DGNN: A Neural PDE Solver Induced by Discontinuous Galerkin Methods

Partial Differential Equations (PDEs) are fundamental to the mathematical modeling of a wide range of physical, biological, and engineering systems. These systems span diverse fields, such as fluid dynamics, heat transfer, structural mechanics, and population dynamics. Traditionally, numerical methods such as Finite Difference Method (FDM), Finite Volume Method (FVM), Finite Element Method (FEM) [Smith, 1985, Zienkiewicz et al., 2005], and spectral methods have been employed to obtain approximate solutions to PDEs. Although these methods are highly effective, they ultimately reduce to solving linear systems of equations. This often leads to several challenges, such as when solving high-dimensional problems or performing adaptive solutions for complex domains.