通过从训练的深度神经网络导出动力学对称性来进行可解释的守恒定律估计
Interpretable Conservation Law Estimation by Deriving the Symmetries of Dynamics from Trained Deep Neural Networks
摘要：
用简化模型来理解复杂系统是科学活动的中心任务之一。虽然物理学在物理学家的物理见解下得到了极大的发展，但有时仅基于见解来构建这种复杂系统的简化模型是具有挑战性的。我们提出了一个新的框架，可以从已经用系统的物理数据训练的深度神经网络（DNN）中推断出复杂系统的隐藏守恒律。该框架的目的不是用深度学习分析物理数据，而是从训练好的DNN中提取可解释的物理信息。利用Noether定理和一种有效的采样方法，该框架通过从训练的DNN中提取动力学的对称性来推断守恒律。建议的框架是通过推导时间序列数据集的流形结构和Noether定理的必要条件之间的关系来开发的。所提出的框架的可行性已在一些原始的情况下，守恒定律是众所周知的。我们还将所提出的框架应用于一个更实际的情况下，即一个大规模的集体运动系统的亚稳态的守恒定律估计，我们得到了与以前的研究一致的结果。
英文摘要：
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.
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