Keywords: symmetry discovery, equivariance, group convolutional neural network, inductive bias, group equivariant architecture, Lie group, Lie algebra, scientific machine learning, physics
TL;DR: Lie algebra simplify and unify encoding of continuous groups into neural nets, and relate them to models in physics
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.
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