Discontinuity-capturing random feature method for interface problems

Interface problems in computational science present significant challenges for traditional numerical methods, particularly in handling complex geometries and interface conditions. While neural networks offer a mesh-free alternative, they suffer from costly optimization and local optima convergence. To address these limitations, we propose the Discontinuity-capturing random feature method (DC-RFM), a novel framework for interface problems. DC-RFM employs augmented variables to distinguish subdomains separated by prescribed interfaces and explicitly capture discontinuities. The governing equations are discretized at collocation points, forming a linear system where a least-squares loss function enforces partial differential equation residuals, initial/boundary conditions, and interface jump conditions. This approach simplifies geometric complexity to point sampling, preserving the robustness of mesh-free methods. Numerical experiments, spanning elliptic problems, Stokes flow, elasticity, evolving interface, multi-interface, and anisotropic cases, demonstrate the effectiveness and robustness of DC-RFM. Notably, DC-RFM achieves an order-of-magnitude reduction in degrees of freedom compared to traditional methods for discontinuous solutions, while decoupling the number of subdomains from computational cost to efficiently resolve multi-interface problems.