Multigrid methods are one of the most efficient techniques for solving linear systems arising from Partial Differential Equations (PDEs) and graph Laplacians from machine learning applications. One of the key components of multigrid is smoothing, which aims at reducing high-frequency errors on each grid level. However, finding optimal smoothing algorithms is problem-dependent and can impose challenges for many problems. In this paper, we propose an efficient adaptive framework for learning optimized smoothers from operator stencils in the form of convolutional neural networks (CNNs). The CNNs are trained on small-scale problems from a given type of PDEs based on a supervised loss function derived from multigrid convergence theories, and can be applied to large-scale problems of the same class of PDEs. Numerical results on anisotropic rotated Laplacian problems demonstrate improved convergence rates and solution time compared with classical hand-crafted relaxation methods.

翻译：多格方法是解决来自机器学习应用的局部差异(PDEs)和图形 Laplacecians产生的线性系统的最有效技术之一。多格方法的关键组成部分之一是平滑,目的是减少每个网格层次的高频错误。然而,找到最佳的平滑算法取决于问题,可能会给许多问题带来挑战。在本文件中,我们提出了一个高效的适应框架,以革命性神经网络的形式,从操作员中学习最优化的平滑器。有线电视新闻网根据多格融合理论产生的监督性损失函数,就某类PDE的小规模问题进行了培训,并可以应用于同一类PDE的大规模问题。厌食性旋转拉巴卡问题的数字结果表明,与古典手制放松方法相比,趋同率和溶解时间有所改善。