Algebraic Multigrid

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Graph Neural Multilevel Preconditioners for Iterative Solvers
Jul 30, 2026Zechen Zhang, Rui Peng Li, Yousef SaadAlgebraic Multigrid Spectral Preconditioning
University of Minnesota Minneapolis, MN, USA · CASA, Lawrence Livermore National Laboratory Livermore, CA, USA
McMg: A Learned Phase-Space Multi-channel Multigrid Preconditioner for Helmholtz Equation
Jun 29, 2026Jiwei Jia, Xinliang Liu, Juntao Wang +1Algebraic Multigrid
School of Mathematics, Jilin University, Changchun 130012, China · Shenzhen Loop Area Institute, Shenzhen 518038, China · Ocean University of China, Qingdao 266100, China +1
Multigrid Training for Molecular Generation using Graph Neural Networks
Jun 21, 2026Zixuan Ling, Paula Mercurio, Di LiuMolecular Generation Algebraic Multigrid
Department of Mathematics, Michigan State University, East Lansing, MI. · Department of Mathematics, Hamline University, Saint Paul, MN.
Acceleration of an algebraic multigrid pressure solver using graph neural networks
Jun 17, 2026Eric Chillón, Artur K. Lidtke, Nguyen Anh Khoa Doan +1Algebraic Multigrid Graph Neural Networks
Faculty of Mechanical Engineering, Delft University of Technology, The Netherlands · Maritime Research Institute Netherlands, The Netherlands · Department of Aeronautics, Imperial College London, United Kingdom
RAPNet: Accelerating Algebraic Multigrid with Learned Sparse Corrections
May 26, 2026Yali Fink, Ido Ben-Yair, Lars Ruthotto +1Algebraic Multigrid Graph Neural Networks
Institute for Interdisciplinary Computational Sciences, Faculty of Computer and Information Science, Ben-Gurion University of the Negev, Be’er Sheva, Israel · Department of Mathematics and Computer Science, Emory University, Atlanta, GA, USA.
Full-field prediction for engineering-scale three-dimensional aircraft with multigrid-hierarchical learning
May 26, 2026Yunfei Liu, Hao Wang, Yuhang Qi +8Computational Fluid Dynamic Queryable Sector Flow Fields
Computational Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang, 621000,China. · Laboratory of Digitizing Software for Frontier Equipment,National University of Defense Technology,Changsha,410073,China. · National Key Laboratory of Parallel and Distributed Computing, National University of Defense Technology, Changsha,410073,China. +1
IV-Net: A neural network for elliptic PDEs with random and highly varying coefficients
May 24, 2026Shan Zhong, George BirosFourier Neural Operators Algebraic Multigrid
Oden Institute for Computational Science and Engineering, The University of Texas at Austin, 201 E 24th St, Austin, 78712, TX, USA · Walker Department of Mechanical Engineering, The University of Texas at Austin, 3012026 E Dean Keeton St, Austin, 78712, TX, USA
Affine Tracing: A New Paradigm for Probabilistic Linear Solvers
May 11, 2026Disha Hegde, Marvin Pförtner, Jon CockayneSolver Iterations Affine
University of Southampton · AI Center, University of Tübingen
NSPOD: Accelerating Krylov solvers via DeepONet-learned POD subspaces
May 8, 2026Francesc Levrero-Florencio, Youngkyu Lee, Jay Pathak +1Algebraic Multigrid Solver Iterations
Synopsys UK Ltd, UK · Division of Applied Mathematics, Brown University, Rhode Island, USA · Synopsys Inc, California, USA
CO $_2$ sequestration hybrid solver using isogeometric alternating-directions and collocation-based robust variational physics informed neural networks (IGA-ADS-CRVPINN)
Apr 22, 2026Askold Vilkha, Tomasz Służalec, Marcin Łoś +1Neural Solvers Algebraic Multigrid
AGH University of Krakow, Faculty of Computer Science, Al. Mickiewicza 2026 30, Kraków, 30-059, Poland
Hybrid coupling with operator inference and the overlapping Schwarz alternating method
Nov 20, 2025Irina Tezaur, Eric Parish, Anthony Gruber +3Reduced-Order Models Algebraic Multigrid
Sandia National Laboratories, Livermore, CA and Albuquerque, NM, USA · Rice University, Houston, TX, USA

Jul 30, 2026, math.NAJ/K move · Enter open · S save

Graph Neural Multilevel Preconditioners for Iterative Solvers

Zechen Zhang, Rui Peng Li, Yousef Saad

University of Minnesota Minneapolis, MN, USA · CASA, Lawrence Livermore National Laboratory Livermore, CA, USA

Algebraic Multigrid Spectral Preconditioning Solver Iterations Graph Neural Networks Preconditioner Solvers

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Solving large, sparse linear systems is a core task in scientific computing, and efficient iterative solvers rely critically on effective and robust preconditioning. While classical methods such as algebraic multigrid (AMG) are highly scalable, their robustness can degrade on indefinite or nonsymmetric systems where heuristics originally developed for elliptic PDEs are less reliable. Recently, Graph Neural Networks (GNNs) have emerged as data-driven preconditioners; yet, the practical impact of imposing an AMG-style hierarchy remains underexplored for general sparse matrices. In this work, we propose a Graph Neural Multilevel Preconditioner (GMP) that adopts an AMG hierarchy as a structural prior and learns smoothing, restriction, and interpolation operators in a unified framework. Our method targets general sparse systems and is instantiated as a drop-in preconditioner for standard Krylov solvers. On a benchmark of over 800 sparse matrices, we compare against classical AMG, single-level ILUT, and state-of-the-art GNN preconditioners, and characterize the regimes where multilevel graph neural preconditioning improves convergence or, conversely, introduces overhead relative to strong single-level baselines. These results highlight both the promise and the limitations of enforcing AMG-style multilevel structure in learned preconditioners for large-scale scientific simulations.

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