Dynamically training machine-learning-based force fields for strongly anharmonic materials

TL;DR

The paper tackles the reliability of ML-based interatomic potentials for strongly anharmonic materials by introducing a dynamic training framework that uses Bayesian error estimation to drive adaptive data acquisition. By integrating a CSLD-based force field with trajectory-averaged uncertainty, retraining is performed only when the current Bayesian error significantly exceeds its recent average, enabling efficient exploration of configuration space. The approach yields robust force fields for BAs, Si, and SnSe, achieving low RMSE and enabling convergence assessment without additional ab initio data, demonstrated across materials with increasing anharmonicity. This framework improves transferability and practicality for finite-temperature properties in challenging systems, with clear metrics for convergence based on both average uncertainty and fitting error.

Abstract

Machine learning (ML) force fields have emerged as a powerful tool for computing materials properties at finite temperatures, particularly in regimes where traditional phonon-based perturbation theories fail or cannot be extended beyond the harmonic approximation. These approaches offer accuracy comparable to ab initio molecular dynamics (MD), but at a fraction of the computational cost. However, their reliability critically depends on the quality and representativeness of the training data. In particular, static training datasets often lead to failure when the force field encounters previously unseen atomic configurations during MD simulations. In this work, we present a framework for dynamically training ML force fields and demonstrate its effectiveness across materials with varying degrees of anharmonicity, including cubic boron arsenide (c-BAs), silicon (Si), and tin selenide (SnSe). Our method builds on the conventional lattice dynamics expansion of total energy and incorporates Bayesian error estimation to guide adaptive data acquisition during simulation. Specifically, we show that trajectory-averaged Bayesian errors enable efficient and targeted exploration of the configuration space, significantly enhancing the robustness and transferability of the resulting force fields. We further demonstrate how Bayesian error estimation can be applied to determine the convergence of the dynamic training without requiring additional ab initio data. This proposed framework offers a practical and easily implementable scheme to improve the training process, which is the most critical step in developing reliable ML force fields.

Figures (4)

• Figure 1: The flow-chart for the dynamic force field training method. The idea is to explore the configuration space by using molecular dynamics with a force field trained on some initial training data. Whenever a structure $\mathbf{X}$ that is not well-represented by the training data is encountered during the training, the force field will be updated based on ab initio forces.
• Figure 2: The trajectory average $\mu_k[\varepsilon_{\text{B}}]$ of the Bayesian error (a) and the corresponding standard deviation $\sigma_k[\varepsilon_{\text{B}}]$ during the dynamic training. Both quantities have been evaluated for the $500~\text{ps}$ training trajectories of BAs (black), Si (blue), and SnSe (red) and the averages have been taken over $k = 1000$ molecular dynamics steps.
• Figure 3: Convergence of the RMSE with respect to ab inito forces on a test set (a) and the fitting error $\beta^{-1/2}$ of the linear regression (b) during the dynamic training of the BAs force-field as a function of the number of training structures. The force-fields have been trained using either the $\varepsilon_{\text{min}}$ (black) criterion or the trajectory average criterion (blue). The test set with ab initio forces comprising 50 structures was sampled from a $150~\text{ps}$ NVT trajectory at $300~\text{K}$.
• Figure 4: Distribution of the standardized Bayesian error $z_{\text{B}}$ over a $500~\text{ps}$ molecular dynamics trajectory for BAs (black), Si (blue), and SnSe (red). The employed force fields have been trained including either the full training data set (a) or a subset of the training data corresponding to a twice as large RMSE. An ideal normal distribution with mean $\mu = 0$ and standard deviation $\sigma = 1$ is shown for reference (dashed black).