Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Xinyue Yu, Hayden Schaeffer

TL;DR

This paper tackles operator learning for PDE mappings under noisy data by introducing Regularized Random Fourier Features (RRFF) paired with a Finite Element Recovery Map (RRFF-FEM). It establishes high-probability conditioning bounds for the random feature matrix when frequencies are drawn from multivariate Student's t distributions, showing that N should scale like m log m for stability. The authors provide extensive numerical experiments across advection, Burgers', Darcy, Helmholtz, Navier–Stokes, and structural mechanics problems, demonstrating that RRFF and RRFF-FEM outperform unregularized variants and offer competitive accuracy with reduced training time versus kernel and neural operator baselines. The results highlight improved noise robustness and practical viability of RRFF-FEM for multi-query operator learning in Sobolev-type spaces.

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.

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

TL;DR

This paper tackles operator learning for PDE mappings under noisy data by introducing Regularized Random Fourier Features (RRFF) paired with a Finite Element Recovery Map (RRFF-FEM). It establishes high-probability conditioning bounds for the random feature matrix when frequencies are drawn from multivariate Student's t distributions, showing that N should scale like m log m for stability. The authors provide extensive numerical experiments across advection, Burgers', Darcy, Helmholtz, Navier–Stokes, and structural mechanics problems, demonstrating that RRFF and RRFF-FEM outperform unregularized variants and offer competitive accuracy with reduced training time versus kernel and neural operator baselines. The results highlight improved noise robustness and practical viability of RRFF-FEM for multi-query operator learning in Sobolev-type spaces.

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 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 scales like with the number of training samples , 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.

Paper Structure

This paper contains 16 sections, 1 theorem, 46 equations, 45 figures, 10 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Random Feature Model
  4. Finite Element Recovery Map
  5. Conditioning of Random Feature Matrix
  6. Background
  7. Main Result
  8. Numerical Experiments
  9. Advection Equations i, ii, and iii
  10. Burgers' Equation
  11. Darcy Flow
  12. Helmholtz Equation
  13. Navier-Stokes Equation
  14. Structural Mechanics
  15. Discussion of Results
  16. ...and 1 more sections

Key Result

Theorem 3.1

Let $D \subset \mathbb{R}^d$ be a compact domain. Assume the data $\{\mathbf{x}_j\}_{j\in[m]}\subset D$, the random feature weights $\{\boldsymbol{\omega}_k\}_{k\in[N]}$, and the random feature matrix $\mathbf{A}\in\mathbb{C}^{m\times N}$ satisfy Assumption rf_assumption. Suppose the following condi for some $\delta,\eta \in(0,1)$, where $C>0$ is a universal constant and $K_{\nu/2}(x)$ is the modi

Figures (45)

  • Figure 1: Commutative diagram of the operator learning framework following mhaskar2023localapproximationbartolucci2023representationequivalentneuraloperatorsbatlle2023kernelmethodscompetitiveoperatorliu2025generalizationerrorliao2025cauchyrandomfeaturesoperator.
  • Figure 2: Advection i: (left to right) An example of (a) training input with 5% noise, (b) training output with 5% noise, (c) test example and predictions of $R_\mathcal{V}\circ\hat{f}$ using RFF-FEM-$\infty$ and RRFF-FEM-$\infty$, (d) pointwise errors for predictions of $R_\mathcal{V}\circ\hat{f}$ using RFF-FEM-$\infty$ and RRFF-FEM-$\infty$.
  • Figure 3: Advection i: (left to right) (a) test example and predictions of $R_\mathcal{V}\circ\hat{f}$ using RFF-FEM-2 and RRFF-FEM-2, (b) pointwise errors for predictions of $R_\mathcal{V}\circ\hat{f}$ using RFF-FEM-2 and RRFF-FEM-2, (c) test example and predictions of $R_\mathcal{V}\circ\hat{f}$ using RFF-FEM-3 and RRFF-FEM-3, (d) pointwise errors for predictions of $R_\mathcal{V}\circ\hat{f}$ using RFF-FEM-3 and RRFF-FEM-3.
  • Figure 4: Advection ii: (left to right) An example of (a) training input with 5% noise, (b) training output with 5% noise, (c) test example and predictions of $R_\mathcal{V}\circ\hat{f}$ using RFF-FEM-$\infty$ and RRFF-FEM-$\infty$, (d) pointwise errors for predictions of $R_\mathcal{V}\circ\hat{f}$ using RFF-FEM-$\infty$ and RRFF-FEM-$\infty$.
  • Figure 5: Advection ii: (left to right) (a) test example and predictions of $R_\mathcal{V}\circ\hat{f}$ using RFF-FEM-2 and RRFF-FEM-2, (b) pointwise errors for predictions of $R_\mathcal{V}\circ\hat{f}$ using RFF-FEM-2 and RRFF-FEM-2, (c) test example and predictions of $R_\mathcal{V}\circ\hat{f}$ using RFF-FEM-3 and RRFF-FEM-3, (d) pointwise errors for predictions of $R_\mathcal{V}\circ\hat{f}$ using RFF-FEM-3 and RRFF-FEM-3.
  • ...and 40 more figures

Theorems & Definitions (3)

  • Remark 1
  • Theorem 3.1: Concentration Property of Random Feature Matrix
  • proof : Proof of Theorem \ref{['thm:concentration']}