[2603.03621] Extending Neural Operators: Robust Handling of Functions Beyond the Training Set

Computer Science > Machine Learning arXiv:2603.03621 (cs) [Submitted on 4 Mar 2026] Title:Extending Neural Operators: Robust Handling of Functions Beyond the Training Set Authors:Blaine Quackenbush, Paul J. Atzberger View a PDF of the paper titled Extending Neural Operators: Robust Handling of Functions Beyond the Training Set, by Blaine Quackenbush and Paul J. Atzberger View PDF HTML (experimental) Abstract: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. Comments: Subjects: Machine Learning (cs.LG); Computer Vision and Pattern Recognition (cs.CV); Numerical Analysis (...