Artificial neural network methods for the solution of second order boundary value problems

Cosmin Anitescu; Elena Atroshchenko; Naif Alajlan; Timon Rabczuk

doi:10.32604/cmc.2019.06641

Artificial neural network methods for the solution of second order boundary value problems

Cosmin Anitescu
, Elena Atroshchenko
, Naif Alajlan
, Timon Rabczuk^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

672 Citations (Scopus)

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.

Original language	English
Pages (from-to)	345-359
Number of pages	15
Journal	Computers, Materials and Continua
Volume	59
Issue number	1
DOIs	https://doi.org/10.32604/cmc.2019.06641
Publication status	Published - 2019

Bibliographical note

Funding Information:
Acknowledgements: N. Alajlan and T. Rabczuk acknowledge the Distinguished Scientist Fellowship Program (DSFP) at King Saud University for supporting this work.

Publisher Copyright:
Copyright © 2019 Tech Science Press.

Keywords

Adaptive collocation
Artificial neural networks
Deep learning
Inverse problems

ASJC Scopus subject areas

Biomaterials
Modelling and Simulation
Mechanics of Materials
Computer Science Applications
Electrical and Electronic Engineering

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10.32604/cmc.2019.06641

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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.",

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AU - Anitescu, Cosmin

AU - Atroshchenko, Elena

AU - Alajlan, Naif

AU - Rabczuk, Timon

N1 - Funding Information: Acknowledgements: N. Alajlan and T. Rabczuk acknowledge the Distinguished Scientist Fellowship Program (DSFP) at King Saud University for supporting this work. Publisher Copyright: Copyright © 2019 Tech Science Press.

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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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KW - Deep learning

KW - Inverse problems

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IS - 1

ER -

Artificial neural network methods for the solution of second order boundary value problems

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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