Adaptive feature capture method for solving partial differential equations with near singular solutions

Abstract

Partial differential equations (PDEs) with near singular solutions pose significant challenges for traditional numerical methods, particularly in complex geometries where mesh generation and adaptive refinement become computationally expensive. Although deep-learning-based approaches, such as Physics-Informed Neural Networks (PINNs) and the Random Feature Method (RFM), offer mesh-free alternatives, they often lack adaptive resolution in critical regions, limiting their accuracy for solutions with steep gradients or singularities. In this work, we propose the Adaptive Feature Capture Method (AFCM), a novel machine learning framework that adaptively redistributes neurons and collocation points in high-gradient regions to enhance local expressive power. Inspired by adaptive moving mesh techniques, AFCM uses the gradient norm of an approximate solution as a monitor function to guide the reinitialization of feature function parameters. This ensures that partition hyperplanes and collocation points cluster where they are most needed, achieving higher resolution without increasing computational overhead. The AFCM extends the capabilities of RFM to handle PDEs with near-singular solutions while preserving its mesh-free efficiency. Numerical experiments demonstrate the method's effectiveness in accurately resolving near-singular problems with a performance that is better than that of the traditional finite element method in terms of accuracy and efficiency. AFCM offers a robust and scalable approach to solving challenging PDEs in scientific and engineering applications.