基于深度神经网络的复杂区域偏微分方程求解

兰州理工大学学报 ›› 2022, Vol. 48 ›› Issue (6): 149-157.

基于深度神经网络的复杂区域偏微分方程求解

郭晓斌¹, 袁冬芳², 曹富军^*2

1.内蒙古科技大学信息工程学院, 内蒙古包头 014010;
2.内蒙古科技大学理学院, 内蒙古包头 014010

收稿日期:2021-10-08 出版日期:2022-12-28 发布日期:2023-03-21
通讯作者: 曹富军(1984-),男,宁夏中卫人,副教授.Email:caofujun@imust.edu.cn
基金资助:
国家自然科学基金(11801287,12161067,12261067),内蒙古自然科学基金(2018BS01002,2018LH01008,2020MS06010,2021LHMS01006,2022MS01008),内
蒙古自治区高等学校科研项目(NJZZ18140),内蒙古自治区青年科技英才支持计划项目(NJYT20B15)

Solving partial differential equation with complex geometries based on deep neural network

GUO Xiao-bin¹, YUAN Dong-fang², CAO Fu-jun²

1. School of Information Engineering, Inner Mongolia University of Science and Technology, Baotou 014010, China;
2. School of Science, Inner Mongolia University of Science and Technology, Baotou 014010, China

Received:2021-10-08 Online:2022-12-28 Published:2023-03-21

摘要/Abstract

摘要： 基于深度神经网络求解复杂区域的椭圆型偏微分方程,通过实现深度前馈人工神经网络,构造合适的损失函数和神经网络求解策略,并且提出针对椭圆型偏微分方程的精确、有效的策略和数值方法.该方法只需要在边界和内部上分别选取少量样本点作为训练集,经过迭代学习神经网络的参数使其逼近椭圆型偏微分方程的解.与传统数值方法相比,本方法具有无网格特点,无需生成计算网格,便于处理任意复杂区域问题.数值算例表明此方法可以求解具有复杂区域的微分方程问题且具有较好的数值精度.

关键词: 深度神经网络, 偏微分方程, 损失函数, 梯度下降法

Abstract: Based on the deep neural network, this paper solved elliptic partial differential equations in complex regions. By realizing the deep feedforward artificial neural network, the appropriate loss function and neural network solution strategy was constructred, and the accurate and effective strategies as well as numerical methods for elliptic partial differential equations were put forward. This method simply needed to select a small number of sample points on the boundary and inside region as the training set, and the parameters of the neural network were learned iteratively to approximate the solution of the elliptic partial differential equation. Compared with the traditional numerical method, this method is mesh free and does not need to generate computational grid, so it is easy to deal with any complex computational region. Numerical examples show that this method can solve differential equation problems with complex regions and has good numerical accuracy.

Key words: deep neural networks, partial differential equation, loss function, gradient descent method

中图分类号:

O241.82

郭晓斌, 袁冬芳, 曹富军. 基于深度神经网络的复杂区域偏微分方程求解[J]. 兰州理工大学学报, 2022, 48(6): 149-157.

GUO Xiao-bin, YUAN Dong-fang, CAO Fu-jun. Solving partial differential equation with complex geometries based on deep neural network[J]. Journal of Lanzhou University of Technology, 2022, 48(6): 149-157.

参考文献

[1] ARORA G,JOSHI V.A computational approach for solution of one dimensional parabolic partial differential equation with application in biological processes [J].Ain Shams Engineering Journal,2018,9(4):1141-1150.
[2] ALIYI K A.Numerical solution for one dimensional linear types of parabolic partial differential equation and application to heat equation [J].Mathematics and Computer Science,2020,5(4):76-85.
[3] CHERNYSHENKO A Y,OLSHANSKII M A.An adaptive octree finite element method for PDEs posed on surfaces [J].Computer Methods in Applied Mechanics and Engineering,2015,291(1):146-172.
[4] BASDEVANT C,DEVILLE M,HALDENWANG P,et al.Spectral and finite difference solutions of Burgers equation [J].Computers & Fluids,1986,14(1):23-41.
[5] CHATZIPANTELIDIS P.Finite volume methods for ellipticPDE’s:a new approach [J].ESAIM:Mathematical Modelling and Numerical Analysis,2002,36(2):307-324.
[6] HORNIK K,STINCHCOMBE M,WHITE H.Multilayer feedforward networks are universal approximators [J].Neural networks,1989,2(5):359-366.
[7] LAGARIS I E,LIKAS A.Artificial neural networks for solving ordinary and partial differential equations [J].IEEE Transactions on Neural Networks,1998,9(5):987-1000.
[8] MCFALL K S,MAHAN J R.Artificial neural network method for solution of boundary value problems with exact satisfaction of arbitrary boundary conditions [J].IEEE Transactions on Neural Networks,2009,20(8):1221-1233.
[9] E W N,HAN J Q,JENTZEN A.Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations [J].Communications in Mathematics and Statistics,2017,5(4):349-380.
[10] E W N,YU B.The deep ritz method:A deep learning-based numerical algorithm for solving variational problems [J].Communications in Mathematics & Statistics,2018,6(1):1-12.
[11] KHOO Y,LU J,YING L.Solving parametric PDE problems with artificial neural networks [J].European Journal of Applied Mathematics,2021,32(3):421-435.
[12] NABIAN M A,MEIDANI H.A deep learning solution approach for high-dimensional random differential equations [J].Probabilistic Engineering Mechanics,2019,57:14-25.
[13] PERDIKARIS P,RAISSI M,KARNIADAKIS G E.Machine learning of linear differential equations using Gaussian processes [J].Journal of Computational Physics,2017,348:683-693.
[14] RAISSI M,PERDIKARIS P,KARNIADAKIS G E.Physics-informed neural networks:A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations [J].Journal of Computational Physics,2019,378:686-707.
[15] OWHADI H.Bayesian numerical homogenization [J].Multiscale Modeling & Simulation,2015,13(3):812-828.
[16] BERG J,NYSTROM K.A unified deep artificial neural network approach to partial differential equations in complex geometries [J].Neurocomputing,2018,317:28-41.
[17] DWIVEDI V,SRINIVASAN B.Physics informed extreme learning machine (PIELM)-A rapid method for the numerical solution of partial differential equations [J].Neurocomputing,2020,391:96-118.
[18] ANITESCU C,ATROSHCHENKO E,ALAJLAN N,et al.Artificial neural network methods for the solution of second order boundary value problems [J].Computers and Materials and Continua,2019,59(1):345-359.
[19] SUZUKI Y.Neural network-based discretization of nonlinear differential equations [J].Neural Computing and Applications,2019,31(7):3023-3038.
[20] BERTRAND F,DEMKOWICZ L,GOPALAKRISHNAN J,et al.Recent advances in least-squares and discontinuous Petrov-Galerkin finite element methods [J].Computational Methods in Applied Mathematics,2019,19(3):395-397.
[21] UTE H,DIETRICH S.Unbiased stereological estimation of the surface area of gradient surface processes [J].Advances in Applied Probability,1998,30(4):904-920.
[22] BAYDIN A G,PEARLMUTTER B A,RADUL A A,et al.Automatic differentiation in machine learning:a survey [J].Journal of Marchine Learning Research,2018,18:1-43.
[23] CHO H,HAN H,LEE B,et al.A second-order boundary condition capturing method for solving the elliptic interface problems on irregular domains [J].Journal of Scientific Computing,2019,81(1):217-251.