  Machine learning is used to develop closure terms for coarse grained model of
two-dimensional turbulent flow directly from the coarse grained data by adding
a source term to the Navier-Stokes equations to ensure that the coarse-grained
flow evolves in the correct way. The source term is related to the average flow
using a Neural Network with a relatively simple structure and smoothed slightly
to prevent instabilities in a posteriori test. The time dependent coarse
grained flow field is generated by filtering fully resolved results and the
predicted coarse field evolution agrees well with the filtered results, both
for the flow used to learn the closure terms and for flows not used for the
learning.
