# Atomic-scale representation and statistical learning of tensorial properties

## TL;DR

This paper presents a framework for incorporating three-dimensional symmetries into Gaussian process regression models to accurately predict tensorial properties of atomic structures, enhancing the modeling of molecular polarizability and electron density.

## Contribution

It introduces a symmetry-adapted Gaussian process regression approach with specialized representations and kernels for tensorial properties, extending existing methods to account for geometric covariance.

## Key findings

- Effective learning of molecular polarizability.
- Accurate prediction of ground-state electron density.
- Framework generalizes smooth overlap representations.

## Abstract

This chapter discusses the importance of incorporating three-dimensional symmetries in the context of statistical learning models geared towards the interpolation of the tensorial properties of atomic-scale structures. We focus on Gaussian process regression, and in particular on the construction of structural representations, and the associated kernel functions, that are endowed with the geometric covariance properties compatible with those of the learning targets. We summarize the general formulation of such a symmetry-adapted Gaussian process regression model, and how it can be implemented based on a scheme that generalizes the popular smooth overlap of atomic positions representation. We give examples of the performance of this framework when learning the polarizability and the ground-state electron density of a molecule.