Learning Contractive Integral Operators with Fredholm Integral Neural Operators
This work addresses the need for interpretable and mathematically rigorous operators in scientific machine learning and numerical analysis, offering a novel framework for solving integral equations and PDEs, though it is incremental in extending existing neural network methods.
The authors tackled the problem of learning contractive integral operators for Fredholm Integral Equations by proposing Fredholm Integral Neural Operators (FREDINOs), which are proven to be universal approximators and guarantee contractivity, leading to high-accuracy approximations in benchmark problems including linear and non-linear equations in arbitrary dimensions and a 2D elliptic PDE.
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.