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.

翻译：本文提出了一种将局部随机神经网络（LRNNs）与杂交化间断Petrov-Galerkin（HDPG）方法相结合的新型数值方法，用于求解耦合流体流动问题。该方法将计算域划分为若干子域，并在每个子域上构建一个LRNN。随后，利用HDPG格式对各LRNN进行耦合，以逼近未知函数。我们基于速度-应力公式发展了LRNN-HDPG方法，用于求解两类问题：模拟多孔介质流与自由流的Stokes-Darcy问题以及Brinkman方程。我们设计了一种简单有效的方式处理Stokes-Darcy问题中的界面条件，无需在数值格式中添加额外项。我们进行了广泛的数值实验，证明了所提方法的稳定性、高效性和鲁棒性。数值结果表明，LRNN-HDPG方法能够在较少自由度下实现高精度。