  Machine learning for scientific applications faces the challenge of limited
data. We propose a framework that leverages a priori known physics to reduce
overfitting when training on relatively small datasets. A deep neural network
is embedded in a partial differential equation (PDE) that expresses the known
physics and learns to describe the corresponding unknown or unrepresented
physics from the data. Crafted as such, the neural network can also provide
corrections for erroneously represented physics, such as discretization errors
associated with the PDE's numerical solution. Once trained, the deep learning
PDE model (DPM) can make out-of-sample predictions for new physical parameters,
geometries, and boundary conditions.
  Our approach optimizes over the functional form of the PDE. Estimating the
embedded neural network requires optimizing over the entire PDE, which itself
is a function of the neural network. Adjoint partial differential equations are
used to efficiently calculate the high-dimensional gradient of the objective
function with respect to the neural network parameters. A stochastic adjoint
method (SAM), similar in spirit to stochastic gradient descent, further
accelerates training.
  The approach is demonstrated and evaluated for turbulence predictions using
large-eddy simulation (LES), a filtered version of the Navier--Stokes equation
containing unclosed sub-filter-scale terms. The DPM outperforms the widely-used
constant-coefficient and dynamic Smagorinsky models, even for filter sizes so
large that these established models become qualitatively incorrect. It also
significantly outperforms a priori trained models, which do not account for the
full PDE. A relaxation of the discrete enforcement of the divergence-free
constraint is also considered, instead allowing the DPM to approximately
enforce incompressibility physics.
