Regularized Random Fourier Features and Finite Element Reconstruction for Operator Learning in Sobolev Space
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Abstract (arXiv)
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.
Key facts. Cited by 2 papers in the map (data through 14 Oct 2026). Builds on 19 of its 37 references that are in the map. First posted on arXiv 19 Dec 2025.
Builds on
19 of 37 references.
References · 19
- 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
- 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
- 2020The Random Feature Model for Input-Output Maps between Banach Spacescited by 90
- 2020DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problemscited by 62
- 2023Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learningcited by 57
- 2023Kernel Methods are Competitive for Operator Learningcited by 50
- 2023Error Bounds for Learning with Vector-Valued Random Featurescited by 23
- 2021Generalization Bounds for Sparse Random Feature Expansionscited by 22
- 2024Operator Learning Using Random Features: A Tool for Scientific Computingcited by 21
- 2024Operator Learning with Gaussian Processescited by 18
- 2024Neural Scaling Laws of Deep ReLU and Deep Operator Network: A Theoretical Studycited by 14
- 2022HARFE: Hard-Ridge Random Feature Expansioncited by 12
7 more
- 2024Generalization Error Guaranteed Auto-Encoder-Based Nonlinear Model Reduction for Operator Learningcited by 12
- 2022Random Feature Models for Learning Interacting Dynamical Systemscited by 10
- 2022Concentration of Random Feature Matrices in High-Dimensionscited by 9
- 2025A Deep Learning Framework for Multi-Operator Learning: Architectures and Approximation Theorycited by 9
- 2022Local approximation of operatorscited by 8
- 2025Cauchy Random Features for Operator Learning in Sobolev Spacecited by 5
- 2024Differentially Private Random Feature Modelcited by 2
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Citing papers per year: 2026: 2.
2 papers.