# Learning Green's Function Efficiently Using Low-Rank Approximations

arXiv: https://arxiv.org/abs/2308.00350 · HTML: https://mlsanity.com/paper/2308.00350/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 31 references that are in the map.
- First posted on arXiv 1 Aug 2023.

## Builds on (most cited first, 16 of 16)

- [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)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [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)
- [Separable Physics-Informed Neural Networks](https://mlsanity.com/paper/2306.15969/) (2023)
- [DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems](https://mlsanity.com/paper/2101.07206/) (2020)
- [Meta-learning PINN loss functions](https://mlsanity.com/paper/2107.05544/) (2021)
- [Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality](https://mlsanity.com/paper/2110.06897/) (2021)
- [Learning elliptic partial differential equations with randomized linear algebra](https://mlsanity.com/paper/2102.00491/) (2021)
- [Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activations](https://mlsanity.com/paper/2206.09527/) (2022)
- [Learning Green's functions associated with time-dependent partial differential equations](https://mlsanity.com/paper/2204.12789/) (2022)
- [MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEs](https://mlsanity.com/paper/2107.03673/) (2021)
- [Bounding the Rademacher Complexity of Fourier neural operators](https://mlsanity.com/paper/2209.05150/) (2022)
- [Learning Green's Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solver](https://mlsanity.com/paper/2105.11045/) (2021)

## Built on by (most cited first, 1 of 1)

- [Learning Singularity-Encoded Green's Functions with Application to Iterative Methods](https://mlsanity.com/paper/2509.11580/) (2025)

## Abstract (arXiv)

Learning the Green's function using deep learning models enables to solve different classes of partial differential equations. A practical limitation of using deep learning for the Green's function is the repeated computationally expensive Monte-Carlo integral approximations. We propose to learn the Green's function by low-rank decomposition, which results in a novel architecture to remove redundant computations by separate learning with domain data for evaluation and Monte-Carlo samples for integral approximation. Using experiments we show that the proposed method improves computational time compared to MOD-Net while achieving comparable accuracy compared to both PINNs and MOD-Net.