# Green Multigrid Network

arXiv: https://arxiv.org/abs/2407.03593 · HTML: https://mlsanity.com/paper/2407.03593/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 67 references that are in the map.
- First posted on arXiv 4 Jul 2024.

## Builds on (most cited first, 30 of 36)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [A Simple Convergence Proof of Adam and Adagrad](https://mlsanity.com/paper/2003.02395/) (2020)
- [Learning Operators with Coupled Attention](https://mlsanity.com/paper/2201.01032/) (2022)
- [Improved architectures and training algorithms for deep operator networks](https://mlsanity.com/paper/2110.01654/) (2021)
- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [Rational neural networks](https://mlsanity.com/paper/2004.01902/) (2020)
- [Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modeling](https://mlsanity.com/paper/2203.08205/) (2022)
- [DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems](https://mlsanity.com/paper/2101.07206/) (2020)
- [Data-driven discovery of Green's functions with human-understandable deep learning](https://mlsanity.com/paper/2105.00266/) (2021)
- [MgNO: Efficient Parameterization of Linear Operators via Multigrid](https://mlsanity.com/paper/2310.19809/) (2023)
- [Learning elliptic partial differential equations with randomized linear algebra](https://mlsanity.com/paper/2102.00491/) (2021)
- [Elliptic PDE learning is provably data-efficient](https://mlsanity.com/paper/2302.12888/) (2023)
- [MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEs](https://mlsanity.com/paper/2107.03673/) (2021)
- [Transformer Meets Boundary Value Inverse Problems](https://mlsanity.com/paper/2209.14977/) (2022)

## Built on by (most cited first, 3 of 3)

- [Orthogonal greedy algorithm for linear operator learning with shallow neural network](https://mlsanity.com/paper/2501.02791/) (2025)
- [Learning Singularity-Encoded Green's Functions with Application to Iterative Methods](https://mlsanity.com/paper/2509.11580/) (2025)
- [IV-Net: A neural network for elliptic PDEs with random and highly varying coefficients](https://mlsanity.com/paper/2605.24876/) (2026)

## Abstract (arXiv)

GreenLearning networks (GL) directly learn Green's function in physical space, making them an interpretable model for capturing unknown solution operators of partial differential equations (PDEs). For many PDEs, the corresponding Green's function exhibits asymptotic smoothness. In this paper, we propose a framework named Green Multigrid networks (GreenMGNet), an operator learning algorithm designed for a class of asymptotically smooth Green's functions. Compared with the pioneering GL, the new framework presents itself with better accuracy and efficiency, thereby achieving a significant improvement. GreenMGNet is composed of two technical novelties. First, Green's function is modeled as a piecewise function to take into account its singular behavior in some parts of the hyperplane. Such piecewise function is then approximated by a neural network with augmented output(AugNN) so that it can capture singularity accurately. Second, the asymptotic smoothness property of Green's function is used to leverage the Multi-Level Multi-Integration (MLMI) algorithm for both the training and inference stages. Several test cases of operator learning are presented to demonstrate the accuracy and effectiveness of the proposed method. On average, GreenMGNet achieves 3.8% to 39.15% accuracy improvement. To match the accuracy level of GL, GreenMGNet requires only about 10% of the full grid data, resulting in a 55.9% and 92.5% reduction in training time and GPU memory cost for one-dimensional test problems, and a 37.7% and 62.5% reduction for two-dimensional test problems.