# Operator Learning: A Statistical Perspective

arXiv: https://arxiv.org/abs/2504.03503 · HTML: https://mlsanity.com/paper/2504.03503/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 53 of its 122 references that are in the map.
- First posted on arXiv 4 Apr 2025.

## Builds on (most cited first, 30 of 53)

- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [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)
- [Scaling Vision Transformers](https://mlsanity.com/paper/2106.04560/) (2021)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [When Do Neural Nets Outperform Boosted Trees on Tabular Data?](https://mlsanity.com/paper/2305.02997/) (2023)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behavior](https://mlsanity.com/paper/2306.00258/) (2023)
- [Artificial Intelligence for Science in Quantum, Atomistic, and Continuum Systems](https://mlsanity.com/paper/2307.08423/) (2023)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modeling](https://mlsanity.com/paper/2203.08205/) (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)
- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [Kernel Methods are Competitive for Operator Learning](https://mlsanity.com/paper/2304.13202/) (2023)
- [Deep Nonparametric Estimation of Operators between Infinite Dimensional Spaces](https://mlsanity.com/paper/2201.00217/) (2022)
- [Data-driven discovery of Green's functions with human-understandable deep learning](https://mlsanity.com/paper/2105.00266/) (2021)
- [Operator learning with PCA-Net: upper and lower complexity bounds](https://mlsanity.com/paper/2303.16317/) (2023)
- [A Library for Learning Neural Operators](https://mlsanity.com/paper/2412.10354/) (2024)
- [Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equation](https://mlsanity.com/paper/2301.11509/) (2023)
- [Plasma Surrogate Modelling using Fourier Neural Operators](https://mlsanity.com/paper/2311.05967/) (2023)

## Built on by (most cited first, 17 of 17)

- [Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning](https://mlsanity.com/paper/2506.10973/) (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)
- [Continuum Transformers Perform In-Context Learning by Operator Gradient Descent](https://mlsanity.com/paper/2505.17838/) (2025)
- [Learning where to learn: Training data distribution optimization for scientific machine learning](https://mlsanity.com/paper/2505.21626/) (2025)
- [Method of Manufactured Learning for Solver-free Training of Neural Operators](https://mlsanity.com/paper/2511.12890/) (2025)
- [A Kernel-based Stochastic Approximation Framework for Nonlinear Operator Learning](https://mlsanity.com/paper/2509.11070/) (2025)
- [Is Zero-Shot Super-Resolution Possible in Operator Learning?](https://mlsanity.com/paper/2606.00296/) (2026)
- [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)
- [Optimal Multiscale Learning of Linear Operators](https://mlsanity.com/paper/2606.16913/) (2026)
- [Droplet-LNO: Physics-Informed Laplace Neural Operators for Accurate Prediction of Droplet Spreading Dynamics on Complex Surfaces](https://mlsanity.com/paper/2604.20993/) (2026)
- [Learning solution operators of PDEs with sparse approximation methods](https://mlsanity.com/paper/2606.06046/) (2026)
- [From Spectral Methods to Sample Complexity Bounds for Fourier Neural Operators](https://mlsanity.com/paper/2607.00320/) (2026)
- [Near-Optimal Learning of Gaussian Sobolev Operators](https://mlsanity.com/paper/2607.11921/) (2026)
- [Two-Step MV-DeepONet: Probabilistic Operator Learning for Uncertainty Propagation Driven by Random Input Fields](https://mlsanity.com/paper/2608.09071/) (2026)
- [MnemoDyn: Learning Resting State Dynamics from 40K FMRI sequences](https://mlsanity.com/paper/2608.23936/) (2026)
- [Efficient Neural Surrogates for Linear Radiation Transport on the Lattice and Hohlraum benchmarks](https://mlsanity.com/paper/2610.04665/) (2026)

## Abstract (arXiv)

Operator learning has emerged as a powerful tool in scientific computing for approximating mappings between infinite-dimensional function spaces. A primary application of operator learning is the development of surrogate models for the solution operators of partial differential equations (PDEs). These methods can also be used to develop black-box simulators to model system behavior from experimental data, even without a known mathematical model. In this article, we begin by formalizing operator learning as a function-to-function regression problem and review some recent developments in the field. We also discuss PDE-specific operator learning, outlining strategies for incorporating physical and mathematical constraints into architecture design and training processes. Finally, we end by highlighting key future directions such as active data collection and the development of rigorous uncertainty quantification frameworks.