Latent Space Symmetry Discovery

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.

Generative Adversarial Symmetry Discovery

Despite the success of equivariant neural networks in scientific applications, they require knowing the symmetry group a priori. However, it may be difficult to know the right symmetry to use as an inductive bias in practice and enforcing the wrong symmetry could hurt the performance. In this paper, we propose a framework, LieGAN, to automatically discover equivariances from a dataset using a paradigm akin to generative adversarial training. Specifically, a generator learns a group of transformations applied to the data, which preserves the original distribution and fools the discriminator. LieGAN represents symmetry as interpretable Lie algebra basis and can discover various symmetries such as rotation group $\mathrm{SO}(n)$ and restricted Lorentz group $\mathrm{SO}(1,3)^+$ in trajectory prediction and top quark tagging tasks. The learned symmetry can also be readily used in several existing equivariant neural networks to improve accuracy and generalization in prediction.