# Deep Operator Learning Lessens the Curse of Dimensionality for PDEs

arXiv: https://arxiv.org/abs/2301.12227 · HTML: https://mlsanity.com/paper/2301.12227/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 60 references that are in the map.
- First posted on arXiv 28 Jan 2023.

## Builds on (most cited first, 27 of 27)

- [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)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Speech Recognition with Deep Recurrent Neural Networks](https://mlsanity.com/paper/1303.5778/) (2013)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017)
- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [Deep Network Approximation for Smooth Functions](https://mlsanity.com/paper/2001.03040/) (2020)
- [Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionality](https://mlsanity.com/paper/1810.08033/) (2018)
- [Deep Learning on Image Denoising: An overview](https://mlsanity.com/paper/1912.13171/) (2019)
- [Deep Network Approximation Characterized by Number of Neurons](https://mlsanity.com/paper/1906.05497/) (2019)
- [The Random Feature Model for Input-Output Maps between Banach Spaces](https://mlsanity.com/paper/2005.10224/) (2020)
- [Deep ReLU network approximation of functions on a manifold](https://mlsanity.com/paper/1908.00695/) (2019)
- [Nonparametric Regression on Low-Dimensional Manifolds using Deep ReLU Networks : Function Approximation and Statistical Recovery](https://mlsanity.com/paper/1908.01842/) (2019)
- [Deep Neural Networks for Estimation and Inference](https://mlsanity.com/paper/1809.09953/) (2018)
- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [Generative Adversarial Neural Operators](https://mlsanity.com/paper/2205.03017/) (2022)
- [Deep Nonparametric Estimation of Operators between Infinite Dimensional Spaces](https://mlsanity.com/paper/2201.00217/) (2022)
- [Elementary superexpressive activations](https://mlsanity.com/paper/2102.10911/) (2021)
- [SwitchNet: a neural network model for forward and inverse scattering problems](https://mlsanity.com/paper/1810.09675/) (2018)
- [Besov Function Approximation and Binary Classification on Low-Dimensional Manifolds Using Convolutional Residual Networks](https://mlsanity.com/paper/2109.02832/) (2021)
- [A framework for data-driven solution and parameter estimation of PDEs using conditional generative adversarial networks](https://mlsanity.com/paper/2105.13136/) (2021)
- [Learning to Synthesize: Robust Phase Retrieval at Low Photon counts](https://mlsanity.com/paper/1907.11713/) (2019)
- [IAE-Net: Integral Autoencoders for Discretization-Invariant Learning](https://mlsanity.com/paper/2203.05142/) (2022)
- [A deep network construction that adapts to intrinsic dimensionality beyond the domain](https://mlsanity.com/paper/2008.02545/) (2020)
- [Deep Generative Models that Solve PDEs: Distributed Computing for Training Large Data-Free Models](https://mlsanity.com/paper/2007.12792/) (2020)

## Built on by (most cited first, 18 of 18)

- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Elliptic PDE learning is provably data-efficient](https://mlsanity.com/paper/2302.12888/) (2023)
- [Fourier neural operator for learning solutions to macroscopic traffic flow models: Application to the forward and inverse problems](https://mlsanity.com/paper/2308.07051/) (2023)
- [Neural Operator: Is data all you need to model the world? An insight into the paradigm of data-driven scientific ML](https://mlsanity.com/paper/2301.13331/) (2023)
- [An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learning](https://mlsanity.com/paper/2408.13222/) (2024)
- [Operator learning for hyperbolic partial differential equations](https://mlsanity.com/paper/2312.17489/) (2023)
- [Quantitative Approximation for Neural Operators in Nonlinear Parabolic Equations](https://mlsanity.com/paper/2410.02151/) (2024)
- [Micrometer: Micromechanics Transformer for Predicting Mechanical Responses of Heterogeneous Materials](https://mlsanity.com/paper/2410.05281/) (2024)
- [Deep Neural ODE Operator Networks for PDEs](https://mlsanity.com/paper/2510.15651/) (2025)
- [On the Approximation of Bi-Lipschitz Maps by Invertible Neural Networks](https://mlsanity.com/paper/2308.09367/) (2023)
- [Algorithmically Designed Artificial Neural Networks (ADANNs): Higher order deep operator learning for parametric partial differential equations](https://mlsanity.com/paper/2302.03286/) (2023)
- [Neural Network Approximation for Pessimistic Offline Reinforcement Learning](https://mlsanity.com/paper/2312.11863/) (2023)
- [Curse of Dimensionality in Neural Network Optimization](https://mlsanity.com/paper/2502.05360/) (2025)
- [Rates and architectures for learning geometrically non-trivial operators](https://mlsanity.com/paper/2512.09376/) (2025)
- [Multi-Scale Finite Expression Method for PDEs with Oscillatory Solutions on Complex Domains](https://mlsanity.com/paper/2510.22497/) (2025)
- [Learning Biomolecular Motion: The Physics-Informed Machine Learning Paradigm](https://mlsanity.com/paper/2511.06585/) (2025)
- [Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEs](https://mlsanity.com/paper/2605.10277/) (2026)
- [Approximation Theory of Laplacian-Based Neural Operators for Reaction-Diffusion System](https://mlsanity.com/paper/2605.12025/) (2026)

## Abstract (arXiv)

Deep neural networks (DNNs) have achieved remarkable success in numerous domains, and their application to PDE-related problems has been rapidly advancing. This paper provides an estimate for the generalization error of learning Lipschitz operators over Banach spaces using DNNs with applications to various PDE solution operators. The goal is to specify DNN width, depth, and the number of training samples needed to guarantee a certain testing error. Under mild assumptions on data distributions or operator structures, our analysis shows that deep operator learning can have a relaxed dependence on the discretization resolution of PDEs and, hence, lessen the curse of dimensionality in many PDE-related problems including elliptic equations, parabolic equations, and Burgers equations. Our results are also applied to give insights about discretization-invariance in operator learning.