Ensemble gradient for learning turbulence models from indirect observations

TL;DR

This paper introduces an ensemble-based approach to approximate sensitivities in training turbulence models from indirect velocity observations, offering a practical alternative to adjoint methods with specific advantages and limitations.

Contribution

It presents a novel ensemble approximation method for RANS sensitivities, enabling training of turbulence models without explicit adjoint computations.

Findings

Ensemble approximation can effectively replace adjoint sensitivities in turbulence model training.

The method improves velocity predictions in learned turbulence models.

Limitations arise when sensitivity to the model becomes very small.

Abstract

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.