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Artificial neural networks for solving elliptic differential equations with boundary layer
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  • Dongfang Yuan,
  • Wenhui Liu,
  • Yongbin Ge,
  • Guimei Cui,
  • Lin Shi,
  • Fujun Cao
Dongfang Yuan
Inner Mongolia University of Science and Technology
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Wenhui Liu
Inner Mongolia University of Science and Technology
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Yongbin Ge
Ningxia University
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Guimei Cui
Inner Mongolia University of Science and Technology
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Lin Shi
Inner Mongolia University of Science and Technology
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Fujun Cao
Inner Mongolia University of Science and Technology
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Abstract

In this paper, we consider the artificial neural networks for solving the 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 theoretical analyzing the changing rate of the weights of 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.

Peer review status:IN REVISION

17 Dec 2020Submitted to Mathematical Methods in the Applied Sciences
23 Dec 2020Submission Checks Completed
23 Dec 2020Assigned to Editor
26 Dec 2020Reviewer(s) Assigned
24 Feb 2021Review(s) Completed, Editorial Evaluation Pending
25 Feb 2021Editorial Decision: Revise Major
23 Apr 20211st Revision Received
23 Apr 2021Assigned to Editor
23 Apr 2021Submission Checks Completed