# MgNet: A Unified Framework of Multigrid and Convolutional Neural Network

arXiv: https://arxiv.org/abs/1901.10415 · HTML: https://mlsanity.com/paper/1901.10415/

## Key facts

- Cited by 41 papers in the map (data through 14 Oct 2026).
- Placed in Latent predictive world models, one of 11 directions in World models.
- Builds on 0 of its 58 references that are in the map.
- First posted on arXiv 29 Jan 2019.

## Placed in

- World models / [Latent predictive world models](https://mlsanity.com/wm/latent-predictive-world-models/)

## Built on by (most cited first, 30 of 41)

- [Understanding World or Predicting Future? A Comprehensive Survey of World Models](https://mlsanity.com/paper/2411.14499/) (2024)
- [Overview frequency principle/spectral bias in deep learning](https://mlsanity.com/paper/2201.07395/) (2022)
- [Blending Neural Operators and Relaxation Methods in PDE Numerical Solvers](https://mlsanity.com/paper/2208.13273/) (2022)
- [MgNO: Efficient Parameterization of Linear Operators via Multigrid](https://mlsanity.com/paper/2310.19809/) (2023)
- [OneForecast: A Universal Framework for Global and Regional Weather Forecasting](https://mlsanity.com/paper/2502.00338/) (2025)
- [Inducing Point Operator Transformer: A Flexible and Scalable Architecture for Solving PDEs](https://mlsanity.com/paper/2312.10975/) (2023)
- [Operator Learning with Gaussian Processes](https://mlsanity.com/paper/2409.04538/) (2024)
- [PottsMGNet: A Mathematical Explanation of Encoder-Decoder Based Neural Networks](https://mlsanity.com/paper/2307.09039/) (2023)
- [Probabilistic Deep Learning for Real-Time Large Deformation Simulations](https://mlsanity.com/paper/2111.01867/) (2021)
- [Turb-L1: Achieving Long-term Turbulence Tracing By Tackling Spectral Bias](https://mlsanity.com/paper/2505.19038/) (2025)
- [NOWS: Neural Operator Warm Starts for Accelerating Iterative Solvers](https://mlsanity.com/paper/2511.02481/) (2025)
- [Variational operator learning: A unified paradigm marrying training neural operators and solving partial differential equations](https://mlsanity.com/paper/2304.04234/) (2023)
- [Double-well Net for Image Segmentation](https://mlsanity.com/paper/2401.00456/) (2023)
- [An Interpretive Constrained Linear Model for ResNet and MgNet](https://mlsanity.com/paper/2112.07441/) (2021)
- [M2NO: An Efficient Multi-Resolution Operator Framework for Dynamic Multi-Scale PDE Solvers](https://mlsanity.com/paper/2406.04822/) (2024)
- [An Enhanced V-cycle MgNet Model for Operator Learning in Numerical Partial Differential Equations](https://mlsanity.com/paper/2302.00938/) (2023)
- [A Mathematical Explanation of UNet](https://mlsanity.com/paper/2410.04434/) (2024)
- [Orthogonal greedy algorithm for linear operator learning with shallow neural network](https://mlsanity.com/paper/2501.02791/) (2025)
- [Multiscale Training of Convolutional Neural Networks](https://mlsanity.com/paper/2501.12739/) (2025)
- [Self-composing neural operators for high-frequency and multiscale PDE surrogates](https://mlsanity.com/paper/2508.20650/) (2025)
- [FAS-UNet: A Novel FAS-driven Unet to Learn Variational Image Segmentation](https://mlsanity.com/paper/2210.15164/) (2022)
- [A Neural-preconditioned Poisson Solver for Mixed Dirichlet and Neumann Boundary Conditions](https://mlsanity.com/paper/2310.00177/) (2023)
- [Solutions to Elliptic and Parabolic Problems via Finite Difference Based Unsupervised Small Linear Convolutional Neural Networks](https://mlsanity.com/paper/2311.00259/) (2023)
- [Adaptive Multilevel Neural Networks for Parametric PDEs with Error Estimation](https://mlsanity.com/paper/2403.12650/) (2024)
- [A MgNO Method for Multiphase Flow in Porous Media](https://mlsanity.com/paper/2407.02505/) (2024)
- [Two-Dimensional Deep ReLU CNN Approximation for Korobov Functions: A Constructive Approach](https://mlsanity.com/paper/2503.07976/) (2025)
- [A Mathematical Explanation of Transformers](https://mlsanity.com/paper/2510.03989/) (2025)
- [Flowers: A Warp Drive for Neural PDE Solvers](https://mlsanity.com/paper/2603.04430/) (2026)
- [Metriplector: From Field Theory to Neural Architecture](https://mlsanity.com/paper/2603.29496/) (2026)
- [Error Analysis of Physics-Informed Neural Networks for Approximating Dynamic PDEs of Second Order in Time](https://mlsanity.com/paper/2303.12245/) (2023)

## Abstract (arXiv)

We develop a unified model, known as MgNet, that simultaneously recovers some convolutional neural networks (CNN) for image classification and multigrid (MG) methods for solving discretized partial differential equations (PDEs). This model is based on close connections that we have observed and uncovered between the CNN and MG methodologies. For example, pooling operation and feature extraction in CNN correspond directly to restriction operation and iterative smoothers in MG, respectively. As the solution space is often the dual of the data space in PDEs, the analogous concept of feature space and data space (which are dual to each other) is introduced in CNN. With such connections and new concept in the unified model, the function of various convolution operations and pooling used in CNN can be better understood. As a result, modified CNN models (with fewer weights and hyper parameters) are developed that exhibit competitive and sometimes better performance in comparison with existing CNN models when applied to both CIFAR-10 and CIFAR-100 data sets.