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

Abstract

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

CLC Number:

O241.82

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.

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