想给稀疏矩阵求解提速的话，这篇论文把AMG层级和图神经网络结合了，在800多个矩阵上实测，可以看到它什么时候比AMG好、什么时候反而更慢。
该论文提出图神经网络多层级预条件器（GMP），将代数多重网格（AMG）层级结构作为先验，统一学习平滑、限制与插值算子。在超800个稀疏矩阵基准上，对比经典AMG、单层ILUT及现有GNN预条件器，分析了多层级图神经预条件在收敛性和计算开销上的优劣。结果表明GMP可作为标准Krylov求解器的即插即用预条件器，但在某些场景下不及强单层基线。
Graph Neural Multilevel Preconditioners for Iterative Solvers
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.