DGNN: A Neural PDE Solver Induced by Discontinuous Galerkin Methods
This work addresses PDE solving for computational science and engineering, presenting an incremental hybrid method.
The paper tackles solving partial differential equations (PDEs) by proposing DGNN, a neural network framework inspired by discontinuous Galerkin methods, and demonstrates improved accuracy and training efficiency in numerical examples, including handling high perturbations and discontinuous solutions.
We propose a general framework for the Discontinuous Galerkin-induced Neural Network (DGNN), inspired by the Interior Penalty Discontinuous Galerkin Method (IPDGM). In this approach, the trial space consists of piecewise neural network space defined over the computational domain, while the test function space is composed of piecewise polynomials. We demonstrate the advantages of DGNN in terms of accuracy and training efficiency across several numerical examples, including stationary and time-dependent problems. Specifically, DGNN easily handles high perturbations, discontinuous solutions, and complex geometric domains.