  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 which symmetry to use as an inductive bias in practice.
Enforcing the wrong symmetry could even 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
preserve the original distribution and fool the discriminator. LieGAN
represents symmetry as interpretable Lie algebra basis and can discover various
symmetries such as the rotation group $\mathrm{SO}(n)$, 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.
