Deep Adaptive Numerical Scheme for Nonlinear Hammerstein Integral Equations with Automatic Error Control

Abstract

This paper presents the Deep Adaptive Collocation Method (DACM) for nonlinear Hammerstein integral equations.  DACM uses both traditional discretisation and a neural module that adaptively refines the mesh based on local residuals. This increases node density in areas where the solution changes quickly.  This gets more accurate with fewer nodes, and a theoretical analysis shows that it is consistent, stable, and convergent.A number of numerical experiments show that the proposed DACM reduces the maximum error by up to an order of magnitude compared to uniform collocation, while still being efficient in terms of computation. The results confirm that neural-guided adaptivity provides a powerful mechanism for enhancing classical numerical algorithms applied to nonlinear integral equations.