Accurate prediction of tensorial spectra using equivariant graph neural network

Abstract

Optical spectroscopies provide a powerful means to probe light-matter interactions and complex electronic features that are crucial for the development and optimization of optoelectronic devices, where performance is closely tied to the underlying electronic spectrum. However, realistic modeling of tensor optical responses in materials remains computationally demanding and challenging. Here we introduce the Tensorial Spectra Equivariant Neural Network (TSENN), an equivariant graph neural network architecture that maps crystal structures directly to their full photon-frequency-dependent optical tensors. By encoding isotropic sequential scalar components and anisotropic sequential tensor components into spherical tensor representations, TSENN ensures symmetry-aware predictions consistent with crystalline symmetry constraints. Trained on frequency-dependent dielectric tensors of 1,432 bulk semiconductors, the model achieves a mean absolute error of 0.127, demonstrating its potential for efficient and general modeling of optical properties. Our framework opens new avenues for data-driven design of anisotropic optical responses to accelerate materials discovery for optoelectronic applications.

Modeling full tensorial optical responses of crystalline materials is computationally demanding. Here, the authors introduce TSENN, an equivariant graph neural network that predicts symmetry-consistent, frequency-dependent optical tensors directly from crystal structures.

Introduction

Optical spectroscopies, which involve excitation of a material with external electromagnetic fields, provide a powerful tool for probing properties of materials. The two key frequency-dependent response functions involved here are the dielectric function and optical conductivity, which together form the foundation for interpreting a wide range of optical phenomena in solid-state systems^1–3. Dielectric function captures the energy storage and dissipation mechanisms in the material, while optical conductivity characterizes the current response to varying external fields. Refractive index, extinction coefficient, and absorption coefficient also play important roles in specific applications such as the LEDs, integrated circuits, and solar cells^4–8. Optical responses are often divided into various frequency regions, where the spectra carry fingerprints of distinct types of quasiparticles in materials^9–11.

Materials with low-symmetry crystal structures exhibit pronounced anisotropies, which require tensors with nonzero off-diagonal components and a basis for new strategies for controlling their optical properties. For example, direction-dependent optical responses of van der Waals heterostructures can be tailored for polarization-sensitive photodetectors and developing thermal management schemes, among other applications^12,13. These heterostructures are also promising materials platforms for developing ultrafast lasers and artificial synaptic devices, and other transformational emerging optoelectronic technologies^14–16.

Graph neural networks (GNNs) have recently shown strong capabilities in capturing structure–property relationships and efficiently predicting complex functional properties of materials^17–20. In particular, Euclidean neural networks (E(3)NN)^21–23, which explicitly incorporate spatial symmetry constraints into their architecture, can, in principle, capture the full symmetry of a crystal. This enables high-fidelity predictions with relatively small training datasets (~10³ examples). E(3)NNs have demonstrated success in a variety of applications, including the prediction of interatomic potentials²⁴ and the construction of machine-learning Hamiltonians^25,26.

Equivariant GNNs have also been extended to predict sequential material properties such as phonon density of states, optical conductivities, and absorption spectra^18,27–29. For instance, GNNOpt employs ensemble embeddings of aggregated node features to predict the trace of dielectric functions and derived properties, enabling the discovery of promising solar–cell materials and the estimation of quantum weights in materials³⁰. Similarly, Optimate and its extension were initially trained on 9915 semiconductors and insulators using dielectric functions within the independent-particle approximation (IPA), and subsequently fine-tuned on random-phase approximation (RPA) spectra to enhance their correspondence with experimental observations^31,32. In parallel, several works have proposed equivariant architectures for predicting tensor quantities such as dielectric, piezoelectric, and elastic tensors^33,34. For example, AnisoNet leverages spherical-harmonic decomposition in the ℓ = 0 and ℓ = 2 channels to directly predict the dielectric tensor, facilitating high-anisotropy materials discovery through database screening³⁵, while DTNet adopts a transfer-learning approach that incorporates interatomic potentials as optimized latent embeddings to construct the 3 × 3 static dielectric constant using node features and atomic positions³⁶. Notably, most existing studies focus exclusively on the dielectric constant. In contrast, the joint prediction of tensor structure and spectral behavior–specifically, the full frequency-dependent dielectric tensor–remains largely unexplored. Prior work has investigated tensorial spectra indirectly through equivariant machine-learning Hamiltonians, but the Brillouin-zone integrations involved continue to pose a computational bottleneck. Our approach addresses this gap by leveraging the expressive power of higher-order tensor channels within the equivariant GNNs to directly predict the full, frequency-dependent dielectric tensor from crystal structure, thereby providing a comprehensive framework for modeling anisotropic optical responses in crystalline materials.

Here, we propose the Tensorial Spectra Equivariant Neural Network (TSENN) whose architecture is based on E(3)NN, which is equivariant under translation, rotation, and inversion symmetries. These symmetry-aware capabilities of E(3)NN allow TSENN to model complex directional dependencies of functional properties of materials accurately. Our method utilizes sequential ℓ = 0 components to capture isotropic parts that are associated with quantities such as the total permittivity. The sequential ℓ = 2 components are employed to capture anisotropic parts associated with quantities such as birefringence and direction-dependent refractive index. We demonstrate the capabilities of TSENN by considering the frequency-dependent dielectric tensors of a dataset of 1432 nonmagnetic semiconductors calculated via first-principles methods using OpenMX^37–40. By training TSENN on the aforementioned dataset, we achieve a mean absolute error (MAE) of 0.127 in the predicted imaginary part of the dielectric tensor. The high prediction accuracy enables reconstruction of the real part of the dielectric tensor via the Kramers–Kronig (K–K) relations, allowing us to compute additional optical properties derived from the spectra. Accurately predicting the full tensor while preserving its symmetry opens new opportunities for designing materials with tailored anisotropic responses, with applications in photovoltaics, optical modulators, and sensing technologies.

Results

Dataset

With our focus on small bandgap semiconductors in mind, we selected materials from the Materials Project⁴¹ with bandgaps ranging from 0.3 to 3 eV (photon wavelengths of 4100 nm (infrared) to 413 nm (violet)) that are appropriate for optoelectronic applications^42,43. To maintain structural stability, we imposed an energy-above-hull threshold of 0.02 eV/atom. For computational efficiency, we limited our selection to materials with fewer than three elements and fewer than ten atoms per unit cell. Additionally, we excluded magnetic materials as well as materials containing elements with f electrons due to the lack of availability of accurate, predefined pseudo-potential sets. These constraints yielded a dataset of 1432 materials, whose composition is summarized in Fig. 1. Importantly, we did not impose symmetry constraints, such as filtering for specific space groups. As a result, our dataset includes anisotropic materials, whose tensor properties are not intrinsically diagonal. For each entry, we first symmetrized the structure by following the procedure of ref. ⁴⁴ and performed first-principles calculations using OpenMX. The dielectric tensors were computed using the Kubo formula for photon energies ranging from 0 to 30 eV, with an increment of 0.01 eV, resulting in 1432 frequency-dependent dielectric tensors satisfying the symmetry constraints of the crystal systems considered^45–47. In this work, the complex dielectric tensor refers to the frequency-dependent relative dielectric tensor, expressed as ${\epsilon }^{\alpha \beta }\left(\omega \right)={\epsilon }_1^{\alpha \beta }\left(\omega \right)+i{\epsilon }_2^{\alpha \beta }\left(\omega \right)$, where ${\epsilon }_1^{\alpha \beta }$ and ${\epsilon }_2^{\alpha \beta }$ denote the real and imaginary parts, respectively. The superscripts αβ index the directional dependence of the spectra. For simplicity, we omit the explicit frequency dependence (ω) hereafter.

Fig. 1. Overview of the diversity of materials in our dataset.

a Histogram showing the frequency of materials across different crystal symmetries. b Distribution of bandgaps (eV). c Periodic table highlighting the frequency of elements in the dataset, with the color gradient ranging from blue (low frequency) to yellow (high frequency). The Lanthanide and Actinide series are filtered out and not displayed.

To predict the dielectric tensor using an equivariant neural network, we express it in the spherical-harmonics basis, which naturally encodes rotational symmetry. The dielectric tensor is symmetric under the permutation of Cartesian indices, ε^ij = ε^ji, and therefore admits the decomposition

$${\epsilon }^{ij}=T^{\alpha \beta }{Y}_1^{\alpha }{Y}_1^{\beta }={\epsilon }^{\left(0\right)}\oplus {\epsilon }^{\left(2\right)}={\sum }_{\ell \in \left\{0,2\right\}}{\sum }_{m=-\ell }^{\ell }{\epsilon }_{\left(\ell \right)}^m{Y}_{\ell }^m,$$ 1

which separates the scalar trace component (ℓ = 0) from the traceless symmetric tensor component (ℓ = 2). Details of the decomposition procedure are provided in SI Section 6. This approach transforms the tensors from the Cartesian basis into the spherical-harmonics basis, making it easier to work with tensor properties in systems with rotational and inversion symmetries. In Eq. (1), note that a rank-2 Cartesian tensor can be decomposed into ℓ = 0 and ℓ = 2 channels, which involve 2ℓ + 1 different magnetic quantum numbers m ∈ { − ℓ, …, ℓ} that form a spherical tensor of order ℓ. Since we are dealing with frequency-dependent tensors, we take the Cartesian tensor at each photon energy, perform this change-of-basis operation, and obtain the frequency-dependent spherical-harmonic coefficients as our target, as shown in Fig. 2. The change of basis and the evaluation of the Wigner 3j coefficients were carried out using the e3nn package²¹.

Fig. 2. Dataset visualization.

Left panels illustrate the frequency-dependent dielectric tensors in Cartesian coordinates, with insets detailing the corresponding crystal systems, tensor formulas, and the non-vanishing components after ref. ⁴⁷. Right panels depict the transformed spectra into the spherical-harmonic basis, serving as the ground truth for model training. a Triclinic system with all tensor components present. b Hexagonal system with non-vanishing components satisfying xx = yy ≠ zz, where the $Y_0^0$ component arises from the xx + yy + zz component and the $Y_0^2$ arises from the $-\frac{1}{2}\left(xx+yy\right)+zz$ component.

Figure 2 illustrates the spherical-harmonics decomposition of the dielectric tensor for two representative materials from our dataset, which are shown both in the Cartesian and spherical-harmonics basis. It can be shown through a crystallographic analysis that there are five distinct cases of non-vanishing dielectric tensor components across the seven crystal systems. We highlight two examples. (i) Triclinic compound Br₂PdSe₆, which lacks any symmetry operations, so that all components of the dielectric tensor are non-vanishing. Consequently, the decomposition of its Cartesian dielectric tensor yields nonzero contributions across all irreducible spherical components, including the scalar channel $Y_0^0$ and the five rank-2 spherical-harmonic components $Y_m^2$ for m ∈ { − 2, − 1, 0, 1, 2}. And (ii) the hexagonal compound AsBaLi, in which the higher crystal symmetry imposes constraints on the tensor. Specifically, the sixfold rotational symmetry enforces the presence of diagonal components only with the relationships ε^xx = ε^yy ≠ ε^zz. The xx and yy components (blue and red curves in Fig. 2) are visually indistinguishable within numerical accuracy. All off-diagonal components vanish, consistent with the symmetry constraints summarized in the inset of Fig. 2. As a result, the spherical-harmonics decomposition becomes sparse, with only the $Y_0^0$ and $Y_0^2$ components remaining nonzero. Origin of these spherical channels can be traced back to the specific linear combinations of Cartesian components: the scalar component $Y_0^0$ arises from the trace xx + yy + zz, while the rank-2 spherical-harmonic components correspond to the combination $-\frac{1}{2}\left(xx+yy\right)+zz$⁴⁸.

Equivariant graph neural network architecture

Our TSENN model extends the e3nn and GNNOpt frameworks, both of which have demonstrated strong performance in sequential prediction tasks such as electron and phonon densities of states, as well as the traces of optical spectra^29,30,49. While TSENN is built on the foundational principles of equivariant GNNs as implemented in e3nn, it incorporates key extensions that distinguish it from GNNOpt. In particular, TSENN leverages higher-order spherical-harmonic components to better capture the anisotropy of tensorial properties (Fig. 3). The effectiveness of the output representation, model architecture, and comparison with state-of-the-art methods is further detailed in the Supplementary Information (SI). Accordingly, we construct a GNN that represents each material as a crystal graph, where the atomic and structural information is encoded through node and edge features. Each atomic node i is initially represented by a one-hot encoding of its atomic species $z_{\mathrm{type},i}=\mathrm{onehot}\left(\mathrm{atomic\; number}\right)\in {\mathbb{R}}^{118},$ corresponding to the 118 known chemical elements⁵⁰. These atomic types are embedded via a multilayer perceptron to produce initial node-level features f_i.

Fig. 3. Schematic architecture of TSENN.

The model takes a periodic crystal graph as input and encodes atom types via one-hot embeddings⁵⁰. Edge vectors r_ij are expanded using radial basis functions and spherical-harmonics. Features pass through multiple gated equivariant convolution layers, followed by output convolution and mean pooling. Dropout is incorporated after each convolution block to prevent overfitting. The prediction is split into ℓ = 0 (isotropic) and ℓ = 2 (anisotropic) channels and trained using the composite loss function: MAE(ℓ = 0) + MAE(ℓ = 2).

TSENN enforces equivariance to 3D Euclidean transformations by adopting the formalism of tensor field networks²². Translational equivariance is achieved through the use of relative atomic positions in message passing, while rotational equivariance is enforced by constraining intermediate features to lie within irreducible representations of the rotation group SO(3). These are indexed by rotation order ℓ ∈ {0, 1, 2, …} and parity p ∈ {−1, 1}. Each feature tensor takes the form $V_{cm}^{\left(\ell ,p\right)}$, where c indexes feature channels and m ∈ [−ℓ, ℓ] indexes spherical-harmonic components. Thus, a representation of degree ℓ occupies a tensor space of dimension ${\mathbb{R}}^{N_{\mathrm{channels}}\times \left(2\ell +1\right)}$.

These representations are combined using equivariant tensor product operations  ⊗ to construct the pointwise convolution layers. This operation can be applied iteratively L times, following the Clebsch–Gordan decomposition as implemented in e3nn:

$${\left({\mathbf{U}}^{\left({\ell }_1,p_1\right)}\otimes {\mathbf{V}}^{\left({\ell }_2,p_2\right)}\right)}_{c{m}_0}^{\left({\ell }_0,p_0\right)}={\sum }_{m_1=-{\ell }_1}^{{\ell }_1}{\sum }_{m_2=-{\ell }_2}^{{\ell }_2}{C}_{\left({\ell }_1,m_1\right)\left({\ell }_2,m_2\right)}^{\left({\ell }_0,m_0\right)}{U}_{c{m}_1}^{\left({\ell }_1,p_1\right)}{V}_{c{m}_2}^{\left({\ell }_2,p_2\right)},$$ 2

where ${\ell }_0\in \left[\mid {\ell }_1-{\ell }_2\mid ,{\ell }_1+{\ell }_2\right]$ and p₀ = p₁p₂. For brevity, we omit the indices ℓ, p, c, and m when referring to these tensors hereafter.

After the final pointwise convolution layer, we apply mean pooling over all nodes to obtain a global representation of the material. The network output is then partitioned into two parts: a scalar component corresponding to ℓ = 0 and a tensor component corresponding to ℓ = 2. To train the model, we define a composite loss function $\mathcal{L}$, which separately evaluates the scalar and tensor components of the predicted optical spectra. Let N denote the number of materials and n_ω the number of frequency points. For each material entry i, ${\widehat{y}}_{0e}^{\left(i\right)}\left(\omega \right)$ and ${\widehat{y}}_{2e}^{\left(i\right)}\left(\omega ,m\right)$ denote the predicted ℓ = 0 and ℓ = 2 components, respectively, and $y_{0e}^{\left(i\right)}\left(\omega \right)$ and $y_{2e}^{\left(i\right)}\left(\omega ,m\right)$ denote the corresponding ground-truth values. Here, the hat symbol denotes the predicted tensor, unless explicitly specified otherwise, distinguishing it from the ground-truth tensor obtained via the Kubo formula. Since there are two objectives (${\mathcal{L}}_1\left(\ell =0\right)$ and ${\mathcal{L}}_2\left(\ell =2\right)$) with different magnitudes, we adopt the loss balancing approach for multi-objective training, which allows the network to determine the optimal weighting between objectives dynamically⁵¹. The effectiveness of this loss balancing is discussed in SI Section 2.2. The total loss is defined as:

$$\mathcal{L}=\frac{1}{2{\sigma }_1^2}{\mathcal{L}}_1\left(\ell =0\right)+\frac{1}{2{\sigma }_2^2}{\mathcal{L}}_2\left(\ell =2\right)+\log {\sigma }_1+\log {\sigma }_2$$ 3

where σ₁ and σ₂ are learnable parameters that represent the relative uncertainty of each task, with component-wise definitions:

$${\mathcal{L}}_1\left(\ell =0\right)=\frac{1}{N{n}_{\omega }}{\sum }_{i=1}^N{\sum }_{\omega }\mid {\widehat{y}}_{0e}^{\left(i\right)}\left(\omega \right)-y_{0e}^{\left(i\right)}\left(\omega \right)\mid ,$$ 4

$${\mathcal{L}}_2\left(\ell =2\right)=\frac{1}{N{n}_{\omega }{M}_{\ell =2}}{\sum }_{i=1}^N{\sum }_{\omega }{\sum }_{m=-2}^2\mid {\widehat{y}}_{2e}^{\left(i\right)}\left(\omega ,m\right)-y_{2e}^{\left(i\right)}\left(\omega ,m\right)\mid ,$$ 5

where M_ℓ = 2 = 2 × 2 + 1 = 5 is the number of components for the rank-2 irreducible representation. The model used for prediction was selected based on the best performance according to the validation loss. After obtaining the predicted spherical tensor components from the best model, we transformed them back into the Cartesian tensor representation.

To evaluate the model’s performance, the most direct measure is the MAE of the dielectric tensor,

$$\mathrm{MAE}=\frac{1}{6n_{\omega }}{\sum }_{\omega }{\sum }_{\alpha \le \beta }∣{\widehat{\epsilon }}^{\alpha \beta }\left(\omega \right)-{\epsilon }^{\alpha \beta }\left(\omega \right)∣.$$ 6

This ensures that peak position, width, and sign are faithfully captured in the evaluation, without cancellation across frequencies or components. In addition, we compute the per-component errors MAE^αβ, as well as a Normalized MAE (NMAE^αβ), defined by normalizing MAE^αβ to the characteristic magnitude of each tensor component. This scale–aware metric highlights how the prediction errors compare to the intrinsic strength of the typical dielectric response, thereby providing additional physical interpretability.

To isolate anisotropic behavior, we remove the isotropic contribution from the tensor,

$${\epsilon }_{\mathrm{aniso}}\left(\omega \right)=\epsilon \left(\omega \right)-\frac{1}{3}\mathrm{Tr}\left[\epsilon \left(\omega \right)\right]I.$$ 7

This decomposition yields the component-wise anisotropic contributions ${\epsilon }_{\mathrm{aniso}}^{\alpha \beta }\left(\omega \right)$ and their overall magnitudes ${∥{\epsilon }_{\mathrm{aniso}}\left(\omega \right)∥}_F$. From these, we define per-component errors ${\mathrm{MAE}}_{\mathrm{aniso}}^{\alpha \beta }$, the related component–averaged value MAE^aniso, and the Frobenius-norm errors ${\mathrm{MAE}}_{\mathrm{aniso}}^{\mathrm{norm}}$, to quantify errors in the overall anisotropy strength. For completeness, we also report additional shape-sensitive metrics such as the first-derivative MAE of individual components ${\mathrm{MAE}}^{\prime ,\alpha \beta }$, and the KL divergence for the diagonal components. Detailed definitions of these metrics are provided in the SI Section 1.

Prediction results

Table 1 summarizes error metrics for dielectric tensor predictions across chemically and structurally diverse test systems. The overall tensor MAE is 0.127 (0.097 median), establishing TSENN’s robust baseline accuracy. When decomposed into isotropic and anisotropic contributions, the averaged anisotropic error MAE_aniso is 0.096 (0.074 median), demonstrating reliable capture of directional variations. By contrast, the Frobenius-norm anisotropy error ${\mathrm{MAE}}_{\mathrm{aniso}}^{\mathrm{norm}}$ is larger at 0.141 (0.111 median), reflecting the stricter requirement of reproducing the total anisotropy magnitude rather than individual Cartesian components. We emphasize that this norm-based measure is the most physically meaningful measure, since it is rotationally-invariant and directly corresponds to the ℓ = 2 spherical-harmonics channel. This invariance complements the model’s equivariance, which guarantees the correct transformation of anisotropic components as a rank-2 tensor under rotations.

Table 1.

Summary of full-tensor and per-component error metrics for dielectric tensor predictions (mean values with median in parentheses)

Full-tensor metrics		Per-component metrics
Metric	Value	αβ	MAEαβ	NMAEαβ (%)	${\mathrm{MAE}}^{\prime ,\alpha \beta }$	KL divergence	${\mathrm{MAE}}_{\mathrm{aniso}}^{\alpha \beta }$
MAE	0.127 (0.097)	xx	0.251 (0.176)	2.9 (2.8)	0.099 (0.084)	0.119 (0.047)	0.092 (0.066)
MAEaniso	0.096 (0.074)	yy	0.237 (0.177)	2.8 (2.9)	0.094 (0.082)	0.106 (0.047)	0.083 (0.064)
${\mathrm{MAE}}_{\mathrm{aniso}}^{\mathrm{norm}}$	0.141 (0.111)	zz	0.259 (0.206)	3.0 (3.2)	0.105 (0.092)	0.161 (0.067)	0.125 (0.099)
		xya	0.082 (0.072)	5.4 (8.4)	0.034 (0.032)	–	0.082 (0.072)
		xza	0.091 (0.089)	7.3 (7.4)	0.037 (0.034)	–	0.091 (0.089)
		yza	0.105 (0.098)	7.0 (8.0)	0.039 (0.039)	–	0.105 (0.098)

^aOff-diagonal values are reported only for symmetry-allowed systems. Since the isotropic trace contributes only to diagonal elements, ${\mathrm{MAE}}^{\alpha \beta }={\mathrm{MAE}}_{\mathrm{aniso}}^{\alpha \beta }$ for off-diagonal terms. The KL divergence is not reported for off-diagonal components because these spectra are not positive definite, and the normalization required for KL is ill-defined.

At the component level, the diagonal elements (xx, yy, zz) exhibit higher errors (0.237–0.259, with NMAE ~3%) compared to the off-diagonal terms (xy, xz, yz) (0.082–0.105, with NMAE ~5–7%). This hierarchy is physically natural: the diagonal responses are much stronger in magnitude and receive contributions from both the isotropic (ℓ = 0) and anisotropic (ℓ = 2) channels, whereas the off-diagonal terms arise purely from the ℓ = 2 sector and vanish in many high-symmetry systems. Note that we do not apply per-component normalization or weighting: equivariance permits only blockwise scalings that are constant on each SO(3) irreducible subspace. Any differential weighting inside the ℓ = 2 block (e.g., different weights for m = ±2, ±1, 0 or per–Cartesian rescaling) would break rotational equivariance.

Beyond absolute and relative magnitudes, values of first-derivative MAEs (0.094–0.105 for diagonal and 0.034–0.039 for off-diagonal terms) and KL divergences (0.106-0.161 for diagonal terms) confirm that the model reproduces not only overall intensities but also the line shapes and peak positions. The close agreement between the mean and median values across all metrics further indicates that performance is uniform across diverse systems, without being skewed by outliers. These results demonstrate that TSENN achieves high fidelity in predicting the full dielectric tensor, capturing both the dominant diagonal responses and the more subtle anisotropic features with accuracy and physical consistency.

To visualize the predictions, we stratified the test set based on MAE values and selected representative materials from each quartile, excluding cubic lattices since their isotropic dielectric response is already well understood. This sampling highlights systems with nontrivial anisotropies [Fig. 4a]. With the accurately predicted imaginary parts, we used K–K relations to obtain the real parts of the dielectric tensors. Since K–K relations are model-independent, they provide a rigorous test of the internal consistency of the predicted spectra. In Fig. 4b, we compare the real parts obtained via K–K relations from the predicted imaginary spectra against the reference real parts computed directly using the Kubo formula. An exemplar monoclinic system in the test set is used (Se₄Te₂), where, due to symmetry constraints, only four components of the dielectric tensor ${\widehat{\epsilon }}^{ij}$ are nonzero: ${\widehat{\epsilon }}_2^{xx}$, ${\widehat{\epsilon }}_2^{yy}$, ${\widehat{\epsilon }}_2^{zz}$, and ${\widehat{\epsilon }}_2^{xz}$, while the off-diagonal components ${\widehat{\epsilon }}_2^{xy}$ and ${\widehat{\epsilon }}_2^{yz}$ vanish. As shown in the figure, the reconstructed components exhibit excellent agreement with the Kubo-derived results, demonstrating that both the off-diagonal elements (e.g., ${\widehat{\epsilon }}_1^{xz}$) and the anisotropic diagonal elements (${\widehat{\epsilon }}_1^{xx}$, ${\widehat{\epsilon }}_1^{yy}$, ${\widehat{\epsilon }}_1^{zz}$) are faithfully reproduced. The other off-diagonal components, which are not shown, are found to be negligible in keeping with symmetry constraints. This analysis validates not only the predictive accuracy of the imaginary dielectric spectrum but also the physical plausibility and self-consistency of the full complex tensor. These results demonstrate the model’s potential for downstream applications, including photonic materials discovery, symmetry-aware property screening, and the inverse design of materials with tailored optical anisotropy.

Fig. 4. Prediction and reconstruction accuracy for the full complex dielectric tensor.

a Distribution of full-tensor MAE values across the test set (left) and representative spectra obtained by stratified sampling across quartiles (right). From the cumulative kernel-density-estimator (KDE) plot, we randomly selected three systems in each quartile (Q₁–Q₄), resulting in 12 representative examples that span the full error distribution. In these spectra, the horizontal axis is photon energy and the vertical axis is ε₂(ω). Cubic lattices were excluded, as their isotropic spectra are already well studied. For each sampled material, the target and predicted tensor components are shown side by side, with the corresponding MAE reported in each panel. This stratified sampling strategy ensures that both the high- and low-error cases are represented, providing a balanced and unbiased view of the model’s performance across crystal systems and anisotropy strengths. b Real parts of the dielectric tensor components for the monoclinic system Se₄Te₂. Black curves give the ab initio dielectric spectra based on the IPA using the Kubo formula, while red curves denote spectra reconstructed from the predicted imaginary part via the K–K relation. MAE^αβ is reported in each panel, with values ranging from 0.082 to 0.232, demonstrating strong agreement across all non-vanishing components (ε^xx, ε^yy, ε^zz, ε^xz). Notably, both the off-diagonal element (ε^xz) and the anisotropic diagonal terms are faithfully recovered.

To visualize the model’s performance and its ability to preserve symmetry across different crystal systems, we randomly selected five materials from the test set, each representing a distinct crystal class. These materials are arranged according to increasing symmetry, from triclinic (minimum symmetry) to cubic (maximum symmetry), see Fig. 5. In the Cartesian tensor representation, the predicted dielectric tensor exhibits symmetry-consistent behavior, correctly reproducing the expected nonzero tensor components in agreement with the ground truth.

Fig. 5. Symmetry preservation across different crystal systems.

Comparison of predicted and target dielectric tensors in the Cartesian and spherical-harmonics basis in five different crystal systems, ordered by increasing symmetry. In the triclinic system, all tensor components are non-vanishing. The monoclinic system retains four non-vanishing components: xx, yy, zz, and xz. The orthorhombic system exhibits anisotropy with xx ≠ yy ≠ zz, while the hexagonal system satisfies xx = yy ≠ zz. Finally, the cubic system represents the highest symmetry with xx = yy = zz. Left panel shows the pairwise comparison under the spherical tensor representation, while the right panel shows the comparison under the Cartesian tensor. Inset in the right panel provides the chemical formula, crystal system, and MAE.

To further confirm symmetry preservation, we applied a symmetry-based masking procedure to the predicted tensor at each photon energy (see Methods). While equivariance enforces rotational symmetry at the representation level, the network is not explicitly given point-group information. In practice, small numerical deviations or approximate symmetries in the input structures can lead to residual off-diagonal terms. The masking procedure ensures that the predicted spectra conform to the exact crystal symmetry across the entire spectral range. These distinct symmetries are also reflected in the spherical-harmonics decomposition. Each crystal system preserves different spherical channels that depend on its symmetry. This feature is observable in the spherical tensor plots of Fig. 5, where the colormap indicates the magnitude of each spherical-harmonic coefficients, providing a compact, symmetry-adapted view of the underlying Cartesian dielectric tensor. The prominent red region in the $Y_0^0$ channel corresponds to a strong isotropic contribution, originating from the trace of the tensor, i.e., ${\widehat{\epsilon }}^{xx}+{\widehat{\epsilon }}^{yy}+{\widehat{\epsilon }}^{zz}$. In contrast, strong features in the $Y_0^2$ channel (red or blue in Fig. 5) indicate significant anisotropic contributions from the diagonal components (e.g., $-\frac{1}{2}\left({\widehat{\epsilon }}^{xx}+{\widehat{\epsilon }}^{yy}\right)+{\widehat{\epsilon }}^{zz}$). A detailed breakdown of these contributions can be found in ref. ⁴⁸. For example, in a hexagonal material, such as Br₆Cd₂Cs₂, a visible peak appears in the $Y_0^2$ channel, highlighted in red and blue in Fig. 5, which reflects the anisotropic nature of the diagonal components due to the symmetry condition (${\widehat{\epsilon }}^{xx}={\widehat{\epsilon }}^{yy}\ne {\widehat{\epsilon }}^{zz}$). This illustrates how the spherical-harmonics basis captures not only the symmetry but also the complexity of the directional responses.

As already pointed out, in the spherical tensor representation, the dielectric tensor can be decomposed into ℓ = 0 and ℓ = 2 channels. Since there are no additional irreducible components at this order, the ℓ = 0 channel remains decoupled as it cannot mix with higher-order terms. Improper mixing between the channels could introduce unphysical features in the predicted spectra. However, because both the input crystal structure and the corresponding dielectric tensor are inherently symmetry-consistent, the model preserves the underlying symmetry constraints during training. As a result, it outputs only the physically allowed tensor components, maintaining the correct form of the spherical decomposition. This symmetry-preserving behavior ensures that no spurious mixing occurs between channels and that the reconstructed Cartesian tensor contains only the symmetry-permitted elements. Consequently, the model captures the intrinsic symmetry of materials without violating crystallographic constraints.

Discussion

A major strength of our TSENN model lies in its equivariant architecture, which preserves the symmetries of the input crystal structure. We validate this feature of TSENN through a symmetry-mask-based analysis of the predicted dielectric tensors, particularly in the triclinic systems where all components, including the off-diagonal terms, are expected to be nonzero. As a result, the model effectively captures directional couplings (e.g., x → z, x → y) that are often missed or underestimated by the GNN in the absence of the equivariance architecture. Compared to first-principles methods, TSENN offers an efficient data-driven approach, enabling rapid screening of materials with pronounced anisotropic optical responses. In addition to maintaining a low MAE, the model respects both physical and symmetry constraints. For instance, all predicted spectra decay toward zero as the photon energy approaches 30 eV, a behavior consistent with the asymptotic vanishing of interband transitions at high frequencies. This trend ensures compliance with the f-sum rule, which dictates that the integral over the optical conductivity (or dielectric function) must satisfy a conserved quantity related to the total number of electrons. Such behavior enables the evaluation of derived quantities governed by the f-sum rule, including the generalized quantum weights⁵². For respecting symmetry constraints, the bipartite channel design of our model ensures that the trace (ℓ = 0) as well as the traceless tensor part (ℓ = 2) remains strictly separated, as validated in SI Section 2.3. This separation reflects the irreducible decomposition of a symmetric rank-2 tensor under SO(3), where the scalar ℓ = 0 channel is invariant under rotations and the ℓ = 2 channel transforms as a rank-2 tensor. As a result, the model faithfully captures the input symmetry: for example, in hexagonal systems, rotational invariance within the plane perpendicular to the c-axis enforces ${\widehat{\epsilon }}^{xx}={\widehat{\epsilon }}^{yy}$, while allowing ${\widehat{\epsilon }}^{zz}$ to differ. More generally, the model reproduces symmetry-enforced relations across various crystal classes without imposing external symmetry priors.

To further investigate the model’s treatment of symmetry, we systematically examined the importance of the input structure in satisfying crystal symmetry constraints. Since the model is not explicitly trained on space or point-group information, and it is not symmetry-restricted in its architecture, we conducted a series of controlled tests to assess its behavior (Fig. 6). Specifically, we selected a cubic material absent from the training set and applied uniaxial strain along the c-axis, systematically varying the strain from 0 to 5% in increments of 1%, thereby transforming the structure into a tetragonal system, as illustrated in Fig. 6a. From symmetry considerations, the unstrained cubic structure enforces the constraint ${\widehat{\epsilon }}^{xx}={\widehat{\epsilon }}^{yy}={\widehat{\epsilon }}^{zz}$ on the predicted dielectric tensor ${\widehat{\epsilon }}^{ij}$, while the strained tetragonal structure satisfies ${\widehat{\epsilon }}^{xx}={\widehat{\epsilon }}^{yy}\ne {\widehat{\epsilon }}^{zz}$. At zero strain, the predicted ${\widehat{\epsilon }}^{zz}$ and ${\widehat{\epsilon }}^{xx}$ components are identical, resulting in a vanishing frequency-averaged anisotropy, $\overline{{\widehat{\mathrm{\Delta }}}^{xx-zz}\left(\omega \right)}$, between them, which reflects the preservation of cubic symmetry. As the strain increases, an increasing difference between ${\widehat{\epsilon }}^{zz}$ and ${\widehat{\epsilon }}^{xx}$ emerges, consistent with the onset of anisotropy expected for a tetragonal system. By computing the frequency-averaged anisotropy components across all strained structures, we observe a highly linear dependence on the strain magnitude, with a R² = 0.999. We also performed additional ab initio calculations to verify that the predicted trends are physically consistent (see SI, Section 5). These results demonstrate that the model faithfully captures the progressive symmetry breaking induced by strain while preserving the underlying physical constraints.

Fig. 6. Symmetry breaking under strain: tensile and shear validation.

Validation of the model’s symmetry-preserving behavior under tensile and shear strain. a Uniaxial strain along the c-axis is applied to a cubic material, which is not included in the training set, varying from 0 to 5% in 1% increments, thereby transforming the structure into a tetragonal system. At zero strain, cubic symmetry imposes ${\widehat{\epsilon }}^{xx}={\widehat{\epsilon }}^{yy}={\widehat{\epsilon }}^{zz}$, resulting in a vanishing mean anisotropy that grows between ${\widehat{\epsilon }}^{zz}$ and ${\widehat{\epsilon }}^{xx}$. As strain increases, the mean anisotropy grows linearly with strain, yielding R² = 0.999. b Shear strain applied to an orthorhombic material by varying the lattice angle β from 90° to 100°. In the undistorted case (β = 90°), the off-diagonal dielectric tensor element ${\widehat{\epsilon }}^{xz}\left(\omega \right)$ vanishes. As β deviates from 90°, a finite ${\widehat{\epsilon }}^{xz}\left(\omega \right)$ emerges, and its mean magnitude $\mid {\widehat{\overline{\epsilon }}}^{xz}\mid$ (averaged over the photon energy grid) increases with the shear angle. In the small-strain regime (< 10°), the dependence is nearly linear (R² = 0.995), while a sinusoidal fit provides a better fit (R² = 1.000), consistent with the expected angle dependence. These results confirm that the model captures the onset of symmetry breaking with high fidelity and reproduces the correct physical response across both perturbative and large-angle regimes.

We also evaluated the model’s response to shear strain perturbations by modifying the lattice angles, as shown in Fig. 6b. Starting from an orthorhombic material (a ≠ b ≠ c, α = β = γ = 90°), we introduced a monoclinic distortion by incrementally varying β from 90° to 100° in 2° steps. The essential distinction between the orthorhombic and monoclinic systems is that in the latter, β deviates from 90°, allowing the emergence of a nonzero ${\widehat{\epsilon }}^{xz}\left(\omega \right)$ component in the dielectric tensor. When no perturbation is applied (β = 90°), ${\widehat{\epsilon }}^{xz}\left(\omega \right)$ vanishes, yielding a zero mean magnitude $\mid {\widehat{\overline{\epsilon }}}^{xz}\mid$. As the angular deviation increases, a finite ${\widehat{\epsilon }}^{xz}\left(\omega \right)$ progressively emerges, producing a nonzero $\mid {\widehat{\overline{\epsilon }}}^{xz}\mid$. We quantified this emergent behavior as a function of the shear angle and observed an almost perfect linear relationship in the small-strain regime (R² = 0.995). As expected, the emergence of the off-diagonal component is directly coupled to the applied β-angle distortion; consequently, a sine fit provides an even better description, with R² = 1.0. At larger distortions, the dependence smoothly evolves toward a sinusoidal form, reflecting higher-order effects of the angular perturbation. Ab initio calculations confirm this crossover behavior: linear scaling in the perturbative regime and sinusoidal dependence at larger angles (see SI Section 5). The model thus reproduces both regimes with high fidelity, demonstrating that it not only respects the symmetry of the input structure but also quantitatively matches the physical response to both small and large shear distortions.

While the primary focus of this work is on predicting the imaginary part of the dielectric tensor, we also demonstrate that the real part can be accurately reconstructed by using K–K relations. This yields a complete complex dielectric tensor, which in turn enables the derivation of a broad range of optical response functions containing rich information, such as the direct extraction of optical bandgaps (see SI Section 5). This predictive capability is valuable for designing materials for photovoltaics, sensors, and nonlinear optical applications. Triclinic crystals, for example, with both ${\widehat{\epsilon }}^{xz}$ and ${\widehat{\epsilon }}^{xy}$ components, offer multidirectional optical control, which is desirable for heterostructures and chiral photonic devices.

An important challenge revealed by our analysis is that predictive errors nearly double in the low-frequency regime (ω < ω_p), where the dielectric response is dominated by rich, material-specific absorption features with strong oscillator strengths. Here, ω_p denotes the plasma frequency, which marks the crossover between collective electronic oscillations and the asymptotic high-frequency regime. The low frequency regime is both physically meaningful—since it encodes the distinct fingerprints of each compound—and difficult to capture accurately. One might consider introducing an energy-resolved weighting scheme during training to place more emphasis on this region; however, such an approach would require prior knowledge of ω_p for each material and would introduce material-dependent biases at inference time, thereby compromising generalizability. To prevent this, we chose to retain an energy-independent training loss and instead provide a detailed frequency-resolved benchmarking in SI Section 4, which compares the error behavior above and below the plasma frequency.

It is appropriate to acknowledge the intrinsic limitations of the underlying ab initio data, which is based on the IPA rather than the RPA, so that the training of our model neglects local-field effects. Although the RPA provides some improvement by narrowing the discrepancy between theory and experiment, the computationally far more demanding Bethe–Salpeter approach⁵³ is needed to capture excitonic effects properly. The Kohn–Sham DFT-based eigenvalues systematically underestimate bandgaps that can be corrected through more expensive GW calculations. Despite these limitations, TSENN, trained on a relatively modest dataset of approximately 10³ materials, achieves strong data efficiency in learning high-rank tensorial responses. Recent work shows that machine-learning models trained on IPA spectra can be fine-tuned or transferred to RPA-level data to incorporate local-field effects³². In addition to these physical approximations, the dataset itself is dominated by materials with weak optical anisotropy, leading to intrinsically small off-diagonal components of the dielectric tensor and an imbalance in the distribution of tensor magnitudes. To more accurately assess model performance on genuinely anisotropic systems, we curated an auxiliary dataset composed exclusively of materials with finite off-diagonal responses. This benchmark not only targets applications that depend on off-diagonal tensor elements, but also reveals that our model attains substantially improved accuracy on these components (see SI Section 8). Building on this enhanced capability to predict off-diagonal optical responses, we further demonstrate the potential applicability of our framework by considering the magneto-optical Kerr effect and magnetic circular dichroism; see SI Section 9.

In summary, TSENN advances the prediction of complex frequency-dependent tensorial optical properties by accurately resolving full-tensor components across various symmetry classes, providing a practical tool for accelerating materials discovery for next-generation optoelectronic technologies. Our approach is inherently general as it proceeds by decomposing Cartesian tensors into spherical-harmonics components, and it would thus be naturally extendable to treat higher-rank tensors such as the piezoelectric and elastic tensors through the inclusion of higher-order ℓ channels. Furthermore, it holds promise for modeling sequential higher-order properties such as the shift current⁵⁴ and second harmonic generation (see SI Section 10), making it a versatile framework for learning symmetry-constrained anisotropic physical phenomena.

Methods

Ab initio calculations

DFT calculations were performed using OpenMX^37–40, which employs numerical pseudo-atomic orbitals as the basis set to expand the one-particle Kohn–Sham wave functions. The high-throughput calculations involved three main steps: generating input files, submitting jobs, and monitoring their progress. The input files were generated using the ASE package, which provides a predefined lookup table for optimized basis selection. However, this table has not been updated since 2013, leading to missing entries for some elements with heavy f-orbital character. We used the PBE-GGA functional in all calculations. We use a custom workflow for job submission and monitoring, implemented with Atomate⁵⁵. Self-consistent field (SCF) convergence was ensured to an accuracy of 3.67 × 10⁻⁸ Hartree or 10⁻⁶ eV. All other settings used the default OpenMX parameters. The SCF k-point grid was generated using the pymatgen automatic density function with a target density of 2000 k-points per atom (kppa). This method selects Γ-centered meshes for hexagonal and face-centered lattices, and Monkhorst–Pack grids for all other crystal systems, ensuring consistent and resolution-adaptive Brillouin-zone sampling.

After achieving SCF convergence with OpenMX, we extracted the Hamiltonian, overlap matrices, and position operators to evaluate the tensorial relative permittivity. The dielectric tensor was computed within the IPA using the Kubo formula:

$${\epsilon }_{\mathrm{r}}^{\alpha \beta }\left(\omega \right)=\frac{{\epsilon }^{\alpha \beta }\left(\omega \right)}{{\epsilon }_0}={\chi }^{\alpha \beta }\left(\omega \right)+{\delta }^{\alpha \beta },{\chi }^{\alpha \beta }\left(\omega \right)=\frac{e^2}{{\epsilon }_0\hslash }\int \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}{\sum }_{n,m}\frac{f_{nm}{r}_{nm}^{\alpha }{r}_{mn}^{\beta }}{{\omega }_{mn}-\omega -i\eta }.$$ 8

In this work, we denote the real and imaginary parts of the ${\epsilon }_{\mathrm{r}}^{\alpha \beta }$ as ${\epsilon }_1^{\alpha \beta }$ and ${\epsilon }_2^{\alpha \beta }$, respectively. Here ℏω_nm = E_n−E_m denotes the interband energy difference, f_nm = f(E_n)−f(E_m) is the difference of Fermi–Dirac occupations, and $r_{nm}^{\alpha }$ is the Berry connection, defined as $r_{nm}^{\alpha }=i⟨u_{n\mathbf{k}}\mid {\partial }_{\alpha }{u}_{m\mathbf{k}}⟩,$ with ∂_α = ∂/∂k_α the derivative with respect to the Cartesian momentum coordinate α ∈ {x, y, z}, where the Hamiltonian and position operator are projected onto a localized basis and the Brillouin-zone integrals are evaluated efficiently^56–60. A uniform k-point mesh was constructed with reciprocal-space resolution of 0.05 Å⁻¹ along each reciprocal lattice vector, ensuring consistent Brillouin-zone sampling across all materials. The smearing parameter η, introduced to improve numerical convergence of the optical response, was fixed at 0.05 eV in all calculations. This value corresponds to typical carrier relaxation times (~10 fs) in semiconductors^61–63. This procedure is equivalent to constructing a Wannier-based tight-binding representation of the Hamiltonian, which provides a localized framework for efficient evaluation of the dielectric response. The present calculations are performed within the IPA, i.e., based on Kohn–Sham eigenvalues and wave functions without including quasiparticle (GW) corrections or electron–hole interaction (excitonic) effects⁶⁴. Consequently, discrepancies between the computed dielectric spectra and the corresponding experimental results are expected.

Machine learning

The spectra were computed with a resolution of 0.01 eV, with a total of 3001 points. Including all points in the training process would significantly increase computational cost. To make training more efficient, we performed downsampling by interpolating the data along the photon energy grid to 300 points, which still captures the spectral response reasonably. We used an 80:10:10 split for training, validation, and testing datasets. Given our focus on crystal symmetries, we implemented balanced loading by analyzing the distribution of crystal systems across all splits to ensure even representation, see SI Section 7 for details.

Group theoretical analysis reveals five distinct cases based on crystalline symmetries. This enables us to construct a mask M that identifies the symmetry-allowed, non-vanishing components of the tensorial spectra. The mask is derived by initializing a 3 × 3 matrix of ones, 1_3 × 3, and applying the set of rotation operations $\mathcal{O}=\left\{R_1,R_2,\dots ,R_{N_R}\right\}$ associated with the space group symmetry of the crystal. The final mask is computed by averaging over the transformed matrices:

$$\mathbf{M}=\frac{1}{N_R}{\sum }_{R_i\in \mathcal{O}}{R}_i^{\top }{\mathbf{1}}_{3\times 3}{R}_i.$$ 9

For example, in a monoclinic system, this process yields the following mask:

$$\mathbf{M}=\left(\begin{array}{ccc}\hfill 1& \hfill 0\hfill & 1\hfill \\ \hfill 0& \hfill 1\hfill & 0\hfill \\ \hfill 1& \hfill 0\hfill & 1\hfill \end{array}\right)$$ 10

This mask highlights the non-vanishing tensor components, such as xx, yy, zz, and xz, as dictated by the symmetry constraints. To identify any symmetry violations, we define the residual mask as

$$\mathrm{Residual\; mask}=ϵ\left({\mathbf{1}}_{3\times 3}-\mathbf{M}\right)=\left(\begin{array}{ccc}\hfill 0& \hfill ϵ\hfill & 0\hfill \\ \hfill ϵ& \hfill 0\hfill & ϵ\hfill \\ \hfill 0& \hfill ϵ\hfill & 0\hfill \end{array}\right),$$ 11

where ϵ = 10⁻⁶ serves as the detection threshold. We apply this residual mask across the photon energy grid to verify the input tensors before training and the predicted tensors after training. Our results confirm that all tensors respect the imposed symmetry constraints, demonstrating that the model maintains equivariance throughout training without symmetry breaking.

We performed a comprehensive hyperparameter search using Optuna⁶⁵ to optimize the performance of the model. The multiplicity of irreducible representations m used in the convolution layers was varied from 16 to 64, with 32 selected as the optimal value. The number of pointwise convolution layers L was tuned between 1 and 4, with the final model using 4. The embedding feature vector length was explored from 16 to 160, with 128 yielding the best performance. Dropout probability was tested at 0, 0.1, 0.2, and 0.4 with 0.4 selected as the optimal choice. Batch sizes from 1 to 16 were systematically evaluated, and a size of 8 was selected as the best tradeoff between computational efficiency and training stability. Smaller batches (e.g., size 1) led to unstable optimization due to insufficient statistical averaging per update. For very large batches (>32), the anisotropy was found to become less pronounced compared to smaller ones, likely because the optimization tends to get trapped in local minima and averages out sharper anisotropic signals through stochastic smoothing. The hyperparameters of the AdamW optimizer were carefully tuned. Learning rates between 5 × 10⁻⁴ and 5 × 10⁻² were explored, with 1 × 10⁻² achieving the lowest validation loss. To further enhance convergence, we applied PyTorch’s CosineAnnealingWarmRestarts scheduler, which gradually reduces the learning rate according to a cosine decay schedule. The initial number of iterations before the first restart was set to T₀ = 10, with subsequent settings kept at their default values. This smooth, non-monotonic decay enables larger updates during the early training stages and finer adjustments at later stages, helping the model escape shallow minima and converge more robustly⁶⁶.

To contrast the computational efficiency, we benchmarked our workflow against a representative ab initio calculation. For this purpose, we recalculated the dielectric tensor of Ag₃AsS₄ (Materials Project ID: mp-9538, 16 atoms per unit cell). The DFT self-consistent calculation converged in 119 s using a hybrid MPI/OpenMP setup on 2 AMD EPYC 7763 CPUs (256 threads in total) with default workflow settings. The subsequent dielectric tensor evaluation, carried out with Brillouin-zone integration parallelized over 128 processes, required 22 min 9 s. Including workflow orchestration overhead, the full pipeline–from job launch to data storage–took 26 min 32 s. By contrast, the GNN-based workflow was orders of magnitude faster: graph construction required 6.96 ± 0.01 ms, forward inference 695.20 ± 0.01 ms, and K–K relations took 570.04 ± 0.01 ms, giving a total runtime of 1.18 s on a single NVIDIA RTX 4090 GPU. In other words, our approach reduces a half-hour DFT calculation to just over 1 s.

Author contributions

T.-W.H., Z.F., and Q.Y. conceived the study. T.-W.H. performed the calculations and trained the models. Z.F. assisted with data analysis. A.B. and Q.Y. supervised the project. The manuscript was written with contributions from all authors, and all authors approved the final version.

Peer review

Peer review information

Nature Communications thanks Can Ataca, Erich Runge, Rui Zu, and the other anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

All datasets and optimized model weights used in this study are openly available at https://github.com/qmatyanlab/TSENN. The archived release is available on Zenodo⁶⁷. The repository also includes reproducible Jupyter notebooks demonstrating data usage and reproducing the figures presented in this work.

Code availability

The third-party code OPENMX used in this study is publicly available at https://www.openmx-square.org. The custom workflows developed to compute the dielectric functions are accessible at https://github.com/qmatyanlab/OpenMX-workflowand were implemented using the JULIA programming language (https://julialang.org).

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-69159-9.