Sampling and resolution characteristics in reduced order models of shallow water equations: Intrusive vs nonintrusive
Abstract
We investigate the sensitivity of reduced order models (ROMs) to training data spatial resolution as well as sampling rate. In particular, we consider proper orthogonal decomposition (POD), coupled with Galerkin projection (POD-GP), as an intrusive reduced order modeling technique. For non-intrusive ROMs, we consider two frameworks. The first is using dynamic mode decomposition (DMD), and the second is based on artificial neural networks (ANNs). For ANN, we utilized a residual deep neural network, and for DMD we have studied two versions of DMD approaches; one with hard thresholding and the other with sorted bases selection. Also, we highlight the differences between mean-subtracting the data (centering) and using the data without mean-subtraction. We tested these ROMs using a system of 2D shallow water equations for four different numerical experiments, adopting combinations of sampling rates and spatial resolutions. For these cases, we found that the DMD basis obtained with hard threshodling is sensitive to sampling rate. The sorted DMD algorithm helps to mitigate this problem and yields more stabilized and converging solution. Furthermore, we demonstrate that both DMD approaches without mean subtraction provide significantly more accurate results than DMD with mean-subtracting the data. On the other hand, POD is relatively insensitive tomore »
- Authors:
-
[1];
[1];
[3]
- Oklahoma State Univ., Stillwater, OK (United States)
- Univ. “Politechnica” of Timisoara, Hunedoara (Romania)
- Florida State Univ., Tallahassee, FL (United States)
- Publication Date:
- Research Org.:
- Oklahoma State Univ., Stillwater, OK (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
- OSTI Identifier:
- 1737500
- Alternate Identifier(s):
- OSTI ID: 1602300
- Grant/Contract Number:
- SC0019290
- Resource Type:
- Accepted Manuscript
- Journal Name:
- International Journal for Numerical Methods in Fluids
- Additional Journal Information:
- Journal Volume: 92; Journal Issue: 8; Journal ID: ISSN 0271-2091
- Publisher:
- Wiley
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 42 ENGINEERING; Artificial neural network; dynamic mode decomposition; proper orthogonal decomposition; reduced order modeling; resolution; sampling rat
Citation Formats
Ahmed, Shady E., San, Omer, Bistrian, Diana A., and Navon, Ionel M. Sampling and resolution characteristics in reduced order models of shallow water equations: Intrusive vs nonintrusive. United States: N. p., 2020.
Web. doi:10.1002/fld.4815.
Ahmed, Shady E., San, Omer, Bistrian, Diana A., & Navon, Ionel M. Sampling and resolution characteristics in reduced order models of shallow water equations: Intrusive vs nonintrusive. United States. https://doi.org/10.1002/fld.4815
Ahmed, Shady E., San, Omer, Bistrian, Diana A., and Navon, Ionel M. Fri .
"Sampling and resolution characteristics in reduced order models of shallow water equations: Intrusive vs nonintrusive". United States. https://doi.org/10.1002/fld.4815. https://www.osti.gov/servlets/purl/1737500.
@article{osti_1737500,
title = {Sampling and resolution characteristics in reduced order models of shallow water equations: Intrusive vs nonintrusive},
author = {Ahmed, Shady E. and San, Omer and Bistrian, Diana A. and Navon, Ionel M.},
abstractNote = {We investigate the sensitivity of reduced order models (ROMs) to training data spatial resolution as well as sampling rate. In particular, we consider proper orthogonal decomposition (POD), coupled with Galerkin projection (POD-GP), as an intrusive reduced order modeling technique. For non-intrusive ROMs, we consider two frameworks. The first is using dynamic mode decomposition (DMD), and the second is based on artificial neural networks (ANNs). For ANN, we utilized a residual deep neural network, and for DMD we have studied two versions of DMD approaches; one with hard thresholding and the other with sorted bases selection. Also, we highlight the differences between mean-subtracting the data (centering) and using the data without mean-subtraction. We tested these ROMs using a system of 2D shallow water equations for four different numerical experiments, adopting combinations of sampling rates and spatial resolutions. For these cases, we found that the DMD basis obtained with hard threshodling is sensitive to sampling rate. The sorted DMD algorithm helps to mitigate this problem and yields more stabilized and converging solution. Furthermore, we demonstrate that both DMD approaches without mean subtraction provide significantly more accurate results than DMD with mean-subtracting the data. On the other hand, POD is relatively insensitive to sampling rate and yields better representation of the flow field. Meanwhile, spatial resolution has little effect on both POD and DMD performances. Here, the numerical results reveal that an ANN on POD subspace (POD-ANN) performs remarkably better than POD-GP and DMD in capturing system dynamics, even with a small number of modes.},
doi = {10.1002/fld.4815},
journal = {International Journal for Numerical Methods in Fluids},
number = 8,
volume = 92,
place = {United States},
year = {Fri Jan 24 00:00:00 EST 2020},
month = {Fri Jan 24 00:00:00 EST 2020}
}
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Works referenced in this record:
Deep Fluids: A Generative Network for Parameterized Fluid Simulations
journal, May 2019
- Kim, Byungsoo; Azevedo, Vinicius C.; Thuerey, Nils
- Computer Graphics Forum, Vol. 38, Issue 2
Non-linear model reduction for uncertainty quantification in large-scale inverse problems
journal, January 2009
- Galbally, D.; Fidkowski, K.; Willcox, K.
- International Journal for Numerical Methods in Engineering
An improved algorithm for the shallow water equations model reduction: Dynamic Mode Decomposition vs POD: AN IMPROVED ALGORITHM FOR DYNAMIC MODE DECOMPOSITION VS POD
journal, April 2015
- Bistrian, D. A.; Navon, I. M.
- International Journal for Numerical Methods in Fluids, Vol. 78, Issue 9
Efficiency of a POD-based reduced second-order adjoint model in 4D-Var data assimilation
journal, January 2007
- Daescu, D. N.; Navon, I. M.
- International Journal for Numerical Methods in Fluids, Vol. 53, Issue 6
A dynamic closure modeling framework for model order reduction of geophysical flows
journal, April 2019
- Rahman, Sk. M.; Ahmed, S. E.; San, O.
- Physics of Fluids, Vol. 31, Issue 4
Optimal control of the cylinder wake in the laminar regime by trust-region methods and POD reduced-order models
journal, August 2008
- Bergmann, M.; Cordier, L.
- Journal of Computational Physics, Vol. 227, Issue 16
Optimized Dynamic Mode Decomposition via Non-Convex Regularization and Multiscale Permutation Entropy
journal, February 2018
- Dang, Zhang; Lv, Yong; Li, Yourong
- Entropy, Vol. 20, Issue 3
The Weather Research and Forecasting Model: Overview, System Efforts, and Future Directions
journal, August 2017
- Powers, Jordan G.; Klemp, Joseph B.; Skamarock, William C.
- Bulletin of the American Meteorological Society, Vol. 98, Issue 8
A proper orthogonal decomposition approach to approximate balanced truncation of infinite dimensional linear systems
journal, February 2009
- Singler, John R.; Batten, Belinda A.
- International Journal of Computer Mathematics, Vol. 86, Issue 2
Uber Die Beste Annaherung Von Funktionen Einer Gegebenen Funktionenklasse
journal, January 1936
- Kolmogoroff, A.
- The Annals of Mathematics, Vol. 37, Issue 1
Total variation diminishing Runge-Kutta schemes
journal, January 1998
- Gottlieb, Sigal; Shu, Chi-Wang
- Mathematics of Computation of the American Mathematical Society, Vol. 67, Issue 221
Koopman-mode decomposition of the cylinder wake
journal, June 2013
- Bagheri, Shervin
- Journal of Fluid Mechanics, Vol. 726
Receptive fields, binocular interaction and functional architecture in the cat's visual cortex
journal, January 1962
- Hubel, D. H.; Wiesel, T. N.
- The Journal of Physiology, Vol. 160, Issue 1
Use of Reduced-order Models for Improved Data Assimilation within an EnKF Context
conference, April 2013
- He, Jincong; Sarma, Pallav; Durlofsky, Louis J.
- SPE Reservoir Simulation Symposium
Efficiency of randomised dynamic mode decomposition for reduced order modelling
journal, March 2018
- Bistrian, D. A.; Navon, I. M.
- International Journal of Computational Fluid Dynamics, Vol. 32, Issue 2-3
Variable Projection Methods for an Optimized Dynamic Mode Decomposition
journal, January 2018
- Askham, Travis; Kutz, J. Nathan
- SIAM Journal on Applied Dynamical Systems, Vol. 17, Issue 1
A reduced-order approach for optimal control of fluids using proper orthogonal decomposition
journal, January 2000
- Ravindran, S. S.
- International Journal for Numerical Methods in Fluids, Vol. 34, Issue 5
Spectral proper orthogonal decomposition
journal, March 2016
- Sieber, Moritz; Paschereit, C. Oliver; Oberleithner, Kilian
- Journal of Fluid Mechanics, Vol. 792
Galerkin v. least-squares Petrov–Galerkin projection in nonlinear model reduction
journal, February 2017
- Carlberg, Kevin; Barone, Matthew; Antil, Harbir
- Journal of Computational Physics, Vol. 330
Multi-scale proper orthogonal decomposition of complex fluid flows
journal, May 2019
- Mendez, M. A.; Balabane, M.; Buchlin, J. -M.
- Journal of Fluid Mechanics, Vol. 870
Implications of the selection of a particular modal decomposition technique for the analysis of shallow flows
journal, March 2018
- Higham, J. E.; Brevis, W.; Keylock, C. J.
- Journal of Hydraulic Research, Vol. 56, Issue 6
Data Assimilation Using an Ensemble Kalman Filter Technique
journal, March 1998
- Houtekamer, P. L.; Mitchell, Herschel L.
- Monthly Weather Review, Vol. 126, Issue 3
Hybrid Reduced-Order Integration with Proper Orthogonal Decomposition and Dynamic Mode Decomposition
journal, January 2013
- Williams, Matthew O.; Schmid, Peter J.; Kutz, J. Nathan
- Multiscale Modeling & Simulation, Vol. 11, Issue 2
Fluid control using the adjoint method
conference, January 2004
- McNamara, Antoine; Treuille, Adrien; Popović, Zoran
- ACM SIGGRAPH 2004 Papers on - SIGGRAPH '04
POD-Galerkin approximations in PDE-constrained optimization
journal, October 2010
- Sachs, Ekkehard W.; Volkwein, Stefan
- GAMM-Mitteilungen, Vol. 33, Issue 2
Reduced Order Galerkin Models of flow Around Naca‐0012 Airfoil
journal, March 2008
- Stankiewicz, W.; Morzyński, M.; Noack, B. R.
- Mathematical Modelling and Analysis, Vol. 13, Issue 1
On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high-Reynolds-number flow over an Ahmed body
journal, April 2014
- Östh, Jan; Noack, Bernd R.; Krajnović, Siniša
- Journal of Fluid Mechanics, Vol. 747
2D Burgers equation with large Reynolds number using POD/DEIM and calibration: 2D BURGERS EQUATIONS USING POD/DEIM AND TIKHONOV REGULARIZATION
journal, June 2016
- Wang, Yuepeng; Navon, Ionel M.; Wang, Xinyue
- International Journal for Numerical Methods in Fluids, Vol. 82, Issue 12
Pressure velocity coupling in a subsonic round jet
journal, June 2000
- Picard, C.; Delville, J.
- International Journal of Heat and Fluid Flow, Vol. 21, Issue 3
Optimal control of vortex shedding using low-order models. Part I?open-loop model development
journal, March 1999
- Graham, W. R.; Peraire, J.; Tang, K. Y.
- International Journal for Numerical Methods in Engineering, Vol. 44, Issue 7
Data-driven POD-Galerkin reduced order model for turbulent flows
journal, September 2020
- Hijazi, Saddam; Stabile, Giovanni; Mola, Andrea
- Journal of Computational Physics, Vol. 416
Model reduction for compressible flows using POD and Galerkin projection
journal, February 2004
- Rowley, Clarence W.; Colonius, Tim; Murray, Richard M.
- Physica D: Nonlinear Phenomena, Vol. 189, Issue 1-2
On dynamic mode decomposition: Theory and applications
journal, December 2014
- Kutz, J. Nathan; Brunton, Steven L.; Luchtenburg, Dirk M.
- Journal of Computational Dynamics, Vol. 1, Issue 2
Spectral analysis of nonlinear flows
journal, November 2009
- Rowley, Clarence W.; MeziĆ, Igor; Bagheri, Shervin
- Journal of Fluid Mechanics, Vol. 641
Neural networks and principal component analysis: Learning from examples without local minima
journal, January 1989
- Baldi, Pierre; Hornik, Kurt
- Neural Networks, Vol. 2, Issue 1
The method of dynamic mode decomposition in shallow water and a swirling flow problem: THE DMD METHOD IN SHALLOW WATER AND A SWIRLING FLOW PROBLEM
journal, June 2016
- Bistrian, Diana A.; Navon, Ionel M.
- International Journal for Numerical Methods in Fluids, Vol. 83, Issue 1
Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis
journal, May 2018
- Towne, Aaron; Schmidt, Oliver T.; Colonius, Tim
- Journal of Fluid Mechanics, Vol. 847
Turbulence and the dynamics of coherent structures. I. Coherent structures
journal, January 1987
- Sirovich, Lawrence
- Quarterly of Applied Mathematics, Vol. 45, Issue 3
Spectral Properties of Dynamical Systems, Model Reduction and Decompositions
journal, August 2005
- Mezić, Igor
- Nonlinear Dynamics, Vol. 41, Issue 1-3
Super-resolution reconstruction of turbulent flows with machine learning
journal, May 2019
- Fukami, Kai; Fukagata, Koji; Taira, Kunihiko
- Journal of Fluid Mechanics, Vol. 870
Optimal control of vortex shedding using low-order models. Part II?model-based control
journal, March 1999
- Graham, W. R.; Peraire, J.; Tang, K. Y.
- International Journal for Numerical Methods in Engineering, Vol. 44, Issue 7
Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders
journal, November 2019
- Lee, Kookjin; Carlberg, Kevin T.
- Journal of Computational Physics
Linearly Recurrent Autoencoder Networks for Learning Dynamics
journal, January 2019
- Otto, Samuel E.; Rowley, Clarence W.
- SIAM Journal on Applied Dynamical Systems, Vol. 18, Issue 1
A Reduced-Order Method for Simulation and Control of Fluid Flows
journal, July 1998
- Ito, K.; Ravindran, S. S.
- Journal of Computational Physics, Vol. 143, Issue 2
Stochastic approaches to uncertainty quantification in CFD simulations
journal, March 2005
- Mathelin, Lionel; Hussaini, M. Yousuff; Zang, Thomas A.
- Numerical Algorithms, Vol. 38, Issue 1-3
Dynamic mode decomposition of numerical and experimental data
journal, July 2010
- Schmid, Peter J.
- Journal of Fluid Mechanics, Vol. 656
Artificial viscosity proper orthogonal decomposition
journal, January 2011
- Borggaard, Jeff; Iliescu, Traian; Wang, Zhu
- Mathematical and Computer Modelling, Vol. 53, Issue 1-2
Hamiltonian Systems and Transformation in Hilbert Space
journal, May 1931
- Koopman, B. O.
- Proceedings of the National Academy of Sciences, Vol. 17, Issue 5
On the stability and extension of reduced-order Galerkin models in incompressible flows: A numerical study of vortex shedding
journal, June 2009
- Akhtar, Imran; Nayfeh, Ali H.; Ribbens, Calvin J.
- Theoretical and Computational Fluid Dynamics, Vol. 23, Issue 3
A Sequential Ensemble Kalman Filter for Atmospheric Data Assimilation
journal, January 2001
- Houtekamer, P. L.; Mitchell, Herschel L.
- Monthly Weather Review, Vol. 129, Issue 1
LIII. On lines and planes of closest fit to systems of points in space
journal, November 1901
- Pearson, Karl
- The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Vol. 2, Issue 11
Recursive dynamic mode decomposition of transient and post-transient wake flows
journal, November 2016
- Noack, Bernd R.; Stankiewicz, Witold; Morzyński, Marek
- Journal of Fluid Mechanics, Vol. 809
Principal component analysis in linear systems: Controllability, observability, and model reduction
journal, February 1981
- Moore, B.
- IEEE Transactions on Automatic Control, Vol. 26, Issue 1
A modified proper orthogonal decomposition method for flow dynamic analysis
journal, March 2019
- Zhao, Yijia; Zhao, Ming; Li, Xiaojian
- Computers & Fluids, Vol. 182
Machine learning closures for model order reduction of thermal fluids
journal, August 2018
- San, Omer; Maulik, Romit
- Applied Mathematical Modelling, Vol. 60
An artificial neural network framework for reduced order modeling of transient flows
journal, October 2019
- San, Omer; Maulik, Romit; Ahmed, Mansoor
- Communications in Nonlinear Science and Numerical Simulation, Vol. 77
Sparse reduced-order modelling: sensor-based dynamics to full-state estimation
journal, April 2018
- Loiseau, Jean-Christophe; Noack, Bernd R.; Brunton, Steven L.
- Journal of Fluid Mechanics, Vol. 844
Model Reduction for Fluids, Using Balanced Proper Orthogonal Decomposition
journal, March 2005
- Rowley, C. W.
- International Journal of Bifurcation and Chaos, Vol. 15, Issue 03
Reduced-order modeling for cardiac electrophysiology. Application to parameter identification: REDUCED-ORDER MODELING IN CARDIAC ELECTROPHYSIOLOGY
journal, March 2012
- Boulakia, M.; Schenone, E.; Gerbeau, J-F.
- International Journal for Numerical Methods in Biomedical Engineering, Vol. 28, Issue 6-7
Multiresolution Dynamic Mode Decomposition
journal, January 2016
- Kutz, J. Nathan; Fu, Xing; Brunton, Steven L.
- SIAM Journal on Applied Dynamical Systems, Vol. 15, Issue 2
Randomized dynamic mode decomposition for nonintrusive reduced order modelling: RANDOMIZED DYNAMIC MODE DECOMPOSITION FOR NIROM
journal, February 2017
- Bistrian, Diana Alina; Navon, Ionel Michael
- International Journal for Numerical Methods in Engineering, Vol. 112, Issue 1
A Survey of Finite-Difference Schemes for the Primitive Equations for a Barotropic Fluid
journal, May 1969
- Grammeltvedt, Arne
- Monthly Weather Review, Vol. 97, Issue 5
A deep learning enabler for nonintrusive reduced order modeling of fluid flows
journal, August 2019
- Pawar, S.; Rahman, S. M.; Vaddireddy, H.
- Physics of Fluids, Vol. 31, Issue 8
A reduced-order approach to four-dimensional variational data assimilation using proper orthogonal decomposition
journal, January 2007
- Cao, Yanhua; Zhu, Jiang; Navon, I. M.
- International Journal for Numerical Methods in Fluids, Vol. 53, Issue 10
Balanced POD for model reduction of linear PDE systems: convergence theory
journal, November 2011
- Singler, John R.
- Numerische Mathematik, Vol. 121, Issue 1
POD/DEIM nonlinear model order reduction of an ADI implicit shallow water equations model
journal, March 2013
- Ştefănescu, R.; Navon, I. M.
- Journal of Computational Physics, Vol. 237
Stable Galerkin reduced order models for linearized compressible flow
journal, April 2009
- Barone, Matthew F.; Kalashnikova, Irina; Segalman, Daniel J.
- Journal of Computational Physics, Vol. 228, Issue 6
On the hidden beauty of the proper orthogonal decomposition
journal, August 1991
- Aubry, Nadine
- Theoretical and Computational Fluid Dynamics, Vol. 2, Issue 5-6
Dynamic mode decomposition for multiscale nonlinear physics
journal, June 2019
- Dylewsky, Daniel; Tao, Molei; Kutz, J. Nathan
- Physical Review E, Vol. 99, Issue 6
POD/DEIM Reduced-Order Modeling of Time-Fractional Partial Differential Equations with Applications in Parameter Identification
journal, April 2017
- Fu, Hongfei; Wang, Hong; Wang, Zhu
- Journal of Scientific Computing, Vol. 74, Issue 1
Reduced Basis Method for Optimal Control of Unsteady Viscous Flows
journal, October 2001
- Ito, K.; Ravindran, S. S.
- International Journal of Computational Fluid Dynamics, Vol. 15, Issue 2
pyROM: A computational framework for reduced order modeling
journal, January 2019
- Puzyrev, Vladimir; Ghommem, Mehdi; Meka, Shiv
- Journal of Computational Science, Vol. 30
Proper orthogonal decomposition closure models for turbulent flows: A numerical comparison
journal, September 2012
- Wang, Zhu; Akhtar, Imran; Borggaard, Jeff
- Computer Methods in Applied Mechanics and Engineering, Vol. 237-240
Neural network closures for nonlinear model order reduction
journal, January 2018
- San, Omer; Maulik, Romit
- Advances in Computational Mathematics, Vol. 44, Issue 6
Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics
journal, January 2009
- Najm, Habib N.
- Annual Review of Fluid Mechanics, Vol. 41, Issue 1
Spatiotemporal analysis of complex signals: Theory and applications
journal, August 1991
- Aubry, Nadine; Guyonnet, R�gis; Lima, Ricardo
- Journal of Statistical Physics, Vol. 64, Issue 3-4
Stable architectures for deep neural networks
journal, December 2017
- Haber, Eldad; Ruthotto, Lars
- Inverse Problems, Vol. 34, Issue 1
Closed-Loop Turbulence Control: Progress and Challenges
journal, August 2015
- Brunton, Steven L.; Noack, Bernd R.
- Applied Mechanics Reviews, Vol. 67, Issue 5
Multilayer feedforward networks are universal approximators
journal, January 1989
- Hornik, Kurt; Stinchcombe, Maxwell; White, Halbert
- Neural Networks, Vol. 2, Issue 5
Truncated-Newton training algorithm for neurocomputational viscoplastic model
journal, May 2003
- Al-Haik, M. S.; Garmestani, H.; Navon, I. M.
- Computer Methods in Applied Mechanics and Engineering, Vol. 192, Issue 19
Physically constrained data‐driven correction for reduced‐order modeling of fluid flows
journal, October 2018
- Mohebujjaman, M.; Rebholz, L. G.; Iliescu, T.
- International Journal for Numerical Methods in Fluids, Vol. 89, Issue 3
Certain Topics in Telegraph Transmission Theory
journal, April 1928
- Nyquist, H.
- Transactions of the American Institute of Electrical Engineers, Vol. 47, Issue 2
POD-spectral decomposition for fluid flow analysis and model reduction
journal, February 2013
- Cammilleri, A.; Gueniat, F.; Carlier, J.
- Theoretical and Computational Fluid Dynamics, Vol. 27, Issue 6
Neural Network Modeling for Near Wall Turbulent Flow
journal, October 2002
- Milano, Michele; Koumoutsakos, Petros
- Journal of Computational Physics, Vol. 182, Issue 1
Dynamic Mode Decomposition
book, January 2016
- Kutz, J. Nathan; Brunton, Steven L.; Brunton, Bingni W.
- Society for Industrial and Applied Mathematics
Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow. Part I
journal, August 1966
- Arakawa, Akio
- Journal of Computational Physics, Vol. 1, Issue 1
Dynamic mode decomposition for large and streaming datasets
journal, November 2014
- Hemati, Maziar S.; Williams, Matthew O.; Rowley, Clarence W.
- Physics of Fluids, Vol. 26, Issue 11
Sparsity-promoting dynamic mode decomposition
journal, February 2014
- Jovanović, Mihailo R.; Schmid, Peter J.; Nichols, Joseph W.
- Physics of Fluids, Vol. 26, Issue 2
Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition
journal, February 2016
- Dawson, Scott T. M.; Hemati, Maziar S.; Williams, Matthew O.
- Experiments in Fluids, Vol. 57, Issue 3
The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows
journal, January 1993
- Berkooz, G.; Holmes, P.; Lumley, J. L.
- Annual Review of Fluid Mechanics, Vol. 25, Issue 1
Analysis of Fluid Flows via Spectral Properties of the Koopman Operator
journal, January 2013
- Mezić, Igor
- Annual Review of Fluid Mechanics, Vol. 45, Issue 1
A Hybrid Approach for Model Order Reduction of Barotropic Quasi-Geostrophic Turbulence
journal, October 2018
- Rahman, Sk.; San, Omer; Rasheed, Adil
- Fluids, Vol. 3, Issue 4
Variants of Dynamic Mode Decomposition: Boundary Condition, Koopman, and Fourier Analyses
journal, April 2012
- Chen, Kevin K.; Tu, Jonathan H.; Rowley, Clarence W.
- Journal of Nonlinear Science, Vol. 22, Issue 6, p. 887-915
Stability Properties of POD-Galerkin Approximations for the Compressible Navier-Stokes Equations
journal, March 2000
- Iollo, A.; Lanteri, S.; Désidéri, J. -A.
- Theoretical and Computational Fluid Dynamics, Vol. 13, Issue 6
On linear and nonlinear aspects of dynamic mode decomposition: Linear and Nonlinear Aspects of Dynamic Mode Decomposition
journal, February 2016
- Alekseev, A. K.; Bistrian, D. A.; Bondarev, A. E.
- International Journal for Numerical Methods in Fluids, Vol. 82, Issue 6
Data Assimilation for Numerical Weather Prediction: A Review
book, January 2009
- Navon, Ionel M.
- Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications
Identification and control of dynamical systems using neural networks
journal, March 1990
- Narendra, K. S.; Parthasarathy, K.
- IEEE Transactions on Neural Networks, Vol. 1, Issue 1
Nonlinear Model Order Reduction via Dynamic Mode Decomposition
journal, January 2017
- Alla, Alessandro; Kutz, J. Nathan
- SIAM Journal on Scientific Computing, Vol. 39, Issue 5
Deep learning in fluid dynamics
journal, January 2017
- Kutz, J. Nathan
- Journal of Fluid Mechanics, Vol. 814
De-biasing the dynamic mode decomposition for applied Koopman spectral analysis of noisy datasets
journal, April 2017
- Hemati, Maziar S.; Rowley, Clarence W.; Deem, Eric A.
- Theoretical and Computational Fluid Dynamics, Vol. 31, Issue 4
Balanced Model Reduction via the Proper Orthogonal Decomposition
journal, November 2002
- Willcox, K.; Peraire, J.
- AIAA Journal, Vol. 40, Issue 11
A deep learning enabler for non-intrusive reduced order modeling of fluid flows
text, January 2019
- Pawar, S.; Rahman, S. M.; Vaddireddy, H.
- arXiv