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Discontinuity-capturing random feature method for interface problems
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AI
DOI：10.1016/j.cma.2026.118841.png)
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Abstract
En 中文
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.
Keywords:
Discontinuity-capturing
Random feature method
Interface problems
Mesh-free methods
Least-squares loss
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