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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Nima Dehmamy is a research professor at the Center for Science of Science and Innovation at Northwestern University. He received his PhD in physics from Boston University in 2016 and his BSc and MSc in physics from Sharif University, Iran. His work is at the intersection of physics, machine learning and complex systems, including machine learning for science, graph learning, optimization and computational social science. His works on properties of 3D embedded graphs were featured on the covers of Nature magazine and Nature Physics. In the past, he has worked on a broad spectrum of topics such as black hole physics, financial networks and condensed matter physics.
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