Volume 30, Issue 5
Ensemble Gradient for Learning Turbulence Models from Indirect Observations

Carlos A. Michelén Ströfer, Xin-Lei Zhang & Heng Xiao

Commun. Comput. Phys., 30 (2021), pp. 1269-1289.

Published online: 2021-10

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  • 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.

  • Keywords

Ensemble methods, turbulence modeling, deep learning.

  • AMS Subject Headings

76F99, 76M99, 65Z05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-30-1269, author = {Ströfer , Carlos A. Michelén and Zhang , Xin-Lei and Xiao , Heng}, title = {Ensemble Gradient for Learning Turbulence Models from Indirect Observations}, journal = {Communications in Computational Physics}, year = {2021}, volume = {30}, number = {5}, pages = {1269--1289}, 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0082}, url = {http://global-sci.org/intro/article_detail/cicp/19929.html} }
TY - JOUR T1 - Ensemble Gradient for Learning Turbulence Models from Indirect Observations AU - Ströfer , Carlos A. Michelén AU - Zhang , Xin-Lei AU - Xiao , Heng JO - Communications in Computational Physics VL - 5 SP - 1269 EP - 1289 PY - 2021 DA - 2021/10 SN - 30 DO - http://doi.org/10.4208/cicp.OA-2021-0082 UR - https://global-sci.org/intro/article_detail/cicp/19929.html KW - Ensemble methods, turbulence modeling, deep learning. AB -

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.

Carlos A. Michelén Ströfer, Xin-Lei Zhang & Heng Xiao. (2021). Ensemble Gradient for Learning Turbulence Models from Indirect Observations. Communications in Computational Physics. 30 (5). 1269-1289. doi:10.4208/cicp.OA-2021-0082
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