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A Novel ANN-Based Radial Basis Function Collocation Method for Solving Elliptic Boundary Value Problems

Author

Listed:
  • Chih-Yu Liu

    (Department of Civil Engineering, National Central University, Taoyuan 320317, Taiwan)

  • Cheng-Yu Ku

    (School of Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan)

Abstract

Elliptic boundary value problems (BVPs) are widely used in various scientific and engineering disciplines that involve finding solutions to elliptic partial differential equations subject to certain boundary conditions. This article introduces a novel approach for solving elliptic BVPs using an artificial neural network (ANN)-based radial basis function (RBF) collocation method. In this study, the backpropagation neural network is employed, enabling learning from training data and enhancing accuracy. The training data consist of given boundary data from exact solutions and the radial distances between exterior fictitious sources and boundary points, which are used to construct RBFs, such as multiquadric and inverse multiquadric RBFs. The distinctive feature of this approach is that it avoids the discretization of the governing equation of elliptic BVPs. Consequently, the proposed ANN-based RBF collocation method offers simplicity in solving elliptic BVPs with only given boundary data and RBFs. To validate the model, it is applied to solve two- and three-dimensional elliptic BVPs. The results of the study highlight the effectiveness and efficiency of the proposed method, demonstrating its capability to deliver accurate solutions with minimal data input for solving elliptic BVPs while relying solely on given boundary data and RBFs.

Suggested Citation

  • Chih-Yu Liu & Cheng-Yu Ku, 2023. "A Novel ANN-Based Radial Basis Function Collocation Method for Solving Elliptic Boundary Value Problems," Mathematics, MDPI, vol. 11(18), pages 1-19, September.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:18:p:3935-:d:1241300
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    References listed on IDEAS

    as
    1. Ku, Cheng-Yu & Xiao, Jing-En & Liu, Chih-Yu & Lin, Der-Guey, 2021. "On solving elliptic boundary value problems using a meshless method with radial polynomials," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 185(C), pages 153-173.
    2. Chih-Yu Liu & Cheng-Yu Ku, 2022. "A Simplified Radial Basis Function Method with Exterior Fictitious Sources for Elliptic Boundary Value Problems," Mathematics, MDPI, vol. 10(10), pages 1-23, May.
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    Cited by:

    1. Dmitry Stenkin & Vladimir Gorbachenko, 2024. "Mathematical Modeling on a Physics-Informed Radial Basis Function Network," Mathematics, MDPI, vol. 12(2), pages 1-11, January.
    2. Chih-Yu Liu & Cheng-Yu Ku & Wei-Da Chen, 2024. "A Spacetime RBF-Based DNNs for Solving Unsaturated Flow Problems," Mathematics, MDPI, vol. 12(18), pages 1-25, September.

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