the five samples where GotenNetOpt most outperforms OptiMate3B as measured by ∆SC between the two models on the RPA test set. Top 5 OptiMate3B wins as measured by ∆SC. Figure 8 shows the predicted and target dielectric spectra for the five samples where OptiMate3B most outperforms GotenNetOpt as measured by ∆SC. Top 5 samples where both models fail. Figure 9 shows the predicted and target dielectric spectra for the five samples with the lowest average SC across both models, i.e. the cases where both models perform worst. Top 5 samples where both models succeed. Figure 10 shows the predicted and target dielectric spectra for the five samples with the highest average SC across both models, i.e. the cases where both models perform best.
15
Equivariant Graph Neural Networks Improve Optical Spectra Prediction for Materials Screening
()
()
GeSe SC=+0.251
40
Target GotenNetOpt (SC=0.911) OptiMate3B (SC=0.660)
30 20
20 10
0
0
KAs4Br SC=+0.249
0
500
750
1000 1250 1500 1750 2000
0
250
500
750
1000 1250 1500 1750 2000
0
250
500
750
1000 1250 1500 1750 2000
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250
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1000 1250 1500 1750 2000
0
250
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750
1000 1250 1500 1750 2000
0
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500
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1000 1250 1500 1750 2000
8
6
6
4
4
2 0
Be2C SC=+0.200
250
2 0 0
250
500
750
1000 1250 1500 1750 2000
20
15
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10
10
5
5
0
0
S SC=+0.190
0
500
750
1000 1250 1500 1750 2000
3
2
2 1 1 0
Tl2HgO SC=+0.182
250
0
250
500
750
1000 1250 1500 1750 2000
8
10
6 5
4 2
0 0
250
500
750
1000 1250 1500 1750 2000
Frequency index
Frequency index
Figure 7. Predicted of GotenNetOpt (dashed, red) and OptiMate3B (dashed, green) vs. target (solid, blue) dielectric spectra for the top 5 wins of GotenNetOpt over OptiMate3B as measured by ∆SC between the two models on the RPA test set.
16
Equivariant Graph Neural Networks Improve Optical Spectra Prediction for Materials Screening
()
Kr SC=-0.361
2.0
()
1.5
3
1.0
2
0.5 1
0.0
Kr SC=-0.352
0
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1.5
3
1.0
2
0
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0
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0
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0
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0
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0.5 1
0.0 0
Kr SC=-0.305
Target GotenNetOpt (SC=0.407) OptiMate3B (SC=0.769)
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1000 1250 1500 1750 2000
1.5
3
1.0
2
0.5 1 0.0
Kr SC=-0.293
0
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500
750
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3
1.5 1.0
2
0.5 1 0.0 0
250
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KBe4P3 SC=-0.183
30
40
20 20 10 0
0 0
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Frequency index
Frequency index
Figure 8. Predicted spectra of GotenNetOpt (dashed, red) and OptiMate3B (dashed, green) vs. target (solid, blue) dielectric spectra for the top 5 wins of OptiMate3B over GotenNetOpt as measured by ∆SC on the RPA test set.
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Equivariant Graph Neural Networks Improve Optical Spectra Prediction for Materials Screening
()
() 15
BeAlHO SC=+0.127
15
Target GotenNetOpt (SC=0.343) OptiMate3B (SC=0.216)
10
10
5 5 0 0
Rb2BrCl5 SC=-0.064
0
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0
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0
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0
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0
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0
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500
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15
15
10
10
5 5
0
0 0
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Sr2AuCl SC=-0.136
40 40
30 20
20
10 0
0 0
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KBe4P3 SC=-0.183
30
20 10 0
InAsSe2 SC=+0.091
40
20
0 0
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30 20
20
10 0
10 0 0
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Frequency index
Frequency index
Figure 9. Predicted spectra of GotenNetOpt (dashed, red) and OptiMate3B (dashed, green) vs. target (solid, blue) for the 5 samples with the lowest average SC across both models (SC < 0.80) on the RPA test set, illustrating failure cases shared by both models.
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Equivariant Graph Neural Networks Improve Optical Spectra Prediction for Materials Screening
KRbMgCl4 SC=+0.014
()
()
2
3
1
2 1
0
KRb2SeBr SC=+0.011
0
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0
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0
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0
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0
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0
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2.5 1.5 2.0
1.0
1.5
0.5 0.0
1.0 0
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500
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2.0
KRbSrBr4 SC=+0.020
Target GotenNetOpt (SC=0.994) OptiMate3B (SC=0.980)
3
1.5 2
1.0 0.5
1
0.0 0
250
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Sr4PSe2Cl SC=+0.013
6 4 3
4
2
2
1 0
0 0
250
500
750
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K2SrI4 SC=+0.013
4 2
3 2
1
1 0
0
250
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750
1000 1250 1500 1750 2000
Frequency index
Frequency index
Figure 10. Predicted spectra of GotenNetOpt (dashed, red) and OptiMate3B (dashed, green) vs. target (solid, blue) for the 5 samples with the highest average SC across both models (SC > 0.95) on the RPA test set, illustrating representative cases where both models achieve high fidelity.
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