A derivative-free method for solving elliptic partial differential equations with deep neural networks

JOURNAL OF COMPUTATIONAL PHYSICS

卷:419

Article

Han, Jihun¹  Nica, Mihai²  Stinchcombe, Adam R.¹ 

[1] Univ Toronto, Dept Math, 40 St George St, Toronto, ON M5S 2E4, Canada

[2] Univ Guelph, Dept Math & Stat, 50 Stone Rd East, Guelph, ON N1G 2W1, Canada

关键词: Numerical method;    Partial differential equation;    Neural network;    Brownian motion;    Reinforcement learning;    Bellman equation;   

DOI  :  10.1016/j.jcp.2020.109672

来源: Elsevier

【 摘 要 】

We introduce a deep neural network based method for solving a class of elliptic partial differential equations. We approximate the solution of the PDE with a deep neural network which is trained under the guidance of a probabilistic representation of the PDE in the spirit of the Feynman-Kac formula. The solution is given by an expectation of a martingale process driven by a Brownian motion. As Brownian walkers explore the domain, the deep neural network is iteratively trained using a form of reinforcement learning. Our method is a 'Derivative-Free Loss Method' since it does not require the explicit calculation of the derivatives of the neural network with respect to the input neurons in order to compute the training loss. The advantages of our method are showcased in a series of test problems: a corner singularity problem, a high-dimensional Poisson's equation, an interface problem, and an application to a chemotaxis population model. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free