We propose a convenient matrix-free neural architecture for the multigrid
method. The architecture is simple enough to be implemented in less than fifty
lines of code, yet it encompasses a large number of distinct multigrid solvers.
We argue that a fixed neural network without dense layers can not realize an
efficient iterative method. Because of that, standard training protocols do not
lead to competitive solvers. To overcome this difficulty, we use parameter
sharing and serialization of layers. The resulting network can be trained on
linear problems with thousands of unknowns and retains its efficiency on
problems with millions of unknowns. From the point of view of numerical linear
algebra network's training corresponds to finding optimal smoothers for the
geometric multigrid method. We demonstrate our approach on a few second-order
elliptic equations. For tested linear systems, we obtain from two to five times
smaller spectral radius of the error propagation matrix compare to a basic
linear multigrid with Jacobi smoother.

我们提出了一种简便的无矩阵神经网络结构用于多重网格方法。该结构简单到可以在不到五十行的代码中实现，但包含许多不同的多重网格求解器。我们认为，固定的神经网络没有密集层不能实现高效的迭代方法。因此，标准的训练协议不能生成竞争优势的求解器。为了克服这个困难，我们使用参数共享和层序列化。所得到的网络可以在数以千计未知元的线性问题上进行训练，并在百万未知元的问题上保持其效率。从数值线性代数网络的训练角度来看，它对应于找到几何多重网格方法的最佳平滑器。我们在几个二阶椭圆方程上演示了我们的方法。对于测试的线性系统，与基本线性多重网格方法的 Jacobi 平滑器相比，我们得到的误差传播矩阵的谱半径较小，是其 2 到 5 倍。