  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.
