Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

DGNN: A Neural PDE Solver Induced by Discontinuous Galerkin Methods

Guanyu Chen, Shengze Xu, Dong Ni, Tieyong Zeng

TL;DR

DGNN introduces a neural PDE solver that inherits the discontinuous Galerkin philosophy by assigning a local neural surrogate to each mesh element and coupling them via numerically computed fluxes. By optimizing a weak-form loss built from per-element residuals, boundary terms, and penalties, DGNN achieves local conservation, robust performance on irregular geometries, and efficient training on nonuniform meshes. Across 1D/2D Poisson and Burgers tests, DGNN outperforms strong- and weak-form baselines in convergence speed, stability, and accuracy, particularly for high-frequency or discontinuous solutions. This framework offers a scalable, parallelizable, and physics-informed approach to PDE solving with practical impact for complex domains and multiscale problems, and code is available on GitHub.

Abstract

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.

DGNN: A Neural PDE Solver Induced by Discontinuous Galerkin Methods

TL;DR

DGNN introduces a neural PDE solver that inherits the discontinuous Galerkin philosophy by assigning a local neural surrogate to each mesh element and coupling them via numerically computed fluxes. By optimizing a weak-form loss built from per-element residuals, boundary terms, and penalties, DGNN achieves local conservation, robust performance on irregular geometries, and efficient training on nonuniform meshes. Across 1D/2D Poisson and Burgers tests, DGNN outperforms strong- and weak-form baselines in convergence speed, stability, and accuracy, particularly for high-frequency or discontinuous solutions. This framework offers a scalable, parallelizable, and physics-informed approach to PDE solving with practical impact for complex domains and multiscale problems, and code is available on GitHub.

Abstract

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.

Paper Structure

This paper contains 21 sections, 39 equations, 30 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Strong Form Solver
  4. Weak Form Solver
  5. Discontinuous Galerkin Method
  6. Discontinuous Galerkin Neural Network
  7. The Framework of DGNN
  8. Loss Functions
  9. Algorithm and Details
  10. Test Functions: Fewer.
  11. Test domains: Fewer and more flexible.
  12. Training: Fewer parameters and higher parallelism.
  13. Experiments
  14. Experimental setups
  15. 1D Poisson Equation
  16. ...and 6 more sections

Figures (30)

  • Figure 1: The sketch structure of DGNN. (a) The whole surrogate solver. The surrogate solver consists of N modules. Each module has a domain mesh and a block network. (b) Components of Loss Function. With N modules, we can compute N local weak-form equations and corresponding Numerical Flux. (c) An example of a discretized domain. (d)(e) The corresponding local solution and test functions. (f) The local weak forms and numerical flux.
  • Figure 3: Net structure. The left column (a) is the "All in One" solver; The right column (b) is the "Divide and Conquer" solver we proposed. Here, the mesh gate is used to decide which block the estimate point belongs to. Actually, during the training process, the mesh gate is virtualized. And the red-boxed area is a scratch of the parallel linear layer.
  • Figure : (a) 1D Test Functions
  • Figure : MSE vs. Time
  • Figure : MSE vs. Time
  • ...and 25 more figures

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5