基于人工神经网络的椭圆型微分方程数值求解  

Numerical Solution of Differential Equation Based on Artificial Neural Network

作　　者：袁冬芳 刘文慧 崔桂梅[2] 石琳[1] Yuan Dongfang;Liu Wenhui;Cui Guimei;Shi Lin(School of Science,Inner Mongolia University of Science and Technology,Baotou 014010,China;School of Information Engineering,Inner Mongolia University of Science and Technology,Baotou 014010,China)

机构地区：[1]内蒙古科技大学理学院,内蒙古包头014010 [2]内蒙古科技大学信息工程学院,内蒙古包头014010

出　　处：《宁夏大学学报（自然科学版）》2022年第1期6-11,共6页Journal of Ningxia University(Natural Science Edition)

基　　金：国家自然科学基金资助项目(11801287);内蒙古自治区高等学校科研项目(NJZZ18140);内蒙古自然科学基金资助项目(2018BS01002,2018LH01008,2020MS06010);内蒙古自治区青年科技英才支持计划项目(NJYT20B15);内蒙古科技大学创新基金项目(2019YQL02)。

摘　　要：神经网络因其能够无限逼近任意非线性函数的特性,为求解微分方程提供了一种新的思路.通过神经网络训练,得到偏微分方程的近似解是连续函数,且具有足够的精度,因此可以得到解的任意阶导数.该方法的优势在于当问题维数增大时,计算量和存储量增加相对较小,可以克服维数灾难求解高维问题.同时,具有良好的泛化性和求解复杂区域问题的能力.针对带边界层的对流扩散问题,由于其解的梯度在边界层附近变化剧烈,常规的数值方法和传统的神经网络模型均难求得其精确解.为此,设计了一种新的神经网络构造方法,能够保证优化算法的收敛性,且近似解具有足够的精度.Neural network method has become a new approach to solve partial differential equations because of its infinite approximation to any nonlinear function.The approximation solutions of PDE can be obtained by training the neural network.Because the approximated solution is continuous and has enough accuracy,then any order derivative of the solution can also be calculated analytically.The advantage of using neural network method to solve PDE is that when the dimension of the problem increases,the amount of calculation and storage increases relatively small,which can overcome the dimension disaster and solve high-dimensional problems.Moreover,the method has good generalization and ability to solve complex regional problems.For the convection diffusion problem with boundary layer,it is difficult to obtain its exact solution by conventional numerical method and traditional neural network model because the gradient of its solution varies sharply near the boundary layer.In this paper,a new neural network construction method is designed,which can ensure the convergence of the optimization algorithm and the approximate solution has enough accuracy.

关 键 词：微分方程 神经网络 损失函数构造 

分 类 号：O241.82[理学—计算数学]