Atomic-Scale Representation and Statistical Learning of Tensorial Properties

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.

Electron density learning of non-covalent systems

“…Gaussian process regression (GPR) can be extended to encode all the fundamental symmetries of the O(3) group, effectively allowing machine-learning of all the molecular properties that transform as spherical tensors under rotation and inversion operations. 57,74 In the specic case of the electron density, the scheme relies upon the decomposition of the eld into additive, atom-centered contributions and the subsequent prediction of the corresponding expansion coefficients. 58 In SA-GPR, each molecule is represented as a collection of atom-centered environments, whose relationships and similarities are measured by symmetry adapted kernels.…”

Multi-scale approach for the prediction of atomic scale properties

“…which we indicate in what follows using the shorthand notation |ρ ⊗ν i ⊗V ⊗ν i ; σ ; λ µ , omitting the σ ; λ µ indices when considering invariant features (σ = 1, λ = 0). Within this construction, the ket |λ µ has the role of making the resulting features transform as a Y µ λ spherical harmonic 39,68 , while |σ indicates the parity of the features under inversion ‡ . Eq.…”

“…Written in this basis, arlm|ρ i expresses the decomposition of the density in independent angular momentum channels, evaluated at a distance r from the central atom. In practical implementations we use a basis of Gaussian type orbitals to also discretize the radial component 68 . This is the form that is usually chosen to write and compute the local SOAP features 16 , that can be evaluated as…”

Representations and descriptors unifying the study of molecular and bulk systems

“…Their successful application was then demonstrated across molecular and bulk systems of a variety of compositions. Their use is not limited to force and energy predictions, however, extending to tensorial properties, pattern recognition for ice phases, and atomic mobility of defects and grain boundaries . The generality of such a descriptor has also allowed it to be applied in unsupervised schemes to determine similarities among previously challenging‐to‐compare systems for the case of molecular systems in both crystalline and gas phases .…”

“…Furthermore, while atom‐density descriptors have been remarkably accurate when applied to local property challenges, they cannot be of help in resolving nonlocal problems, for example, predicting the energy of systems governed by long‐range electrostatic interaction such as charged dimers. A recent advance in this area has shown how an equivalent long‐range description, remapped as a feature vector defined locally and equivariant in O(3), could hold the solution …”