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

A Deep Learning algorithm to accelerate Algebraic Multigrid methods in Finite Element solvers of 3D elliptic PDEs