Large sparse linear systems of equations are ubiquitous in science and engineering, such as those arising from discretizations of partial differential equations. Algebraic multigrid (AMG) methods are one of the most common methods of solving such linear systems, with an extensive body of underlying mathematical theory. A system of linear equations defines a graph on the set of unknowns and each level of a multigrid solver requires the selection of an appropriate coarse graph along with restriction and interpolation operators that map to and from the coarse representation. The efficiency of the multigrid solver depends critically on this selection and many selection methods have been developed over the years. Recently, it has been demonstrated that it is possible to directly learn the AMG interpolation and restriction operators, given a coarse graph selection. In this paper, we consider the complementary problem of learning to coarsen graphs for a multigrid solver, a necessary step in developing fully learnable AMG methods. We propose a method using a reinforcement learning (RL) agent based on graph neural networks (GNNs), which can learn to perform graph coarsening on small planar training graphs and then be applied to unstructured large planar graphs, assuming bounded node degree. We demonstrate that this method can produce better coarse graphs than existing algorithms, even as the graph size increases and other properties of the graph are varied. We also propose an efficient inference procedure for performing graph coarsening that results in linear time complexity in graph size.
翻译:巨大的线性方程式系统在科学和工程学上是无处不在的,例如部分差异方程式的离散化所产生的系统。 代数多格( AMG) 方法是解决这种线性系统的最常见方法之一, 具有广泛的基本数学理论。 一个线性方程式系统定义了一组未知和多格数求解器的每个级别上的图表, 需要选择一个适当的粗缩图, 以及地图和粗略表达式之间的限制和内插操作器。 多格解运算器的效率关键取决于此选择, 许多选择方法多年来已经开发出来。 最近, 已经证明可以直接学习 AMG 干涉和限制操作器( AMGM GM), 并且有一个粗略的图操作器。 我们考虑的是, 学习多格解析图的图图的补补补补补问题, 这是开发完全可学习的 AMG方法的一个必要步骤。 我们建议一种方法, 以图形神经网络( GNNP) 为基础, 它可以在小平面的图表中进行直线性直径分析, 直径分析过程的计算结果, 也可以在小平面图的平面图中进行更精确的直径化的平面图形分析过程中, 。 我们的平面图的平面图的平面的平面图则可以用来显示更细的平面的直径直图。