Special Session 39: Recent Developments in Gradient Flows: Theory, Numerics, and Applications

Adaptive feature capture method for solving partial differential equations with near singular solutions
Xiaoping Wang
The Chinese University of Hong Kong (Shenzhen) /Shenzhen International Center for Industrial and Applied Mathematics
Peoples Rep of China
Co-Author(s):    Xiaoping Wang
Abstract:
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. AFCM employs 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.