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

## TL;DR

MgNet unifies multigrid methods and convolutional neural networks, revealing their deep connections, leading to improved CNN models with fewer parameters that perform well on image classification tasks.

## Contribution

This work introduces MgNet, a unified framework linking CNNs and multigrid methods, providing new insights and improved models with fewer weights.

## Key findings

- Modified CNNs with fewer parameters perform competitively on CIFAR datasets.
- The model uncovers direct correspondences between CNN operations and multigrid components.
- Enhanced understanding of CNN functions through the lens of multigrid theory.

## Abstract

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.