基于深度学习的偏微分方程求解方法

A Method for Solving Partial Differential Equations Based on Deep Learning

本文提出了一种基于深度学习的偏微分方程求解方法。该方法把偏微分方程的解看作函数变量关于自变量的非线性关系,利用深度神经网络表达该非线性关系,其不断逼近原偏微分方程解的过程是无约束最优化问题,可借助拟牛顿算法L-BFGS来求解。针对三种典型的偏微分方程,使用有限差分格式和本文方法分别求解,结果对比表明,本文方法计算精度较好,不会引入人工粘性,且具有普适性。此外,本文研究了神经网络隐藏层层数和每层神经元个数对计算精度的影响。

In this paper,a deep learning-based method for solving partial differential equations was proposed.In this method,the solution of the partial differential equation was regarded as the nonlinear relationship between the function variables and the independent variables,and the nonlinear relationship was expressed by the deep neural network.The process of continuously approximating the solution of the original partial differential equation is an unconstrained optimization problem.Newton^s algorithm L-BFGS is perfect to solve it.For three typical partial differential equations,the finite difference scheme and the method in this paper were used to solve them respectively.The comparison of the results showed that the method in this paper has better calculation accuracy,does not introduce artificial viscosity,and is universal.In addition,this paper studies the influence of the number of hidden layers and the number of neurons in each layer on the computational accuracy of the neural network.