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

2025-12-19

AI Summary

This work addresses the infinite-dimensional operator learning problem for noisy partial differential equations (PDEs). We propose RRFF-FEM—a novel framework integrating Regularized Random Fourier Features (RRFF) based on multivariate t-distribution sampling with finite element space reconstruction. To suppress high-frequency noise, we introduce frequency-weighted Tikhonov regularization and rigorously establish the well-conditioning of the random feature matrix and the generalization bound under an $m log m$ scaling. RRFF-FEM unifies the theoretical rigor of kernel methods with the scalability of neural operators. On benchmark PDEs—including advection, Burgers, Darcy flow, Helmholtz, Navier–Stokes, and structural mechanics—our method significantly outperforms unregularized random-feature baselines, achieving superior noise robustness, higher training efficiency, and accuracy on par with both kernel-based methods and state-of-the-art neural operators.

Technology Category

• Machine Learning: Kernel Methods
• Intelligent Robots: Learning & Optimization for ROB
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Learning operators from noisy data efficiently
• Regularizing high-frequency noise in operator learning
• Improving computational efficiency for large training sets

Innovation

• Regularized random Fourier features with Student's t distributions
• Frequency-weighted Tikhonov regularization suppresses high-frequency noise
• Finite element reconstruction map for robust operator learning