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 methods to nonlinear PDEs like KdV and Burgers equations, introducing adaptive strategies for improved efficiency and accuracy.

Contribution

The paper develops a novel extension of LRNN-DG methods for nonlinear PDEs, incorporating adaptive domain decomposition and characteristic direction techniques.

Findings

High accuracy with fewer degrees of freedom

Adaptive domain decomposition improves efficiency

Characteristic direction approach enhances computational performance

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.