Equivariant Manifold Flows

TL;DR

This paper introduces equivariant manifold flows, a theoretical framework for learning symmetry-invariant distributions on arbitrary manifolds, demonstrated through gauge-invariant densities in quantum field theory.

Contribution

It provides the first theoretical foundation for symmetry-respecting density learning on manifolds using equivariant flows, applicable to complex scientific domains.

Findings

Successfully learned gauge-invariant densities over SU(n)

Established theoretical basis for equivariant flows on manifolds

Demonstrated applicability in quantum field theory

Abstract

Tractably modelling distributions over manifolds has long been an important goal in the natural sciences. Recent work has focused on developing general machine learning models to learn such distributions. However, for many applications these distributions must respect manifold symmetries -- a trait which most previous models disregard. In this paper, we lay the theoretical foundations for learning symmetry-invariant distributions on arbitrary manifolds via equivariant manifold flows. We demonstrate the utility of our approach by using it to learn gauge invariant densities over $SU(n)$ in the context of quantum field theory.