Cluster expansion constructed over Jacobi-Legendre polynomials for accurate force fields

TL;DR

This paper presents a novel cluster expansion method using Jacobi and Legendre polynomials to create accurate, flexible, and interpretable machine-learning force fields that incorporate physical constraints and symmetries.

Contribution

The introduction of a Jacobi-Legendre polynomial-based cluster expansion that reduces parameters and enforces physical behaviors in machine-learning force fields.

Findings

Achieved highly accurate force fields for carbon including graphite, diamond, and amorphous forms.

Demonstrated the method's ability to impose physical constraints like repulsive tails naturally.

Compared favorably with existing machine-learning potential schemes.

Abstract

We introduce a compact cluster expansion method, constructed over Jacobi and Legendre polynomials, to generate highly accurate and flexible machine-learning force fields. The constituent many-body contributions are separated, interpretable and adaptable to replicate the physical knowledge of the system. In fact, the flexibility introduced by the use of the Jacobi polynomials allows us to impose, in a natural way, constrains and symmetries to the cluster expansion. This has the effect of reducing the number of parameters needed for the fit and of enforcing desired behaviours of the potential. For instance, we show that our Jacobi-Legendre cluster expansion can be designed to generate potentials with a repulsive tail at short inter-atomic distances, without the need of imposing any external function. Our method is here continuously compared with available machine-learning potential schemes, such as the atomic cluster expansion and potentials built over the bispectrum. As an example we construct a Jacobi-Legendre potential for carbon, by training a slim and accurate model capable of describing crystalline graphite and diamond, as well as liquid and amorphous elemental carbon.