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

2026-03-03

AI Summary

Existing neural operators lack reliability and theoretical guarantees when handling out-of-distribution input functions. This work proposes an extended framework grounded in reproducing kernel Hilbert spaces (RKHS), leveraging kernel approximation techniques to achieve robust approximation of both out-of-distribution functions and their derivatives. The key innovation lies in establishing a theoretical connection between kernel selection and Sobolev eigenfunction spaces, thereby providing predictable guarantees on generalization error and derivative accuracy for neural operators. When applied to solving elliptic partial differential equations—particularly on manifolds represented as point clouds—the method demonstrates significantly enhanced geometric awareness, improved extrapolation accuracy, and greater computational efficiency.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Mixed Discrete/Continuous Search
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• neural operators
• out-of-distribution
• function extension
• RKHS
• Sobolev spaces

Innovation

• neural operators
• out-of-distribution generalization
• Reproducing Kernel Hilbert Space
• Sobolev Native Spaces
• kernel approximation