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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| Published in: | Computers, materials & continua Vol. 59; no. 1; pp. 345 - 359 |
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| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Henderson
Tech Science Press
2019
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| Subjects: | |
| ISSN: | 1546-2226, 1546-2218, 1546-2226 |
| Online Access: | Get full text |
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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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| AbstractList | 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. |
| Author | Atroshchenko, Elena Alajlan, Naif Rabczuk, Timon Anitescu, Cosmin |
| Author_xml | – sequence: 1 givenname: Cosmin surname: Anitescu fullname: Anitescu, Cosmin – sequence: 2 givenname: Elena surname: Atroshchenko fullname: Atroshchenko, Elena – sequence: 3 givenname: Naif surname: Alajlan fullname: Alajlan, Naif – sequence: 4 givenname: Timon surname: Rabczuk fullname: Rabczuk, Timon |
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| Title | Artificial Neural Network Methods for the Solution of Second Order Boundary Value Problems |
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