  Recent advances in the field of machine learning open a new era in high
performance computing. Applications of machine learning algorithms for the
development of accurate and cost-efficient surrogates of complex problems have
already attracted major attention from scientists. Despite their powerful
approximation capabilities, however, surrogates cannot produce the `exact'
solution to the problem. To address this issue, this paper exploits up-to-date
ML tools and delivers customized iterative solvers of linear equation systems,
capable of solving large-scale parametrized problems at any desired level of
accuracy. Specifically, the proposed approach consists of the following two
steps. At first, a reduced set of model evaluations is performed and the
corresponding solutions are used to establish an approximate mapping from the
problem's parametric space to its solution space using deep feedforward neural
networks and convolutional autoencoders. This mapping serves a means to obtain
very accurate initial predictions of the system's response to new query points
at negligible computational cost. Subsequently, an iterative solver inspired by
the Algebraic Multigrid method in combination with Proper Orthogonal
Decomposition, termed POD-2G, is developed that successively refines the
initial predictions towards the exact system solutions. The application of
POD-2G as a standalone solver or as preconditioner in the context of
preconditioned conjugate gradient methods is demonstrated on several numerical
examples of large scale systems, with the results indicating its superiority
over conventional iterative solution schemes.
