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.

翻译：我们提出了一种便捷的无矩阵神经架构，用于实现多重网格方法。该架构足够简洁，可用不到五十行代码实现，却涵盖了多种不同的多重网格求解器。我们论证了不含密集层的固定神经网络无法实现高效的迭代方法，因此标准训练协议无法产生具有竞争力的求解器。为克服这一困难，我们采用参数共享与层序列化技术。由此得到的网络可在包含数千未知量的线性问题上进行训练，并保持其在数百万未知量问题上的求解效率。从数值线性代数角度看，网络训练对应于求解几何多重网格方法中的最优平滑器。我们通过若干二阶椭圆方程验证了该方法。对于所测试的线性系统，与使用雅可比平滑器的基本线性多重网格相比，我们获得的误差传播矩阵谱半径降低了2至5倍。