Local randomized neural networks with hybridized discontinuous Petrov–Galerkin methods for Stokes–Darcy flows

This paper introduces a new numerical approach that integrates local randomized neural networks (LRNNs) and the hybridized discontinuous Petrov–Galerkin (HDPG) method for solving coupled fluid flow problems. The proposed method partitions the domain of interest into several subdomains and constructs an LRNN on each subdomain. Then, the HDPG scheme is used to couple the LRNNs to approximate the unknown functions. We develop local randomized neural networks with hybridized discontinuous Petrov–Galerkin (LRNN-HDPG) methods based on velocity-stress formulation to solve two types of problems: Stokes–Darcy problems and Brinkman equations, which model the flow in porous media and free flow. We devise a simple and effective way to deal with the interface conditions in the Stokes–Darcy problems without adding extra terms to the numerical scheme. We conduct extensive numerical experiments to demonstrate the stability, efficiency, and robustness of the proposed method. The numerical results show that the LRNN-HDPG method can achieve high accuracy with a small number of degrees of freedom.