Deep Neural Network for Solving Differential Equations Motivated by Legendre-Galerkin Approximation

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.