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 linear symmetries in their search space and cannot
handle the complexity of symmetries in real-world, often high-dimensional data.
We propose a novel generative model, Latent LieGAN (LaLiGAN), which can
discover nonlinear symmetries from data. It learns a mapping from data 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 certain conditions. Experimentally, our
method can capture the intrinsic symmetry in high-dimensional observations,
which results in a well-structured latent space that is useful for other
downstream tasks. We demonstrate the use cases for LaLiGAN in improving
equation discovery and long-term forecasting for various dynamical systems.

提出了一种能够从数据中发现非线性对称性的新颖生成模型 LaLiGAN，该模型可以将数据映射到特征空间，其中对称性变得线性，并同时在特征空间中发现对称性，理论上表明在某些条件下可以表达任何非线性对称性。实验结果显示，该方法可以捕捉到高维观测中的内在对称性，从而得到一个有用于其他下游任务的结构良好的特征空间。在各种动态系统的方程式发现和长期预测中展示了 LaLiGAN 的应用案例。