In many numerical schemes, the computational complexity scales non-linearly with the problem size. Solving a linear system of equations using direct methods or most iterative methods is a typical example. Algebraic multi-grid (AMG) methods are numerical methods used to solve large linear systems of equations efficiently. One of the main differences between AMG methods is how the coarser grid is constructed from a given fine grid. There are two main classes of AMG methods; graph and aggregation based coarsening methods. Here we propose an aggregation-based coarsening framework leveraging graph representation learning and clustering algorithms. Our method introduces the power of machine learning into the AMG research field and opens a new perspective for future researches. The proposed method uses graph representation learning techniques to learn latent features of the graph obtained from the underlying matrix of coefficients. Using these extracted features, we generated a coarser grid from the fine grid. The proposed method is highly capable of parallel computations. Our experiments show that the proposed method's efficiency in solving large systems is closely comparable with other aggregation-based methods, demonstrating the high capability of graph representation learning in designing multi-grid solvers.

中文翻译：

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GL-Coarsener：为 AMG 求解器构建粗网格层次结构的图表示学习框架

在许多数值方案中，计算复杂度与问题大小呈非线性关系。使用直接方法或大多数迭代方法求解线性方程组是一个典型的例子。代数多重网格 (AMG) 方法是用于有效求解大型线性方程组的数值方法。AMG 方法之间的主要区别之一是粗网格是如何从给定的细网格构建的。AMG方法有两大类；基于图和聚合的粗化方法。在这里，我们提出了一种利用图表示学习和聚类算法的基于聚合的粗化框架。我们的方法将机器学习的力量引入 AMG 研究领域，并为未来的研究开辟了新的视角。所提出的方法使用图表示学习技术来学习从基础系数矩阵获得的图的潜在特征。使用这些提取的特征，我们从细网格生成了一个粗网格。所提出的方法具有很强的并行计算能力。我们的实验表明，所提出的方法在解决大型系统方面的效率与其他基于聚合的方法非常相似，证明了图表示学习在设计多网格求解器方面的强大能力。