# Elliptic PDE learning is provably data-efficient

arXiv: https://arxiv.org/abs/2302.12888 · HTML: https://mlsanity.com/paper/2302.12888/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 82 references that are in the map.
- First posted on arXiv 24 Feb 2023.

## Builds on (most cited first, 18 of 18)

- [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)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems](https://mlsanity.com/paper/2101.07206/) (2020)
- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statistics](https://mlsanity.com/paper/1605.00353/) (2016)
- [Data-driven discovery of Green's functions with human-understandable deep learning](https://mlsanity.com/paper/2105.00266/) (2021)
- [Learning elliptic partial differential equations with randomized linear algebra](https://mlsanity.com/paper/2102.00491/) (2021)
- [Learning Green's functions associated with time-dependent partial differential equations](https://mlsanity.com/paper/2204.12789/) (2022)
- [Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity](https://mlsanity.com/paper/1706.02205/) (2017)
- [Deep Operator Learning Lessens the Curse of Dimensionality for PDEs](https://mlsanity.com/paper/2301.12227/) (2023)
- [Unperturbed: spectral analysis beyond Davis-Kahan](https://mlsanity.com/paper/1706.06516/) (2017)
- [A generalization of the randomized singular value decomposition](https://mlsanity.com/paper/2105.13052/) (2021)
- [Error Estimation for Sketched SVD via the Bootstrap](https://mlsanity.com/paper/2003.04937/) (2020)
- [Perturbation Analysis of Randomized SVD and its Applications to Statistics](https://mlsanity.com/paper/2203.10262/) (2022)

## Built on by (most cited first, 26 of 26)

- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [An operator learning perspective on parameter-to-observable maps](https://mlsanity.com/paper/2402.06031/) (2024)
- [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024)
- [Operator Learning: A Statistical Perspective](https://mlsanity.com/paper/2504.03503/) (2025)
- [The Sample Complexity of Learning Lipschitz Operators with respect to Gaussian Measures](https://mlsanity.com/paper/2410.23440/) (2024)
- [Adversarial dynamical systems characterize when data-driven learning succeeds or fails](https://mlsanity.com/paper/2407.06312/) (2024)
- [Operator learning without the adjoint](https://mlsanity.com/paper/2401.17739/) (2024)
- [Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysis](https://mlsanity.com/paper/2503.01036/) (2025)
- [Operator learning for hyperbolic partial differential equations](https://mlsanity.com/paper/2312.17489/) (2023)
- [Controlling Statistical, Discretization, and Truncation Errors in Learning Fourier Linear Operators](https://mlsanity.com/paper/2408.09004/) (2024)
- [PETScML: Second-order solvers for training regression problems in Scientific Machine Learning](https://mlsanity.com/paper/2403.12188/) (2024)
- [Conservation-informed Graph Learning for Spatiotemporal Dynamics Prediction](https://mlsanity.com/paper/2412.20962/) (2024)
- [Green Multigrid Network](https://mlsanity.com/paper/2407.03593/) (2024)
- [Structure-Preserving Operator Learning](https://mlsanity.com/paper/2410.01065/) (2024)
- [Orthogonal greedy algorithm for linear operator learning with shallow neural network](https://mlsanity.com/paper/2501.02791/) (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)
- [Noisy PDE Training Requires Bigger PINNs](https://mlsanity.com/paper/2507.06967/) (2025)
- [Rates and architectures for learning geometrically non-trivial operators](https://mlsanity.com/paper/2512.09376/) (2025)
- [Neural-HSS: Hierarchical Semi-Separable Neural PDE Solver](https://mlsanity.com/paper/2602.18248/) (2026)
- [Optimal Multiscale Learning of Linear Operators](https://mlsanity.com/paper/2606.16913/) (2026)
- [Spectrally Safe Neural Operator Warm-Starts for Large-Scale Newton Solvers](https://mlsanity.com/paper/2606.21828/) (2026)
- [Interpretability and Generalization Bounds for Learning Spatial Physics](https://mlsanity.com/paper/2506.15199/) (2025)
- [Query Efficient Structured Matrix Learning](https://mlsanity.com/paper/2507.19290/) (2025)
- [A zero-one law for one-shot system identification](https://mlsanity.com/paper/2607.15832/) (2026)
- [Conformal Uncertainty Quantification Guarantees for Neural Operators](https://mlsanity.com/paper/2608.28515/) (2026)

## Abstract (arXiv)

PDE learning is an emerging field that combines physics and machine learning to recover unknown physical systems from experimental data. While deep learning models traditionally require copious amounts of training data, recent PDE learning techniques achieve spectacular results with limited data availability. Still, these results are empirical. Our work provides theoretical guarantees on the number of input-output training pairs required in PDE learning. Specifically, we exploit randomized numerical linear algebra and PDE theory to derive a provably data-efficient algorithm that recovers solution operators of 3D uniformly elliptic PDEs from input-output data and achieves an exponential convergence rate of the error with respect to the size of the training dataset with an exceptionally high probability of success.