The Jacobi-Legendre framework for Machine Learning in Materials Investigation and Discovery

Abstract

Machine-learning models have rapidly become fundamental tools in the study of materials properties. In the past few years there has been a surge of interest in the construction of new models and descriptors to accelerate the investigation of known materials and the discovery of new ones, mostly in the framework provided by electronic structure calculations. This thesis revolves around the study of descriptors for machine-learning models, capable of describing the configuration of the system, while encoding the symmetries that are required to efficiently target properties of interest. Several cases will be treated, ranging from systems with magnetic degrees of freedom, to models that could predict potential energy surfaces, from models dedicated to the prediction of the electronic density, to ones for tensor and tensor fields. All of this will be done in the spirit of accelerating ab-initio calculations within a unified framework, denoted with the name "Jacobi-Legendre", here defined and meticulously investigated. In the thesis, we will explore a model for magnetic systems, in which the spin degrees of freedom will be placed on the same footing as the description of the position of the atoms. We will then dedicate the core of this work to the definition and construction of the Jacobi-Legendre framework, starting from a model devoted to the prediction of the potential energy surface of a system. With the formalism in place, we will generalize the descriptors to the prediction of the electron density, proving how the reached accuracy enables to accelerate electronic-structure calculations. We will then define the formalism in full by presenting and exploring methods for the prediction of tensors and tensorial fields. The thesis will be concluded with a thorough study on how the use of multipolar-spherical harmonics can be beneficial in simplifying the definition of descriptors and in exposing limits of current approaches, while proposing new strategies based on the Jacobi-Legendre potentials.