# MgNO: Efficient Parameterization of Linear Operators via Multigrid

arXiv: https://arxiv.org/abs/2310.19809 · HTML: https://mlsanity.com/paper/2310.19809/

## Key facts

- Cited by 40 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 47 references that are in the map.
- First posted on arXiv 16 Oct 2023.

## Builds on (most cited first, 20 of 20)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (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)
- [Solving High-Dimensional PDEs with Latent Spectral Models](https://mlsanity.com/paper/2301.12664/) (2023)
- [MgNet: A Unified Framework of Multigrid and Convolutional Neural Network](https://mlsanity.com/paper/1901.10415/) (2019)
- [A Physics-Guided Neural Operator Learning Approach to Model Biological Tissues from Digital Image Correlation Measurements](https://mlsanity.com/paper/2204.00205/) (2022)
- [Multigrid with rough coefficients and Multiresolution operator decomposition from Hierarchical Information Games](https://mlsanity.com/paper/1503.03467/) (2015)
- [Approximation Properties of Deep ReLU CNNs](https://mlsanity.com/paper/2109.00190/) (2021)
- [An Interpretive Constrained Linear Model for ResNet and MgNet](https://mlsanity.com/paper/2112.07441/) (2021)
- [An Enhanced V-cycle MgNet Model for Operator Learning in Numerical Partial Differential Equations](https://mlsanity.com/paper/2302.00938/) (2023)

## Built on by (most cited first, 30 of 40)

- [OneForecast: A Universal Framework for Global and Regional Weather Forecasting](https://mlsanity.com/paper/2502.00338/) (2025)
- [Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspective](https://mlsanity.com/paper/2404.07200/) (2024)
- [Operator Learning with Gaussian Processes](https://mlsanity.com/paper/2409.04538/) (2024)
- [Harnessing Scale and Physics: A Multi-Graph Neural Operator Framework for PDEs on Arbitrary Geometries](https://mlsanity.com/paper/2411.15178/) (2024)
- [Recurrent Neural Operators: Stable Long-Term PDE Prediction](https://mlsanity.com/paper/2505.20721/) (2025)
- [A hybrid iterative method based on MIONet for PDEs: Theory and numerical examples](https://mlsanity.com/paper/2402.07156/) (2024)
- [Spectral-Refiner: Accurate Fine-Tuning of Spatiotemporal Fourier Neural Operator for Turbulent Flows](https://mlsanity.com/paper/2405.17211/) (2024)
- [Operator Learning at Machine Precision](https://mlsanity.com/paper/2511.19980/) (2025)
- [M2NO: An Efficient Multi-Resolution Operator Framework for Dynamic Multi-Scale PDE Solvers](https://mlsanity.com/paper/2406.04822/) (2024)
- [Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators](https://mlsanity.com/paper/2601.14517/) (2026)
- [Higher Order Approximation Rates for ReLU CNNs in Korobov Spaces](https://mlsanity.com/paper/2501.11275/) (2025)
- [Multi-Level Monte Carlo Training of Neural Operators](https://mlsanity.com/paper/2505.12940/) (2025)
- [Learning solution operators of PDEs defined on varying domains via MIONet](https://mlsanity.com/paper/2402.15097/) (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)
- [Unsupervised operator learning approach for dissipative equations via Onsager principle](https://mlsanity.com/paper/2508.07440/) (2025)
- [Self-composing neural operators for high-frequency and multiscale PDE surrogates](https://mlsanity.com/paper/2508.20650/) (2025)
- [Discovering Physical Directions in Weight Space: Composing Neural PDE Experts](https://mlsanity.com/paper/2605.14546/) (2026)
- [Newton Informed Neural Operator for Computing Multiple Solutions of Nonlinear Partials Differential Equations](https://mlsanity.com/paper/2405.14096/) (2024)
- [A MgNO Method for Multiphase Flow in Porous Media](https://mlsanity.com/paper/2407.02505/) (2024)
- [A deformation-based framework for learning solution mappings of PDEs defined on varying domains](https://mlsanity.com/paper/2412.01379/) (2024)
- [Learn Singularly Perturbed Solutions via Homotopy Dynamics](https://mlsanity.com/paper/2502.00488/) (2025)
- [OpenBreastUS: Benchmarking Neural Operators for Wave Imaging Using Breast Ultrasound Computed Tomography](https://mlsanity.com/paper/2507.15035/) (2025)
- [From Basis to Basis: Gaussian Particle Representation for Interpretable PDE Operators](https://mlsanity.com/paper/2602.21551/) (2026)
- [DeepONet for Solving Nonlinear Partial Differential Equations with Physics-Informed Training](https://mlsanity.com/paper/2410.04344/) (2024)
- [Noise-adapted Neural Operator for Robust Non-Line-of-Sight Imaging](https://mlsanity.com/paper/2508.09655/) (2025)
- [Statistical Learning Guarantees for Group-Invariant Barron Functions](https://mlsanity.com/paper/2509.23474/) (2025)
- [Efficient High-Accuracy PDEs Solver with the Linear Attention Neural Operator](https://mlsanity.com/paper/2510.16816/) (2025)
- [Learning from History: A Retrieval-Augmented Framework for Spatiotemporal Prediction](https://mlsanity.com/paper/2510.24049/) (2025)

## Abstract (arXiv)

In this work, we propose a concise neural operator architecture for operator learning. Drawing an analogy with a conventional fully connected neural network, we define the neural operator as follows: the output of the i-th neuron in a nonlinear operator layer is defined by Oi(u) = σleft( sumj Wij u + Bijright). Here,  Wij denotes the bounded linear operator connecting j-th input neuron to i-th output neuron, and the bias  Bij takes the form of a function rather than a scalar. Given its new universal approximation property, the efficient parameterization of the bounded linear operators between two neurons (Banach spaces) plays a critical role. As a result, we introduce MgNO, utilizing multigrid structures to parameterize these linear operators between neurons. This approach offers both mathematical rigor and practical expressivity. Additionally, MgNO obviates the need for conventional lifting and projecting operators typically required in previous neural operators. Moreover, it seamlessly accommodates diverse boundary conditions. Our empirical observations reveal that MgNO exhibits superior ease of training compared to other CNN-based models, while also displaying a reduced susceptibility to overfitting when contrasted with spectral-type neural operators. We demonstrate the efficiency and accuracy of our method with consistently state-of-the-art performance on different types of partial differential equations (PDEs).