Learning Contractive Integral Operators with Fredholm Integral Neural Operators

TL;DR

This paper introduces FREDINOs, a neural operator framework for learning contractive integral operators from Fredholm equations, with proven universality and applications to PDEs, ensuring convergence and high accuracy.

Contribution

The paper develops FREDINOs, a novel neural operator architecture that guarantees contractivity and universality for solving integral equations and PDEs.

Findings

FREDINOs are universal approximators of integral operators.

They are guaranteed to be contractive, ensuring convergence.

Numerical experiments show high accuracy on benchmark problems.

Abstract

We generalize the framework of Fredholm Neural Networks, to learn non-expansive integral operators arising in Fredholm Integral Equations (FIEs) of the second kind in arbitrary dimensions. We first present the proposed Fredholm Integral Neural Operators (FREDINOs), for FIEs and prove that they are universal approximators of linear and non-linear integral operators and corresponding solution operators. We furthermore prove that the learned operators are guaranteed to be contractive, thereby strictly satisfying the mathematical property required for the convergence of the fixed point scheme. Finally, we also demonstrate how FREDINOs can be used to learn the solution operator of non-linear elliptic PDEs, via a Boundary Integral Equation (BIE) formulation. We assess the proposed methodology numerically, via several benchmark problems: linear and non-linear FIEs in arbitrary dimensions, as well as a non-linear elliptic PDE in 2D. Built on tailored mathematical/numerical analysis theory, FREDINOs offer high-accuracy approximations and interpretable schemes, making them well suited for scientific machine learning/numerical analysis computations.