Automatic Symmetry Discovery with Lie Algebra Convolutional Network
Automatic Symmetry Discovery with Lie Algebra Convolutional Network
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发表时间：
2021-09
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通讯作者：
Nima Dehmamy;R. Walters;Yanchen Liu-;Dashun Wang;Rose Yu
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作者：
Nima Dehmamy;R. Walters;Yanchen Liu-;Dashun Wang;Rose Yu
现有的等变神经网络需要对称群的先验知识以及对连续群进行离散化。我们提议使用李代数（无穷小生成元）而非李群。我们的模型——李代数卷积网络（L - conv）能够自动发现对称性，且不需要对群进行离散化。我们表明L - conv可作为构建模块来构建任何群等变前馈架构。卷积神经网络和图卷积网络都可以用适当的群表示为L - conv。我们发现了L - conv与物理学之间的直接联系：（1）群不变损失泛化了场论；（2）欧拉 - 拉格朗日方程衡量了鲁棒性；（3）等变性导致守恒定律和诺特定理流。这些联系为设计更通用的等变网络并将其应用于物理科学中的重要问题开辟了新的途径。
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