# Learning optimal multigrid smoothers via neural networks

arXiv: https://arxiv.org/abs/2102.12071 · HTML: https://mlsanity.com/paper/2102.12071/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 39 references that are in the map.
- First posted on arXiv 24 Feb 2021.

## Builds on (most cited first, 13 of 13)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [Learning to Control PDEs with Differentiable Physics](https://mlsanity.com/paper/2001.07457/) (2020)
- [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 Neural PDE Solvers with Convergence Guarantees](https://mlsanity.com/paper/1906.01200/) (2019)
- [Multi-level Residual Networks from Dynamical Systems View](https://mlsanity.com/paper/1710.10348/) (2017)
- [Learning Algebraic Multigrid Using Graph Neural Networks](https://mlsanity.com/paper/2003.05744/) (2020)
- [A machine learning framework for data driven acceleration of computations of differential equations](https://mlsanity.com/paper/1807.09519/) (2018)
- [General solutions for nonlinear differential equations: a rule-based self-learning approach using deep reinforcement learning](https://mlsanity.com/paper/1805.07297/) (2018)
- [Optimizing Geometric Multigrid Methods with Evolutionary Computation](https://mlsanity.com/paper/1910.02749/) (2019)

## Built on by (most cited first, 17 of 17)

- [Spectral-Refiner: Accurate Fine-Tuning of Spatiotemporal Fourier Neural Operator for Turbulent Flows](https://mlsanity.com/paper/2405.17211/) (2024)
- [Gauge-equivariant neural networks as preconditioners in lattice QCD](https://mlsanity.com/paper/2302.05419/) (2023)
- [Gauge-equivariant pooling layers for preconditioners in lattice QCD](https://mlsanity.com/paper/2304.10438/) (2023)
- [Leveraging Operator Learning to Accelerate Convergence of the Preconditioned Conjugate Gradient Method](https://mlsanity.com/paper/2508.00101/) (2025)
- [Distributed Multigrid Neural Solvers on Megavoxel Domains](https://mlsanity.com/paper/2104.14538/) (2021)
- [Evolving Generalizable Multigrid-Based Helmholtz Preconditioners with Grammar-Guided Genetic Programming](https://mlsanity.com/paper/2204.12846/) (2022)
- [Learning Relaxation for Multigrid](https://mlsanity.com/paper/2207.11255/) (2022)
- [Reducing operator complexity in Algebraic Multigrid with Machine Learning Approaches](https://mlsanity.com/paper/2307.07695/) (2023)
- [A Neural-preconditioned Poisson Solver for Mixed Dirichlet and Neumann Boundary Conditions](https://mlsanity.com/paper/2310.00177/) (2023)
- [Mitigating distribution shift in machine learning-augmented hybrid simulation](https://mlsanity.com/paper/2401.09259/) (2024)
- [Algorithmically Designed Artificial Neural Networks (ADANNs): Higher order deep operator learning for parametric partial differential equations](https://mlsanity.com/paper/2302.03286/) (2023)
- [Towards Automated Algebraic Multigrid Preconditioner Design Using Genetic Programming for Large-Scale Laser Beam Welding Simulations](https://mlsanity.com/paper/2412.08186/) (2024)
- [Graph Lineages and Skeletal Graph Products](https://mlsanity.com/paper/2508.00197/) (2025)
- [A Greedy PDE Router for Blending Neural Operators and Classical Methods](https://mlsanity.com/paper/2509.24814/) (2025)
- [A level-wise training scheme for learning neural multigrid smoothers with application to integral equations](https://mlsanity.com/paper/2603.01064/) (2026)
- [Automated Grammar-based Algebraic Multigrid Design With Evolutionary Algorithms](https://mlsanity.com/paper/2603.17641/) (2026)
- [Acceleration of an algebraic multigrid pressure solver using graph neural networks](https://mlsanity.com/paper/2606.19251/) (2026)

## Abstract (arXiv)

Multigrid methods are one of the most efficient techniques for solving linear systems arising from Partial Differential Equations (PDEs) and graph Laplacians from machine learning applications. One of the key components of multigrid is smoothing, which aims at reducing high-frequency errors on each grid level. However, finding optimal smoothing algorithms is problem-dependent and can impose challenges for many problems. In this paper, we propose an efficient adaptive framework for learning optimized smoothers from operator stencils in the form of convolutional neural networks (CNNs). The CNNs are trained on small-scale problems from a given type of PDEs based on a supervised loss function derived from multigrid convergence theories, and can be applied to large-scale problems of the same class of PDEs. Numerical results on anisotropic rotated Laplacian problems demonstrate improved convergence rates and solution time compared with classical hand-crafted relaxation methods.