Local randomized neural networks with finite difference methods for interface problems

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 (RaNNs) and finite difference methods (FDM), which avoid optimization solvers during training, making them more efficient than traditional deep neural networks. Our approach, called Local Randomized Neural Networks with finite difference methods (LRaNN-FDM), uses different RaNNs 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. Unlike automatic differentiation for partial derivative calculations, the finite difference approach offers significantly faster computation. For time-dependent interface problems, we use a space-time approach based on LRaNNs. We show the effectiveness and robustness of the LRaNN-FDM 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.