We present a novel Deep Learning-based algorithm to accelerate - through the use of Artificial Neural Networks (ANNs) - the convergence of Algebraic Multigrid (AMG) methods for the iterative solution of the linear systems of equations stemming from Finite Element discretizations of Partial Differential Equations. We show that ANNs can be be successfully used to predict the strong connection parameter that enters in the construction of the sequence of increasingly smaller matrix problems standing at the basis of the AMG algorithm, so as to maximize the corresponding convergence factor of the AMG scheme. To demonstrate the practical capabilities of the proposed algorithm, which we call AMG-ANN, we consider the iterative solution via the AMG method of the algebraic system of equations stemming from Finite Element discretizations of a two-dimensional elliptic equation with a highly heterogeneous diffusion coefficient. We train (off-line) our ANN with a rich data-set and present an in-depth analysis of the effects of tuning the strong threshold parameter on the convergence factor of the resulting AMG iterative scheme.
翻译:我们提出了一个新型的深学习算法,通过使用人工神经网络(ANNs)来加速代数多格格丽德(AMG)方法的融合,以迭代解决部分差异方程式的精度分解产生的等式的线性系统;我们表明,可以成功地利用非格朗法来预测在构筑以AMG算法为基础的越来越小的矩阵问题序列过程中出现的强有力的连接参数,以便最大限度地实现AMG办法的相应趋同系数;为了证明我们称之为AMG-ANN的拟议的算法的实际能力,我们考虑通过AMG法的迭代解决办法,即由二维离子分解的单方程式的精度分解法,并具有高度混合的传播系数;我们用丰富的数据集来培训我们的ANN(脱线)我们的ANN,并对调整强阈值参数对由此形成的AMG迭接法的聚合系数的影响进行深入分析。