# Learning Green's functions associated with time-dependent partial differential equations

arXiv: https://arxiv.org/abs/2204.12789 · HTML: https://mlsanity.com/paper/2204.12789/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 67 references that are in the map.
- First posted on arXiv 27 Apr 2022.

## Builds on (most cited first, 23 of 23)

- [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)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Deep learning to represent sub-grid processes in climate models](https://mlsanity.com/paper/1806.04731/) (2018)
- [AI Feynman 2.0: Pareto-optimal symbolic regression exploiting graph modularity](https://mlsanity.com/paper/2006.10782/) (2020)
- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [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)
- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [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)
- [Reduced operator inference for nonlinear partial differential equations](https://mlsanity.com/paper/2102.00083/) (2021)
- [Meta-learning Pseudo-differential Operators with Deep Neural Networks](https://mlsanity.com/paper/1906.06782/) (2019)
- [On the Complexity of Robust PCA and ℓ1-norm Low-Rank Matrix Approximation](https://mlsanity.com/paper/1509.09236/) (2015)
- [A generalization of the randomized singular value decomposition](https://mlsanity.com/paper/2105.13052/) (2021)

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

- [Scalable Transformer for PDE Surrogate Modeling](https://mlsanity.com/paper/2305.17560/) (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)
- [Elliptic PDE learning is provably data-efficient](https://mlsanity.com/paper/2302.12888/) (2023)
- [Minimax Optimal Kernel Operator Learning via Multilevel Training](https://mlsanity.com/paper/2209.14430/) (2022)
- [Interpreting and generalizing deep learning in physics-based problems with functional linear models](https://mlsanity.com/paper/2307.04569/) (2023)
- [A Deep Learning Framework for Multi-Operator Learning: Architectures and Approximation Theory](https://mlsanity.com/paper/2510.25379/) (2025)
- [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)
- [Neural Control of Parametric Solutions for High-dimensional Evolution PDEs](https://mlsanity.com/paper/2302.00045/) (2023)
- [Operator learning for hyperbolic partial differential equations](https://mlsanity.com/paper/2312.17489/) (2023)
- [Latent Neural PDE Solver: a reduced-order modelling framework for partial differential equations](https://mlsanity.com/paper/2402.17853/) (2024)
- [Solving Partial Differential Equations in Different Domains by Operator Learning method Based on Boundary Integral Equations](https://mlsanity.com/paper/2406.02298/) (2024)
- [Structure-Preserving Operator Learning](https://mlsanity.com/paper/2410.01065/) (2024)
- [Generalization Bounds and Statistical Guarantees for Multi-Task and Multiple Operator Learning with MNO Networks](https://mlsanity.com/paper/2604.01961/) (2026)
- [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)
- [Optimal Multiscale Learning of Linear Operators](https://mlsanity.com/paper/2606.16913/) (2026)
- [Intrinsic Green's Learning: Supervised Learning on Manifolds via Inverse PDE](https://mlsanity.com/paper/2607.07034/) (2026)
- [Approximation of Solution Operators for High-dimensional PDEs](https://mlsanity.com/paper/2401.10385/) (2024)
- [Adaptive Error-Bounded Hierarchical Matrices for Efficient Neural Network Compression](https://mlsanity.com/paper/2409.07028/) (2024)
- [Learning convolution operators on compact Abelian groups](https://mlsanity.com/paper/2501.05279/) (2025)
- [A transition-density-based operator learning method for Fokker-Planck equations with various initial conditions](https://mlsanity.com/paper/2606.09434/) (2026)
- [Learning PDE Dynamics between Submanifolds Using Green's Observation Operators](https://mlsanity.com/paper/2610.01697/) (2026)
- [Green-Routed Neural Operators: Physics Determines Where the Network Reads](https://mlsanity.com/paper/2610.05337/) (2026)

## Abstract (arXiv)

Neural operators are a popular technique in scientific machine learning to learn a mathematical model of the behavior of unknown physical systems from data. Neural operators are especially useful to learn solution operators associated with partial differential equations (PDEs) from pairs of forcing functions and solutions when numerical solvers are not available or the underlying physics is poorly understood. In this work, we attempt to provide theoretical foundations to understand the amount of training data needed to learn time-dependent PDEs. Given input-output pairs from a parabolic PDE in any spatial dimension n≥ 1, we derive the first theoretically rigorous scheme for learning the associated solution operator, which takes the form of a convolution with a Green's function G. Until now, rigorously learning Green's functions associated with time-dependent PDEs has been a major challenge in the field of scientific machine learning because G may not be square-integrable when n>1, and time-dependent PDEs have transient dynamics. By combining the hierarchical low-rank structure of G together with randomized numerical linear algebra, we construct an approximant to G that achieves a relative error of smashO(Gammaε-1/2ε) in the L1-norm with high probability by using at most smashO(ε^-fracn+22log(1/ε)) input-output training pairs, where Gammaε is a measure of the quality of the training dataset for learning G, and ε>0 is sufficiently small.