# Learning Contractive Integral Operators with Fredholm Integral Neural Operators

arXiv: https://arxiv.org/abs/2604.03034 · HTML: https://mlsanity.com/paper/2604.03034/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 25 references that are in the map.
- First posted on arXiv 3 Apr 2026.

## Builds on (most cited first, 12 of 12)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [RiemannONets: Interpretable Neural Operators for Riemann Problems](https://mlsanity.com/paper/2401.08886/) (2024)
- [Neural Integral Equations](https://mlsanity.com/paper/2209.15190/) (2022)
- [RandONet: Shallow-Networks with Random Projections for learning linear and nonlinear operators](https://mlsanity.com/paper/2406.05470/) (2024)
- [Enabling Local Neural Operators to perform Equation-Free System-Level Analysis](https://mlsanity.com/paper/2505.02308/) (2025)
- [A Recursively Recurrent Neural Network (R2N2) Architecture for Learning Iterative Algorithms](https://mlsanity.com/paper/2211.12386/) (2022)
- [Operator Learning at Machine Precision](https://mlsanity.com/paper/2511.19980/) (2025)
- [Systematic construction of continuous-time neural networks for linear dynamical systems](https://mlsanity.com/paper/2403.16215/) (2024)
- [Self-Supervised Multi-Scale Network for Blind Image Deblurring via Alternating Optimization](https://mlsanity.com/paper/2409.00988/) (2024)
- [Fredholm Neural Networks for inverse problems in elliptic PDEs](https://mlsanity.com/paper/2507.06038/) (2025)

## Abstract (arXiv)

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.