Ensemble Gradient for Learning Turbulence Models from Indirect Observations (Communications in Computational Physics, 2021)
Training data-driven turbulence models with high fidelity Reynolds stresscan be impractical and recently such models have been trained with velocity and pressure measurements. For gradient-based optimization, such as training deep learningmodels, this requires evaluating the sensitivities of the RANS equations. This paperexplores the use of an ensemble approximation of the sensitivities of the RANS equations in training data-driven turbulence models with indirect observations. A deepneural network representing the turbulence model is trained using the network’s gradients obtained by backpropagation and the ensemble approximation of the RANSsensitivities. Different ensemble approximations are explored and a method based onexplicit 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 turbulencemodels from velocity observations. In all cases, the learned model predicts improvedvelocities. However, it was observed that once the sensitivity of the velocity to theunderlying model becomes small, the approximate nature of the ensemble gradienthinders further optimization of the underlying model. The benefits and limitations ofthe ensemble gradient approximation are discussed, in particular as compared to theadjoint equations.