TY - JOUR
T1 - Deep Neural Network for Solving Differential Equations Motivated by Legendre-Galerkin Approximation 
AU - Chudomelka , Bryce
AU - Hong , Youngjoon
AU - Morgan , John
AU - Kim , Hyunwoo
AU - Park , Jinyoung
JO - International Journal of Numerical Analysis and Modeling
VL - 5
SP - 652
EP - 673
PY - 2024
DA - 2024/10
SN - 21
DO - http://doi.org/10.4208/ijnam2024-1026
UR - https://global-sci.org/intro/article_detail/ijnam/23447.html
KW - Deep learning, neural network, spectral element method, Legendre-Galerkin method, data driven numerical method.
AB - <p style="text-align: justify;">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.</p>