# Operator Learning Using Random Features: A Tool for Scientific Computing

arXiv: https://arxiv.org/abs/2408.06526 · HTML: https://mlsanity.com/paper/2408.06526/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 65 of its 147 references that are in the map.
- First posted on arXiv 12 Aug 2024.

## Builds on (most cited first, 30 of 65)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Generalization Bounds of Stochastic Gradient Descent for Wide and Deep Neural Networks](https://mlsanity.com/paper/1905.13210/) (2019)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [Generalization Properties of Learning with Random Features](https://mlsanity.com/paper/1602.04474/) (2016)
- [Survey of multifidelity methods in uncertainty propagation, inference, and optimization](https://mlsanity.com/paper/1806.10761/) (2018)
- [On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions](https://mlsanity.com/paper/1502.06800/) (2015)
- [Generalization error of random features and kernel methods: hypercontractivity and kernel matrix concentration](https://mlsanity.com/paper/2101.10588/) (2021)
- [On the Power and Limitations of Random Features for Understanding Neural Networks](https://mlsanity.com/paper/1904.00687/) (2019)
- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification](https://mlsanity.com/paper/1802.00850/) (2018)
- [A Theoretical Analysis of Deep Neural Networks and Parametric PDEs](https://mlsanity.com/paper/1904.00377/) (2019)
- [A Mean-Field Optimal Control Formulation of Deep Learning](https://mlsanity.com/paper/1807.01083/) (2018)
- [Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2205.14027/) (2022)
- [DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems](https://mlsanity.com/paper/2101.07206/) (2020)
- [Operator-valued Kernels for Learning from Functional Response Data](https://mlsanity.com/paper/1510.08231/) (2015)
- [A physics-informed operator regression framework for extracting data-driven continuum models](https://mlsanity.com/paper/2009.11992/) (2020)
- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [Approximation Bounds for Random Neural Networks and Reservoir Systems](https://mlsanity.com/paper/2002.05933/) (2020)
- [Machine Learning from a Continuous Viewpoint](https://mlsanity.com/paper/1912.12777/) (2019)
- [Kernel Methods are Competitive for Operator Learning](https://mlsanity.com/paper/2304.13202/) (2023)

## Built on by (most cited first, 21 of 21)

- [Operator Learning with Gaussian Processes](https://mlsanity.com/paper/2409.04538/) (2024)
- [Operator Learning: A Statistical Perspective](https://mlsanity.com/paper/2504.03503/) (2025)
- [Physics-informed kernel learning](https://mlsanity.com/paper/2409.13786/) (2024)
- [Polynomial Chaos Expansion for Operator Learning](https://mlsanity.com/paper/2508.20886/) (2025)
- [Cauchy Random Features for Operator Learning in Sobolev Space](https://mlsanity.com/paper/2503.00300/) (2025)
- [Towards Sharp Minimax Risk Bounds for Operator Learning](https://mlsanity.com/paper/2512.17805/) (2025)
- [A short tour of operator learning theory: Convergence rates, statistical limits, and open questions](https://mlsanity.com/paper/2603.00819/) (2026)
- [Coefficient-to-Basis Network: A Fine-Tunable Operator Learning Framework for Inverse Problems with Adaptive Discretizations and Theoretical Guarantees](https://mlsanity.com/paper/2503.08642/) (2025)
- [Learning where to learn: Training data distribution optimization for scientific machine learning](https://mlsanity.com/paper/2505.21626/) (2025)
- [Bilevel optimization for learning hyperparameters: Application to solving PDEs and inverse problems with Gaussian processes](https://mlsanity.com/paper/2510.05568/) (2025)
- [Differentially Private Random Feature Model](https://mlsanity.com/paper/2412.04785/) (2024)
- [Regularized Random Fourier Features and Finite Element Reconstruction for Operator Learning in Sobolev Space](https://mlsanity.com/paper/2512.17884/) (2025)
- [DeepONet Augmented by Randomized Neural Networks for Efficient Operator Learning in PDEs](https://mlsanity.com/paper/2503.00317/) (2025)
- [Graph-Based Operator Learning from Limited Data on Irregular Domains](https://mlsanity.com/paper/2505.18923/) (2025)
- [Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regime](https://mlsanity.com/paper/2606.08654/) (2026)
- [Kernel-based Operator Learning: Error Analysis, Budget Allocation, and a Physics-Informed Extension](https://mlsanity.com/paper/2607.06287/) (2026)
- [Reduced Order Models and Conditional Expectation -- Analysing Parametric Low-Order Approximations](https://mlsanity.com/paper/2412.19836/) (2024)
- [Prescribed-Basis Coefficient-to-Coefficient Neural Operator for Partial Differential Equations](https://mlsanity.com/paper/2510.10350/) (2025)
- [Learning solution operators of PDEs with sparse approximation methods](https://mlsanity.com/paper/2606.06046/) (2026)
- [Spectral Convergence of Random Feature Method in Multiple Dimensions](https://mlsanity.com/paper/2609.03401/) (2026)
- [Structured Extrema Errors in Classical Surrogates for Viscous Burgers: A Physics-Consistent Interpretation](https://mlsanity.com/paper/2609.07952/) (2026)

## Abstract (arXiv)

Supervised operator learning centers on the use of training data, in the form of input-output pairs, to estimate maps between infinite-dimensional spaces. It is emerging as a powerful tool to complement traditional scientific computing, which may often be framed in terms of operators mapping between spaces of functions. Building on the classical random features methodology for scalar regression, this paper introduces the function-valued random features method. This leads to a supervised operator learning architecture that is practical for nonlinear problems yet is structured enough to facilitate efficient training through the optimization of a convex, quadratic cost. Due to the quadratic structure, the trained model is equipped with convergence guarantees and error and complexity bounds, properties that are not readily available for most other operator learning architectures. At its core, the proposed approach builds a linear combination of random operators. This turns out to be a low-rank approximation of an operator-valued kernel ridge regression algorithm, and hence the method also has strong connections to Gaussian process regression. The paper designs function-valued random features that are tailored to the structure of two nonlinear operator learning benchmark problems arising from parametric partial differential equations. Numerical results demonstrate the scalability, discretization invariance, and transferability of the function-valued random features method.