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Local Randomized Neural Networks with Discontinuous Galerkin Methods for KdV-type and Burgers Equations

Jingbo Sun, Fei Wang

TL;DR

This paper extends Local Randomized Neural Networks with Discontinuous Galerkin (LRNN-DG) to nonlinear PDEs, applying a space-time DG framework to the KdV-type and Burgers equations. It introduces two coupling schemes, LRNN-DG and LRNN-$C^1$DG, to integrate local RNNs across space-time subdomains, and augments the approach with adaptive domain decomposition and a characteristic-direction mesh strategy. The methods couple randomized neural bases with least-squares training for the output layer and Newton/Picard iterations for nonlinear terms, achieving high accuracy with relatively few degrees of freedom. Results on KdV solitons, double soliton collisions, and Burgers equations show robust performance, with adaptive and direction-aware meshes significantly improving efficiency and accuracy, suggesting strong potential for time-dependent nonlinear PDEs in multi-physics contexts.

Abstract

The Local Randomized Neural Networks with Discontinuous Galerkin (LRNN-DG) methods, introduced in [42], were originally designed for solving linear partial differential equations. In this paper, we extend the LRNN-DG methods to solve nonlinear PDEs, specifically the Korteweg-de Vries (KdV) equation and the Burgers equation, utilizing a space-time approach. Additionally, we introduce adaptive domain decomposition and a characteristic direction approach to enhance the efficiency of the proposed methods. Numerical experiments demonstrate that the proposed methods achieve high accuracy with fewer degrees of freedom, additionally, adaptive domain decomposition and a characteristic direction approach significantly improve computational efficiency.

Local Randomized Neural Networks with Discontinuous Galerkin Methods for KdV-type and Burgers Equations

TL;DR

This paper extends Local Randomized Neural Networks with Discontinuous Galerkin (LRNN-DG) to nonlinear PDEs, applying a space-time DG framework to the KdV-type and Burgers equations. It introduces two coupling schemes, LRNN-DG and LRNN-DG, to integrate local RNNs across space-time subdomains, and augments the approach with adaptive domain decomposition and a characteristic-direction mesh strategy. The methods couple randomized neural bases with least-squares training for the output layer and Newton/Picard iterations for nonlinear terms, achieving high accuracy with relatively few degrees of freedom. Results on KdV solitons, double soliton collisions, and Burgers equations show robust performance, with adaptive and direction-aware meshes significantly improving efficiency and accuracy, suggesting strong potential for time-dependent nonlinear PDEs in multi-physics contexts.

Abstract

The Local Randomized Neural Networks with Discontinuous Galerkin (LRNN-DG) methods, introduced in [42], were originally designed for solving linear partial differential equations. In this paper, we extend the LRNN-DG methods to solve nonlinear PDEs, specifically the Korteweg-de Vries (KdV) equation and the Burgers equation, utilizing a space-time approach. Additionally, we introduce adaptive domain decomposition and a characteristic direction approach to enhance the efficiency of the proposed methods. Numerical experiments demonstrate that the proposed methods achieve high accuracy with fewer degrees of freedom, additionally, adaptive domain decomposition and a characteristic direction approach significantly improve computational efficiency.
Paper Structure (11 sections, 45 equations, 10 figures, 9 tables)

This paper contains 11 sections, 45 equations, 10 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Network Structure and Notation
  3. Randomized Neural Networks
  4. Notation for Space-Time DG Formulation
  5. LRNN-DG Methods for KdV Equations
  6. LRNN-DG Formulation
  7. LRNN-$C^1$DG Formulation
  8. LRNN-DG Methods for the Burgers Equation
  9. Adaptive Domain Decomposition
  10. Numerical Examples
  11. Summary

Figures (10)

  • Figure 1: The structure of a randomized neural network
  • Figure 2: Errors of LRNN-$C^1$DG methods on uniform and adaptive meshes in Example \ref{['ex_gKdV']}.
  • Figure 3: The performances of the adaptive LRNN-$C^1$DG method in Example \ref{['ex_gKdV']}.
  • Figure 4: Performances of the LRNN-DG method on the characteristic mesh in Example \ref{['ex_gKdV']}.
  • Figure 5: Performances of the LRNN-$C^1$DG method on the characteristic mesh in Example \ref{['ex_gKdV']}.
  • ...and 5 more figures

Theorems & Definitions (6)

  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Example 6.1: Generalized KdV Equation
  • Example 6.2: KdV Equation with Double Solitons Collision
  • Example 6.3: 2D Burgers Equation