  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 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.
