Accurate modeling of complex physical problems, such as fluid-structure interaction, requires multiphysics coupling across the interface, which often has intricate geometry and dynamic boundaries. Conventional numerical methods face challenges in handling interface conditions. Deep neural networks offer a mesh-free and flexible alternative, but they suffer from drawbacks such as time-consuming optimization and local optima. In this paper, we propose a mesh-free approach based on Randomized Neural Networks (RNNs), which avoid optimization solvers during training, making them more efficient than traditional deep neural networks. Our approach, called Local Randomized Neural Networks (LRNNs), uses different RNNs to approximate solutions in different subdomains. We discretize the interface problem into a linear system at randomly sampled points across the domain, boundary, and interface using a finite difference scheme, and then solve it by a least-square method. For time-dependent interface problems, we use a space-time approach based on LRNNs. We show the effectiveness and robustness of the LRNNs methods through numerical examples of elliptic and parabolic interface problems. We also demonstrate that our approach can handle high-dimension interface problems. Compared to conventional numerical methods, our approach achieves higher accuracy with fewer degrees of freedom, eliminates the need for complex interface meshing and fitting, and significantly reduces training time, outperforming deep neural networks.

翻译：复杂物理问题（如流固耦合）的精确建模需要跨越界面进行多物理场耦合，而界面通常具有复杂几何形状和动态边界。传统数值方法在处理界面条件时面临挑战。深度神经网络提供了一种无网格且灵活的替代方案，但其存在优化耗时和易陷入局部最优等缺陷。本文提出一种基于随机神经网络（RNN）的无网格方法，该方法在训练过程中无需优化求解器，因此比传统深度神经网络更高效。我们提出的方法称为局部随机神经网络（LRNN），它使用不同RNN近似不同子域内的解。通过有限差分格式，我们在域内、边界和界面上随机采样点处将界面问题离散化为线性系统，并采用最小二乘法求解。对于时间依赖的界面问题，我们采用基于LRNN的时空方法。通过椭圆和抛物线型界面问题的数值算例，我们展示了LRNN方法的有效性和鲁棒性。同时，我们证明了该方法能够处理高维界面问题。与传统数值方法相比，本方法以更少的自由度实现了更高精度，无需复杂的界面网格划分和拟合，且显著缩减了训练时间，性能优于深度神经网络。