  Recent research has used deep learning to develop partial differential
equation (PDE) models in science and engineering. The functional form of the
PDE is determined by a neural network, and the neural network parameters are
calibrated to available data. Calibration of the embedded neural network can be
performed by optimizing over the PDE. Motivated by these applications, we
rigorously study the optimization of a class of linear elliptic PDEs with
neural network terms. The neural network parameters in the PDE are optimized
using gradient descent, where the gradient is evaluated using an adjoint PDE.
As the number of parameters become large, the PDE and adjoint PDE converge to a
non-local PDE system. Using this limit PDE system, we are able to prove
convergence of the neural network-PDE to a global minimum during the
optimization. Finally, we use this adjoint method to train a neural network
model for an application in fluid mechanics, in which the neural network
functions as a closure model for the Reynolds-averaged Navier--Stokes (RANS)
equations. The RANS neural network model is trained on several datasets for
turbulent channel flow and is evaluated out-of-sample at different Reynolds
numbers.
