Local Randomized Neural Networks Methods for Interface Problems

TL;DR

This paper introduces Local Randomized Neural Networks (LRNNs), a mesh-free, efficient, and accurate method for solving complex interface problems in physics, outperforming traditional neural networks and numerical methods.

Contribution

The paper proposes LRNNs, a novel mesh-free neural network approach that avoids optimization, handles high-dimensional and dynamic interface problems, and improves efficiency and accuracy over existing methods.

Findings

LRNNs outperform traditional neural networks in accuracy and training time.

LRNNs effectively handle high-dimensional and dynamic interface problems.

Numerical examples demonstrate robustness and efficiency of LRNNs.

Abstract

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.