Automatic Symmetry Discovery with Lie Algebra Convolutional Network

Abstract

Existing equivariant neural networks for continuous groups require discretization or group representations. All these approaches require detailed knowledge of the group parametrization and cannot learn entirely new symmetries. We propose to work with the Lie algebra (infinitesimal generators) instead of the Lie group. Our model, the Lie algebra convolutional network (L-conv) can learn potential 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 architecture. We discuss how CNNs and Graph Convolutional Networks are related to and can be expressed as L-conv with appropriate groups. We also derive the MSE loss for a single L-conv layer and find a deep relation with Lagrangians used in physics, with some of the physics aiding in defining generalization and symmetries in the loss landscape. Conversely, L-conv could be used to propose more general equivariant ansätze for scientific machine learning.Another approach taken in Finzi et al. ( 2020) approximates the integral over the Lie group by Monte Carlo sampling. However, their approach is quite involved, using the matrix logarithm and exponential to define a distance function used to obtain a sampling neighborhood for each point. Finzi et al.(2021) provides a unified approach for encoding continuous groups multi-layer perceptrons (MLP).Preprint. Under review.