Abstract

In this work, we consider the problem of learning a reduced-order model of a high-dimensional stochastic nonlinear system with control inputs from noisy data. In particular, we develop a hybrid parametric/nonparametric model that learns the “average” linear dynamics in the data using dynamic mode decomposition with control (DMDc) and the nonlinearities and model uncertainties using Gaussian process (GP) regression and compare it with total least-squares dynamic mode decomposition (tlsDMD), extended here to systems with control inputs (tlsDMDc). The proposed approach is also compared with existing methods, such as DMDc-only and GP-only models, in two tasks: controlling the stochastic nonlinear Stuart–Landau equation and predicting the flowfield induced by a jet-like body force field in a turbulent boundary layer using data from large-scale numerical simulations.

References

1.
Lumey
,
J. L.
,
1970
,
Stochastic Tools in Turbulence
,
Elsevier Science
,
New York
.
2.
Sirovich
,
L.
,
1987
, “
Turbulence and the Dynamics of Coherent Structures. I. Coherent Structures
,”
Q. Appl. Math.
,
45
(
3
), pp.
561
571
.10.1090/qam/910462
3.
Willcox
,
K.
, and
Peraire
,
J.
,
2002
, “
Balanced Model Reduction Via the Proper Orthogonal Decomposition
,”
AIAA J.
,
40
(
11
), pp.
2323
2330
.10.2514/2.1570
4.
Rowley
,
C. W.
,
Mezić
,
I.
,
Bagheri
,
S.
,
Schlatter
,
P.
, and
Henningson
,
D. S.
,
2009
, “
Spectral Analysis of Nonlinear Flows
,”
J. Fluid Mech.
,
641
, pp.
115
127
.10.1017/S0022112009992059
5.
Schmid
,
P. J.
,
2010
, “
Dynamic Mode Decomposition of Numerical and Experimental Data
,”
J. Fluid Mech.
,
656
, pp.
5
28
.10.1017/S0022112010001217
6.
Proctor
,
J. L.
,
Brunton
,
S. L.
, and
Kutz
,
J. N.
,
2016
, “
Dynamic Mode Decomposition With Control
,”
SIAM J. Appl. Dyn. Sys
t.,
15
(
1
), pp.
142
161
.10.1137/15M1013857
7.
Tsolovikos
,
A.
,
Bakolas
,
E.
,
Suryanarayanan
,
S.
, and
Goldstein
,
D.
,
2021
, “
Estimation and Control of Fluid Flows Using Sparsity-Promoting Dynamic Mode Decomposition
,”
IEEE Control Syst. Lett.
,
5
(
4
), pp.
1145
1150
.10.1109/LCSYS.2020.3015776
8.
Hemati
,
M. S.
,
Rowley
,
C. W.
,
Deem
,
E. A.
, and
Cattafesta
,
L. N.
,
2017
, “
De-Biasing the Dynamic Mode Decomposition for Applied Koopman Spectral Analysis of Noisy Datasets
,”
Theor. Comput. Fluid Dyn.
,
31
(
4
), pp.
349
368
.10.1007/s00162-017-0432-2
9.
Dawson
,
S.
,
Hemati
,
M. S.
,
Williams
,
M. O.
, and
Rowley
,
C. W.
,
2016
, “
Characterizing and Correcting for the Effect of Sensor Noise in the Dynamic Mode Decomposition
,”
Exp. Fluids
,
57
(
3
), pp.
1
19
.10.1007/s00348-016-2127-7
10.
Jovanović
,
M. R.
,
Schmid
,
P. J.
, and
Nichols
,
J. W.
,
2014
, “
Sparsity-Promoting Dynamic Mode Decomposition
,”
Phys. Fluids
,
26
(
2
), p.
024103
.10.1063/1.4863670
11.
Mezić
,
I.
,
2013
, “
Analysis of Fluid Flows Via Spectral Properties of the Koopman Operator
,”
Annu. Rev. Fluid Mech.
,
45
, pp.
357
378
.10.1146/annurev-fluid-011212-140652
12.
Korda
,
M.
, and
Mezić
,
I.
,
2018
, “
Linear Predictors for Nonlinear Dynamical Systems: Koopman Operator Meets Model Predictive Control
,”
Automatica
,
93
, pp.
149
160
.10.1016/j.automatica.2018.03.046
13.
Abraham
,
I.
, and
Murphey
,
T. D.
,
2019
, “
Active Learning of Dynamics for Data-Driven Control Using Koopman Operators
,”
IEEE Trans. Rob.
,
35
(
5
), pp.
1071
1083
.10.1109/TRO.2019.2923880
14.
Williams
,
M. O.
,
Hemati
,
M. S.
,
Dawson
,
S. T.
,
Kevrekidis
,
I. G.
, and
Rowley
,
C. W.
,
2016
, “
Extending Data-Driven Koopman Analysis to Actuated Systems
,”
IFAC-PapersOnLine
,
49
(
18
), pp.
704
709
.10.1016/j.ifacol.2016.10.248
15.
Li
,
Q.
,
Dietrich
,
F.
,
Bollt
,
E. M.
, and
Kevrekidis
,
I. G.
,
2017
, “
Extended Dynamic Mode Decomposition With Dictionary Learning: A Data-Driven Adaptive Spectral Decomposition of the Koopman Operator
,”
Chaos: Interdiscip. J. Nonlinear Sci.
,
27
(
10
), p.
103111
.10.1063/1.4993854
16.
Williams
,
M. O.
,
Rowley
,
C. W.
, and
Kevrekidis
,
I. G.
,
2015
, “
A Kernel-Based Approach to Data-Driven Koopman Spectral Analysis
,”
J. Comput. Nonlinear Dyn.
, 2(2), pp.
247
252
.10.3934/jcd.2015005
17.
Lusch
,
B.
,
Kutz
,
J. N.
, and
Brunton
,
S. L.
,
2018
, “
Deep Learning for Universal Linear Embeddings of Nonlinear Dynamics
,”
Nat. Commun.
,
9
(
1
), pp.
1
10
.10.1038/s41467-018-07210-0
18.
Yeung
,
E.
,
Kundu
,
S.
, and
Hodas
,
N.
,
2019
, “
Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems
,”
2019 American Control Conference
, Philadelphia, PA, July 10–12 , pp.
4832
4839
.10.23919/ACC.2019.8815339
19.
Brunton
,
S. L.
,
Proctor
,
J. L.
, and
Kutz
,
J. N.
,
2016
, “
Discovering Governing Equations From Data by Sparse Identification of Nonlinear Dynamical Systems
,”
Proc. Natl. Acad. Sci.
,
113
(
15
), pp.
3932
3937
.10.1073/pnas.1517384113
20.
Benner
,
P.
,
Goyal
,
P.
,
Kramer
,
B.
,
Peherstorfer
,
B.
, and
Willcox
,
K.
,
2020
, “
Operator Inference for Non-Intrusive Model Reduction of Systems With Non-Polynomial Nonlinear Terms
,”
Comput. Methods Appl. Mech. Eng.
,
372
, p.
113433
.10.1016/j.cma.2020.113433
21.
Rasmussen
,
C. E.
,
Williams
,
C. K. I.
,
2006
, “
Gaussian Processes for Machine Learning
,”
The MIT Press
,
Cambridge, MA
.
22.
Quinonero-Candela
,
J.
, and
Rasmussen
,
C. E.
,
2005
, “
A Unifying View of Sparse Approximate Gaussian Process Regression
,”
J. Mach. Learn. Res.
,
6
, pp.
1939
1959
.10.5555/1046920.1194909
23.
Titsias
,
M.
,
2009
, “
Variational Learning of Inducing Variables in Sparse Gaussian Processes
,”
Artificial Intelligence and Statistics
, Clearwater Beach, FL, Apr. 16–18, pp.
567
574
.https://proceedings.mlr.press/v5/titsias09a.html
24.
Hoffman
,
M. D.
,
Blei
,
D. M.
,
Wang
,
C.
, and
Paisley
,
J.
,
2013
, “
Stochastic Variational Inference
,”
J. Mach. Learn. Res.
,
14
(
5
),
1303
1347
.10.48550/arXiv.1206.7051
25.
Hensman
,
J.
,
Fusi
,
N.
, and
Lawrence
,
N. D.
,
2013
, “
Gaussian Processes for Big Data
,”
Proceedings of the Twenty-Ninth Conference on Uncertainty in Artificial Intelligence
, Bellevue, WA, Aug. 11–15, pp.
282
290
.https://auai.org/uai2013/prints/papers/244.pdf
26.
Grimes
,
D. B.
,
Chalodhorn
,
R.
, and
Rao
,
R. P.
,
2006
, “
Dynamic Imitation in a Humanoid Robot Through Nonparametric Probabilistic Inference
,”
Robotics: Science and Systems
,
Philadelphia, PA
, Aug. 16–19, pp.
199
206
.10.15607/RSS.2006.II.026
27.
Ko
,
J.
,
Klein
,
D. J.
,
Fox
,
D.
, and
Haehnel
,
D.
,
2007
, “
Gaussian Processes and Reinforcement Learning for Identification and Control of an Autonomous Blimp
,”
Proceedings 2007 IEEE International Conference on Robotics and Automation
, Rome, Italy, Apr. 10–14, pp.
742
747
.10.1109/ROBOT.2007.363075
28.
Pan
,
Y.
, and
Theodorou
,
E. A.
,
2015
, “
Data-Driven Differential Dynamic Programming Using Gaussian Processes
,” 2015 American Control Conference (
ACC)
, Chicago, IL, July 1–3, pp.
4467
4472
.
29.
Hewing
,
L.
,
Kabzan
,
J.
, and
Zeilinger
,
M. N.
,
2020
, “
Cautious Model Predictive Control Using Gaussian Process Regression
,”
IEEE Trans. Control Syst. Technol.
,
28
(
6
), pp.
2736
2743
.10.1109/TCST.2019.2949757
30.
Tsolovikos
,
A.
, and
Bakolas
,
E.
,
2021
, “
Cautious Nonlinear Covariance Steering Using Variational Gaussian Process Predictive Models
,”
IFAC-PapersOnLine
,
54
(
20
), pp.
59
64
.10.1016/j.ifacol.2021.11.153
31.
Xiao
,
M.
,
Breitkopf
,
P.
,
Filomeno Coelho
,
R.
,
Knopf-Lenoir
,
C.
,
Sidorkiewicz
,
M.
, and
Villon
,
P.
,
2010
, “
Model Reduction by CPOD and Kriging
,”
Struct. Multidiscip. Optim.
,
41
(
4
), pp.
555
574
.10.1007/s00158-009-0434-9
32.
Guo
,
M.
, and
Hesthaven
,
J. S.
,
2018
, “
Reduced Order Modeling for Nonlinear Structural Analysis Using Gaussian Process Regression
,”
Comput. Methods Appl. Mech. Eng.
,
341
, pp.
807
826
.10.1016/j.cma.2018.07.017
33.
Chang
,
Y.-H.
,
Zhang
,
L.
,
Wang
,
X.
,
Yeh
,
S.-T.
,
Mak
,
S.
,
Sung
,
C.-L.
,
Jeff Wu
,
C.
, and
Yang
,
V.
,
2019
, “
Kernel-Smoothed Proper Orthogonal Decomposition-Based Emulation for Spatiotemporally Evolving Flow Dynamics Prediction
,”
AIAA J.
,
57
(
12
), pp.
5269
5280
.10.2514/1.J057803
34.
Ortali
,
G.
,
Demo
,
N.
, and
Rozza
,
G.
,
2022
, “
A Gaussian Process Regression Approach Within a Data-Driven POD Framework for Engineering Problems in Fluid Dynamics
,”
Math. Eng.
,
4
(
3
), pp.
1
16
.10.3934/mine.2022021
35.
Masuda
,
A.
,
Susuki
,
Y.
,
Martínez-Ramón
,
M.
,
Mammoli
,
A.
, and
Ishigame
,
A.
,
2019
, “
Application of Gaussian Process Regression to Koopman Mode Decomposition for Noisy Dynamic Data
,” e-print
arXiv:1911.01143
.10.48550/arXiv.1911.01143
36.
Maulik
,
R.
,
Botsas
,
T.
,
Ramachandra
,
N.
,
Mason
,
L. R.
, and
Pan
,
I.
,
2021
, “
Latent-Space Time Evolution of Non-Intrusive Reduced-Order Models Using Gaussian Process Emulation
,”
Phys. D: Nonlinear Phenom.
,
416
, p.
132797
.10.1016/j.physd.2020.132797
37.
Bonilla
,
E. V.
,
Chai
,
K.
, and
Williams
,
C.
,
2007
, “
Multi-Task Gaussian Process Prediction
,”
Advances in Neural Information Processing Systems
, Vancouver, BC, Dec. 3–4.https://papers.nips.cc/paper_files/paper/2007/file/66368270ffd51418ec58bd793f2d9b1b-Paper.pdf
38.
Noack
,
B. R.
,
Afanasiev
,
K.
,
Morzyński
,
M.
,
Tadmor
,
G.
, and
Thiele
,
F.
,
2003
, “
A Hierarchy of Low-Dimensional Models for the Transient and Post-Transient Cylinder Wake
,”
J. Fluid Mech.
,
497
, pp.
335
363
.10.1017/S0022112003006694
39.
Borrelli
,
F.
,
Bemporad
,
A.
, and
Morari
,
M.
,
2017
,
Predictive Control for Linear and Hybrid Systems
,
Cambridge University Press
,
Cambridge, UK
.
40.
Tsolovikos
,
A.
,
Suryanarayanan
,
S.
,
Bakolas
,
E.
, and
Goldstein
,
D.
,
2021
, “
Model Predictive Control of Material Volumes With Application to Vortical Structures
,”
AIAA J.
,
59
(
10
), pp.
4057
4070
.10.2514/1.J060413
41.
Tsolovikos
,
A.
,
Jariwala
,
A.
,
Suryanarayanan
,
S.
,
Bakolas
,
E.
, and
Goldstein
,
D.
,
2023
, “
Separation Delay in Turbulent Boundary Layers Via Model Predictive Control of Large-Scale Motions
,”
Phys. Fluids
,
35
(
11
), p.
115118
.10.1063/5.0169138
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