In this paper, we investigate the combination of multigrid methods and neural networks, starting from a Finite Element discretization of an elliptic PDE. Multigrid methods use interpolation operators to transfer information between different levels of approximation. These operators are crucial for fast convergence of multigrid, but they are generally unknown. We propose Deep Neural Network models for learning interpolation operators and we build a multilevel hierarchy based on the output of the network. We investigate the accuracy of the interpolation operator predicted by the Neural Network, testing it with different network architectures. This Neural Network approach for the construction of grid operators can then be extended for an automatic definition of multilevel solvers, allowing a portable solution in scientific computing

中文翻译：

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为多级求解器构建网格算子：一种神经网络方法

在本文中，我们从椭圆偏微分方程的有限元离散化开始，研究多重网格方法和神经网络的组合。多重网格方法使用插值算子在不同的近似级别之间传递信息。这些算子对于多重网格的快速收敛至关重要，但它们通常是未知的。我们提出了用于学习插值算子的深度神经网络模型，并基于网络的输出构建了一个多级层次结构。我们研究了神经网络预测的插值算子的准确性，并使用不同的网络架构对其进行了测试。这种用于构建网格算子的神经网络方法可以扩展为自动定义多级求解器，从而在科学计算中提供可移植的解决方案