基于人工神经网络数值求解Helmholtz方程  

Based on Artificial Neural Network to Calculate the Numerical Solution of the Helmholtz Equation

作　　者：邱国强 马云云 QIU Guoqiang;MA Yunyun(School of Computer Science and Technology,Dongguan University of Technology,Dongguan 523808,China)

机构地区：[1]东莞理工学院计算机科学与技术学院,广东东莞523808

出　　处：《东莞理工学院学报》2023年第5期44-53,共10页Journal of Dongguan University of Technology

基　　金：国家自然科学基金(11901085)。

摘　　要：本文主要研究运用无网格化的神经网络方法数值求解有界域上的Helmholtz方程边值问题。与基于网格化的传统数值方法相比,无网格化的神经网络方法不会因为精密的网格剖分带来巨大的计算开销和存储开销。首先针对单位正方形区域,提出满足边值条件的神经网络,将其用于Helmholtz方程的数值求解。对于一般矩形区域上的Helmholtz方程边值问题,将通过线性变换,映射成单位正方形区域上的二阶偏微分方程边值问题,再利用上述神经网络数值求解相关问题。最后,通过数值算例,讨论神经网络方法数值求解有界域上Helmholtz方程边值问题的特点,并验证方法的有效性。In this paper,a meshless neural network method is used to solve the boundary value problem of Helmholtz equation.Compared with the conventional numerical methods based on meshing,the meshless neural network method won’t bring huge computing and storage costs due to the precise mesh generation.Firstly,a neural network satisfying the boundary condition is proposed for the unit square region,and it is used to calculate the numerical solution of Helmholtz equation.For the boundary value problem of Helmholtz equation in a general rectangular region,it will be mapped into a boundary value problem of second-order partial differential equation in a unit square region through a linear transformation,and then the neural network will be used to solve the problem.Finally,through numerical examples,we will discuss the characteristics of the above neural network method,and verify the effectiveness of the method.

关 键 词：神经网络 HELMHOLTZ方程 边值问题 DIRICHLET边界条件 

分 类 号：TP391[自动化与计算机技术—计算机应用技术]