# Learning elliptic partial differential equations with randomized linear algebra arXiv: https://arxiv.org/abs/2102.00491 · HTML: https://mlsanity.com/paper/2102.00491/ ## Key facts - Cited by 29 papers in the map (data through 14 Oct 2026). - Builds on 16 of its 75 references that are in the map. - First posted on arXiv 31 Jan 2021. ## Builds on (most cited first, 16 of 16) - [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020) - [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016) - [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017) - [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018) - [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017) - [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017) - [PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs](https://mlsanity.com/paper/1909.10145/) (2019) - [Error bounds for approximations with deep ReLU neural networks in Ws,p norms](https://mlsanity.com/paper/1902.07896/) (2019) - [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018) - [Rational neural networks](https://mlsanity.com/paper/2004.01902/) (2020) - [DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems](https://mlsanity.com/paper/2101.07206/) (2020) - [On the folded normal distribution](https://mlsanity.com/paper/1402.3559/) (2014) - [Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity](https://mlsanity.com/paper/1706.02205/) (2017) - [Neural-net-induced Gaussian process regression for function approximation and PDE solution](https://mlsanity.com/paper/1806.11187/) (2018) - [Meta-learning Pseudo-differential Operators with Deep Neural Networks](https://mlsanity.com/paper/1906.06782/) (2019) - [Stability selection enables robust learning of partial differential equations from limited noisy data](https://mlsanity.com/paper/1907.07810/) (2019) ## Built on by (most cited first, 29 of 29) - [Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behavior](https://mlsanity.com/paper/2306.00258/) (2023) - [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024) - [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023) - [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) - [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) - [Learning Green's functions associated with time-dependent partial differential equations](https://mlsanity.com/paper/2204.12789/) (2022) - [Elliptic PDE learning is provably data-efficient](https://mlsanity.com/paper/2302.12888/) (2023) - [Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEs](https://mlsanity.com/paper/2208.01565/) (2022) - [Error Bounds for Learning with Vector-Valued Random Features](https://mlsanity.com/paper/2305.17170/) (2023) - [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024) - [Operator learning without the adjoint](https://mlsanity.com/paper/2401.17739/) (2024) - [A generalization of the randomized singular value decomposition](https://mlsanity.com/paper/2105.13052/) (2021) - [Operator learning for hyperbolic partial differential equations](https://mlsanity.com/paper/2312.17489/) (2023) - [A randomized algorithm to solve reduced rank operator regression](https://mlsanity.com/paper/2312.17348/) (2023) - [Green Multigrid Network](https://mlsanity.com/paper/2407.03593/) (2024) - [Structure-Preserving Operator Learning](https://mlsanity.com/paper/2410.01065/) (2024) - [An explainable operator approximation framework under the guideline of Green's function](https://mlsanity.com/paper/2412.16644/) (2024) - [Score Operator Newton transport](https://mlsanity.com/paper/2305.09792/) (2023) - [Learning Green's Function Efficiently Using Low-Rank Approximations](https://mlsanity.com/paper/2308.00350/) (2023) - [chebgreen: Learning and Interpolating Continuous Empirical Green's Functions from Data](https://mlsanity.com/paper/2501.18715/) (2025) - [Rates and architectures for learning geometrically non-trivial operators](https://mlsanity.com/paper/2512.09376/) (2025) - [Supervised and Unsupervised Neural Network Solver for First Order Hyperbolic Nonlinear PDEs](https://mlsanity.com/paper/2601.06388/) (2026) - [Optimal Multiscale Learning of Linear Operators](https://mlsanity.com/paper/2606.16913/) (2026) - [A unified error analysis for randomized low-rank approximation with application to data assimilation](https://mlsanity.com/paper/2405.04811/) (2024) - [An adaptive data sampling strategy for stabilizing dynamical systems via controller inference](https://mlsanity.com/paper/2506.01816/) (2025) - [Query Efficient Structured Matrix Learning](https://mlsanity.com/paper/2507.19290/) (2025) - [Cost-Accuracy Trade-offs: Neural Operator vs Classical Numerical Solver](https://mlsanity.com/paper/2609.24021/) (2026) ## Abstract (arXiv) Given input-output pairs of an elliptic partial differential equation (PDE) in three dimensions, we derive the first theoretically-rigorous scheme for learning the associated Green's function G. By exploiting the hierarchical low-rank structure of G, we show that one can construct an approximant to G that converges almost surely and achieves a relative error of O(Gammaε-1/2log3(1/ε)ε) using at most O(ε-6log4(1/ε)) input-output training pairs with high probability, for any 0<ε<1. The quantity 0