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 address two challenging missing physics problems in dynamical systems theory: dynamics discovery and parameter estimation. The presented methods encode available information relating to the system dynamics into deep learning architectures, incorporating different assumptions on the known inputs and desired outputs in each case. Results demonstrate the effectiveness of the proposed approaches in making data-driven model predictions given corrupt system observations 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.

翻译：将先验物理知识融入机器学习可提升算法的鲁棒性与可解释性。本研究将深度学习技术与微分方程经典数值方法相结合，解决动力系统理论中两个具有挑战性的缺失物理问题：动力学发现与参数估计。所提出的方法将系统动力学相关的可用信息编码到深度学习架构中，针对不同情形对已知输入与期望输出引入相应的假设。在包含振荡与混沌动力学的测试问题集上，实验结果表明所提方法能在系统观测存在噪声的情况下实现数据驱动的模型预测。对比龙格-库塔法及线性多步法等不同数值方案性能时发现，通过合理选择空间/时间离散格式与数值方法阶数，本方法在预测系统动力学及估计物理参数方面展现出显著优势。