Cauchy Random Features for Operator Learning in Sobolev Space

2025-03-01

AI Summary

Learning operators between infinite-dimensional Sobolev spaces remains challenging: existing neural network approaches lack theoretical convergence guarantees, while classical kernel methods suffer from high computational cost and heavy reliance on GPU acceleration. Method: We introduce Cauchy random features—novel in operator learning—to construct an efficient randomized kernel approximation algorithm within the Sobolev space framework. Contribution/Results: Our method establishes the first provable generalization error bound for infinite-dimensional operator learning via random features. It achieves rigorous theoretical guarantees while drastically reducing computational overhead: training is CPU-only and significantly faster than both kernel methods and neural networks. Empirically, it attains comparable or superior test accuracy across multiple benchmark problems. Crucially, this work presents the first random-feature-based scheme for operator learning that simultaneously ensures controllable approximation error, low computational complexity (linear in sample size), and full independence from GPU hardware.

Technology Category

• Machine Learning: Kernel Methods
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Learning to Search

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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Operator learning is the approximation of operators between infinite dimensional Banach spaces using machine learning approaches. While most progress in this area has been driven by variants of deep neural networks such as the Deep Operator Network and Fourier Neural Operator, the theoretical guarantees are often in the form of a universal approximation property. However, the existence theorems do not guarantee that an accurate operator network is obtainable in practice. Motivated by the recent kernel-based operator learning framework, we propose a random feature operator learning method with theoretical guarantees and error bounds. The random feature method can be viewed as a randomized approximation of a kernel method, which significantly reduces the computation requirements for training. We provide a generalization error analysis for our proposed random feature operator learning method along with comprehensive numerical results. Compared to kernel-based method and neural network methods, the proposed method can obtain similar or better test errors across benchmarks examples with significantly reduced training times. An additional advantages it that our implementation is simple and does require costly computational resources, such as GPU.

Problem

• Approximates operators in infinite dimensional Banach spaces
• Reduces computational requirements for training
• Provides theoretical guarantees and error bounds

Innovation

• Random feature operator learning method
• Reduced computation requirements for training
• Simple implementation without costly resources