  Training data-driven turbulence models with high fidelity Reynolds stress can
be impractical and recently such models have been trained with velocity and
pressure measurements. For gradient-based optimization, such as training deep
learning models, this requires evaluating the sensitivities of the RANS
equations. This paper explores the use of an ensemble approximation of the
sensitivities of the RANS equations in training data-driven turbulence models
with indirect observations. A deep neural network representing the turbulence
model is trained using the network's gradients obtained by backpropagation and
the ensemble approximation of the RANS sensitivities. Different ensemble
approximations are explored and a method based on explicit projection onto the
sample space is presented. As validation, the gradient approximations from the
different methods are compared to that from the continuous adjoint equations.
The ensemble approximation is then used to learn different turbulence models
from velocity observations. In all cases, the learned model predicts improved
velocities. However, it was observed that once the sensitivity of the velocity
to the underlying model becomes small, the approximate nature of the ensemble
gradient hinders further optimization of the underlying model. The benefits and
limitations of the ensemble gradient approximation are discussed, in particular
as compared to the adjoint equations.
