Algebraic multigrid (AMG) methods are among the most efficient solvers for linear systems of equations and they are widely used for the solution of problems stemming from the discretization of Partial Differential Equations (PDEs). The most severe limitation of AMG methods is the dependence on parameters that require to be fine-tuned. In particular, the strong threshold parameter is the most relevant since it stands at the basis of the construction of successively coarser grids needed by the AMG methods. We introduce a novel Deep Learning algorithm that minimizes the computational cost of the AMG method when used as a finite element solver. We show that our algorithm requires minimal changes to any existing code. The proposed Artificial Neural Network (ANN) tunes the value of the strong threshold parameter by interpreting the sparse matrix of the linear system as a black-and-white image and exploiting a pooling operator to transform it into a small multi-channel image. We experimentally prove that the pooling successfully reduces the computational cost of processing a large sparse matrix and preserves the features needed for the regression task at hand. We train the proposed algorithm on a large dataset containing problems with a highly heterogeneous diffusion coefficient defined in different three-dimensional geometries and discretized with unstructured grids and linear elasticity problems with a highly heterogeneous Young's modulus. When tested on problems with coefficients or geometries not present in the training dataset, our approach reduces the computational time by up to 30%.

翻译：代数多重网格（AMG）方法是最高效的线性方程组求解器之一，广泛应用于偏微分方程（PDE）离散化所引发问题的求解。AMG方法最严重的局限性在于其对需要精细调参的参数依赖。其中，强阈值参数最为关键，因为它构成了AMG方法中连续粗化网格构造的基础。我们提出了一种新颖的深度学习算法，该算法最小化了AMG方法在作为有限元求解器时的计算成本。我们证明，该算法对任何现有代码的改动需求极小。该人工神经网络（ANN）通过将线性系统的稀疏矩阵解释为黑白图像，并利用池化算子将其转换为小型多通道图像，从而调节强阈值参数的值。我们通过实验证明，池化成功降低了处理大型稀疏矩阵的计算成本，并保留了当前回归任务所需的特征。我们在一个包含高度异质扩散系数（定义于不同三维几何体并采用非结构化网格离散化的问题）以及高度异质杨氏模量的线弹性问题的大型数据集上对算法进行训练。当对训练数据集中未出现的系数或几何体的问题进行测试时，我们的方法能将计算时间减少高达30%。