# Learning Algebraic Multigrid Using Graph Neural Networks

arXiv: https://arxiv.org/abs/2003.05744 · HTML: https://mlsanity.com/paper/2003.05744/

## Key facts

- Cited by 54 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Mar 2020.

## Built on by (most cited first, 30 of 54)

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- [A Deep Conjugate Direction Method for Iteratively Solving Linear Systems](https://mlsanity.com/paper/2205.10763/) (2022)
- [MG-GNN: Multigrid Graph Neural Networks for Learning Multilevel Domain Decomposition Methods](https://mlsanity.com/paper/2301.11378/) (2023)
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- [Gauge-equivariant neural networks as preconditioners in lattice QCD](https://mlsanity.com/paper/2302.05419/) (2023)
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- [NOWS: Neural Operator Warm Starts for Accelerating Iterative Solvers](https://mlsanity.com/paper/2511.02481/) (2025)
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- [Gauge-equivariant pooling layers for preconditioners in lattice QCD](https://mlsanity.com/paper/2304.10438/) (2023)
- [A Deep Learning algorithm to accelerate Algebraic Multigrid methods in Finite Element solvers of 3D elliptic PDEs](https://mlsanity.com/paper/2304.10832/) (2023)
- [Graph Neural Networks and Applied Linear Algebra](https://mlsanity.com/paper/2310.14084/) (2023)
- [UGrid: An Efficient-And-Rigorous Neural Multigrid Solver for Linear PDEs](https://mlsanity.com/paper/2408.04846/) (2024)
- [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)
- [Optimized Sparse Matrix Operations for Reverse Mode Automatic Differentiation](https://mlsanity.com/paper/2212.05159/) (2022)
- [Accurate and scalable deep Maxwell solvers using multilevel iterative methods](https://mlsanity.com/paper/2509.03622/) (2025)
- [SymMaP: Improving Computational Efficiency in Linear Solvers through Symbolic Preconditioning](https://mlsanity.com/paper/2510.24170/) (2025)
- [AI-enhanced iterative solvers for accelerating the solution of large scale parametrized systems](https://mlsanity.com/paper/2207.02543/) (2022)
- [PEARL: Preconditioner Enhancement through Actor-critic Reinforcement Learning](https://mlsanity.com/paper/2501.10750/) (2025)

## Abstract (arXiv)

Efficient numerical solvers for sparse linear systems are crucial in science and engineering. One of the fastest methods for solving large-scale sparse linear systems is algebraic multigrid (AMG). The main challenge in the construction of AMG algorithms is the selection of the prolongation operator -- a problem-dependent sparse matrix which governs the multiscale hierarchy of the solver and is critical to its efficiency. Over many years, numerous methods have been developed for this task, and yet there is no known single right answer except in very special cases. Here we propose a framework for learning AMG prolongation operators for linear systems with sparse symmetric positive (semi-) definite matrices. We train a single graph neural network to learn a mapping from an entire class of such matrices to prolongation operators, using an efficient unsupervised loss function. Experiments on a broad class of problems demonstrate improved convergence rates compared to classical AMG, demonstrating the potential utility of neural networks for developing sparse system solvers.