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Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction

Xiang Fu, Brandon M. Wood, Luis Barroso-Luque, Daniel S. Levine, Meng Gao, Misko Dzamba, C. Lawrence Zitnick

arXiv:2502.12147v2physics.comp-phcs.LG

TL;DR

Held-out test errors do not always translate into accurate downstream physical-property predictions for MLIPs. The paper evaluates practical energy conservation in molecular dynamics, identifies design choices affecting smoothness and conservation, and develops eSEN, which achieves state-of-the-art results across several property-prediction tasks.

  • Problem

    Static test-set accuracy does not always improve downstream physical-property prediction, creating a gap between conventional MLIP evaluation and practical applications.

  • Method

    The paper tests practical energy conservation in molecular dynamics and analyzes MLIP design choices affecting conservative forces and smoothly varying, bounded energy derivatives.

  • Results

    eSEN achieves state-of-the-art performance across materials stability, phonon, thermal-conductivity, geometry-optimization, and force/energy prediction tasks.

  • Takeaways & Limitations

    For models that pass energy-conservation tests, test errors may serve as a proxy for physical-property performance during development.

  • Takeaways & Limitations

    Direct comparison for one phonon result is avoided because the compared models use different levels of theory.

Abstract

from arXiv · show

Machine learning interatomic potentials (MLIPs) have become increasingly effective at approximating quantum mechanical calculations at a fraction of the computational cost. However, lower errors on held out test sets do not always translate to improved results on downstream physical property prediction tasks. In this paper, we propose testing MLIPs on their practical ability to conserve energy during molecular dynamic simulations. If passed, improved correlations are found between test errors and their performance on physical property prediction tasks. We identify choices which may lead to models failing this test, and use these observations to improve upon highly-expressive models. The resulting model, eSEN, provides state-of-the-art results on a range of physical property prediction tasks, including materials stability prediction, thermal conductivity prediction, and phonon calculations.

1 Introduction

MLIPs can achieve strong static test-set accuracy without reliably improving downstream physical-property predictions. The paper proposes energy conservation in molecular dynamics as a practical test and introduces eSEN, which achieves leading results across several benchmarks.

  • Static energy and force errors do not always predict performance on complex physical-property tasks.The paper highlights this gap because downstream workflows may require thousands to millions of force evaluations.
  • The paper tests whether MLIPs practically conserve energy during molecular-dynamics simulations and finds stronger test-error/property correlations among models that pass.The proposed criterion reflects conservative forces and continuous, bounded energy derivatives.
  • eSEN achieves a leading F1 score of 0.831 and κSRME of 0.340 on the compliant Matbench-Discovery benchmark.Previous models are reported to excel in only one of these metrics.
  • eSEN reaches an F1 score of 0.925 and κSRME of 0.170 on the non-compliant Matbench-Discovery category.
  • eSEN achieves state-of-the-art results on the MDR Phonon benchmark and the highest test accuracy on SPICE-MACE-OFF.

2 Preliminaries

MLIPs approximate DFT-derived energies, forces, and stresses for workflows that include relaxation, molecular dynamics, phonon calculations, and thermal-conductivity prediction. Standard held-out errors alone may not track performance on these downstream tasks.

  • MLIPs are commonly evaluated on held-out DFT calculations using MAE or RMSE for energies, forces, and stresses.
  • Geometry optimization/relaxation: Geometry relaxation iteratively updates atomic positions using predicted forces to locate stable configurations corresponding to minima of the potential-energy surface.Because many properties are evaluated near equilibrium, relaxation is usually an early workflow step.
  • MD simulations: Molecular dynamics uses predicted per-atom forces to integrate Newton’s equations of motion and can support macroscopic-property calculations.The paper focuses on the microcanonical NVE ensemble.
  • Phonon and thermal conductivity calculations: Phonon and thermal-conductivity calculations require accurate higher-order derivatives of the learned potential-energy surface, including second and third derivatives.These predictions relate to dynamical stability, thermal stability, thermal conductivity, and optoelectronic behavior.

3 Desideratum for physical property prediction

The paper characterizes useful MLIPs by energy-conserving forces and a smoothly varying potential-energy surface with well-behaved higher derivatives. It proposes practical MD energy-conservation testing while accounting for numerical-integration effects and derivative regularity.

  • 3.1 Conservative forces: An MLIP is energy conserving when its forces produce zero work around any closed path.The paper states that this property holds when forces are calculated as the negative derivative of the potential-energy surface.
  • 3.1 Conservative forces: Direct-force models can be efficient and accurate on static benchmarks but may be non-conservative and produce larger errors in some property-prediction tasks.They avoid an additional backpropagation step required when forces are obtained from energy derivatives.
  • 3.1 Conservative forces: Conservative forces are necessary but not sufficient for practical energy conservation in MD because finite-step numerical integration introduces truncation errors.The paper discusses the second-order Verlet integrator used for NVE simulations.
  • 3.2 Bounded energy derivatives: Direct-force models are technically force fields rather than true potentials.
  • 3.2 Bounded energy derivatives: Long-time energy conservation requires the potential-energy surface and sufficiently high-order derivatives to be continuously differentiable with bounded derivatives.The required derivative order and simulation-time bound arise from the numerical-integrator analysis.

4 eSEN

eSEN is an equivariant message-passing MLIP that combines edgewise and nodewise processing to predict total potential energy and derive forces and stress. Its design modifies existing architectures to improve energy conservation.

  • Architecture: eSEN is a message-passing neural network with repeated edgewise and nodewise processing blocks.The model begins with multichannel spherical-harmonic node representations.
  • Energy and derivatives: Final-layer L = 0 features predict nodewise energies, which are summed into total potential energy E; forces and stress come from back-propagation.
  • Architecture: Each eSEN block applies edgewise convolution followed by nodewise feed-forward processing, normalization, and residual connections.
  • Architectural choices: Compared with eSCN and EquiformerV2, eSEN avoids discretizing spherical-harmonic node representations during nodewise processing.Its edgewise blocks concatenate source and target embeddings, use two SO2 convolutions, and add an envelope function.
  • Architectural choices: The paper argues that eSEN’s architectural choices improve its ability to conserve energy.

5 Design choices for enhancing physical property prediction

The paper evaluates MLIP design choices by testing energy conservation during fixed-step molecular dynamics, linking smooth, conservative potential-energy surfaces to physical-property prediction. It finds that several choices barely affect test errors but strongly affect practical conservation, and uses these findings to improve training and architecture.

  • Evaluation: Energy conservation during fixed-step MD is used to assess whether an MLIP has a continuous, smoothly varying potential-energy surface.The evaluation uses controlled ablations and out-of-distribution inorganic-material and organic-molecule simulations.
  • Direct-force prediction: Conservative forces are computed as the negative gradient of the potential-energy surface, unlike direct-force predictions that can be non-conservative.Direct-force models may be efficient, but their formulation produces significant energy drift in MD simulations.
  • Direct-force prediction: Direct-force pre-training followed by conservative fine-tuning reaches lower validation loss after 40 epochs than from-scratch conservative training after 100 epochs.The strategy also reduces wallclock training time by 40%.
  • Equivariance and nonlinearities: Grid-based spherical-harmonic nonlinearities can introduce sampling errors that break strict equivariance and energy conservation.eSEN instead uses a SiLU-based equivariant gated non-linearity directly in the spherical-harmonic representation.
  • Neighbor selection and cutoffs: Limiting graphs to a maximum number of neighbors can create PES discontinuities when nearest-K neighbors change under small position perturbations.eSEN keeps all neighbors within a 6Å distance cutoff instead.
  • Neighbor selection and cutoffs: Envelope functions make edge messages and their position derivatives decay to zero near the cutoff, while omitting them causes conservation failure.
  • Radial basis functions: Using many radial basis functions can break conservation on TM23 but conserve energy on MD22, whereas Bessel versus Gaussian bases do not affect conservation in either task.The default eSEN setting uses 10 basis functions rather than 512.
  • Test errors and practical performance: Many studied architecture choices have negligible test-error effects but dramatically change practical conservativeness; among conservative models, test errors correlate more strongly with property-prediction performance.

6 Experiments

The experiments evaluate eSEN on materials stability, thermal conductivity, phonon, and molecular-property tasks, emphasizing energy conservation and smoothness. eSEN achieves strong benchmark performance, including leading compliant and non-compliant Matbench-Discovery results and convergent phonon predictions.

  • eSEN is evaluated on materials stability, thermal conductivity, phonon calculations, and correlations between test energy MAE and physical-property performance.
  • Materials stability prediction: 0.831 F1 is achieved by compliant eSEN-30M-MP on Matbench-Discovery, the highest among compliant models, alongside the lowest relaxed-structure RMSD.
  • Thermal conductivity prediction: 0.340 κSRME is achieved under the default protocol, while eSEN leads both F1 and κSRME; previous models reach state-of-the-art performance on only one metric.
  • Materials stability prediction: 0.925 F1, 0.170 κSRME, and 0.0608 RMSD are reported for non-compliant eSEN-30M-OAM, the best-performing non-compliant model.
  • MDR phonon benchmark: Conservative MLIPs significantly outperform direct-force models for phonon prediction at 0.01 Å displacement, while larger displacement can improve direct-force metrics with caveats.
  • MDR phonon benchmark: eSEN phonon bands converge as displacement decreases, unlike direct-force eqV2-S-DeNS, which shows missing acoustic branches and spurious imaginary frequencies.
  • Scope caveat: OMat-trained models without sAlex/MPTrj fine-tuning are not directly compared because their DFT level of theory differs.
  • Error correlations: Among eSEN variants that pass the MD energy-conservation test, test errors strongly correlate with κSRME and vibrational-entropy MAE.

7 Related works

Related work has advanced symmetry-preserving and increasingly expressive MLIP architectures while also motivating scalable models that may not enforce energy conservation. The literature therefore increasingly emphasizes evaluation on realistic physical-property tasks, not only static prediction accuracy.

  • MLIP architectures increasingly use expressive atom-environment embeddings while commonly preserving physical symmetries.
  • Scalable architectures may sacrifice physical principles such as energy conservation while achieving strong accuracy, scalability, and relaxation performance.
  • Physical-property benchmarks now cover geometry optimization, molecular dynamics, vibrational analysis, and phonon calculations relevant to scientific discovery.

8 Discussion

Conservative forces and a smoothly varying potential energy surface are identified as important for reliable physical property prediction. eSEN uses these insights to achieve state-of-the-art results across several downstream tasks and may make test error more useful after energy-conservation validation.

  • Conservative forces and a smoothly-varying PES are important properties for consistent physical property prediction.
  • eSEN achieves state-of-the-art performance in force/energy prediction, geometry optimization, phonon calculations, and thermal conductivity prediction.
  • Energy-conservation tests may allow test error to serve as a proxy for model performance during development.This could reduce reliance on time-consuming physical-property benchmarking when evaluating models.

A Experimental details

The experiments evaluate MLIPs on out-of-distribution molecular-dynamics tasks in inorganic and organic domains, measuring energy drift over 100-ps simulations. Models are trained and compared using specified datasets, simulation protocols, and matched test-set energy errors.

  • OOD MD tasks cover inorganic crystals and organic molecules to test energy conservation on systems unseen during training.Inorganic tasks use TM23 defects absent from MPTrj training, while organic tasks use larger MD22 molecules than those in SPICE-MACE-OFF.
  • Inorganic materials: Inorganic models use MPTrj for training and TM23 for evaluation, which contains 81 combinations of 27 vacancy-defect metal systems and temperatures.
  • Organic molecules: Organic models use SPICE-MACE-OFF for training and MD22 for evaluation, whose seven molecules are considerably larger than training molecules.
  • All ML-based MD simulations use Velocity-Verlet with ASE and measure energy-conservation error across 100-ps simulations.
  • eSEN models use two layers, Lmax = 2, Mmax = 2, and 3.2M trainable parameters.
  • A random 5000-sample sAlex subset provides a fair comparison of test-set energy MAEs because MPTrj lacks an official test split.

B.1 Correlation of test-set energy errors and vibrational property errors

The paper examines whether test-set energy errors correlate with vibrational-property errors across architectures and eSEN variants. Improved correlations are observed among models that conserve energy, while design choices affect different properties unevenly.

  • Improved correlations between test-set energy MAE and three phonon-related tasks occur among energy-conserving models.The comparison spans different model architectures and different eSEN variants.
  • Failing the conservation test affects physical properties to varying degrees depending on model design choices.
  • Neighbor limit, envelope function, and basis-function count substantially affect κSRME but have relatively minor influence on MDR Phonon properties.
  • Representation discretization affects vibrational entropy and heat capacity but not the other evaluated properties.

B.2 Displacement values and their relation to phonon band structure predictions

Phonon metrics can improve with larger displacements even when the underlying band structure remains inaccurate. The comparisons show that conservative, smoothly varying models are less vulnerable to these artifacts, while broader evaluation metrics are still needed.

  • At displacement 0.2 Å, eqV2-S-DeNS phonon benchmark MAEs become comparable to those of conservative force models.Unlike eSEN and MACE, eqV2-S-DeNS shows a notable decrease in MAE as displacement increases.
  • Improved thermodynamic-property accuracy at larger displacement for direct-force models does not imply accurate phonon bands or DOS.The issue reflects inaccurate dispersion relations and other underlying phonon-structure errors.
  • Small displacement values commonly produce imaginary frequencies, indicating a rough learned energy landscape near minima.
  • Enforcing net zero force restores accurate acoustic modes but does not resolve phonon-band errors or imaginary frequencies at small displacement.
  • eSEN direct-force predictions improve phonon bands and DOS relative to eqV2-S-DeNS, with fewer imaginary-frequency artifacts.The comparison supports the importance of a smoothly varying model even without conservative forces.
  • A more comprehensive MLIP assessment would evaluate phonon dispersion across all modes and frequencies and thermodynamic properties across temperatures.

C Inference efficiency

The models’ inference speed was benchmarked against similarly sized MACE-OFF-L under a standardized GPU and software environment. Across all system sizes, eSEN-3.2M showed comparable inference efficiency to MACE-OFF-L.

  • Benchmark setup: Inference speed was benchmarked against the similarly sized 4.7M-parameter MACE-OFF-L model on a single 80GB Nvidia A-100 GPU.The MACE-OFF-L benchmark used the authors’ published benchmark code with mace-torch v0.3.6.
  • Benchmark setup: The comparison used diamond systems with variable supercell sizes and identical benchmark environments for all models.Benchmarks used Python 3.12, PyTorch 2.4.0, and CUDA 12.1 without compile or TorchScript.
  • Result: Across all system sizes, eSEN-3.2M had comparable inference efficiency to MACE-OFF-L.
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