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.

中文翻译：

---

通过神经网络学习最佳的多网格平滑器

多重网格方法是解决由偏微分方程（PDE）产生的线性系统和来自机器学习应用程序的图拉普拉斯算子的最有效技术之一。多重网格的关键组成部分之一是平滑，其目的是减少每个网格级别的高频误差。但是，找到最佳平滑算法取决于问题，并且可能对许多问题构成挑战。在本文中，我们提出了一种有效的自适应框架，用于以卷积神经网络（CNN）的形式从算子模具中学习优化的平滑器。基于从多网格收敛理论导出的监督损失函数，对CNN进行给定类型PDE的小规模问题的训练，并且可以将其应用于同一类PDE的大范围问题。