  Equivariant neural networks require explicit knowledge of the symmetry group.
Automatic symmetry discovery methods aim to relax this constraint and learn
invariance and equivariance from data. However, existing symmetry discovery
methods are limited to simple linear symmetries and cannot handle the
complexity of real-world data. We propose a novel generative model, Latent
LieGAN (LaLiGAN), which can discover symmetries of nonlinear group actions. It
learns a mapping from the data space to a latent space where the symmetries
become linear and simultaneously discovers symmetries in the latent space.
Theoretically, we show that our method can express any nonlinear symmetry under
some conditions about the group action. Experimentally, we demonstrate that our
method can accurately discover the intrinsic symmetry in high-dimensional
dynamical systems. LaLiGAN also results in a well-structured latent space that
is useful for downstream tasks including equation discovery and long-term
forecasting.
