# Learning Relaxation for Multigrid

arXiv: https://arxiv.org/abs/2207.11255 · HTML: https://mlsanity.com/paper/2207.11255/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 56 references that are in the map.
- First posted on arXiv 25 Jul 2022.

## Builds on (most cited first, 15 of 15)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [A unified deep artificial neural network approach to partial differential equations in complex geometries](https://mlsanity.com/paper/1711.06464/) (2017)
- [Learning to Optimize Multigrid PDE Solvers](https://mlsanity.com/paper/1902.10248/) (2019)
- [Learning Algebraic Multigrid Using Graph Neural Networks](https://mlsanity.com/paper/2003.05744/) (2020)
- [EikoNet: Solving the Eikonal equation with Deep Neural Networks](https://mlsanity.com/paper/2004.00361/) (2020)
- [Geodesic Convolutional Shape Optimization](https://mlsanity.com/paper/1802.04016/) (2018)
- [A machine learning framework for data driven acceleration of computations of differential equations](https://mlsanity.com/paper/1807.09519/) (2018)
- [Learning optimal multigrid smoothers via neural networks](https://mlsanity.com/paper/2102.12071/) (2021)
- [General solutions for nonlinear differential equations: a rule-based self-learning approach using deep reinforcement learning](https://mlsanity.com/paper/1805.07297/) (2018)
- [Tuning Multigrid Methods with Robust Optimization](https://mlsanity.com/paper/2001.00887/) (2020)

## Built on by (most cited first, 2 of 2)

- [Reducing operator complexity in Algebraic Multigrid with Machine Learning Approaches](https://mlsanity.com/paper/2307.07695/) (2023)
- [Acceleration of an algebraic multigrid pressure solver using graph neural networks](https://mlsanity.com/paper/2606.19251/) (2026)

## Abstract (arXiv)

During the last decade, Neural Networks (NNs) have proved to be extremely effective tools in many fields of engineering, including autonomous vehicles, medical diagnosis and search engines, and even in art creation. Indeed, NNs often decisively outperform traditional algorithms. One area that is only recently attracting significant interest is using NNs for designing numerical solvers, particularly for discretized partial differential equations. Several recent papers have considered employing NNs for developing multigrid methods, which are a leading computational tool for solving discretized partial differential equations and other sparse-matrix problems. We extend these new ideas, focusing on so-called relaxation operators (also called smoothers), which are an important component of the multigrid algorithm that has not yet received much attention in this context. We explore an approach for using NNs to learn relaxation parameters for an ensemble of diffusion operators with random coefficients, for Jacobi type smoothers and for 4Color GaussSeidel smoothers. The latter yield exceptionally efficient and easy to parallelize Successive Over Relaxation (SOR) smoothers. Moreover, this work demonstrates that learning relaxation parameters on relatively small grids using a two-grid method and Gelfand's formula as a loss function can be implemented easily. These methods efficiently produce nearly-optimal parameters, thereby significantly improving the convergence rate of multigrid algorithms on large grids.