Steerable Partial Differential Operators for Equivariant Neural Networks

TL;DR

This paper develops a theory of steerable partial differential operators for equivariant neural networks, providing new tools for designing symmetry-preserving models and bridging deep learning with physics concepts.

Contribution

It derives a complete G-steerability constraint for PDOs, solves it for key groups, and introduces a unifying framework connecting convolutions and differential operators.

Findings

Equivariant PDOs can replace convolutions in neural networks.

The G-steerability constraint is fully characterized for several groups.

A unified framework for equivariant maps using Schwartz distributions is proposed.

Abstract

Recent work in equivariant deep learning bears strong similarities to physics. Fields over a base space are fundamental entities in both subjects, as are equivariant maps between these fields. In deep learning, however, these maps are usually defined by convolutions with a kernel, whereas they are partial differential operators (PDOs) in physics. Developing the theory of equivariant PDOs in the context of deep learning could bring these subjects even closer together and lead to a stronger flow of ideas. In this work, we derive a $G$-steerability constraint that completely characterizes when a PDO between feature vector fields is equivariant, for arbitrary symmetry groups $G$. We then fully solve this constraint for several important groups. We use our solutions as equivariant drop-in replacements for convolutional layers and benchmark them in that role. Finally, we develop a framework for equivariant maps based on Schwartz distributions that unifies classical convolutions and differential operators and gives insight about the relation between the two.