Efficient computations for linear feedback control problems for target velocity matching of Navier-Stokes flows via POD and LSTM-ROM

Hyung-Chun Lee; Hyung-Chun Lee

doi:10.3934/era.2020128

Electronic Research Archive

2021, Volume 29, Issue 3: 2533-2552. doi: 10.3934/era.2020128

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Efficient computations for linear feedback control problems for target velocity matching of Navier-Stokes flows via POD and LSTM-ROM

Hyung-Chun Lee ^,

Department of Mathematics, Ajou University, Suwon, 16499, Republic of Korea

Received: 01 August 2020 Revised: 01 September 2020 Published: 14 December 2020
Primary: 76D55, 49M25, 49M41, 68T07; Secondary: 65M60

An efficient computing method for a target velocity tracking problem of fluid flows is considered. We first adopts the Lagrange multipliers method to obtain the optimality system, and then designs a simple and effective feedback control law based on the relationship between the control $ {{\boldsymbol f}} $ and the adjoint variable $ {{\boldsymbol w}} $ in the optimality system. We consider a reduced order modeling (ROM) of this problem for real-time computing. In order to improve the existing ROM method, the deep learning technique, which is currently being actively researched, is applied. We review previous research results and some computational results are presented.
- Navier-Stokes equations,
- finite element,
- reduced order method,
- singular value decomposition,
- deep neural network,
- LSTM
Citation: Hyung-Chun Lee. Efficient computations for linear feedback control problems for target velocity matching of Navier-Stokes flows via POD and LSTM-ROM[J]. Electronic Research Archive, 2021, 29(3): 2533-2552. doi: 10.3934/era.2020128

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Abstract

An efficient computing method for a target velocity tracking problem of fluid flows is considered. We first adopts the Lagrange multipliers method to obtain the optimality system, and then designs a simple and effective feedback control law based on the relationship between the control $ {{\boldsymbol f}} $ and the adjoint variable $ {{\boldsymbol w}} $ in the optimality system. We consider a reduced order modeling (ROM) of this problem for real-time computing. In order to improve the existing ROM method, the deep learning technique, which is currently being actively researched, is applied. We review previous research results and some computational results are presented.

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