  We propose to accelerate a high order discontinuous Galerkin solver using
neural networks. We include a corrective forcing to a low polynomial order
simulation to enhance its accuracy. The forcing is obtained by training a deep
fully connected neural network, using a high polynomial order simulation but
only for a short time frame. With this corrective forcing, we can run the low
polynomial order simulation faster (with large time steps and low cost per time
step) while improving its accuracy.
  We explored this idea for a 1D Burgers' equation in (Marique and Ferrer, CAF
2022), and we have extended this work to the 3D Navier-Stokes equations, with
and without a Large Eddy Simulation closure model. We test the methodology with
the turbulent Taylor Green Vortex case and for various Reynolds numbers (30,
200 and 1600). In addition, the Taylor Green Vortex evolves with time and
covers laminar, transitional, and turbulent regimes, as time progresses.
  The proposed methodology proves to be applicable to a variety of flows and
regimes. The results show that the corrective forcing is effective in all
Reynolds numbers and time frames (excluding the initial flow development). We
can train the corrective forcing with a polynomial order of 8, to increase the
accuracy of simulations from a polynomial order 3 to 6, when correcting outside
the training time frame. The low order correct solution is 4 to 5 times faster
than a simulation with comparable accuracy (polynomial order 6).
  Additionally, we explore changes in the hyperparameters and use transfer
learning to speed up the training. We observe that it is not useful to train a
corrective forcing using a different flow condition. However, an already
trained corrective forcing can be used to initialise a new training (at the
correct flow conditions) to obtain an effective forcing with only a few
training iterations.
