Diagonal Symmetrization of Neural Network Solvers for the Many-Electron Schr\"odinger Equation

TL;DR

This paper explores methods to incorporate diagonal group symmetries into neural network solvers for the many-electron Schrödinger equation, finding that post hoc averaging improves stability and performance over in-training symmetrization.

Contribution

It introduces and compares different symmetry incorporation techniques, revealing the limitations of in-training symmetrization and proposing post hoc averaging as a robust alternative.

Findings

In-training symmetrization can destabilize training and worsen performance.

Post hoc averaging is effective and less sensitive to computational-statistical tradeoffs.

Diagonal invariance methods improve neural network solvers for quantum many-body problems.

Abstract

Incorporating group symmetries into neural networks has been a cornerstone of success in many AI-for-science applications. Diagonal groups of isometries, which describe the invariance under a simultaneous movement of multiple objects, arise naturally in many-body quantum problems. Despite their importance, diagonal groups have received relatively little attention, as they lack a natural choice of invariant maps except in special cases. We study different ways of incorporating diagonal invariance in neural network ans\"atze trained via variational Monte Carlo methods, and consider specifically data augmentation, group averaging and canonicalization. We show that, contrary to standard ML setups, in-training symmetrization destabilizes training and can lead to worse performance. Our theoretical and numerical results indicate that this unexpected behavior may arise from a unique computational-statistical tradeoff not found in standard ML analyses of symmetrization. Meanwhile, we demonstrate that post hoc averaging is less sensitive to such tradeoffs and emerges as a simple, flexible and effective method for improving neural network solvers.