基于图神经网络与深度学习的AMG参数优化研究
Research on AMG parameter optimization based on graph neural networks and deep learning
刘高显 1纪执玥 1柳元1
作者信息
- 1. 天津师范大学电子与通信工程学院，天津 300387
- 折叠
摘要
针对大规模稀疏线性方程组求解过程中代数多重网格（Algebraic Multigrid, AMG）预处理器的V循环次数通常依赖经验设定、难以在不同问题上保持稳定高效这一问题，本文提出一种基于图注意力网络（Graph Attention Network, GAT）与多层感知机（Multilayer Perceptron, MLP）协同的自适应优化框架。该框架将线性方程组的系数矩阵表示为图结构，利用 GAT 自动提取矩阵非零模式所蕴含的拓扑与代数特征，并通过MLP对当前问题对应的 V 循环次数进行分类预测。本文在二维与三维扩散方程离散生成的线性系统上开展数值实验，并将所提方法与固定 V 循环次数策略进行比较。实验结果表明，所提方法能够有效降低GMRES迭代步数，从而提升 AMG 预处理求解器的整体计算效率。
Abstract
For large-scale sparse linear systems, the performance of Algebraic Multigrid (AMG) preconditioners is highly sensitive to the choice of the number of V-cycles, which is usually determined by empirical rules and may not remain efficient across different problems. To address this issue, this paper proposes an adaptive optimization framework based on the collaboration of a Graph Attention Network (GAT) and a Multilayer Perceptron (MLP). In the proposed framework, the coefficient matrix of a linear system is represented as a graph, and GAT is employed to automatically extract topological and algebraic features embedded in the nonzero pattern of the matrix. The extracted features are then fed into an MLP to predict the appropriate number of V-cycles in a classification manner. Numerical experiments are conducted on linear systems generated from the discretization of two-dimensional and three-dimensional diffusion equations. Compared with fixed V-cycle strategies, the proposed method can effectively reduce the number of GMRES iterations and improve the overall computational efficiency of AMG-preconditioned solvers.关键词
人工智能，图注意网络，多层感知机，代数多重网格法Key words
Artificial Intelligence/Graph Attention Network/ Multi-Layer Perceptron/ Algebraic Multigrid Method引用本文复制引用
刘高显,纪执玥,柳元.基于图神经网络与深度学习的AMG参数优化研究[EB/OL].(2026-03-31)[2026-04-03].http://www.paper.edu.cn/releasepaper/content/202603-309.学科分类
计算技术、计算机技术
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