Interpretable conservation law estimation by deriving the symmetries of dynamics from trained deep neural networks

Understanding complex systems with their reduced model is one of the central roles in scientific activities. Although physics has greatly been developed with the physical insights of physicists, it is sometimes challenging to build a reduced model of such complex systems on the basis of insight alone. We propose a framework that can infer hidden conservation laws of a complex system from deep neural networks (DNNs) that have been trained with physical data of the system. The purpose of the proposed framework is not to analyze physical data with deep learning but to extract interpretable physical information from trained DNNs. With Noether's theorem and by an efficient sampling method, the proposed framework infers conservation laws by extracting the symmetries of dynamics from trained DNNs. The proposed framework is developed by deriving the relationship between a manifold structure of a time-series data set and the necessary conditions for Noether's theorem. The feasibility of the proposed framework has been verified in some primitive cases in which the conservation law is well known. We also apply the proposed framework to conservation law estimation for a more practical case, that is, a large-scale collective motion system in the metastable state, and we obtain a result consistent with that of a previous study.