A Unified Framework to Enforce, Discover, and Promote Symmetry in Machine Learning

Samuel E. Otto; Nicholas Zolman; J. Nathan Kutz; Steven L. Brunton

Symmetry is present throughout nature and continues to play an increasingly central role in machine learning. In this paper, we provide a unifying theoretical and methodological framework for incorporating Lie group symmetry into machine learning models in three ways: 1. enforcing known symmetry when training a model; 2. discovering unknown symmetries of a given model or data set; and 3. promoting symmetry during training by learning a model that breaks symmetries within a user-specified candidate group only when the data provide sufficient evidence. We show that these tasks can be cast within a common mathematical framework whose central object is the Lie derivative. We extend and unify several existing results by showing that enforcing and discovering symmetry are linear-algebraic tasks that are dual under the bilinear pairing induced by the Lie derivative. We also propose a novel way to promote symmetry by introducing a class of convex regularizers, built from the Lie derivative with a nuclear-norm relaxation, that penalizes symmetry breaking during training. We explain how these ideas can be applied to a wide range of machine learning models including basis-function regression, dynamical-systems discovery, neural networks, and neural operators acting on fields.

A Unified Framework to Enforce, Discover, and Promote Symmetry in Machine Learning

Abstract