Artificial neural networks for solving elliptic differential equations with boundary layer

In this paper, we consider the artificial neural networks for solving the elliptic differential equation with boundary layer, in which the gradient of the solution changes sharply near the boundary layer. The solution of the boundary layer problems poses a huge challenge to both traditional numerica...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences 2022-07, Vol.45 (11), p.6583-6598
Main Authors: Yuan, Dongfang, Liu, Wenhui, Ge, Yongbin, Cui, Guimei, Shi, Lin, Cao, Fujun
Format: Article
Language:English
Subjects:
artificial neural network
Artificial neural networks
boundary layer problems
Boundary layers
Convection-diffusion equation
Diffusion layers
elliptic differential equation
Elliptic differential equations
Elliptic functions
Numerical methods
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Summary:In this paper, we consider the artificial neural networks for solving the elliptic differential equation with boundary layer, in which the gradient of the solution changes sharply near the boundary layer. The solution of the boundary layer problems poses a huge challenge to both traditional numerical methods and artificial neural network methods. By theoretically analyzing the changing rate of the weights of the first hidden layer near the boundary layer, a mapping strategy is added in traditional neural network to improve the convergence of the loss function. Numerical examples are carried out for the 1D and 2D convection–diffusion equation with boundary layer. The results demonstrate that the modified neural networks significantly improve the ability in approximating the solutions with sharp gradient.
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.8192