Interpretable Conservation Law Estimation by Deriving the Symmetries of Dynamics from Trained Deep Neural Networks

As deep neural networks (DNN) have the ability to model the distribution of datasets as a low-dimensional manifold, we propose a method to extract the coordinate transformation that makes a dataset distribution invariant by sampling DNNs using the replica exchange Monte-Carlo method. In addition, we derive the relation between the canonical transformation that makes the Hamiltonian invariant (a necessary condition for Noether's theorem) and the symmetry of the manifold structure of the time series data of the dynamical system. By integrating this knowledge with the method described above, we propose a method to estimate the interpretable conservation laws from the time-series data. Furthermore, we verified the efficiency of the proposed methods in primitive cases and large scale collective motion in metastable state.