# Building nonparametric $n$-body force fields using Gaussian process regression

## TL;DR

This paper introduces a Bayesian framework using Gaussian process regression to construct nonparametric force fields for atomic systems, enabling automatic selection of interaction order and efficient modeling of complex materials.

## Contribution

It develops a novel method to encode physical properties into GP kernels and automatically determine the optimal interaction order for force fields.

## Key findings

- Models agree with physical intuition for real materials
- Lower order models preferred with limited data
- Tabulated force fields significantly speed up computations

## Abstract

Constructing a classical potential suited to simulate a given atomic system is a remarkably difficult task. This chapter presents a framework under which this problem can be tackled, based on the Bayesian construction of nonparametric force fields of a given order using Gaussian process (GP) priors. The formalism of GP regression is first reviewed, particularly in relation to its application in learning local atomic energies and forces. For accurate regression it is fundamental to incorporate prior knowledge into the GP kernel function. To this end, this chapter details how properties of smoothness, invariance and interaction order of a force field can be encoded into corresponding kernel properties. A range of kernels is then proposed, possessing all the required properties and an adjustable parameter $n$ governing the interaction order modelled. The order $n$ best suited to describe a given system can be found automatically within the Bayesian framework by maximisation of the marginal likelihood. The procedure is first tested on a toy model of known interaction and later applied to two real materials described at the DFT level of accuracy. The models automatically selected for the two materials were found to be in agreement with physical intuition. More in general, it was found that lower order (simpler) models should be chosen when the data are not sufficient to resolve more complex interactions. Low $n$ GPs can be further sped up by orders of magnitude by constructing the corresponding tabulated force field, here named "MFF".