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Learning Contractive Integral Operators with Fredholm Integral Neural Operators

arXiv 2604.03034 ↗First posted 3 Apr 2026math.NA cs.LG cs.NAcited by 0

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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.

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References · 12

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  3. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 349
  4. 2024RiemannONets: Interpretable Neural Operators for Riemann Problemscited by 28
  5. 2022Neural Integral Equationscited by 23
  6. 2024RandONet: Shallow-Networks with Random Projections for learning linear and nonlinear operatorscited by 16
  7. 2025Enabling Local Neural Operators to perform Equation-Free System-Level Analysiscited by 10
  8. 2022A Recursively Recurrent Neural Network (R2N2) Architecture for Learning Iterative Algorithmscited by 8
  9. 2025Operator Learning at Machine Precisioncited by 7
  10. 2024Systematic construction of continuous-time neural networks for linear dynamical systemscited by 1
  11. 2024Self-Supervised Multi-Scale Network for Blind Image Deblurring via Alternating Optimizationcited by 1
  12. 2025Fredholm Neural Networks for inverse problems in elliptic PDEscited by 1

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