Deep Neural Network for Solving Differential Equations Motivated by Legendre-Galerkin Approximation
Year: 2024
Author: Bryce Chudomelka, Youngjoon Hong, John Morgan, Hyunwoo Kim, Jinyoung Park
International Journal of Numerical Analysis and Modeling, Vol. 21 (2024), Iss. 5 : pp. 652–673
Abstract
In this paper, we propose the Legendre-Galerkin Network (LGNet), a novel machine learning-based numerical solver for parametric partial differential equations (PDEs) using spectral methods. Spectral methods leverage orthogonal function expansions, such as Fourier series and Legendre polynomials, to achieve highly accurate solutions with a reduced number of grid points. Our framework combines the advantages of spectral methods, including accuracy, efficiency, and generalization, with the capabilities of deep neural networks. By integrating deep neural networks into the spectral framework, our approach reduces computational costs that enable real-time predictions. The mathematical foundation of the LGNet solver is robust and reliable, incorporating a well-developed loss function derived from the weak formulation. This ensures precise approximation of solutions while maintaining consistency with boundary conditions. The proposed LGNet solver offers a compelling solution that harnesses the strengths of both spectral methods and deep neural networks, providing an effective tool for solving parametric PDEs.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ijnam2024-1026
International Journal of Numerical Analysis and Modeling, Vol. 21 (2024), Iss. 5 : pp. 652–673
Published online: 2024-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 22
Keywords: Deep learning neural network spectral element method Legendre-Galerkin method data driven numerical method.
Author Details
Bryce Chudomelka Email
Youngjoon Hong Email
John Morgan Email
Hyunwoo Kim Email
Jinyoung Park Email