基于深度学习方法求解偏微分方程的应用举例

The Application of Solving Partial Differential Equations Based on Deep Learning Method

泊松方程和Hamilton-Jacobi-Bellman方程是两类工程上常用的偏微分方程。笔者实现了一种基于深度神经网络求解这两类方程的机器学习方法。与传统的数值方法不同,该算法不需要任何插值和坐标变换。针对低维和高维系统,数值算例展示了该算法的性能以及求解偏微分方程的可行性和有效性。

Poisson equation and Hamilton-Jacobi-Bellman equation are two kinds of engineering commonly used partial differential equations. In this paper, a machine learning method based on deep neural network is implemented to solve these two kinds of equations. Unlike traditional numerical methods, the algorithm does not require any interpolation and coordinate transformation.For low-and high-dimensional systems, numerical studies show the performance of the algorithm and the feasibility and effectiveness of solving partial differential equations.