mlsanityData to 7 Oct 2026

Extending Neural Operators: Robust Handling of Functions Beyond the Training Set

arXiv 2603.03621 ↗First posted 4 Mar 2026cs.LG cs.CV cs.NAcited by 0

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Abstract (arXiv)

We develop a rigorous framework for extending neural operators to handle out-of-distribution input functions. We leverage kernel approximation techniques and provide theory for characterizing the input-output function spaces in terms of Reproducing Kernel Hilbert Spaces (RKHSs). We provide theorems on the requirements for reliable extensions and their predicted approximation accuracy. We also establish formal relationships between specific kernel choices and their corresponding Sobolev Native Spaces. This connection further allows the extended neural operators to reliably capture not only function values but also their derivatives. Our methods are empirically validated through the solution of elliptic partial differential equations (PDEs) involving operators on manifolds having point-cloud representations and handling geometric contributions. We report results on key factors impacting the accuracy and computational performance of the extension approaches.

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References · 16

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,131
  3. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  4. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  5. 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
  6. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 218
  7. 2023Manifold learning: what, how, and whycited by 66
  8. 2016Deep nets for local manifold learningcited by 26
  9. 2021Two-layer neural networks with values in a Banach spacecited by 26
  10. 2025Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learningcited by 11
  11. 2024A Mathematical Analysis of Neural Operator Behaviorscited by 8
  12. 2019Meshfree Methods on Manifolds for Hydrodynamic Flows on Curved Surfaces: A Generalized Moving Least-Squares (GMLS) Approachcited by 7
4 more
  1. 2022GD-VAEs: Geometric Dynamic Variational Autoencoders for Learning Nonlinear Dynamics and Dimension Reductionscited by 7
  2. 2023Fast evaluation of spherical harmonics with sphericartcited by 6
  3. 2025Transferable Foundation Models for Geometric Tasks on Point Cloud Representations: Geometric Neural Operatorscited by 1
  4. 2025Hilbert Neural Operator: Operator Learning in the Analytic Signal Domaincited by 1

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