发表机构
Wuhan University of Technology; Embodied Physical Intelligence Center in Civil Engineering, Wuhan University of Technology; Key Laboratory of Low-Altitude Technology and Smart Urban Renewal, Department of Housing and Urban-Rural Development of Hubei Province, Wuhan University of Technology; Sanya Science and Education Innovation Park of Wuhan University of Technology（武汉理工大学; 武汉理工大学土木工程具身智能中心; 湖北省住房和城乡建设厅低空技术与智慧城市更新重点实验室（武汉理工大学）; 武汉理工大学三亚科教创新园）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出BandPC框架，利用图神经网络预测灵活共轭梯度法的多阶段预条件子组合，通过残差带划分和软标签机制，在混合基准上显著提升迭代次数与求解时间。
AI 中文摘要
共轭梯度（CG）方法是求解稀疏对称正定（SPD）线性系统的经典迭代方法，但其收敛性强烈依赖于系统矩阵的谱性质。预条件处理可以改善这些性质；然而，设计能够泛化到多样且不规则稀疏模式的预条件子仍然具有挑战性。我们提出BandPC（带预条件子组合，其中“带”表示残差范数带），这是一个数据驱动的框架，利用图神经网络（GNNs）为灵活共轭梯度（FCG）方法预测多阶段预条件策略。BandPC将FCG迭代划分为三个基于残差范数的带，并在五个经典预条件子上定义了包含125种候选组合的结构化搜索空间。通过利用FCG在迭代间切换预条件子的能力，BandPC学习直接将矩阵结构映射到有前景的预条件子序列。为了改善训练和泛化，我们引入了一种软标签机制，保留接近最优的组合并将其分数归一化为标签分布，同时采用一种层次化特征表示，捕获节点级属性、边级代数耦合强度和全局矩阵统计信息。在合成SPD问题和SuiteSparse集合中多样矩阵的混合基准上的实验表明，BandPC在41.3%和33.1%的测试矩阵上分别取得了最佳迭代次数和求解时间，即使与每个矩阵单独最优的传统预条件子相比也是如此。这些结果表明，学习残差带预条件子调度可以有效加速灵活CG求解器。
英文摘要
The conjugate gradient (CG) method is a classical iterative solver for sparse symmetric positive definite (SPD) linear systems, but its convergence strongly depends on the spectral properties of the system matrix. Preconditioning can improve these properties; however, designing preconditioners that generalize across diverse and irregular sparsity patterns remains challenging. We propose BandPC (Band Preconditioner Combinations, where "Band" denotes residual-norm bands), a data-driven framework that leverages graph neural networks (GNNs) to predict multi-stage preconditioning strategies for the flexible conjugate gradient (FCG) method. BandPC partitions the FCG iteration into three residual-norm-based bands and defines a structured search space of 125 candidate combinations over five classical preconditioners. By exploiting FCG's ability to switch preconditioners across iterations, BandPC learns to map matrix structure directly to a promising preconditioner sequence. To improve training and generalization, we introduce a soft-labeling mechanism that retains near-optimal combinations and normalizes their scores into a label distribution, together with a hierarchical feature representation that captures node-level attributes, edge-level algebraic coupling strengths, and global matrix statistics. Experiments on a hybrid benchmark of synthetic SPD problems and diverse matrices from the SuiteSparse Collection show that BandPC achieves the best iteration count and solution time on 41.3% and 33.1% of test matrices, respectively, even when compared with each matrix's individually best traditional preconditioner. These results demonstrate that learned residual-band preconditioner scheduling can effectively accelerate flexible CG solvers.
