Deriving the nonlinear symmetries of dynamics from trained deep neural networks

In this talk, we will present a method for extracting interpretable physical laws from deep neural networks (DNNs) trained on dynamical time series data. Specifically, we have developed a new method for inferring hidden conservation laws of systems from DNNs trained on time-series data that can be regarded as the finite degree of freedom classical Hamiltonian dynamical systems. The method inferred the conservation laws by extracting the symmetry of the dynamical system from the DNN based on Noether's theorem and effective sampling methods. Since the only assumption imposed on DNNs by the method is the manifold hypothesis, which is widely believed to hold for DNNs that have been successfully trained, the method can be applied to a wide range of DNN models. On the other hand, the verification of the method was limited to the case where the symmetric transformation is an affine transformation.

In the central force potential system, the Runge-Lenz vector is conserved. This conservation law is given as the symmetry of SO(4) on the phase space of nonlinear transformations. In this talk, we will present the results of investigating whether it is possible to extract the symmetry and estimate the conservation law under such a nonlinear transformation based on our proposed method.