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

Ensemble gradient for learning turbulence models from indirect observations