Incorporating Crystallographic Symmetries into Probabilistic Learning Approaches for Materials—High-Accuracy Many Body Wave Functions and Generative Models for Inorganic Crystalline Solids

Crystallographic (space group) symmetries play a key role in constraining the phase space of material properties, and virtually all direct physical modeling approaches take advantage of them. However it remains challenging to incorporate the symmetries of crystallographic space groups into emerging machine learning or probabilsitic models for inroganic crystalline materials. Unlike the infinite, continuous rotational or translational symmetries that can be described by approaches such as, e.g., equivariant graph neural networks, crystallographic symmetries constitute a discrete but infinite set of symmetry operations. Approaches to embed them into machine learning models face challenges due the complexity of the symmetry operations and the need to make algorithms invariant (or equivariant) to certain transformations. In this presentation, I will share my perspective and our recent work on developing machine learning approaches that aim to incorporate crystallographic symmetries. Two examples that we will cover include (i) embedding many-body symmetries into neural network ansatz approaches to variational quantum Monte Carlo, towards obtaining high accuracy solutions to the many-body Schrodinger equation, and (ii) generative models for inorganic crystalline solids that learn probability distributions parametrized over crystallographic Wyckoff sites. In the first example, we'll compare the performance (optimization and scaling behavior) of neural network approaches for many body wave functions, when crystallographic symmetries are incorporated by group averaging and by explicit invariant neural network layer construction, to reference cases that are not symmetry-aware. In the second example, we will highlight approaches to generative models for crystalline solids that, by learning parametrized distributions over space groups and crystallographic Wyckoff sites, can generate structures with distributions across crystallographic space groups and Wyckoff sites of varying dimensionality.