Neural Operators as Efficient Function Interpolators

2026-05-08

AI Summary

This work addresses the challenges of efficiency and accuracy in interpolating high-dimensional, finite-dimensional functions for scientific computing by systematically reframing neural operators (NOs) as function interpolation tools. The approach introduces an auxiliary basis space, modeling finite-dimensional functions as operators acting on this space and learning their mapping relationships. By integrating tensorized Fourier neural operators (TFNO), multilayer perceptrons (MLPs), and Kolmogorov–Arnold networks, the method matches or exceeds state-of-the-art performance across multiple analytical benchmarks. Notably, in a nuclear mass model correction task, the TFNO-based ensemble achieves an extrapolation root-mean-square error of 198.2 keV with fewer parameters and lower training cost, significantly enhancing both applicability and parameter efficiency in scientific data modeling.

Abstract

Neural operators (NOs) are designed to learn maps between infinite-dimensional function spaces. We propose a novel reframing of their use. By introducing an auxiliary base-space, any finite-dimensional function can be viewed as an operator acting by composition on functions of the base-space. Through a range of benchmarks on analytic functions of increasing complexity and dimensionality, we demonstrate that NOs can match or outperform standard multilayer perceptrons and Kolmogorov--Arnold Networks in accuracy while requiring significantly fewer parameters and training time. As a real-world application, we apply a two-dimensional Tensorized Fourier Neural Operator (TFNO) to the nuclear chart, learning a correction to state-of-the-art nuclear mass models as a partially observed residual field. A TFNO ensemble reaches a held-out root-mean-square error of 198.2 keV, placing it among the best recent neural-network approaches while retaining high parameter efficiency and short training times. More broadly, these results introduce NOs as a scalable framework for finite-dimensional function interpolation, from analytic benchmarks to structured scientific data.

Problem

• function interpolation
• neural operators
• finite-dimensional functions
• scientific data
• parameter efficiency

Innovation

• Neural Operators
• Function Interpolation
• Tensorized Fourier Neural Operator
• Parameter Efficiency
• Scientific Machine Learning