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.
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.
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. https://doi.org/10.32604/cmc.2019.06641
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