  The predictive accuracy of the Navier-Stokes equations is known to degrade at
the limits of the continuum assumption, thereby necessitating expensive and
often highly approximate solutions to the Boltzmann equation. While tractable
in one spatial dimension, their high dimensionality makes multi-dimensional
Boltzmann calculations impractical for all but canonical configurations. It is
therefore desirable to augment the Navier-Stokes equations in these regimes. We
present an application of a deep learning method to extend the validity of the
Navier-Stokes equations to the transition-continuum flows. The technique
encodes the missing physics via a neural network, which is trained directly
from Boltzmann solutions. While standard DL methods can be considered ad-hoc
due to the absence of underlying physical laws, at least in the sense that the
systems are not governed by known partial differential equations, the DL
framework leverages the a-priori known Boltzmann physics while ensuring that
the trained model is consistent with the Navier-Stokes equations. The online
training procedure solves adjoint equations, constructed using algorithmic
differentiation, which efficiently provide the gradient of the loss function
with respect to the learnable parameters. The model is trained and applied to
predict stationary, one-dimensional shock thickness in low-pressure argon.
