  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 insights alone. We
propose a novel framework that can infer the 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
symmetries of dynamics from trained DNNs. The proposed framework is developed
by deriving the relationship between a manifold structure of time-series
dataset and the necessary conditions for Noether's theorem. The feasibility of
the proposed framework has been verified in some primitive cases for 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.

Data Analysis, Statistics and Probability;Machine Learning;Pattern Formation and Solitons;Computational Physics;Machine Learning