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

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.