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August  2021, 29(3): 2533-2552. 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

Hyung-Chun Lee ,

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

* Corresponding author

Received  August 2020 Revised  September 2020 Published  August 2021 Early access  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.

Keywords: Navier-Stokes equations, finite element, reduced order method, singular value decomposition, deep neural network, LSTM.
Mathematics Subject Classification: Primary: 76D55, 49M25, 49M41, 68T07; Secondary: 65M60.
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, 2021, 29 (3) : 2533-2552. doi: 10.3934/era.2020128
References:
[1]

F. Abergal and R. Temam, On some control problems in fluid mechanics, Theoret. Comput. Fluid Dynamics, 1 (1990), 303-325.  doi: 10.1007/BF00271794.  Google Scholar

[2] R. A. Adams, Sobolev Spaces, Academic Press, 1975.   Google Scholar
[3]

S. E. Ahmed, O. San, A. Rasheed and T. Iliescu, A long short-term memory embedding for hybrid uplifted reduced order models, Phys. D, 409 (2020), 132471, 16 pp. doi: 10.1016/j.physd.2020.132471.  Google Scholar

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D. Amsallem, Interpolation on Manifolds of CFD-Based Fluid and Finite Element-Based Structural Reduced-Order Models for On-Line Aeroelastic Predictions, Ph. D. Thesis, Stanford University, 2010.  Google Scholar

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P. Benner, S. Gugercin and K. Willcox, A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Rev., 57 (2015), 483-531.  doi: 10.1137/130932715.  Google Scholar

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G. Berkooz, P. Holmes and J. L. Lumley, The proper orthogonal decomposition in the analysis of turbulent flows, Annual review of fluid mechanics, 25 (1993), 539-575.   Google Scholar

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S. Brunton, B. Noack and P. Koumoutsakos, Machine Learning for Fluid Mechanics, Annu. Rev. Fluid Mech., 52 (2020), 477-508.  doi: 10.1146/annurev-fluid-010719-060214.  Google Scholar

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J. Burkardt, M. Gunzburger and H.-C. Lee, Centroidal voronoi tessellation-based reduced-order modeling of complex systems, SIAM J. Sci. Comput., 28 (2006), 459-484.  doi: 10.1137/5106482750342221x.  Google Scholar

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J. Burkardt, M. Gunzburger and H.-C. Lee, POD and CVT-based reduced-order modeling of Navier-Stokes flows, Comput. Methods Appl. Mech. Engrg., 196 (2006), 337-355.  doi: 10.1016/j.cma.2006.04.004.  Google Scholar

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J. H. Faghmous, A. Banerjeeand, S. Shekharand, M. Steinbach, V. Kumar, A. R. Ganguly and N. Samatova, Theory-guided data science for climate change, Computer, 47 (2014), 74-78.  doi: 10.1109/MC.2014.335.  Google Scholar

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V. Girault and P. Raviart, Navier-Stokes Equations, North-Hollan, Amsterdam, 1979. Google Scholar

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V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.  Google Scholar

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G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins University, Baltimore, 1996.  Google Scholar

[15]

M. D. Gunzburger and L. S. Hou, Finite-dimensional approximation of a class of constrained nonlinear optimal control problems, SIAM J. Control Optim., 34 (1996), 1001-1043.  doi: 10.1137/S0363012994262361.  Google Scholar

[16]

M. Gunzburger and H.-C. Lee, Active control of vortex shedding, J. Appl. Mech., 63 (1996), 828-835.   Google Scholar

[17]

M. D. Gunzburger and S. Manservisi, Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control, SIAM J. Numer. Anal., 37 (2000), 1481-1512.  doi: 10.1137/S0036142997329414.  Google Scholar

[18]

M. D. Gunzburger and S. Manservisi, The velocity tracking problem for Navier-Stokes flows with boundary control, SIAM J. Control Optim., 39 (2000), 594-634.  doi: 10.1137/S0363012999353771.  Google Scholar

[19]

M. D. Gunzburger and S. Manservisi, Analysis and approximation for linear feedback control for tracking the velocity in Navier-Stokes flows, Comput. Methods Appl. Mech. Engrg., 189 (2000), 803-823.  doi: 10.1016/S0045-7825(99)00344-8.  Google Scholar

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F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013.  Google Scholar

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L. S. Hou and Y. Yan, Dynamics for controlled Navier-Stokes systems with distributed controls, SIAM J. Control Optim., 35 (1997), 654-677.  doi: 10.1137/S0363012994274926.  Google Scholar

[22]

L. S. Hou and Y. Yan, Dynamics and approximations of a velocity tracking problem for the Navier-Stokes flows with piecewise distributed controls, SIAM J. Control Optim., 35 (1997), 1847-1885.  doi: 10.1137/S036301299529286X.  Google Scholar

[23]

A. Karpatne, G. Atluri, J. H. Faghmous, M. Steinbach, A. Banerjee, A. Ganguly, S. Shekhar, N. Samatova and V. Kumar, Theory-guided data science: A new paradigm for scientific discovery from data, IEEE Trans. Knowl. Data Eng., 29 (2017), 2318-2331.  doi: 10.1109/TKDE.2017.2720168.  Google Scholar

[24]

A. J. Majda and D. Qi, Strategies for reduced-order models for predicting the statistical responses and uncertainty quantification in complex turbulent dynamical systems, SIAM Rev., 60 (2018), 491-549.  doi: 10.1137/16M1104664.  Google Scholar

[25]

S. Pawar, S. Ahmed, O. San and A. Rasheed, An evolve-then-correct reduced order model for hidden fluid dynamics. Mathematics, Mathematics, 8 (2020), 570. Google Scholar

[26]

S. Pawar, S. E. Ahmed, O. San and A. Rasheed, Data-driven recovery of hidden physics in reduced order modeling of fluid flows, preprint, arXiv: 1910.13909 doi: 10.1063/5.0002051.  Google Scholar

[27]

M. Rahman, S. Pawar, O. San, A. Rasheed and T. Iliescu, A non-intrusive reduced order modeling framework for quasi-geostrophic turbulence, preprint, arXiv: 1906.11617 Google Scholar

[28]

M. Rathinam and L. R. Petzold, A new look at proper orthogonal decomposition, SIAM J. Numer. Anal., 41 (2003), 1893-1925.  doi: 10.1137/S0036142901389049.  Google Scholar

[29]

L. Scarpa, Analysis and optimal velocity control of a stochastic convective Cahn-Hilliard equation, preprint, arXiv: 2007.14735 Google Scholar

[30]

L. Sirovich, Turbulence and the dynamics of coherent structures, part ⅰ: Coherent structures; part ⅱ: symmetries and transformations; part ⅲ: Dynamics and scaling, Quart. Appl. Math., 45 (1987), 561-590.  doi: 10.1090/qam/910464.  Google Scholar

[31]

M. Strazzullo, Z. Zainib, F. Ballarin and G. Rozza, Reduced order methods for parametrized non-linear and time dependent optimal flow control problems, towards applications in biomedical and environmental sciences, preprint, arXiv: 1912.07886 Google Scholar

[32]

Kailai Xu, Bella Shi and Shuyi Yin, Deep Learning for Partial Differential Equations, Stanford University, 2018. Google Scholar

show all references

References:
[1]

F. Abergal and R. Temam, On some control problems in fluid mechanics, Theoret. Comput. Fluid Dynamics, 1 (1990), 303-325.  doi: 10.1007/BF00271794.  Google Scholar

[2] R. A. Adams, Sobolev Spaces, Academic Press, 1975.   Google Scholar
[3]

S. E. Ahmed, O. San, A. Rasheed and T. Iliescu, A long short-term memory embedding for hybrid uplifted reduced order models, Phys. D, 409 (2020), 132471, 16 pp. doi: 10.1016/j.physd.2020.132471.  Google Scholar

[4]

D. Amsallem, Interpolation on Manifolds of CFD-Based Fluid and Finite Element-Based Structural Reduced-Order Models for On-Line Aeroelastic Predictions, Ph. D. Thesis, Stanford University, 2010.  Google Scholar

[5]

P. Benner, S. Gugercin and K. Willcox, A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Rev., 57 (2015), 483-531.  doi: 10.1137/130932715.  Google Scholar

[6]

G. Berkooz, P. Holmes and J. L. Lumley, The proper orthogonal decomposition in the analysis of turbulent flows, Annual review of fluid mechanics, 25 (1993), 539-575.   Google Scholar

[7]

S. Brunton, B. Noack and P. Koumoutsakos, Machine Learning for Fluid Mechanics, Annu. Rev. Fluid Mech., 52 (2020), 477-508.  doi: 10.1146/annurev-fluid-010719-060214.  Google Scholar

[8]

J. Burkardt, M. Gunzburger and H.-C. Lee, Centroidal voronoi tessellation-based reduced-order modeling of complex systems, SIAM J. Sci. Comput., 28 (2006), 459-484.  doi: 10.1137/5106482750342221x.  Google Scholar

[9]

J. Burkardt, M. Gunzburger and H.-C. Lee, POD and CVT-based reduced-order modeling of Navier-Stokes flows, Comput. Methods Appl. Mech. Engrg., 196 (2006), 337-355.  doi: 10.1016/j.cma.2006.04.004.  Google Scholar

[10]

J. H. Faghmous, A. Banerjeeand, S. Shekharand, M. Steinbach, V. Kumar, A. R. Ganguly and N. Samatova, Theory-guided data science for climate change, Computer, 47 (2014), 74-78.  doi: 10.1109/MC.2014.335.  Google Scholar

[11]

J. Ffowcs-Williams and B. Zhao, Active control of vortex shedding, J. Fluids Struct., S3 (1989), 115-122.  doi: 10.1016/S0889-9746(89)90026-1.  Google Scholar

[12]

V. Girault and P. Raviart, Navier-Stokes Equations, North-Hollan, Amsterdam, 1979. Google Scholar

[13]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.  Google Scholar

[14]

G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins University, Baltimore, 1996.  Google Scholar

[15]

M. D. Gunzburger and L. S. Hou, Finite-dimensional approximation of a class of constrained nonlinear optimal control problems, SIAM J. Control Optim., 34 (1996), 1001-1043.  doi: 10.1137/S0363012994262361.  Google Scholar

[16]

M. Gunzburger and H.-C. Lee, Active control of vortex shedding, J. Appl. Mech., 63 (1996), 828-835.   Google Scholar

[17]

M. D. Gunzburger and S. Manservisi, Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control, SIAM J. Numer. Anal., 37 (2000), 1481-1512.  doi: 10.1137/S0036142997329414.  Google Scholar

[18]

M. D. Gunzburger and S. Manservisi, The velocity tracking problem for Navier-Stokes flows with boundary control, SIAM J. Control Optim., 39 (2000), 594-634.  doi: 10.1137/S0363012999353771.  Google Scholar

[19]

M. D. Gunzburger and S. Manservisi, Analysis and approximation for linear feedback control for tracking the velocity in Navier-Stokes flows, Comput. Methods Appl. Mech. Engrg., 189 (2000), 803-823.  doi: 10.1016/S0045-7825(99)00344-8.  Google Scholar

[20]

F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013.  Google Scholar

[21]

L. S. Hou and Y. Yan, Dynamics for controlled Navier-Stokes systems with distributed controls, SIAM J. Control Optim., 35 (1997), 654-677.  doi: 10.1137/S0363012994274926.  Google Scholar

[22]

L. S. Hou and Y. Yan, Dynamics and approximations of a velocity tracking problem for the Navier-Stokes flows with piecewise distributed controls, SIAM J. Control Optim., 35 (1997), 1847-1885.  doi: 10.1137/S036301299529286X.  Google Scholar

[23]

A. Karpatne, G. Atluri, J. H. Faghmous, M. Steinbach, A. Banerjee, A. Ganguly, S. Shekhar, N. Samatova and V. Kumar, Theory-guided data science: A new paradigm for scientific discovery from data, IEEE Trans. Knowl. Data Eng., 29 (2017), 2318-2331.  doi: 10.1109/TKDE.2017.2720168.  Google Scholar

[24]

A. J. Majda and D. Qi, Strategies for reduced-order models for predicting the statistical responses and uncertainty quantification in complex turbulent dynamical systems, SIAM Rev., 60 (2018), 491-549.  doi: 10.1137/16M1104664.  Google Scholar

[25]

S. Pawar, S. Ahmed, O. San and A. Rasheed, An evolve-then-correct reduced order model for hidden fluid dynamics. Mathematics, Mathematics, 8 (2020), 570. Google Scholar

[26]

S. Pawar, S. E. Ahmed, O. San and A. Rasheed, Data-driven recovery of hidden physics in reduced order modeling of fluid flows, preprint, arXiv: 1910.13909 doi: 10.1063/5.0002051.  Google Scholar

[27]

M. Rahman, S. Pawar, O. San, A. Rasheed and T. Iliescu, A non-intrusive reduced order modeling framework for quasi-geostrophic turbulence, preprint, arXiv: 1906.11617 Google Scholar

[28]

M. Rathinam and L. R. Petzold, A new look at proper orthogonal decomposition, SIAM J. Numer. Anal., 41 (2003), 1893-1925.  doi: 10.1137/S0036142901389049.  Google Scholar

[29]

L. Scarpa, Analysis and optimal velocity control of a stochastic convective Cahn-Hilliard equation, preprint, arXiv: 2007.14735 Google Scholar

[30]

L. Sirovich, Turbulence and the dynamics of coherent structures, part ⅰ: Coherent structures; part ⅱ: symmetries and transformations; part ⅲ: Dynamics and scaling, Quart. Appl. Math., 45 (1987), 561-590.  doi: 10.1090/qam/910464.  Google Scholar

[31]

M. Strazzullo, Z. Zainib, F. Ballarin and G. Rozza, Reduced order methods for parametrized non-linear and time dependent optimal flow control problems, towards applications in biomedical and environmental sciences, preprint, arXiv: 1912.07886 Google Scholar

[32]

Kailai Xu, Bella Shi and Shuyi Yin, Deep Learning for Partial Differential Equations, Stanford University, 2018. Google Scholar

19] (left) and our results (right) where $ Re = 1 $">Figure 3.  $ || {\boldsymbol u}(t) - {\boldsymbol U}(t) ||_2 $, Compare between Figure 1 in [19] (left) and our results (right) where $ Re = 1 $
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Figure 1.  Discrepancy of the trajectories between the FOM, ROM and least squares projected solutions after Galerkin projection
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Figure 2.  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 $
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Figure 4.  $ || {\boldsymbol u} - {\boldsymbol U}||_{{\bf L}^2({\Omega})} $ for different control values $ \gamma $
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Figure 5.  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
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Figure 6.  The singular values of the snapshot data matrix (right) for different control values $ \gamma $
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Figure 7.  RMSE of Galerkin ROM (Upper 4 lines) and LS Projection (Lower 4 lines)
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Figure 8.  LSTM Cell (left) and LSTM Training where $ r = 5 $
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Figure 9.  Training and validation loss
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Figure 10.  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 $
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Figure 11.  RMSE of ROMs predicted by LS Projection, GP-ROM and GP-LSTM for $ \gamma = 50 $ using the basis of $ \gamma = 40 $
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Table 1.  The first 8 singular values of the snapshot matrix
singular vales RIC($ \gamma=40 $) singular vales RIC($ \gamma=40 $)
1 4.83907e+02 99.2476% 2 2.41493e+00 99.8174%
3 3.98155e-01 99.9207% 4 2.02669e-01 99.9731%
5 6.01101e-02 99.9909% 6 1.65390e-02 99.9964%
7 6.20453e-03 99.9985% 8 2.51213e-03 99.9994%
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singular vales RIC($ \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%
Table 2.  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
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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
Table 3.  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
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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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