  GMRES is a powerful numerical solver used to find solutions to extremely
large systems of linear equations. These systems of equations appear in many
applications in science and engineering. Here we demonstrate a real-time
machine learning algorithm that can be used to accelerate the time-to-solution
for GMRES. Our framework is novel in that is integrates the deep learning
algorithm in an in situ fashion: the AI-accelerator gradually learns how to
optimizes the time to solution without requiring user input (such as a
pre-trained data set). We describe how our algorithm collects data and
optimizes GMRES. We demonstrate our algorithm by implementing an accelerated
(MLGMRES) solver in Python. We then use MLGMRES to accelerate a solver for the
Poisson equation -- a class of linear problems that appears in may
applications.
  Informed by the properties of formal solutions to the Poisson equation, we
test the performance of different neural networks. Our key takeaway is that
networks which are capable of learning non-local relationships perform well,
without needing to be scaled with the input problem size, making them good
candidates for the extremely large problems encountered in high-performance
computing. For the inputs studied, our method provides a roughly 2$\times$
acceleration.
