用神经网络生成稀疏稳健的粗网格算子，加速科学计算中的线性方程求解。
RAPNet: Accelerating Algebraic Multigrid with Learned Sparse Corrections
- 设计图神经网络逐层学习稀疏粗网格算子，突破传统AMG在稀疏性与收敛性间的权衡。
- 在多种PDE离散和图拉普拉斯问题上优于经典非加伽金方法，尤其适合多查询任务。
- 仅在求解器初始化阶段运行，不影响求解阶段性能，适合大规模场景部署。
大规模稀疏线性系统的高效求解是科学计算与图分析的核心瓶颈。尽管代数多重网格（AMG）具备最优线性复杂度，但其性能受限于粗网格算子的稀疏性与收敛质量之间的权衡。传统AMG启发式方法难以兼顾二者，常以牺牲稳定性或性能换取稀疏性。本文提出RAPNet，一种基于图神经网络（GNN）的框架，可直接从稀疏代数系统中学习生成稀疏且鲁棒的粗网格算子。关键在于层级式训练策略，使模型能从小规模子图中学习，并泛化至百万节点规模，克服了先前神经AMG方法的瓶颈。RAPNet仅在求解器设置阶段执行，确保求解阶段保持优良计算特性。实验表明，该方法在多种PDE离散与图拉普拉斯问题上超越经典非加伽金基线，特别适用于特征值问题、时变模拟及反演/设计等多查询任务。
原文摘要 · Abstract (English)
The scalable solution of large sparse linear systems is a bottleneck in scientific computing and graph analysis. While algebraic multigrid (AMG) offers optimal linear scaling, its performance is severely constrained by the trade-off between the sparsity and convergence quality of coarse-grid operators. Classical AMG heuristics struggle to balance these objectives, often sacrificing stability or performance for sparsity. We propose RAPNet, a graph neural network (GNN) framework that resolves this trade-off by learning to generate sparse, robust coarse operators directly from the sparse algebraic system. Key to our approach is a level-wise training strategy that enables learning from small subgraphs and generalization to million-node domains, bypassing the bottlenecks of prior neural AMG attempts. RAPNet executes exclusively during the solver setup phase, ensuring that the solve phase retains its favorable computational properties. We show that our method outperforms classical non-Galerkin baselines on diverse PDE discretizations and graph Laplacians, making it particularly effective for multi-query tasks such as eigenproblems, time-dependent simulations, and inverse or design problems.
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