发表机构
University of Minnesota; Lawrence Livermore National Laboratory（明尼苏达大学; 劳伦斯利弗莫尔国家实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对大型稀疏线性系统求解问题，提出图神经多重网格预条件子（GMP），在800余个稀疏矩阵基准上验证其性能，明确该方法的收敛提升潜力与开销局限。
AI 中文摘要
求解大型稀疏线性系统是科学计算的核心任务，高效迭代求解器高度依赖有效且鲁棒的预条件技术。经典方法如代数多重网格（AMG）具有高可扩展性，但在不定或非对称系统上，其鲁棒性会下降，因为原本针对椭圆型偏微分方程（PDE）开发的启发式方法可靠性降低。近年来，图神经网络（GNN）已成为数据驱动的预条件子，但针对一般稀疏矩阵，采用AMG式层级结构的实际影响仍未得到充分探索。本研究提出一种图神经多重网格预条件子（GMP），它采用AMG层级作为结构先验，在统一框架中学习光滑、限制和插值算子。该方法针对一般稀疏系统设计，可作为标准Krylov求解器的替代预条件子使用。在包含800余个稀疏矩阵的基准测试中，我们将其与经典AMG、单层ILUT以及最先进的GNN预条件子进行对比，明确了多重网格图神经预条件子在哪些场景下可提升收敛性，或与强单层基线相比引入额外开销。这些结果凸显了在面向大规模科学模拟的学习型预条件子中，引入AMG式多重网格结构的潜力与局限性。
英文摘要
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.
CommentsAccepted at KDD 2026
