Incorporating a priori physics knowledge into machine learning leads to more robust and interpretable algorithms. In this work, we combine deep learning techniques and classic numerical methods for differential equations to solve two challenging problems in dynamical systems theory: dynamics discovery and parameter estimation. Results demonstrate the effectiveness of the proposed approaches on a suite of test problems exhibiting oscillatory and chaotic dynamics. When comparing the performance of various numerical schemes, such as the Runge-Kutta and linear multistep families of methods, we observe promising results in predicting the system dynamics and estimating physical parameters, given appropriate choices of spatial and temporal discretization schemes and numerical method orders.

本研究解决了动力系统理论中的两个关键问题：动力学发现和参数估计，填补了机器学习与经典数值方法之间的空白。通过将深度学习与经典的微分方程数值方法相结合，提出了一种新的方法，结果表明该方法在预测系统动态和估计物理参数方面表现出色，有潜力提升动态系统分析的有效性。

将物理知识引入深度学习与数值方法结合以进行鲁棒动力学发现和参数估计