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E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials

Simon BatznerAlbert MusaelianLixin SunMario GeigerJonathan P. MailoaMordechai KornbluthNicola MolinariTess E. SmidtBoris Kozinsky

article2021Nature Communications2,362 citations

Introduces NequIP, an E(3)-equivariant graph neural network that learns highly accurate interatomic potentials for molecular dynamics while requiring up to three orders of magnitude less training data than existing models.

Computational Research Division and Center for Advanced Mathematics for Energy Research ApplicationsÉcole Polytechnique Fédérale de LausanneHarvard UniversityLawrence Berkeley National LaboratoryMassachusetts Institute of TechnologyRobert Bosch Research and Technology Centergraph neural networkstraining sampleE(3)-equivariant graph neural networksinteratomic potentialsNequIP
arXiv: 2101.03164mir-group/nequip
  • Paper: E(n) Equivariant Graph Neural Networks, Victor Garcia Satorras et al. (2021). It introduces an alternative, highly efficient E(n)-equivariant graph network architecture that achieves coordinate equivariance via scalar distance updates rather than higher-order spherical tensors.
  • Paper: Do Transformers Really Perform Bad for Graph Representation?, Chengxuan Ying et al. (2021). It advances 3D and graph representation learning by adapting the Transformer architecture with spatial and structural encodings for quantum chemistry tasks.
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Abstract

This work presents Neural Equivariant Interatomic Potentials (NequIP), an E(3)-equivariant neural network approach for learning interatomic potentials from ab-initio calculations for molecular dynamics simulations. While most contemporary symmetry-aware models use invariant convolutions and only act on scalars, NequIP employs E(3)-equivariant convolutions for interactions of geometric tensors, resulting in a more information-rich and faithful representation of atomic environments. The method achieves state-of-the-art accuracy on a challenging and diverse set of molecules and materials while exhibiting remarkable data efficiency. NequIP outperforms existing models with up to three orders of magnitude fewer training data, challenging the widely held belief that deep neural networks require massive training sets. The high data efficiency of the method allows for the construction of accurate potentials using high-order quantum chemical level of theory as reference and enables high-fidelity molecular dynamics simulations over long time scales.

Topics

graph neural networkstraining sample

Keywords

E(3)-equivariant graph neural networksinteratomic potentialsNequIPClebsch-Gordan coefficientsatom-wise self-interaction layersirreducible representationsquantum chemical accuracyatomic cluster expansiondata efficiency

Table of Contents

  • I Introduction
  • I.0.1 Related Work
  • II Results
  • II.0.1 Equivariance
  • II.0.2 Neural Equivariant Interatomic Potentials
  • II.1 Experiments
  • II.1.1 MD-17 small molecule dynamics
  • II.1.2 Force training at quantum chemical accuracy
  • II.1.3 Liquid Water and Ice Dynamics
  • II.1.4 Heterogeneous catalysis of formate dehydrogenation
  • II.1.5 Lithium Phosphate Amorphous Glass Formation
  • II.1.6 Lithium Thiophosphate Superionic Transport
  • II.2 Data Efficiency
  • III Discussion
  • IV Methods
  • V Data Availability
  • References
  • VI Acknowledgements
  • VII Author contributions
  • VIII Competing interests
  • IX Appendix A: Long Molecular Dynamics Simulation of Li4P2O7
  • X Appendix B: Learning Curves
  • XI Appendix C: Revised MD-17 data set

Knowls

  1. Knowl 1 — NequIP Neural Network Architecture for Interatomic Potentials

    model/method

    Neural Equivariant Interatomic Potential (NequIP) is an E(3)\mathrm{E}(3)-equivariant graph neural network designed to predict potential energy and atomic forces from 3D atomic coordinates and chemical species. Atomic configurations are represented as graphs where nodes represent individual atoms and directed edges connect all pairs of atoms within a local radial cutoff distance rcr_c.

    The architecture consists of three principal stages:

    1. Atomic Embedding: For each atom ii, its atomic number ZiZ_i (represented as a one-hot vector) is mapped to an initial rotationally invariant scalar feature (l=0l=0, even parity) via a trainable atom-wise linear self-interaction layer.
    2. Interaction Blocks: A series of equivariant interaction layers update node feature representations through message passing. Features are geometric tensors organized as direct sums of irreducible representations of O(3)\mathrm{O}(3) labeled by rotation order l \in \{0, 1, \dots, l_\max\} and parity p∈{1,−1}p \in \{1, -1\}. Each interaction block applies:
      • An E(3)\mathrm{E}(3)-equivariant convolution over neighbor nodes combining radial functions and spherical harmonics via Clebsch-Gordan tensor products.
      • Linear atom-wise self-interaction layers that mix features sharing identical (l,p)(l, p) quantum numbers.
      • A ResNet-style residual update: xk+1=f(xk)+Self-Interaction⁡(xk)x^{k+1} = f(x^k) + \operatorname{Self-Interaction}(x^k), where ff denotes the sequence of self-interaction, convolution, concatenation, and self-interaction, and the residual self-interaction weights are learned per chemical species.
      • Equivariant gated nonlinearities using the Sigmoid Linear Unit (SiLU\mathrm{SiLU}) activation function for even scalars, tanh⁡\tanh for odd scalars, and scalar-gated tensor channels for l>0l > 0.
    3. Output Block: The scalar (l=0l=0, even parity) features from the final interaction block are passed through two successive atom-wise linear self-interaction layers to output a single scalar atomic potential energy Ei,atomicE_{i,\text{atomic}} for each atom ii.

  2. Knowl 2 — E(3)-Equivariant Tensor Convolution and Clebsch-Gordan Coupling

    equation

    In the NequIP architecture, atomic geometric feature tensors Vacmili,piV_{a c m_i}^{l_i, p_i} at central atom aa (with channel index cc, rotation order lil_i, parity pip_i, and representation index mi∈[−li,li]m_i \in [-l_i, l_i]) interact with neighbor atoms b∈Sb \in \mathcal{S} through an E(3)\mathrm{E}(3)-equivariant convolutional layer L\mathcal{L}.

    The convolution mapping input representations (li,pi)(l_i, p_i) with convolutional filters of order (lf,pf)(l_f, p_f) to output representations (lo,po)(l_o, p_o) is defined as:

    Lo,po;lf,pf;li,piacmo(r⃗a,Vbcmili,pi)=∑mf,miC(li,mi),(lf,mf)(lo,mo)1Nˉ∑b∈SR(rab)c,lo,po,lf,pf,li,piYmflf(r^ab)Vbcmili,piL_{o, p_o; l_f, p_f; l_i, p_i}^{a c m_o}(\vec{r}_a, V^{l_i, p_i}_{b c m_i}) = \sum_{m_f, m_i} C_{(l_i, m_i), (l_f, m_f)}^{(l_o, m_o)} \frac{1}{\sqrt{\bar{N}}} \sum_{b \in \mathcal{S}} R(r_{ab})_{c, l_o, p_o, l_f, p_f, l_i, p_i} Y_{m_f}^{l_f}(\hat{r}_{ab}) V_{b c m_i}^{l_i, p_i}

    where:

    • r⃗a\vec{r}_a and r⃗b\vec{r}_b are the 3D position vectors of atoms aa and bb, r⃗ab=r⃗b−r⃗a\vec{r}_{ab} = \vec{r}_b - \vec{r}_a, rab=∥r⃗ab∥r_{ab} = \|\vec{r}_{ab}\|, and r^ab=r⃗ab/rab\hat{r}_{ab} = \vec{r}_{ab} / r_{ab}.
    • Ymflf(r^ab)Y_{m_f}^{l_f}(\hat{r}_{ab}) is the spherical harmonic of degree lfl_f and order mfm_f.
    • R(rab)c,lo,po,lf,pf,li,piR(r_{ab})_{c, l_o, p_o, l_f, p_f, l_i, p_i} is a learnable, rotationally invariant radial weight function computed by a multi-layer perceptron.
    • C(li,mi),(lf,mf)(lo,mo)C_{(l_i, m_i), (l_f, m_f)}^{(l_o, m_o)} are the Clebsch-Gordan coefficients coupling irreducible representations lil_i and lfl_f into output order lol_o, satisfying ∣li−lf∣≤lo≤∣li+lf∣|l_i - l_f| \le l_o \le |l_i + l_f|.
    • Nˉ\bar{N} is the average number of neighbors per atom across the dataset, used as a normalization factor.
    • Tensor outputs with identical (lo,po)(l_o, p_o) across valid paths are concatenated, subject to the parity selection rule po=pipfp_o = p_i p_f and the maximum rotation order constraint l_o \le l_\max.

  3. Knowl 3 — Conservative Atomic Force Prediction and Energy Pooling

    model/method

    To strictly satisfy the conservation of energy and rotation equivariance of forces in molecular dynamics, NequIP predicts total potential energy EpotE_\text{pot} as a sum over atomic energies and derives interatomic forces F⃗i\vec{F}_i analytically via negative spatial gradients:

    Epot=∑i=1NatomsEi,atomicE_\text{pot} = \sum_{i=1}^{N_\text{atoms}} E_{i,\text{atomic}}

    F⃗i=−∇r⃗iEpot\vec{F}_i = -\nabla_{\vec{r}_i} E_\text{pot}

    where NatomsN_\text{atoms} is the total number of atoms in the system, r⃗i∈R3\vec{r}_i \in \mathbb{R}^3 is the position vector of atom ii, and Ei,atomicE_{i,\text{atomic}} is the scalar atomic energy predicted by the neural network for atom ii.

    Because EpotE_\text{pot} is invariant under global E(3)\mathrm{E}(3) transformations (translations, rotations, and reflections), the analytically computed force vectors F⃗i\vec{F}_i are guaranteed to be rotationally and translationally equivariant and curl-free (∇×F⃗=0\nabla \times \vec{F} = 0), preventing unphysical energy drift during long-time molecular dynamics simulations.

  4. Knowl 4 — Radial Basis Embedding and Envelope Function

    equation

    In NequIP, interatomic distances rij=∥r⃗j−r⃗i∥r_{ij} = \|\vec{r}_j - \vec{r}_i\| within a radial cutoff rcr_c are embedded into a basis of dimension NbN_b using radial Bessel functions modulated by a smooth polynomial cutoff envelope:

    B(rij)=2rcsin⁡(bπrcrij)rijfenv(rij;rc)B(r_{ij}) = \frac{2}{r_c} \frac{\sin\left(\frac{b\pi}{r_c} r_{ij}\right)}{r_{ij}} f_\text{env}(r_{ij}; r_c)

    where:

    • b=[1,2,…,Nb]b = [1, 2, \dots, N_b] is a vector of basis indices, with the frequencies bπb\pi initialized to these integer multiples and subsequently optimized via backpropagation during training.
    • fenv(rij;rc)f_\text{env}(r_{ij}; r_c) is a polynomial cutoff envelope defined with p=6p=6 on normalized distances u=rij/rcu = r_{ij} / r_c: fenv(rij;rc)=1−(p+1)(p+2)2up+p(p+2)up+1−p(p+1)2up+2for u≤1,0 for u>1f_\text{env}(r_{ij}; r_c) = 1 - \frac{(p+1)(p+2)}{2} u^p + p(p+2) u^{p+1} - \frac{p(p+1)}{2} u^{p+2} \quad \text{for } u \le 1, \quad 0 \text{ for } u > 1
    • The radial embedding B(rij)∈RNbB(r_{ij}) \in \mathbb{R}^{N_b} serves as the input to a multi-layer perceptron (MLP) producing radial filter weights: R(rij)=Wnσ(…σ(W2σ(W1B(rij))))R(r_{ij}) = W_n \sigma(\dots \sigma(W_2 \sigma(W_1 B(r_{ij})))) where WkW_k are weight matrices and σ\sigma denotes the SiLU activation function.

  5. Knowl 5 — Loss Function Formulation with Size-Invariant Weighting and Per-Species Normalization

    equation

    NequIP models are trained using a joint weighted mean squared error loss on total potential energies and atomic force components:

    L=λE∥E^−E∥2+λF3N∑i=1N∑α=13(−∂E^∂ri,α−Fi,α)2\mathcal{L} = \lambda_E \|\hat{E} - E\|^2 + \frac{\lambda_F}{3N} \sum_{i=1}^N \sum_{\alpha=1}^3 \left( -\frac{\partial \hat{E}}{\partial r_{i,\alpha}} - F_{i,\alpha} \right)^2

    where NN is the number of atoms in the structure, E^\hat{E} is the predicted total potential energy, EE is the reference potential energy, Fi,αF_{i,\alpha} is the α\alpha-th Cartesian component of the reference force on atom ii, and ri,αr_{i,\alpha} is the corresponding coordinate.

    To ensure size-invariance across systems with differing numbers of atoms, the force loss is scaled by 1/(3N)1/(3N), and the relative weighting between force and energy loss terms is set by default to λF/λE=N2\lambda_F / \lambda_E = N^2, reflecting the relationship between global scalar energies and 3N3N local vector force components.

    Predicted total energies are obtained by applying trainable per-species scale (σsi\sigma_{s_i}) and shift (λsi\lambda_{s_i}) parameters to raw atomic scalar outputs E^i\hat{E}_i:

    E^=∑i=1N(σsiE^i+λsi)\hat{E} = \sum_{i=1}^N (\sigma_{s_i} \hat{E}_i + \lambda_{s_i})

    where sis_i indicates the chemical element of atom ii, initialized with σsi=1\sigma_{s_i} = 1 and λsi=0\lambda_{s_i} = 0 (or the average per-atom reference energy for mixed systems). Reference targets are pre-normalized by subtracting the training-set mean energy and dividing energies and forces by the root-mean-square force component over the training set.

  6. Knowl 6 — Power-Law Learning Dynamics and Sample Efficiency of Equivariant Convolutions

    theoretical result

    Learning curves for machine-learned interatomic potentials typically follow a power law of generalization error ϵ\epsilon with respect to training set size MM:

    ϵ∝aMb\epsilon \propto a M^b

    where b<0b < 0 is the learning rate exponent (the slope in log-log space) and aa is an offset constant. In standard invariant interatomic potentials (such as SchNet, sGDML, FCHL19, PhysNet, and invariant l=0l=0 NequIP networks), changing the model architecture only shifts the offset parameter aa without changing the empirical log-log slope bb.

    In contrast, incorporating higher-order tensor features and E(3)\mathrm{E}(3)-equivariant convolutions (l≥1l \ge 1) into NequIP fundamentally alters the log-log slope bb, increasing its absolute magnitude. This enables the model to learn at a systematically faster rate as training data increases. Setting l>1l > 1 (e.g., l=2l=2 or l=3l=3) further shifts the learning curve downward (improving accuracy at all sample sizes) while maintaining the steeper equivariant slope. As a consequence, NequIP achieves accuracy equivalent to or better than invariant neural networks and kernel methods using 10 to 1000 times fewer training configurations.

  7. Knowl 7 — Predictive Accuracy on the Revised MD-17 Dataset

    data/table

    NequIP was evaluated on the revised MD-17 (rMD-17) dataset of small organic molecules at the PBE/def2-SVP level of theory. Models were trained on 950 configurations and validated on 50 configurations, with errors evaluated on the remaining trajectory configurations (up to ~99,000 structures per molecule).

    Molecule Target FCHL19 GAP ANI GemNet-(T/Q) NequIP (l=0l=0) NequIP (l=3l=3)
    Aspirin Energy [meV] 6.2 17.7 16.6 – 25.2 2.3
    Forces [meV/Ä] 20.9 44.9 40.6 9.5 42.2 8.2
    Azobenzene Energy [meV] 2.8 8.5 15.9 – 20.3 0.7
    Forces [meV/Ä] 10.8 24.5 35.4 – 34.4 2.9
    Benzene Energy [meV] 0.3 0.75 3.3 – 3.2 0.04
    Forces [meV/Ä] 2.6 6.0 10.0 0.5 10.3 0.3
    Ethanol Energy [meV] 0.9 3.5 2.5 – 2.0 0.4
    Forces [meV/Ä] 6.2 18.1 13.4 3.6 11.9 2.8
    Malonaldehyde Energy [meV] 1.5 4.8 4.6 – 4.4 0.8
    Forces [meV/Ä] 10.2 26.4 24.5 6.6 23.2 5.1
    Naphthalene Energy [meV] 1.2 3.8 11.3 – 14.7 0.2
    Forces [meV/Ä] 6.5 16.5 29.2 1.9 20.6 1.3
    Paracetamol Energy [meV] 2.9 8.5 11.5 – 17.5 1.4
    Forces [meV/Ä] 12.2 28.9 30.4 – 33.6 5.9
    Salicylic acid Energy [meV] 1.8 5.6 9.2 – 11.4 0.7
    Forces [meV/Ä] 9.5 24.7 29.7 5.3 29.8 4.0
    Toluene Energy [meV] 1.6 4.0 7.7 – 9.7 0.3
    Forces [meV/Ä] 8.8 17.8 24.3 2.2 26.6 1.6
    Uracil Energy [meV] 0.6 3.0 5.1 – 10.0 0.4
    Forces [meV/Ä] 4.2 17.6 21.4 3.8 26.0 3.1

    The data demonstrate that NequIP with tensor rank l=3l=3 outperforms invariant neural networks, kernel methods (FCHL19, GAP), and other graph architectures across all molecules on both energy and force Mean Absolute Error (MAE). Moving from scalar invariant convolutions (l=0l=0) to equivariant tensor convolutions (l=3l=3) decreases force errors by a factor of 3 to 10 and energy errors by more than an order of magnitude.

  8. Knowl 8 — Water and Ice Thermodynamic Phase Generalization with Low Data Regimes

    data/table

    To evaluate data efficiency on extended condensed-phase systems, NequIP was trained on a combined dataset of bulk liquid water (P=1 bar,T=300 KP=1\text{ bar}, T=300\text{ K}) and three ice Ih phases under classical and path-integral ab-initio molecular dynamics at the PBE0-TS level of theory. While the baseline DeepMD model was trained on 133,500 structures, NequIP was trained on only 133 uniformly sampled structures (1000-fold reduction in data).

    System Target NequIP (a) NequIP (b) NequIP (c) DeepMD
    (λF=1,λE=0\lambda_F=1, \lambda_E=0) (λF=100,λE=1\lambda_F=100, \lambda_E=1) (λF=105,λE=1\lambda_F=10^5, \lambda_E=1) (133.5k frames)
    Liquid Water Energy [meV/molecule] – 1.6 1.7 1.0
    Forces [meV/Ä] 11.9 49.4 11.6 40.4
    Ice Ih (b) Energy [meV/molecule] – 2.5 4.3 0.7
    Forces [meV/Ä] 10.2 55.8 9.9 43.3
    Ice Ih (c) Energy [meV/molecule] – 3.9 10.2 0.7
    Forces [meV/Ä] 12.0 27.7 11.7 26.8
    Ice Ih (d) Energy [meV/molecule] – 2.6 12.7 0.8
    Forces [meV/Ä] 9.8 23.2 9.5 25.4

    With three orders of magnitude less training data, NequIP (model c, λF=100,000,λE=1\lambda_F=100{,}000, \lambda_E=1) reduces root-mean-square force errors across all liquid and solid phases by a factor of 2.5 to 4 compared to DeepMD (e.g., 11.6 meV/\AA{} vs 40.4 meV/\AA{} on liquid water), while achieving comparable energy RMSE on the liquid phase (1.7 meV/molecule vs 1.0 meV/molecule).

  9. Knowl 9 — Modeling Structural and Kinetic Properties in Lithium Solid Electrolytes and Glasses

    empirical result

    NequIP was evaluated on structural and kinetic property predictions in complex lithium-based materials:

    1. Superionic Conductor Li6.75P3S11\mathrm{Li}_{6.75}\mathrm{P}_3\mathrm{S}_{11} (LiPS):

      • A NequIP potential trained on 2500 AIMD frames (T=520 KT=520\text{ K}, 83-atom unit cell) achieved an energy MAE of 0.08 meV/atom0.08\text{ meV/atom} and force MAE of 4.7 meV/A˚4.7\text{ meV/\AA}.
      • In 50 ps NVT molecular dynamics simulations, the NequIP potential predicted a Li-ion diffusion coefficient of 1.25×10−5 cm2/s1.25 \times 10^{-5}\text{ cm}^2/\text{s}, matching the reference AIMD diffusion coefficient of 1.37×10−5 cm2/s1.37 \times 10^{-5}\text{ cm}^2/\text{s} within a 9% relative error.
      • Models trained on as few as 10 and 100 structures achieved force MAEs of 97.8 meV/A˚97.8\text{ meV/\AA} and 25.8 meV/A˚25.8\text{ meV/\AA}, respectively.

    2. Amorphous Glass Formation in Li4P2O7\mathrm{Li}_4\mathrm{P}_2\mathrm{O}_7:

      • Trained exclusively on 1000 configurations from a high-temperature molten AIMD trajectory at T=3000 KT=3000\text{ K}, NequIP achieved a force MAE of 34.0 meV/A˚34.0\text{ meV/\AA} on the molten test set and 21.3 meV/A˚21.3\text{ meV/\AA} when transferred to a T=600 KT=600\text{ K} quenched glass trajectory.
      • NequIP-driven MD simulations of the quenched state at 600 K600\text{ K} faithfully reproduced the full radial distribution function (RDF) as well as the P−O−O\mathrm{P}-\mathrm{O}-\mathrm{O} tetrahedral bond angle and O−P−P\mathrm{O}-\mathrm{P}-\mathrm{P} bridging angle angular distribution functions (ADFs) obtained from AIMD.

  10. Knowl 10 — Interatomic Potential for Heterogeneous Catalytic Formate Dehydrogenation on Cu(110)

    empirical result

    NequIP was applied to model the reaction dynamics of catalytic formate decomposition (HCOO∗→H∗+CO2\mathrm{HCOO}^* \to \mathrm{H}^* + \mathrm{CO}_2) on a Cu⟨110⟩\mathrm{Cu}\langle 110 \rangle surface, a heterogeneous system containing mixed covalent/metallic bonding and charge transfer across 52 atoms (48 Cu atoms and 4 adsorbate atoms).

    Using a per-element weighted loss function where Cu, C, H, and total O atoms are weighted equally with total force weight λF=Natoms2=2704\lambda_F = N_\text{atoms}^2 = 2704, a NequIP model (l=2l=2, 32 features, rc=5.0 A˚r_c = 5.0\text{ \AA}) trained on 2500 DFT configurations sampled along nudged elastic band (NEB) reaction pathways and short AIMD trajectories achieved:

    • Force MAE of 19.9 meV/A˚19.9\text{ meV/\AA} on carbon (C)
    • Force MAE of 71.3 meV/A˚71.3\text{ meV/\AA} on oxygen (O)
    • Force MAE of 13.0 meV/A˚13.0\text{ meV/\AA} on hydrogen (H)
    • Force MAE of 47.6 meV/A˚47.6\text{ meV/\AA} on copper (Cu)
    • Average per-species force MAE of 38.4 meV/A˚38.4\text{ meV/\AA}
    • Overall energy MAE of 0.50 meV/atom0.50\text{ meV/atom} on out-of-sample test configurations.

Coverage note — Deliberately omitted the original unrevised MD-17 numerical table and the specific CCSD/CCSD(T) molecule table (which appears in the Supplementary Information), focusing on the primary revised MD-17 benchmark, core architectural equations, and physical validation benchmarks.

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Citation

MLA
Batzner, S., et al. “E(3)-equivariant Graph Neural Networks for Data-efficient and Accurate Interatomic Potentials”. Nature Communications, vol. 13, no. 1, 2022, https://doi.org/10.1038/s41467-022-29939-5.
APA
Batzner, S., Musaelian, A., Sun, L., Geiger, M., Mailoa, J. P., Kornbluth, M., Molinari, N., Smidt, T. E., & Kozinsky, B. (2022). E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials. Nature Communications, 13(1). https://doi.org/10.1038/s41467-022-29939-5
Chicago
Batzner, S., A. Musaelian, L. Sun, et al. 2022. “E(3)-equivariant Graph Neural Networks for Data-efficient and Accurate Interatomic Potentials”. Nature Communications 13 (1). https://doi.org/10.1038/s41467-022-29939-5.
Harvard
Batzner, S. et al. (2022) “E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials”, Nature Communications, 13(1). Available at: https://doi.org/10.1038/s41467-022-29939-5.
Vancouver
1. Batzner S, Musaelian A, Sun L, Geiger M, Mailoa JP, Kornbluth M, Molinari N, Smidt TE, Kozinsky B (2022) E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials. Nature Communications. https://doi.org/10.1038/s41467-022-29939-5

BibTeX

@article{Batzner_2022, title={E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials}, volume={13}, ISSN={2041-1723}, url={http://dx.doi.org/10.1038/s41467-022-29939-5}, DOI={10.1038/s41467-022-29939-5}, number={1}, journal={Nature Communications}, publisher={Springer Science and Business Media LLC}, author={Batzner, Simon and Musaelian, Albert and Sun, Lixin and Geiger, Mario and Mailoa, Jonathan P. and Kornbluth, Mordechai and Molinari, Nicola and Smidt, Tess E. and Kozinsky, Boris}, year={2022}, month=May }
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