Extending Neural Operators: Robust Handling of Functions Beyond the Training Set
This work addresses the robustness of neural operators for scientific computing applications where input functions may deviate from training data, though it appears incremental in extending existing operator learning methods.
The authors tackled the problem of extending neural operators to handle out-of-distribution input functions by developing a rigorous framework using kernel approximation techniques and RKHS theory, with empirical validation on elliptic PDEs involving manifolds represented as point clouds.
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.