# Cauchy Random Features for Operator Learning in Sobolev Space

arXiv: https://arxiv.org/abs/2503.00300 · HTML: https://mlsanity.com/paper/2503.00300/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 54 references that are in the map.
- First posted on arXiv 1 Mar 2025.

## Builds on (most cited first, 30 of 32)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2208.05419/) (2022)
- [Generalization Properties of Learning with Random Features](https://mlsanity.com/paper/1602.04474/) (2016)
- [Generalization error of random features and kernel methods: hypercontractivity and kernel matrix concentration](https://mlsanity.com/paper/2101.10588/) (2021)
- [Random Features for Kernel Approximation: A Survey on Algorithms, Theory, and Beyond](https://mlsanity.com/paper/2004.11154/) (2020)
- [Reliable extrapolation of deep neural operators informed by physics or sparse observations](https://mlsanity.com/paper/2212.06347/) (2022)
- [Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learning](https://mlsanity.com/paper/2305.19913/) (2023)
- [Kernel Methods are Competitive for Operator Learning](https://mlsanity.com/paper/2304.13202/) (2023)
- [Towards a Foundation Model for Partial Differential Equations: Multi-Operator Learning and Extrapolation](https://mlsanity.com/paper/2404.12355/) (2024)
- [Deep Nonparametric Estimation of Operators between Infinite Dimensional Spaces](https://mlsanity.com/paper/2201.00217/) (2022)
- [Scalable Uncertainty Quantification for Deep Operator Networks using Randomized Priors](https://mlsanity.com/paper/2203.03048/) (2022)
- [PROSE-FD: A Multimodal PDE Foundation Model for Learning Multiple Operators for Forecasting Fluid Dynamics](https://mlsanity.com/paper/2409.09811/) (2024)
- [On the Approximation Power of Two-Layer Networks of Random ReLUs](https://mlsanity.com/paper/2102.02336/) (2021)
- [VICON: Vision In-Context Operator Networks for Multi-Physics Fluid Dynamics Prediction](https://mlsanity.com/paper/2411.16063/) (2024)
- [Error Bounds for Learning with Vector-Valued Random Features](https://mlsanity.com/paper/2305.17170/) (2023)
- [Deep Neural Networks Are Effective At Learning High-Dimensional Hilbert-Valued Functions From Limited Data](https://mlsanity.com/paper/2012.06081/) (2020)
- [Generalization Bounds for Sparse Random Feature Expansions](https://mlsanity.com/paper/2103.03191/) (2021)
- [Reduced operator inference for nonlinear partial differential equations](https://mlsanity.com/paper/2102.00083/) (2021)
- [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024)
- [PPDONet: Deep Operator Networks for Fast Prediction of Steady-State Solutions in Disk-Planet Systems](https://mlsanity.com/paper/2305.11111/) (2023)
- [BCAT: A Block Causal Transformer for PDE Foundation Models for Fluid Dynamics](https://mlsanity.com/paper/2501.18972/) (2025)
- [HARFE: Hard-Ridge Random Feature Expansion](https://mlsanity.com/paper/2202.02877/) (2022)
- [Generalization Error Guaranteed Auto-Encoder-Based Nonlinear Model Reduction for Operator Learning](https://mlsanity.com/paper/2401.10490/) (2024)
- [DeepONet as a Multi-Operator Extrapolation Model: Distributed Pretraining with Physics-Informed Fine-Tuning](https://mlsanity.com/paper/2411.07239/) (2024)
- [Random Feature Models for Learning Interacting Dynamical Systems](https://mlsanity.com/paper/2212.05591/) (2022)
- [SHRIMP: Sparser Random Feature Models via Iterative Magnitude Pruning](https://mlsanity.com/paper/2112.04002/) (2021)
- [Concentration of Random Feature Matrices in High-Dimensions](https://mlsanity.com/paper/2204.06935/) (2022)
- [A practical existence theorem for reduced order models based on convolutional autoencoders](https://mlsanity.com/paper/2402.00435/) (2024)

## Built on by (most cited first, 5 of 5)

- [Regularized Random Fourier Features and Finite Element Reconstruction for Operator Learning in Sobolev Space](https://mlsanity.com/paper/2512.17884/) (2025)
- [Graph-Based Operator Learning from Limited Data on Irregular Domains](https://mlsanity.com/paper/2505.18923/) (2025)
- [Near-Optimal Learning of Gaussian Sobolev Operators](https://mlsanity.com/paper/2607.11921/) (2026)
- [Kernel Methods for Learning Operators with Multiple Inputs and Outputs](https://mlsanity.com/paper/2608.11831/) (2026)
- [Physics Informed Random Feature Neural Networks for Solving PDEs](https://mlsanity.com/paper/2609.16406/) (2026)

## Abstract (arXiv)

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.