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Volume 13, Issue 2
A Chebyshev Polynomial Neural Network Solver for Boundary Value Problems of Elliptic Equations

Liujun Meng, Xuelin Zhang & Hanquan Wang

DOI: 10.4208/eajam.2022-064.210722

East Asian J. Appl. Math., 13 (2023), pp. 230-245.

Published online: 2023-04

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  • Abstract

A Chebyshev polynomial neural network for solving boundary value problems for one- and two-dimensional partial differential equations is constructed. In particular, the input parameters are expanded by Chebyshev polynomials and fed into the network. A loss function is constructed, and approximate solutions are determined by minimizing the loss function. Elliptic equations are used to test a Chebyshev polynomial neural network solver. The numerical examples illustrate the high accuracy of the method.

  • Keywords

Chebyshev polynomial, neural network, elliptic equation, deep learning.

  • AMS Subject Headings

35Q68

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-13-230, author = {Meng , LiujunZhang , Xuelin and Wang , Hanquan}, title = {A Chebyshev Polynomial Neural Network Solver for Boundary Value Problems of Elliptic Equations}, journal = {East Asian Journal on Applied Mathematics}, year = {2023}, volume = {13}, number = {2}, pages = {230--245}, abstract = {

A Chebyshev polynomial neural network for solving boundary value problems for one- and two-dimensional partial differential equations is constructed. In particular, the input parameters are expanded by Chebyshev polynomials and fed into the network. A loss function is constructed, and approximate solutions are determined by minimizing the loss function. Elliptic equations are used to test a Chebyshev polynomial neural network solver. The numerical examples illustrate the high accuracy of the method.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.2022-064.210722 }, url = {http://global-sci.org/intro/article_detail/eajam/21646.html} }
TY - JOUR T1 - A Chebyshev Polynomial Neural Network Solver for Boundary Value Problems of Elliptic Equations AU - Meng , Liujun AU - Zhang , Xuelin AU - Wang , Hanquan JO - East Asian Journal on Applied Mathematics VL - 2 SP - 230 EP - 245 PY - 2023 DA - 2023/04 SN - 13 DO - http://doi.org/10.4208/eajam.2022-064.210722 UR - https://global-sci.org/intro/article_detail/eajam/21646.html KW - Chebyshev polynomial, neural network, elliptic equation, deep learning. AB -

A Chebyshev polynomial neural network for solving boundary value problems for one- and two-dimensional partial differential equations is constructed. In particular, the input parameters are expanded by Chebyshev polynomials and fed into the network. A loss function is constructed, and approximate solutions are determined by minimizing the loss function. Elliptic equations are used to test a Chebyshev polynomial neural network solver. The numerical examples illustrate the high accuracy of the method.

Meng , LiujunZhang , Xuelin and Wang , Hanquan. (2023). A Chebyshev Polynomial Neural Network Solver for Boundary Value Problems of Elliptic Equations. East Asian Journal on Applied Mathematics. 13 (2). 230-245. doi:10.4208/eajam.2022-064.210722
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