Machine-precision solution of Fredholm integral equations via barycentric physics-informed method

Abstract

This paper presents a barycentric physics-informed (BPI) computational method for solving second-kind Fredholm integral equations. Unlike conventional Multi-Layer Perceptrons (MLPs) used in Physics-Informed Neural Networks (PINNs) that introduce inherent barriers to higher precision, the BPI method employs barycentric interpolation on Chebyshev nodes as the underlying approximation structure. The method treats function values at Chebyshev nodes as trainable parameters and enforces integral constraints through collocation with Clenshaw-Curtis quadrature for integral evaluation. The approximate solution is constructed using the barycentric interpolation formula with parameters optimized to minimize the integral equation residual across the domain. The BPI method achieves 10 to 25 orders of magnitude improvement over traditional neural networks. Comparative analysis against the classical Nyström method shows that BPI achieves comparable accuracy for smooth kernels and superior performance for some singular kernel. Comprehensive numerical experiments demonstrate the method’s effectiveness across diverse problem classes.