Dielectric tensor of perovskite oxides at finite temperature using equivariant graph neural network potentials

Alex Kutana; Koki Yoshimochi; Ryoji Asahi

REVIEW 5 major objections 5 minor 8 references

Dielectric tensor of perovskite oxides at finite temperature using equivariant graph neural network potentials

T0 review · 5 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A neural network trained on 178 snapshots captures CaTiO3's dielectric curve and phase change.

desk verdict A useful small-data demonstration of GNN potentials for finite-T dielectric/phase behavior, but the headline results rest on unvalidated temperature extrapolation and an ionic-only dielectric model. read the letter →

arxiv 2412.03541 v1 pith:D4NXJJNW submitted 2024-12-04 cond-mat.mtrl-sci physics.comp-ph

Alex Kutana , Koki Yoshimochi , Ryoji Asahi This is my paper · ORCID

classification cond-mat.mtrl-sciphysics.comp-ph

keywords equivariantgraphneuralnetworksmachine-learnedforcefieldsBorneffectivechargesdielectrictensorfinite-temperaturemoleculardynamicscalciumtitanateperovskiteoxidesstructuralphasetransitions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The authors aim to show that finite-temperature molecular dynamics with machine-learned potentials can capture two hard-to-compute properties of a real perovskite oxide: the temperature dependence of the dielectric tensor and the structural phase transition. They train equivariant graph neural networks on only about 178 density-functional-theory snapshots of CaTiO3, with one network supplying zero-field energies and forces and another supplying Born effective charges. Applying a small electric field and reading the polarization from the field-induced ionic displacements through the Born charges, they reproduce the experimentally observed decreasing, bell-shaped dielectric response of CaTiO3 with temperature and an orthorhombic-to-cubic transition near 1280 K on heating. Because such finite-temperature response properties are normally too expensive for direct ab initio simulation, a working small-data recipe would make routine studies of dielectric and phase behavior in perovskite oxides practical.

What carries the argument

The load-bearing mechanism is a combined machine-learned force field assembled from Eq. 1 and Eq. 2: zero-field forces from MACE plus an electric-field term $|e|\sum_\beta \mathcal{E}_\beta Z^*_{\kappa,\beta\alpha}$ from Born effective charges predicted by Equivar. Born effective charges are atomic tensors that measure how the force on an atom responds to an applied electric field (equivalently, how polarization changes per atomic displacement), and in perovskite titanates they are anomalously large because Ti-O hybridization shifts charge along bonds. The polarization is then read out as $P_\alpha = (e/\Omega)\sum_{\kappa,\beta} Z^*_{\kappa,\alpha\beta} u_{\kappa,\beta}$, with the static dielectric tensor obtained from the slope of polarization versus field in finite-temperature MD. Equivariant graph neural networks are the enabler because their outputs transform correctly under rotations and translations, so a model trained on roughly 100-500 perturbed snapshots can carry both forces and tensorial Born charges with close-to-ab initio accuracy.

What would settle it

At a representative temperature (for example, 1000 K), recompute the dielectric tensor from the same MD trajectories but with Born charges evaluated by density-functional perturbation theory for the sampled snapshots instead of by the equivariant model; if the decreasing-with-temperature trend disappears or reverses, the neural-network charge model, rather than the dynamics, is what reproduces the experimental curve.

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Extended reading notes

Core claim

The central claim is that a specialized equivariant graph neural network potential trained on a small, system-specific dataset can replace direct ab initio molecular dynamics for finite-temperature response properties. For CaTiO3, the authors combine a MACE force field for zero-field forces and energies with an Equivar model for Born effective charges, and compute the static dielectric tensor from the field-induced displacements via $P_\alpha = (e/\Omega)\sum_{\kappa,\beta} Z^*_{\kappa,\alpha\beta} u_{\kappa,\beta}$ under an applied field of $5 \times 10^7$ V/m. The resulting tensor decreases with temperature and follows the shape of the experimental curve, although the absolute values are smaller than experiment because of the density-functional training data and the absence of a low-temperature zero-point plateau in classical dynamics. In the same simulations, heating switches CaTiO3 from the orthorhombic to the cubic phase at about 1280 K, qualitatively reproducing the experimental phase transition, albeit without the intermediate tetragonal phase and at a lower temperature. The authors take this as evidence that small-data equivariant potentials can capture the finite-temperature physics of realistic perovskite oxides.

Load-bearing premise

The calculation assumes that the applied electric field acts on the crystal only through the Born effective charges of individual ions, with no separate electronic (clamped-ion) contribution to the polarization, and that this purely ionic, linear, local response can be compared directly with the experimental total dielectric tensor.

Editorial extensions

If this is right

Specialized equivariant GNNs trained on about 178 CaTiO3 snapshots predict forces, energies, and Born charges with errors more than an order of magnitude smaller than a generalist model trained on around 10^6 structures.
The dielectric tensor of CaTiO3 computed from finite-temperature MD decreases with increasing temperature and follows the shape of the experimental curve, so such simulations can be used to screen perovskite dielectrics without expensive ab initio MD.
Heating dynamics of CaTiO3 show an orthorhombic-to-cubic phase transition at roughly 1280 K, demonstrating that structural phase stability can be probed with small-data machine-learned potentials.
Because the classical MD lacks zero-point motion and the training functional suppresses the absolute dielectric value, the computed curve is a qualitative rather than quantitative match; improving the underlying DFT functional or adding quantum nuclear effects should be the next correction.
The same training and field-coupling recipe is presented as applicable to other perovskite oxides and to materials whose dielectric response is dominated by anomalous Born charges.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Not stated in the paper: the missing clamped-ion electronic contribution could be estimated by adding a Born-charge-independent electronic dielectric constant from DFPT to the MD ionic values, and the remaining gap would test whether Eqs. 1-2 are complete.
Not stated in the paper: retraining on PBE rather than PBEsol+U and re-running the heating protocol would directly test whether the 1280 K transition temperature and the missing tetragonal phase are functional errors or model errors.
Not stated in the paper: substituting strain for the electric field in the same coupling scheme could yield finite-temperature piezoelectric or flexoelectric tensors with no new architecture.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a practical and mostly honest paper, but the central claims are softer than the abstract makes them sound. The genuinely new thing is the workflow: train a specialized MACE force field and an Equivar Born-charge predictor on a few hundred MD snapshots, then run finite-field MD to get a temperature-dependent dielectric tensor and a structural phase transition in CaTiO3. The small-data learning results are convincing, especially the comparison with the foundation model MACE-MP-0, and the authors are upfront about zero-point motion and DFT limitations. That part deserves credit.

The soft spots are real and they matter. First, the models are trained on 300–500 K ab initio MD snapshots, then run up to 1800 K for the phase transition and to roughly 1000 K for the dielectric tensor. There is no high-temperature DFT validation at all, so the orthorhombic-to-cubic switch at ~1280 K and the decreasing dielectric trend could be artifacts of the learned potential. The paper's caveat that the transition error might come from the density functional does not address this extrapolation gap. Second, the dielectric computation uses Eqs. 1–2, which couple the field only to Born charges and read out polarization from ionic displacements. That gives the ionic contribution, not the total dielectric response, and Fig. 5's caption says 'ionic part' while the text compares to experimental total values. That is a mismatch, not a minor wording issue. Third, there are no error bars on the MD-averaged dielectric tensor, so the claimed agreement with the experimental curve is hard to assess statistically. The abstract also overstates the phase-transition result: the simulation shows a single orthorhombic-to-cubic transition, while experiments show two transitions; the paper admits this, but the plural 'phase transitions' in the abstract is misleading.

None of this kills the paper. The qualitative trends might well be right, and the workflow is worth publishing so others can build on it. But the evidence as presented does not establish that the high-temperature behavior is predictive rather than coincidental. The authors should provide at least a few DFT checks at 800–1500 K, add error bars, and clarify exactly which dielectric quantity is being compared to experiment.

This deserves a serious referee, not a desk reject. I would send it out, with the expectation that the extrapolation issue and the dielectric definition get addressed before publication.

Referee Report

5 major / 5 minor

Summary. This manuscript reports machine-learned interatomic potentials based on the MACE and Equivar equivariant graph neural network architectures, trained on small DFT datasets (178-502 ab initio MD snapshots) for CaTiO3, BaTiO3, and CaZrO3, including energies, forces, stresses, and Born effective charge tensors. The potentials are used in finite-temperature molecular dynamics simulations with an external electric field coupled to Born charges (Eqs. 1-2), and the resulting polarization is used to extract a temperature-dependent dielectric tensor. In addition, NpT heating runs are used to study structural phase transitions in CaTiO3. The central claim is qualitative agreement with experiment for the temperature dependence of the dielectric tensor and for the orthorhombic-to-cubic structural transition.

Significance. If the claims hold, the paper would demonstrate that a specialized equivariant GNN trained on roughly 178 snapshots can capture subtle temperature-dependent dielectric and phase behavior in a perovskite oxide, and that the specialized model outperforms a general-purpose foundation model on this system. The paper includes useful cross-validation and direct comparisons with DFPT Born charges, which are strengths. The practical relevance for finite-temperature property prediction is clear. However, several gaps in the dielectric model, the lack of error estimates, and the unvalidated extrapolation to temperatures far above the training window currently prevent the paper from fully supporting its central qualitative claim.

major comments (5)

[Methods, Eqs. (1)-(2); Fig. 5] The dielectric tensor shown in Fig. 5 is the ionic (lattice) contribution only, as the caption states, but the text refers to it as the 'dielectric tensor' and compares it with experimental total values. The zero-temperature DFPT points are conventionally total static dielectric constants; if so, the comparison mixes the ionic-only result with the total experimental/DFPT values. The omission of the electronic (clamped-ion) contribution is not merely a density-functional limitation but a model restriction. Please add the electronic contribution (e.g., from a DFPT epsilon_infinity) and compare consistently, or explicitly and consistently restrict all claims and comparisons to the ionic contribution.
[Fig. 5 and Methods] No statistical uncertainties are shown in Fig. 5, and the Methods do not state how many field strengths or independent MD runs were used for each dielectric component. If the tensor is obtained from the slope of P(E), multiple field points and an error estimate are required; if it is obtained from a single finite field, the linear-response assumption should be justified. Without this information, the claim that the simulations reproduce the decreasing trend and shape of the experimental curve is not quantitatively supported.
[Fig. 4 and dataset description] The training datasets are ab initio MD snapshots in the 300-500 K range, while the production runs reach 1000-1800 K. The phase-transition simulation switches from orthorhombic to cubic at approximately 1280 K and does not visit the tetragonal phase, and the dielectric simulations also extend beyond the training temperatures. No validation against DFT energies or forces for high-temperature structures is provided. As a result, the agreement with experiment could be an artifact of the learned potential's extrapolation rather than a property of CaTiO3. Please add direct validation (e.g., single-point DFT on snapshots at several temperatures above 500 K) or otherwise demonstrate that the relevant tilt instabilities and thermal disorder are correctly described at high temperature.
[Methods, Eqs. (1)-(2)] In Eqs. (1)-(2), the Born charges Z* are predicted by Equivar as functions of the local environment and therefore change during the MD simulation. The force in Eq. (1) is not the negative gradient of the energy in Eq. (2) unless Z* is independent of the displacements, because the derivative of Z* with respect to the displacement contributes an additional term. The manuscript does not state whether Z* is held fixed during the field-coupled MD or whether this derivative term is neglected. Since the field-induced forces are central to the dielectric response, please clarify the implementation and, if the term is neglected, estimate its magnitude.
[Fig. 4] The phase-transition claim in Fig. 4 is difficult to evaluate because no order parameter or structural metric is defined. The text says the system 'switched from the orthorhombic to cubic phase and back' before adopting the cubic structure, but without a precise definition of phase identity (e.g., octahedral tilt angles or space-group assignment) and without discussion of hysteresis or heating-rate effects, the reported transition temperature of about 1280 K is not well characterized. Please provide the order-parameter curves and address finite-size and heating-rate dependence.

minor comments (5)

[Conclusions] The final sentence of the Conclusions says 'molecular dynamics simulations at finite temperatures and infinite electric fields are performed'; this should read 'finite electric fields.'
[Methods] The sentence 'Hubbard U correction of 3 eV was applied to d electrons in transition metals, except for Ti, where U=0 was used' is confusing because Ti is a transition metal; please specify that the U=3 eV applies to Zr (or otherwise clarify the treatment for each element).
[Fig. 1 caption] The caption refers to 'Dynamic charges'; for consistency with the text, please use 'Born effective charges' or 'dynamical charges' explicitly.
[Fig. 3] The statement that 'MP-0 FF underestimates the force constants' needs a definition of how force constants are extracted from the P(E) curves, and the temperature at which the P(E) data are computed should be stated.
[Fig. 5] The text says the simulations reproduce the shape of the experimental curve, but the experimental data are not shown in Fig. 5; please add the experimental curve or specify the source and values used for the comparison.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; dielectric tensor and phase transition are emergent MD outputs from DFT-trained GNN models, not fitted inputs.

full rationale

The paper's central claims—the temperature-dependent dielectric tensor and the orthorhombic-to-cubic phase transition in CaTiO3—are not training targets and are not obtained by inverting the training data. The MACE force field and Equivar Born-charge model are trained on DFT energies, forces, stresses, and Z* tensors from 300–500 K AIMD snapshots. The dielectric response is then computed by finite-field MD: the external field couples to the predicted Born charges (Eqs. 1–2), and the polarization is read from the resulting ionic displacements. Neither the dielectric constant nor the transition temperature appears as a fitted parameter, and no equation identifies the output with an input by construction. The phase transition emerges from heating MD, and the dielectric trend emerges from thermal sampling; both are compared with independent experimental data. Self-citations (refs. 11, 19, 20) supply the Equivar architecture and background on anomalous Born charges, but they are not the evidence for the finite-temperature results; Fig. 1 and the cross-validation in Fig. 2 benchmark the models against DFT within the paper. The main limitations—extrapolation beyond the 300–500 K training window and the ionic-only field-coupling model (Eqs. 1–2)—are correctness risks, not circularity.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central result rests on ML potentials fitted to DFT data, a linear Born-charge model for the dielectric response, and classical MD assumptions. There are no invented physical entities. The load-bearing fitted or chosen parameters are the GNN weights and hyperparameters, the Hubbard U correction, the MD field magnitude, the heating rate, and the training set composition.

free parameters (5)

Hubbard U correction for transition metal d electrons = 3 eV for transition metals, 0 eV for Ti
Chosen by hand; sets the DFT training labels and therefore all downstream predictions.
External electric field magnitude in finite-field MD = 5 x 10^7 V/m
Chosen for the MD; the dielectric constant is read from the slope of P vs E, so nonlinearity at this field can bias the slope.
Heating rate in NpT phase transition runs = 1 K/ps
Chosen heating schedule; no convergence check on transition temperature with respect to rate.
Neural network weights and hyperparameters of MACE and Equivar = Not reported in the text
All predictions depend on thousands of fitted weights; architecture details, cutoffs, number of layers and batch sizes are not given in the text.
Training dataset composition and sizes (A, B, C, D) = A: 178, B: 502, C: 191, D: combined
The choice of which DFT snapshots are included and the model regularization (e.g., D larger error) affect accuracy; this is a design choice.

assumptions (6)

domain assumption The response to an applied electric field is fully described by Eq. 1: F = F(E=0) + |e| Z* E, with polarization P = (e/Omega) sum Z* u.
This linear Born-charge coupling omits electronic polarization and nonlinear field effects; the entire dielectric readout relies on it, so if it is incomplete the computed tensor is not physical.
domain assumption Nuclei evolve classically in the MD simulations; quantum zero-point motion is excluded.
The paper states that the low-temperature plateau from zero-point motion is absent; this limits quantitative accuracy at low T but not the qualitative trend.
domain assumption The PBEsol+U DFT reference is an adequate ground truth for training the potentials and for reproducing experimental dielectric and phase behavior.
All labels for energies, forces, stresses and Born charges come from this functional; DFT errors are explicitly invoked to explain the gap with experiment.
ad hoc to paper A local message-passing GNN potential trained on 300-500 K orthorhombic snapshots can extrapolate to the cubic phase at 1300-1800 K.
The phase transition prediction depends on this extrapolation, which is outside the training set and is only indirectly validated by the observed transition.
domain assumption Periodic NpT supercells of 640 and 1280 atoms with a uniform external field yield converged macroscopic dielectric averages.
No finite-size convergence or field-strength convergence study is reported.
ad hoc to paper The training set of 178-502 snapshots is representative of the configurations sampled during the finite-temperature MD.
Training snapshots come from 300-500 K MD; high-temperature cubic configurations are absent, so representativeness is assumed.

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Cite this review

Pith. "Pith review of Dielectric tensor of perovskite oxides at finite temperature using equivariant graph neural network potentials." pith.science (2026). https://pith.science/paper/D4NXJJNW

@misc{pith2026241203541,
  author       = {Pith},
  title        = {Pith review of: Dielectric tensor of perovskite oxides at finite temperature using equivariant graph neural network potentials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D4NXJJNW}},
  note         = {Machine review of arXiv:2412.03541}
}

read the original abstract

Atomistic simulations of properties of materials at finite temperatures are computationally demanding and require models that are more efficient than the ab initio approaches. Machine learning (ML) and artificial intelligence (AI) address this issue by enabling accurate models with close to ab initio accuracy. Here, we demonstrate the utility of ML models in capturing properties of realistic materials by performing finite temperature molecular dynamics simulations of perovskite oxides using a force field based on equivariant graph neural networks. The models demonstrate efficient learning from a small training dataset of energies, forces, stresses, and tensors of Born effective charges. We qualitatively capture the temperature dependence of the dielectric tensor and structural phase transitions in calcium titanate.

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Reference graph

Works this paper leans on

8 extracted references · 7 canonical work pages

[2]

Gilmer,J.,Schoenholz,S.S.,Riley,P.F.,Vinyals,O.&Dahl,G.E.NeuralMessagePassing forQuantumChemistry.inProceedingsof the34thInternational ConferenceonMachine Learning(eds.Precup,D.&Teh,Y.W.)vol.701263–1272(PMLR,2017)

work page 2017
[3]

Q.)vol.482990–2999(PMLR,NewYork,NewYork,USA,2016)

Cohen,T.&Welling,M.GroupEquivariantConvolutionalNetworks.inProceedingsof The 33rdInternational ConferenceonMachineLearning(eds.Balcan,M.F.&Weinberger,K. Q.)vol.482990–2999(PMLR,NewYork,NewYork,USA,2016)

work page 2016
[4]

et al.Tensorfieldnetworks:Rotation-andtranslation-equivariantneural networksfor3Dpointclouds.Preprintathttps://arxiv.org/abs/1802.08219(2018)

Thomas,N. et al.Tensorfieldnetworks:Rotation-andtranslation-equivariantneural networksfor3Dpointclouds.Preprintathttps://arxiv.org/abs/1802.08219(2018)

arXiv 2018
[5]

et al.E(3)-equivariantgraphneuralnetworksfordata-efficientandaccurate interatomicpotentials

Batzner,S. et al.E(3)-equivariantgraphneuralnetworksfordata-efficientandaccurate interatomicpotentials. Nat. Commun. 13,2453(2022)

work page 2022
[6]

Batatia,I.,Kovacs,D.P.,Simm,G.,Ortner,C.&Csanyi,G.MACE:HigherOrder EquivariantMessagePassingNeuralNetworksforFastandAccurateForceFields.in AdvancesinNeural InformationProcessingSystems(eds.Koyejo,S.etal.)vol.35 11423–11436(CurranAssociates,Inc.,2022)

work page 2022
[7]

Schütt,K.,Unke,O.&Gastegger,M.Equivariantmessagepassingforthepredictionof tensorialpropertiesandmolecularspectra.inProceedingsof the38thInternational ConferenceonMachineLearning(eds.Meila,M.&Zhang,T.)vol.1399377–9388(PMLR, 2021)

work page 2021
[8]

et al.VirtualNodeGraphNeuralNetworkforFullPhononPrediction.Preprintat https://arxiv.org/abs/2301.02197(2023)

Okabe,R. et al.VirtualNodeGraphNeuralNetworkforFullPhononPrediction.Preprintat https://arxiv.org/abs/2301.02197(2023)

work page arXiv 2023
[9]

Representing Born effective charges with equivariant graph convolutional neural networks

Cheng,Y. et al.Directpredictionofinelasticneutronscatteringspectrafromthecrystal structure*. Mach. Learn. Sci. Technol. 4,015010(2023). 10.Vijay,S. et al.CoeffNet:predictingactivationbarriersthroughachemically-interpretable, equivariantandphysicallyconstrainedgraphneuralnetwork. ChemSci 15,2923–2936 (2024). 11.Kutana,A.,Shimizu,K.,Watanabe,S.&Asahi,R.Repr...

work page Pith review arXiv doi:10.48550/arxiv.2409.08940 2023

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [2]

Gilmer,J.,Schoenholz,S.S.,Riley,P.F.,Vinyals,O.&Dahl,G.E.NeuralMessagePassing forQuantumChemistry.inProceedingsof the34thInternational ConferenceonMachine Learning(eds.Precup,D.&Teh,Y.W.)vol.701263–1272(PMLR,2017)

work page 2017

[2] [3]

Q.)vol.482990–2999(PMLR,NewYork,NewYork,USA,2016)

Cohen,T.&Welling,M.GroupEquivariantConvolutionalNetworks.inProceedingsof The 33rdInternational ConferenceonMachineLearning(eds.Balcan,M.F.&Weinberger,K. Q.)vol.482990–2999(PMLR,NewYork,NewYork,USA,2016)

work page 2016

[3] [4]

et al.Tensorfieldnetworks:Rotation-andtranslation-equivariantneural networksfor3Dpointclouds.Preprintathttps://arxiv.org/abs/1802.08219(2018)

Thomas,N. et al.Tensorfieldnetworks:Rotation-andtranslation-equivariantneural networksfor3Dpointclouds.Preprintathttps://arxiv.org/abs/1802.08219(2018)

arXiv 2018

[4] [5]

et al.E(3)-equivariantgraphneuralnetworksfordata-efficientandaccurate interatomicpotentials

Batzner,S. et al.E(3)-equivariantgraphneuralnetworksfordata-efficientandaccurate interatomicpotentials. Nat. Commun. 13,2453(2022)

work page 2022

[5] [6]

Batatia,I.,Kovacs,D.P.,Simm,G.,Ortner,C.&Csanyi,G.MACE:HigherOrder EquivariantMessagePassingNeuralNetworksforFastandAccurateForceFields.in AdvancesinNeural InformationProcessingSystems(eds.Koyejo,S.etal.)vol.35 11423–11436(CurranAssociates,Inc.,2022)

work page 2022

[6] [7]

Schütt,K.,Unke,O.&Gastegger,M.Equivariantmessagepassingforthepredictionof tensorialpropertiesandmolecularspectra.inProceedingsof the38thInternational ConferenceonMachineLearning(eds.Meila,M.&Zhang,T.)vol.1399377–9388(PMLR, 2021)

work page 2021

[7] [8]

et al.VirtualNodeGraphNeuralNetworkforFullPhononPrediction.Preprintat https://arxiv.org/abs/2301.02197(2023)

Okabe,R. et al.VirtualNodeGraphNeuralNetworkforFullPhononPrediction.Preprintat https://arxiv.org/abs/2301.02197(2023)

work page arXiv 2023

[8] [9]

Representing Born effective charges with equivariant graph convolutional neural networks

Cheng,Y. et al.Directpredictionofinelasticneutronscatteringspectrafromthecrystal structure*. Mach. Learn. Sci. Technol. 4,015010(2023). 10.Vijay,S. et al.CoeffNet:predictingactivationbarriersthroughachemically-interpretable, equivariantandphysicallyconstrainedgraphneuralnetwork. ChemSci 15,2923–2936 (2024). 11.Kutana,A.,Shimizu,K.,Watanabe,S.&Asahi,R.Repr...

work page Pith review arXiv doi:10.48550/arxiv.2409.08940 2023