MGNet

MultiGrid Methods + Neural Networks = MgNet

tl;dr

MultiGrid methods cool. Repository implements paper with pytorch.

Usage

from mgnet.model import mgnet, MGNet
import torch

supported_smoothers = MGNet.supported_smoothers()
print(f'{supported_smoothers=}')
n_image_chan = 3
batch_size = 1
image_width = 28
image_height = 28
model = mgnet(smoother=supported_smoothers[0],
              pretrained=False,
              progress=False,
              in_channels=n_image_chan,
              n_chan_u=128,
              n_chan_f=128,
              n_classes=10,
              n_iter=[2,2,2,2]
              )
x = torch.rand((batch_size, n_image_chan, image_width, image_height))
y = model(x)
print(f'{y=}')

Background

I first came up with the idea of using multigrid methods in deep learning after attending the seminar algebraic multigrid, in which we studied the publication by Jinchao Xu and Ludmil Zikatanov.

If multigrid methods are one of the fastest solvers for partial differential equality problems by use of grids, then knowing that images are functions on grids, multigrid methods can be used in machine learning, especially in FNNs and CNNs ...

i thought. Unfortunately, Sir Xu beat me up to the idea and published a paper 2 years before.

As I couldn't find a ready-to-use implementation of the paper, the work in this repository came to be.

Theory of operation

This type of network generalizes multiple preexisting models like the famous ResNet and DenseNet into a single configurable system.

The premise is built upon the idea of multigrid: the solution u is smoothened and its errors are projected on layers of various sizes.

A single step consists of three phases:

1. Relaxation: Given a smooth S, the solution of the problem u is computed using S(u) = f several time, to obtain an improved solution v. The residuum of the equation is computed r = f - S(v)
2. Restriction: the residuum r is transferred on a layer of lower size and the approximation of the error A_{r}e_{r} = r is computed
3. Prolongation or interpolation: the error e is projected back on the original layer A_{p}e_{r} = r_{p} and the original solution updated: v = v + e_{r}

---
title: Basic multigrid
---
flowchart TD;
    s[Start: v<sup>h</sup> = 0] --> v1    
    subgraph "Relaxation"
        v1["Solve: S<sup>h</sup> * u<sup>h</sup> = f<sup>h</sup>"]--> v["Compute residuum: r = f - S * v"]
    end
    subgraph Restriction 
        v["Compute residuum: r<sup>h</sup> = f<sup>h</sup> - S<sup>h</sup>v<sup>h</sup>"] --> r1
        r1["Project residuum on coarse layer: G<sup>2h</sup>: r<sup>2h</sup>"] --> r2
    end
    subgraph Prolongation
        r2["Compute error: S<sup>2h</sup> * e<sup>2h</sup>= r<sup>2h</sup>"] --> r3
    end
    r3["Project error on finer layer: G<sup>h</sup>: e<sup>h</sup>"] --> p1["Updated solution: v<sup>h</sub> = v<sup>h</sub> + e<sup>h</sub>"]

Testing

Local:
Install dependencies:

uv sync --dev

Run inference test (default values already set. Use --help option for help):

PYTHONPATH="${PYTHONPATH}:${PWD}" uv run tests/test_inference.py

Docker:
Build image:

docker build . -t mgnet

Run image in container:

docker run mgnet --help

Literatur

• "Algebraic Multigrid Methods", Jinchao Xu, Ludmil T Zikatanov
• "MgNet: A Unified Framework of Multigrid and Convolutional Neural Network", Juncai He, Jinchao Xu
• "Iterative Solution of Large Sparse Systems of Equations", Wolfgang Hackbusch