  We propose a new neural network based large eddy simulation framework for the
incompressible Navier-Stokes equations based on the paradigm "discretize first,
filter and close next". This leads to full model-data consistency and allows
for employing neural closure models in the same environment as where they have
been trained. Since the LES discretization error is included in the learning
process, the closure models can learn to account for the discretization.
  Furthermore, we introduce a new divergence-consistent discrete filter defined
through face-averaging. The new filter preserves the discrete divergence-free
constraint by construction, unlike general discrete filters such as
volume-averaging filters. We show that using a divergence-consistent LES
formulation coupled with a convolutional neural closure model produces stable
and accurate results for both a-priori and a-posteriori training, while a
general (divergence-inconsistent) LES model requires a-posteriori training or
other stability-enforcing measures.
