DeepXDE 使用神经网络来近似偏微分方程的解 ，其中 θ 是网络参数，时间 t，坐标 是网络的输入。 优化目标是最小化对待求方程，初始条件和边界条件的破坏程度。 机器人动力学中的深度拉格朗日网络更接近 Rose Yu 的那个报告，即用深度神经网络替换物理模型中无解析解（甚至无数值解）的那部分，使得物理模型更健壮。

## Physics Informed Deep Learning

Physics Informed Neural Network 是如下这个函数 f：

## 个人思考

Deep Galerkin Method 或 DeepXDE 能够给出复杂偏微分方程的近似解，通过这个解对方程，边界条件和初始条件的破坏，能够判断神经网络近似的程度有多高。在这种情况下，新的基于神经网络的方法应该可以用来验证传统方法的数值误差，比如因差分近似带来的数值粘滞等。我看到同时求解爱因斯坦场方程，麦克斯韦方程，流体力学方程来描述中子星融合释放引力波的曙光。这种方法可能是所有数值解里最简单直接暴力的方法了。

## 参考文献

[1] Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations, 2017, Maziar Raissi,Paris Perdikaris,George Em Karniadakis

[2 ] Physics Informed Deep Learning (Part II): Data-driven Solutions of Nonlinear Partial Differential Equations, 2017, Maziar Raissi,Paris Perdikaris,George Em Karniadakis

[3 ] Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations, 2018, Maziar Raissi

[4 ] Physics-guided Neural Networks (PGNN): An Application in Lake Temperature Modeling, 2017, Anuj Karpatne∗ [email protected] William Watkins† [email protected] Jordan Read† [email protected] Vipin Kumar∗ [email protected], all from University of Minnesota

[5 ] How can Physics Inform Deep Learning Methods in Scientific Problems: Recent Progress and Future Prospects，slides 2017, Anuj Karpatne, PostDoc @ University of Minnesota

[6 ] Papers that cited “Physics informed Deep Learning”

[7 ] Deep Galerkin Method — A deep learning algorithm for solving partial differential equations， 2018， Justin Sirignano∗ and Konstantinos Spiliopoulos

[8 ]  Towards Physics-informed Deep Learning for Turbulent Flow Prediction，2019， Rui Wang,Karthik Kashinath,Mustafa Mustafa,Adrian Albert,Rose Yu

[9 ] Neural Lander: Stable Drone Landing Control using Learned Dynamics，2019， Guanya Shi, Xichen Shi, Michael O’Connell, Rose Yu, Kamyar Azizzadenesheli, Anima Anandkumar, Yisong Yue, Soon-Jo Chung

[10 ] Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning，2019，Michael Lutter, Christian Ritter, Jan Peters