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

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 resi...

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Bibliographic Details
Published in:Computers, materials & continua Vol. 59; no. 1; pp. 345 - 359
Main Authors: Anitescu, Cosmin, Atroshchenko, Elena, Alajlan, Naif, Rabczuk, Timon
Format: Journal Article
Language:English
Published: Henderson Tech Science Press 2019
Subjects:
Acoustics
Artificial neural networks
Boundary value problems
Helmholtz equations
Mathematical analysis
Neural networks
Partial differential equations
Robustness (mathematics)
Training
ISSN:1546-2226, 1546-2218, 1546-2226
Online Access:Get full text
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Summary: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.
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ISSN:1546-2226
1546-2218
1546-2226
DOI:10.32604/cmc.2019.06641