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Artificial Neural Network Methods for the Solution of Second Order Boundary Value Problems

Cosmin Anitescu1, Elena Atroshchenko2, Naif Alajlan3, Timon Rabczuk3,*
Institute of Structural Mechanics, Bauhaus-Universität Weimar, 99423, Weimar, Germany.
School of Civil & Environmental Engineering, University of New South Wales, Sydney, Australia.
Department of Computer Engineering, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia.
* Corresponding Author: Timon Rabczuk. Email: .

Computers, Materials & Continua 2019, 59(1), 345-359. https://doi.org/10.32604/cmc.2019.06641

Abstract

We present a method for solving partial differential equations using artificial neural networks and an adaptive collocation strategy. In this procedure, a coarse grid of training points is used at the initial training stages, while more points are added at later stages based on the value of the residual at a larger set of evaluation points. This method increases the robustness of the neural network approximation and can result in significant computational savings, particularly when the solution is non-smooth. Numerical results are presented for benchmark problems for scalar-valued PDEs, namely Poisson and Helmholtz equations, as well as for an inverse acoustics problem.

Keywords

Deep learning, adaptive collocation, inverse problems, artificial neural networks.

Cite This Article

C. Anitescu, E. Atroshchenko, N. Alajlan and T. Rabczuk, "Artificial neural network methods for the solution of second order boundary value problems," Computers, Materials & Continua, vol. 59, no.1, pp. 345–359, 2019.

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This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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