Physics-Informed Weakly Supervised Learning for Interatomic Potentials

TL;DR

This work tackles the challenge of training accurate and robust interatomic potentials when labeled data are sparse by introducing physics-informed weakly supervised learning (PIWSL). PIWSL couples a Taylor-expansion-based consistency loss (PITC) with a physics-inspired spatial consistency loss (PISC) to enforce physical relationships between energies and conservative forces, even with limited force labels. The approach demonstrates substantial improvements in energy and force predictions across multiple datasets and model architectures, enhanced MD stability, and notable gains in fine-tuning foundation models on highly accurate ab initio data. By leveraging approximate energy labels and perturbation-based consistency, PIWSL reduces data requirements while promoting physically plausible potential-energy surfaces, with broad implications for scalable, reliable atomistic simulations.

Abstract

Machine learning plays an increasingly important role in computational chemistry and materials science, complementing computationally intensive ab initio and first-principles methods. Despite their utility, machine-learning models often lack generalization capability and robustness during atomistic simulations, yielding unphysical energy and force predictions that hinder their real-world applications. We address this challenge by introducing a physics-informed, weakly supervised approach for training machine-learned interatomic potentials (MLIPs). We introduce two novel loss functions, extrapolating the potential energy via a Taylor expansion and using the concept of conservative forces. Our approach improves the accuracy of MLIPs applied to training tasks with sparse training data sets and reduces the need for pre-training computationally demanding models with large data sets. Particularly, we perform extensive experiments demonstrating reduced energy and force errors -- often lower by a factor of two -- for various baseline models and benchmark data sets. Moreover, we demonstrate improved robustness during MD simulations of the MLIP models trained with the proposed weakly supervised loss. Finally, our approach improves the fine-tuning of foundation models on sparse, highly accurate ab initio data. An implementation of our method and scripts for executing experiments are available at https://github.com/nec-research/PICPS-ML4Sci.

Figures (5)

• Figure 1: Schematic illustration of physics-informed weakly supervised losses used in this work. (a) Taylor-expansion-based weak label (WL) loss with approximate labels obtained from reference energies and atomic forces cooper2020efficient. (b) Physics-inspired Taylor-expansion-consistency (PITC) loss with approximate labels obtained from energies and atomic forces predicted by an MLIP. (c) Physics-inspired spatial consistency (PISC) loss with approximate labels obtained from energies and atomic forces predicted by an MLIP. Here, $E(\mathcal{S}; \boldsymbol{\theta})$ and $\mathbf{F}_i(\mathcal{S}; \boldsymbol{\theta})$ denote the potential energy and atomic forces predicted by an MLIP parametrized by $\boldsymbol{\theta}$, $\mathcal{S}$ and $\mathcal{S}_{\delta\mathbf{r}}$ define the original atomic structure and the one perturbed by $\delta\mathbf{r}$.
• Figure 2: (a, b) Relative performance gains for MLIPs trained with PIWSL compared to those trained without it and (c, d) potential energy profiles for a C--H bond of the aspirin molecule. Relative performance gains are evaluated for (a) energy (E-) and (b) force (F-) RMSEs. These results are presented for the ANI-1x data set. Potential energy profiles for a C--H bond of the aspirin molecule are presented for models trained using (c) 100 and (d) 200 configurations. The red and blue arrows indicate the direction from the original structure ($E(\mathcal{S};\boldsymbol{\theta})$) to the perturbed one ($E(\mathcal{S}_{\delta\mathbf{r}};\boldsymbol{\theta})$), as defined by Eq. (\ref{['eq:taylor-wsl']}), for the baseline and PIWSL model predictions, respectively.
• Figure 3: Stability analysis of the MLIP models during MD simulations. Stability during MD simulations is assessed for the baseline MLIP models and those trained with PIWSL. Left: Models with the direct force branch. Right: Models with forces computed as negative gradients of the energy. All results are obtained for the aspirin molecule and MD simulations in the microcanonical (NVE) statistical ensemble. We measure stability during MD simulations according to fu2023forces.
• Figure A1: Analysis of the total energy conservation using MD simulations with MLIP models. The amount of the change of the total energy during MD simulations is assessed for the baseline MLIP models and those trained with PIWSL. The total energy is measured at the initial and final time-step and the difference is normalized by the total energy at the initial time-step. All results are obtained for the aspirin molecule and MD simulations in the microcanonical (N V E) statistical ensemble.
• Figure A2: Stability analysis of the MLIP models during MD simulations. Stability during MD simulations is assessed for the baseline MLIP models and those trained with PIWSL. All results are obtained for the aspirin molecule and MD simulations in the canonical (N V T) statistical ensemble. We measure stability during MD simulations according to fu2023forces.