Regularized Random Fourier Features and Finite Element Reconstruction for Operator Learning in Sobolev Space

TL;DR

This paper introduces a regularized random Fourier feature method combined with finite element reconstruction for operator learning in Sobolev spaces, improving robustness to noise and computational efficiency.

Contribution

It proposes a novel RRFF-FEM approach using Student's t-distributed features and frequency-weighted Tikhonov regularization, with theoretical guarantees and practical advantages over existing methods.

Findings

Robust to noisy data in PDE operator learning

Achieves better performance with less training time

Maintains competitive accuracy compared to kernel and neural methods

Abstract

Operator learning is a data-driven approximation of mappings between infinite-dimensional function spaces, such as the solution operators of partial differential equations. Kernel-based operator learning can offer accurate, theoretically justified approximations that require less training than standard methods. However, they can become computationally prohibitive for large training sets and can be sensitive to noise. We propose a regularized random Fourier feature (RRFF) approach, coupled with a finite element reconstruction map (RRFF-FEM), for learning operators from noisy data. The method uses random features drawn from multivariate Student's $t$ distributions, together with frequency-weighted Tikhonov regularization that suppresses high-frequency noise. We establish high-probability bounds on the extreme singular values of the associated random feature matrix and show that when the number of features $N$ scales like $m \log m$ with the number of training samples $m$, the system is well-conditioned, which yields estimation and generalization guarantees. Detailed numerical experiments on benchmark PDE problems, including advection, Burgers', Darcy flow, Helmholtz, Navier-Stokes, and structural mechanics, demonstrate that RRFF and RRFF-FEM are robust to noise and achieve improved performance with reduced training time compared to the unregularized random feature model, while maintaining competitive accuracy relative to kernel and neural operator tests.