Automatic Symmetry Discovery with Lie Algebra Convolutional Network

TL;DR

This paper introduces L-conv, a neural network that automatically discovers symmetries using Lie algebras, eliminating the need for prior symmetry knowledge or discretization, and connects to physical principles.

Contribution

The paper presents L-conv, a novel neural network framework that learns symmetries directly from data via Lie algebras, unifying various architectures and linking to physical laws.

Findings

L-conv can discover symmetries without prior knowledge.

L-conv encompasses CNNs and GCNs as special cases.

Connections to physics suggest new design principles.

Abstract

Existing equivariant neural networks require prior knowledge of the symmetry group and discretization for continuous groups. We propose to work with Lie algebras (infinitesimal generators) instead of Lie groups. Our model, the Lie algebra convolutional network (L-conv) can automatically discover symmetries and does not require discretization of the group. We show that L-conv can serve as a building block to construct any group equivariant feedforward architecture. Both CNNs and Graph Convolutional Networks can be expressed as L-conv with appropriate groups. We discover direct connections between L-conv and physics: (1) group invariant loss generalizes field theory (2) Euler-Lagrange equation measures the robustness, and (3) equivariance leads to conservation laws and Noether current.These connections open up new avenues for designing more general equivariant networks and applying them to important problems in physical sciences