Machine learning for Green's function methods

We aim to develop a general proof-of-principle scheme for machine learning Green’s function (GF)-like quantities in realistic systems such as molecules and solids. Our objective is to learn GF-based quantities that can yield multiple experimentally relevant observables from a single training on a given dataset. This approach contrasts with the current practice, where multiple independent trainings are required to predict different properties such as energy, occupation numbers, or orbital magnetization.

Traditionally, Green’s functions and self-energies are large, complex objects due to their frequency dependence. Nevertheless, they are invaluable computational tools because they can be directly related to a wide range of solid-state experiments. Recent advances in applied mathematics provide techniques to make their representation more compact, enabling their use in modern AI and machine learning frameworks. Furthermore, we exploit functional derivatives of the Luttinger–Ward[1] functional to iteratively machine learn the desired quantities, which are then used to compute experimentally relevant observables.

In this work, we will employ finite-temperature correlated Green’s functions obtained from ab initio diagrammatic methods such as GW and Green’s function second order (GF2). However, the proposed scheme is general and can be applied to any Green’s function methodology.

[1] Luttinger, J. M.; Ward, J. C. (1960). "Ground-State Energy of a Many-Fermion System. II". Physical Review. 118 (5): 1417–1427. Bibcode:1960PhRv..118.1417L. doi:10.1103/PhysRev.118.1417