Recursive evaluation and iterative contraction of $N$-body equivariant features

TL;DR

This paper introduces a recursive and iterative framework for efficiently generating high-order, symmetry-adapted features for atomistic machine learning, improving the representation of atomic environments.

Contribution

It presents a novel recursive method to compute high-order equivariant features and an automatic feature selection process for systematic, symmetry-aware atomistic representations.

Findings

Efficient computation of high-order $N$-body equivariant features.

Systematic generation of symmetry-adapted, complete atomic environment representations.

Enhanced expressiveness of machine learning models for atomistic systems.

Abstract

Mapping an atomistic configuration to an $N$-point correlation of a field associated with the atomic positions (e.g. an atomic density) has emerged as an elegant and effective solution to represent structures as the input of machine-learning algorithms. While it has become clear that low-order density correlations do not provide a complete representation of an atomic environment, the exponential increase in the number of possible $N$-body invariants makes it difficult to design a concise and effective representation. We discuss how to exploit recursion relations between equivariant features of different orders (generalizations of $N$-body invariants that provide a complete representation of the symmetries of improper rotations) to compute high-order terms efficiently. In combination with the automatic selection of the most expressive combination of features at each order, this approach provides a conceptual and practical framework to generate systematically-improvable, symmetry adapted representations for atomistic machine learning.