Automatic Symmetry Discovery with Lie Algebra Convolutional Network
This addresses the need for more flexible and general equivariant networks in machine learning, particularly for physical sciences, though it appears incremental as it builds on existing equivariant methods.
The paper tackles the problem of requiring prior knowledge of symmetries and discretization for continuous groups in equivariant neural networks by proposing the Lie algebra convolutional network (L-conv), which automatically discovers symmetries without discretization and can construct any group equivariant feedforward architecture, with applications showing connections to physics.
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