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doi: 10.3934/era.2020128

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

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

* Corresponding author

Received  August 2020 Revised  September 2020 Published  December 2020

Fund Project: The author is supported by Basic Science Research Program through the National Research Foundation of Korea NRF-2016R1D1A1B03932219 and NRF-2019R1F1A1050231

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.

Citation: Hyung-Chun Lee. Efficient computations for linear feedback control problems for target velocity matching of Navier-Stokes flows via POD and LSTM-ROM. Electronic Research Archive, doi: 10.3934/era.2020128
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$|| {\boldsymbol u}(t) - {\boldsymbol U}(t) ||_2$, Compare between Figure 1 in [19] (left) and our results (right) where $Re = 1$
Discrepancy of the trajectories between the FOM, ROM and least squares projected solutions after Galerkin projection
Controlled (left) and target (right) flows at $t = 0.0, 0.050,0.057,0.059$ (left pair of columns) and $t = 0.060, 0.061,$ $0.064, 0.5$ (right pair of columns) for $\gamma = 10$
$|| {\boldsymbol u} - {\boldsymbol U}||_{{\bf L}^2({\Omega})}$ for different control values $\gamma$
The POD reduced basis of cardinality 8 ($\gamma = 10$ Upper two rows and $\gamma = 40$ Down two rows). Basis 1, 2, 3, 4 from the left to the right (Top) and Basis 5, 6, 7, 8 from the left to the right (Bottom) in each two rows
The singular values of the snapshot data matrix (right) for different control values $\gamma$
RMSE of Galerkin ROM (Upper 4 lines) and LS Projection (Lower 4 lines)
LSTM Cell (left) and LSTM Training where $r = 5$
Training and validation loss
Temporal evolution of the 8 POD modal coefficients predicted by LS Projection, GP-ROM and GP-LSTM for $\gamma = 50$ using the basis from $\gamma = 40$
RMSE of ROMs predicted by LS Projection, GP-ROM and GP-LSTM for $\gamma = 50$ using the basis of $\gamma = 40$
The first 8 singular values of the snapshot matrix
 singular vales RIC($\gamma=40$) singular vales RIC($\gamma=40$) 1 4.83907e+02 99.2476% 2 2.41493e+00 99.8174% 3 3.98155e-01 99.9207% 4 2.02669e-01 99.9731% 5 6.01101e-02 99.9909% 6 1.65390e-02 99.9964% 7 6.20453e-03 99.9985% 8 2.51213e-03 99.9994%
 singular vales RIC($\gamma=40$) singular vales RIC($\gamma=40$) 1 4.83907e+02 99.2476% 2 2.41493e+00 99.8174% 3 3.98155e-01 99.9207% 4 2.02669e-01 99.9731% 5 6.01101e-02 99.9909% 6 1.65390e-02 99.9964% 7 6.20453e-03 99.9985% 8 2.51213e-03 99.9994%
A list of hyperparameters utilized to train the LSTM network for numerical experiments
 Variables Hyperparameters Number of hidden layers 2 Number of neurons in each hidden layer 120 Number of lookbacks 5 Batch size 32 Epochs 1000 Activation functions in the LSTM layers tanh Validation data set 20% Loss function MSE Optimizer ADAM
 Variables Hyperparameters Number of hidden layers 2 Number of neurons in each hidden layer 120 Number of lookbacks 5 Batch size 32 Epochs 1000 Activation functions in the LSTM layers tanh Validation data set 20% Loss function MSE Optimizer ADAM
CPU time (in second) comparison for the different ROM frameworks investigated in this study and RMSE at $t = 0.25$
 Framework Times RMSE(0.25) FOM $1695.7$ 0 LS-Proj 1.34053e-05 GP-ROM(8) 0.3268 3.77711e-03 GP-ROM (16) 1.7012 1.28362e-03 GP-LSTM(8) 0.6372 3.20970e-04
 Framework Times RMSE(0.25) FOM $1695.7$ 0 LS-Proj 1.34053e-05 GP-ROM(8) 0.3268 3.77711e-03 GP-ROM (16) 1.7012 1.28362e-03 GP-LSTM(8) 0.6372 3.20970e-04
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