Accurate Interatomic Force Fields via Machine Learning with Covariant Kernels

TL;DR

This paper introduces a new machine learning approach using covariant kernels in Gaussian Process Regression to predict atomic forces as vectors, ensuring proper rotational behavior and improved accuracy in material simulations.

Contribution

It develops covariant GP kernels based on rotation group integration, enabling accurate vector force predictions that respect physical symmetries, with analytical solutions in certain cases.

Findings

Accurately predicts quantum-mechanical forces in materials

Kernel integration over finite point groups suffices for high accuracy

Method outperforms previous scalar-based force prediction models

Abstract

We present a novel scheme to accurately predict atomic forces as vector quantities, rather than sets of scalar components, by Gaussian Process (GP) Regression. This is based on matrix-valued kernel functions, on which we impose the requirements that the predicted force rotates with the target configuration and is independent of any rotations applied to the configuration database entries. We show that such covariant GP kernels can be obtained by integration over the elements of the rotation group SO(d) for the relevant dimensionality d. Remarkably, in specific cases the integration can be carried out analytically and yields a conservative force field that can be recast into a pair interaction form. Finally, we show that restricting the integration to a summation over the elements of a finite point group relevant to the target system is sufficient to recover an accurate GP. The accuracy of our kernels in predicting quantum-mechanical forces in real materials is investigated by tests on pure and defective Ni, Fe and Si crystalline systems.