Machine-learning hidden symmetries

We present an automated method for finding hidden symmetries, defined as symmetries that become manifest only in a new coordinate system that must be discovered. Its core idea is to quantify asymmetry as violation of certain partial differential equations, and to numerically minimize such violation over the space of all invertible transformations, parametrized as invertible neural networks. For example, our method rediscovers the famous Gullstrand-Painlevé metric that manifests hidden translational symmetry in the Schwarzschild metric of nonrotating black holes, as well as Hamiltonicity, modularity, and other simplifying traits not traditionally viewed as symmetries.