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.

翻译：文献[42]中提出的局部随机神经网络与间断伽辽金（LRNN-DG）方法最初是为求解线性偏微分方程而设计的。本文将该方法推广至非线性偏微分方程的求解，特别针对Korteweg-de Vries（KdV）方程和Burgers方程，采用了时空一体化求解策略。此外，我们引入了自适应区域分解和特征方向法以提升所提方法的计算效率。数值实验表明，所提方法能够在较少自由度下实现高精度计算，同时自适应区域分解与特征方向法的应用显著提升了计算效率。