Over the past decade, computational atomistic modeling, driven by machine learning (ML), has become indispensable to scientific endeavors, improving our understanding and accelerating the search for compounds with enhanced properties. Traditional atomistic ML schemes are bottom-up surrogate models of macroscopic properties extracted from atomic simulations driven by quantum mechanical (QM) calculations, and aim to provide the accuracy of QM without its prohibitive computational cost. Intermediate outputs of electronic structure, such as wavefunctions and electron densities, have also been recently used as both inputs and targets of integrated machine learning models, leading to a holistic approach to predicting complex molecular behaviors and emergent phenomena.
Unlike many other domains, atomistic ML benefits from a wealth of physical laws constraining the relationship between inputs and outputs. Recent ML approaches leverage this domain knowledge by encoding physical priors in intricate end-to-end architectures or symmetry-adapted inputs to usually simpler models.
This thesis aims to unify these methodologies into a single framework. We begin by examining the mathematical underpinning of symmetrized descriptors that characterize local atomic environments, identifying ingredients such as the functional form of representations and symmetries that contribute to their success in modeling atomic properties or contributions to global observables.
Having deconstructed atom-centered descriptors, we propose a recipe to extend their scope, allowing for the description of structural environments indexed by more than one atomic center. This not only helps address the limitations of atom-centered descriptors but also reveals their connection to geometric equivariant message-passing on atomistic graphs, thus bridging the gap between the two distinct approaches in atomistic ML, which together encompass most existing models.
The development of representations describing multiple atomic centers also enables the prediction of ingredients central to QM, such as the effective single-particle Hamiltonian matrix with elements indexed by atomic orbitals on pairs of atoms. This target offers a compelling advantage as it circumvents the need for costly QM calculations and allows the prediction of other properties based on their established analytical relationship with the Hamiltonian. Integrating physical constraints within this framework establishes a connection between individual surrogate models, which construct target properties based on their geometric relationship with the structure, and integrated models that predict QM intermediates while being agnostic to the properties derived from them. This integration is valuable for the transferability of predictions of derived properties across structural space and promotes their accuracy to higher levels of theory at no additional cost when the prediction of the intermediate Hamiltonian can be optimized according to its physical relationship with the target. This research, drawing on a common thread of physics, systematizes the theory of descriptor-based ML for atomic systems. The rigorous integration of symmetry principles in descriptors, coupled with physical constraints in the model, and a carefully crafted representation of the targets, pave the way to robust and integrated machine-learning models that enable predictive simulations of physical observables and chemical properties of arbitrary complexity.