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Title: Memory embedded non-intrusive reduced order modeling of non-ergodic flows

ORCiD logo [1]; ORCiD logo [1]; ORCiD logo [1]; ORCiD logo [2];  [3]
  1. School of Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, Oklahoma 74078, USA
  2. Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7465 Trondheim, Norway
  3. Department of Scientific Computing, Florida State University, Tallahassee, Florida 32306, USA
Publication Date:
Sponsoring Org.:
OSTI Identifier:
Grant/Contract Number:  
Resource Type:
Publisher's Accepted Manuscript
Journal Name:
Physics of Fluids
Additional Journal Information:
Journal Name: Physics of Fluids Journal Volume: 31 Journal Issue: 12; Journal ID: ISSN 1070-6631
American Institute of Physics
Country of Publication:
United States

Citation Formats

Ahmed, Shady E., Rahman, Sk. Mashfiqur, San, Omer, Rasheed, Adil, and Navon, Ionel M. Memory embedded non-intrusive reduced order modeling of non-ergodic flows. United States: N. p., 2019. Web. doi:10.1063/1.5128374.
Ahmed, Shady E., Rahman, Sk. Mashfiqur, San, Omer, Rasheed, Adil, & Navon, Ionel M. Memory embedded non-intrusive reduced order modeling of non-ergodic flows. United States. doi:10.1063/1.5128374.
Ahmed, Shady E., Rahman, Sk. Mashfiqur, San, Omer, Rasheed, Adil, and Navon, Ionel M. Sun . "Memory embedded non-intrusive reduced order modeling of non-ergodic flows". United States. doi:10.1063/1.5128374.
title = {Memory embedded non-intrusive reduced order modeling of non-ergodic flows},
author = {Ahmed, Shady E. and Rahman, Sk. Mashfiqur and San, Omer and Rasheed, Adil and Navon, Ionel M.},
abstractNote = {},
doi = {10.1063/1.5128374},
journal = {Physics of Fluids},
number = 12,
volume = 31,
place = {United States},
year = {2019},
month = {12}

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Works referenced in this record:

The Digital Twin: Realizing the Cyber-Physical Production System for Industry 4.0
journal, January 2017

Learning long-term dependencies with gradient descent is difficult
journal, March 1994

  • Bengio, Y.; Simard, P.; Frasconi, P.
  • IEEE Transactions on Neural Networks, Vol. 5, Issue 2
  • DOI: 10.1109/72.279181

Nonlinear model order reduction based on local reduced-order bases: NONLINEAR MODEL REDUCTION BASED ON LOCAL REDUCED-ORDER BASES
journal, June 2012

  • Amsallem, David; Zahr, Matthew J.; Farhat, Charbel
  • International Journal for Numerical Methods in Engineering, Vol. 92, Issue 10
  • DOI: 10.1002/nme.4371

Deep learning
journal, May 2015

  • LeCun, Yann; Bengio, Yoshua; Hinton, Geoffrey
  • Nature, Vol. 521, Issue 7553
  • DOI: 10.1038/nature14539

The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows
journal, January 1993

Fluid control using the adjoint method
journal, August 2004

  • McNamara, Antoine; Treuille, Adrien; Popović, Zoran
  • ACM Transactions on Graphics, Vol. 23, Issue 3
  • DOI: 10.1145/1015706.1015744

Hybrid forecasting of chaotic processes: Using machine learning in conjunction with a knowledge-based model
journal, April 2018

  • Pathak, Jaideep; Wikner, Alexander; Fussell, Rebeckah
  • Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 28, Issue 4
  • DOI: 10.1063/1.5028373

An Efficient, Globally Convergent Method for Optimization Under Uncertainty Using Adaptive Model Reduction and Sparse Grids
journal, January 2019

  • Zahr, Matthew J.; Carlberg, Kevin T.; Kouri, Drew P.
  • SIAM/ASA Journal on Uncertainty Quantification, Vol. 7, Issue 3
  • DOI: 10.1137/18m1220996

Non-intrusive reduced order modelling with least squares fitting on a sparse grid: NON-INTRUSIVE REDUCED ORDER MODELLING WITH LEAST SQUARES FITTING
journal, July 2016

  • Lin, Z.; Xiao, D.; Fang, F.
  • International Journal for Numerical Methods in Fluids, Vol. 83, Issue 3
  • DOI: 10.1002/fld.4268

Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
journal, February 2019

Unsteady Fluid Mechanics Applications of Neural Networks
journal, January 1997

  • Faller, William E.; Schreck, Scott J.
  • Journal of Aircraft, Vol. 34, Issue 1
  • DOI: 10.2514/2.2134

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
  • DOI: 10.1002/nme.2746

Application of neural networks to turbulence control for drag reduction
journal, June 1997

  • Lee, Changhoon; Kim, John; Babcock, David
  • Physics of Fluids, Vol. 9, Issue 6
  • DOI: 10.1063/1.869290

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
  • DOI: 10.1002/fld.1316

A new synergetic paradigm in environmental numerical modeling: Hybrid models combining deterministic and machine learning components
journal, January 2006

Model Reduction for Flow Analysis and Control
journal, January 2017

Low-dimensional modelling of a confined three-dimensional wake flow
journal, November 2006

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
  • DOI: 10.1063/1.5093355

Optimal control of the cylinder wake in the laminar regime by trust-region methods and POD reduced-order models
journal, August 2008

2D thermal/isothermal incompressible viscous flows
journal, January 2005

  • Nicolás, Alfredo; Bermúdez, Blanca
  • International Journal for Numerical Methods in Fluids, Vol. 48, Issue 4
  • DOI: 10.1002/fld.895

Modal Analysis of Fluid Flows: An Overview
journal, December 2017

  • Taira, Kunihiko; Brunton, Steven L.; Dawson, Scott T. M.
  • AIAA Journal, Vol. 55, Issue 12
  • DOI: 10.2514/1.j056060

Uber Die Beste Annaherung Von Funktionen Einer Gegebenen Funktionenklasse
journal, January 1936

  • Kolmogoroff, A.
  • The Annals of Mathematics, Vol. 37, Issue 1
  • DOI: 10.2307/1968691

pyMOR -- Generic Algorithms and Interfaces for Model Order Reduction
journal, January 2016

  • Milk, René; Rave, Stephan; Schindler, Felix
  • SIAM Journal on Scientific Computing, Vol. 38, Issue 5
  • DOI: 10.1137/15m1026614

Evaluating Data Assimilation Algorithms
journal, November 2012

Vortex merging in quasi-geostrophic flows
journal, June 2000

Applications of the dynamic mode decomposition
journal, August 2010

  • Schmid, P. J.; Li, L.; Juniper, M. P.
  • Theoretical and Computational Fluid Dynamics, Vol. 25, Issue 1-4, p. 249-259
  • DOI: 10.1007/s00162-010-0203-9

Reduced basis techniques for nonlinear conservation laws
journal, April 2015

  • Taddei, T.; Perotto, S.; Quarteroni, A.
  • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 49, Issue 3
  • DOI: 10.1051/m2an/2014054

Long Short-Term Memory
journal, November 1997

Some Recent Developments in Turbulence Closure Modeling
journal, January 2018

Proper orthogonal decomposition and low-dimensional models for driven cavity flows
journal, July 1998

  • Cazemier, W.; Verstappen, R. W. C. P.; Veldman, A. E. P.
  • Physics of Fluids, Vol. 10, Issue 7
  • DOI: 10.1063/1.869686

Nonlinear Model Reduction via Discrete Empirical Interpolation
journal, January 2010

  • Chaturantabut, Saifon; Sorensen, Danny C.
  • SIAM Journal on Scientific Computing, Vol. 32, Issue 5
  • DOI: 10.1137/090766498

The Bosporus Strait: A Special Place in the History of Oceanography
journal, June 2005

Ergodic properties of highly degenerate 2D stochastic Navier–Stokes equations
journal, December 2004

A domain decomposition non-intrusive reduced order model for turbulent flows
journal, March 2019

On the Relation between Energy-Conserving Low-Order Models and a System of Coupled Generalized Volterra Gyrostats with Nonlinear Feedback
journal, November 2007

Nonergodicity of Euler fluid dynamics on tori versus positivity of the Arnold–Ricci tensor
journal, April 2008

Data-Driven Filtered Reduced Order Modeling of Fluid Flows
journal, January 2018

  • Xie, X.; Mohebujjaman, M.; Rebholz, L. G.
  • SIAM Journal on Scientific Computing, Vol. 40, Issue 3
  • DOI: 10.1137/17m1145136

Data-driven deconvolution for large eddy simulations of Kraichnan turbulence
journal, December 2018

  • Maulik, R.; San, O.; Rasheed, A.
  • Physics of Fluids, Vol. 30, Issue 12
  • DOI: 10.1063/1.5079582

Non-intrusive reduced order modelling of the Navier–Stokes equations
journal, August 2015

  • Xiao, D.; Fang, F.; Buchan, A. G.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 293
  • DOI: 10.1016/j.cma.2015.05.015

Physics of vortex merging
journal, May 2005

  • Meunier, Patrice; Le Dizès, Stéphane; Leweke, Thomas
  • Comptes Rendus Physique, Vol. 6, Issue 4-5
  • DOI: 10.1016/j.crhy.2005.06.003

Principal interval decomposition framework for POD reduced-order modeling of convective Boussinesq flows: PRINCIPAL INTERVAL DECOMPOSITION MODEL REDUCTION FRAMEWORK
journal, February 2015

  • San, O.; Borggaard, J.
  • International Journal for Numerical Methods in Fluids, Vol. 78, Issue 1
  • DOI: 10.1002/fld.4006

POD-Galerkin approximations in PDE-constrained optimization
journal, October 2010

Reduced Order Galerkin Models of flow Around Naca‐0012 Airfoil
journal, March 2008

A reduced-order approach for optimal control of fluids using proper orthogonal decomposition
journal, January 2000

Deep learning algorithm for data-driven simulation of noisy dynamical system
journal, January 2019

Learning to Forget: Continual Prediction with LSTM
journal, October 2000

A coarse-grid projection method for accelerating incompressible flow computations
journal, January 2013

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
  • DOI: 10.3934/jcd.2014.1.391

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
  • DOI: 10.1016/j.physd.2003.03.001

Towards non-intrusive reduced order 3D free surface flow modelling
journal, August 2017

Nonlinear model reduction of unconfined groundwater flow using POD and DEIM
journal, November 2016

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
  • DOI: 10.1002/fld.4257

Feedback Control for Systems with Uncertain Parameters Using Online-Adaptive Reduced Models
journal, January 2017

  • Kramer, Boris; Peherstorfer, Benjamin; Willcox, Karen
  • SIAM Journal on Applied Dynamical Systems, Vol. 16, Issue 3
  • DOI: 10.1137/16m1088958

Ergodicity for the 3D stochastic Navier–Stokes equations
journal, August 2003

Turbulence and the dynamics of coherent structures. I. Coherent structures
journal, January 1987

  • Sirovich, Lawrence
  • Quarterly of Applied Mathematics, Vol. 45, Issue 3
  • DOI: 10.1090/qam/910462

Digital twin-driven product design, manufacturing and service with big data
journal, March 2017

  • Tao, Fei; Cheng, Jiangfeng; Qi, Qinglin
  • The International Journal of Advanced Manufacturing Technology, Vol. 94, Issue 9-12
  • DOI: 10.1007/s00170-017-0233-1

Decay of the Kolmogorov N -width for wave problems
journal, October 2019

Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing
journal, January 2006

A Reduced-Order Method for Simulation and Control of Fluid Flows
journal, July 1998

Data-assisted reduced-order modeling of extreme events in complex dynamical systems
journal, May 2018

Ergodicity of the 2-D Navier-Stokes equation under random perturbations
journal, August 1995

  • Flandoli, Franco; Maslowski, Bohdan
  • Communications in Mathematical Physics, Vol. 172, Issue 1
  • DOI: 10.1007/bf02104513

A neural network approach for the blind deconvolution of turbulent flows
journal, October 2017

Artificial viscosity proper orthogonal decomposition
journal, January 2011

  • Borggaard, Jeff; Iliescu, Traian; Wang, Zhu
  • Mathematical and Computer Modelling, Vol. 53, Issue 1-2
  • DOI: 10.1016/j.mcm.2010.08.015

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
  • DOI: 10.1007/s00162-009-0112-y

Data-driven forecasting of high-dimensional chaotic systems with long short-term memory networks
journal, May 2018

  • Vlachas, Pantelis R.; Byeon, Wonmin; Wan, Zhong Y.
  • Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 474, Issue 2213
  • DOI: 10.1098/rspa.2017.0844

Turbulence Modeling in the Age of Data
journal, January 2019

Non-intrusive reduced order modelling of fluid–structure interactions
journal, May 2016

  • Xiao, D.; Yang, P.; Fang, F.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 303
  • DOI: 10.1016/j.cma.2015.12.029

High-order methods for decaying two-dimensional homogeneous isotropic turbulence
journal, June 2012

A second-order projection method for the incompressible navier-stokes equations
journal, December 1989

A parameterized non-intrusive reduced order model and error analysis for general time-dependent nonlinear partial differential equations and its applications
journal, April 2017

  • Xiao, D.; Fang, F.; Pain, C. C.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 317
  • DOI: 10.1016/j.cma.2016.12.033

Stabilized principal interval decomposition method for model reduction of nonlinear convective systems with moving shocks
journal, September 2018

Complex hybrid models combining deterministic and machine learning components for numerical climate modeling and weather prediction
journal, March 2006

Optimal control of vortex shedding using low-order models. Part I?open-loop model development
journal, March 1999

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
  • DOI: 10.1017/jfm.2018.147

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
  • DOI: 10.1002/cnm.2465

Non-intrusive reduced order modeling of unsteady flows using artificial neural networks with application to a combustion problem
journal, May 2019

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
  • DOI: 10.1063/1.5113494

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
  • DOI: 10.1002/fld.1365

POD/DEIM nonlinear model order reduction of an ADI implicit shallow water equations model
journal, March 2013

Parameterised non-intrusive reduced order methods for ensemble Kalman filter data assimilation
journal, November 2018

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
  • DOI: 10.1016/

POD/DEIM Reduced-Order Modeling of Time-Fractional Partial Differential Equations with Applications in Parameter Identification
journal, April 2017

Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations: POD STRATEGIES FOR 2D SWE
journal, August 2014

  • Ştefănescu, Răzvan; Sandu, Adrian; Navon, Ionel M.
  • International Journal for Numerical Methods in Fluids, Vol. 76, Issue 8
  • DOI: 10.1002/fld.3946

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
  • DOI: 10.1080/10618560108970021

pyROM: A computational framework for reduced order modeling
journal, January 2019

Ergodicity of 2D Navier-Stokes Equations with¶Random Forcing and Large Viscosity
journal, October 1999

  • Mattingly, Jonathan C.
  • Communications in Mathematical Physics, Vol. 206, Issue 2
  • DOI: 10.1007/s002200050706

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
  • DOI: 10.1016/j.cma.2012.04.015

Neural network closures for nonlinear model order reduction
journal, January 2018

Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics
journal, January 2009

Closed-Loop Turbulence Control: Progress and Challenges
journal, August 2015

  • Brunton, Steven L.; Noack, Bernd R.
  • Applied Mechanics Reviews, Vol. 67, Issue 5
  • DOI: 10.1115/1.4031175

An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations
journal, November 2004

  • Barrault, Maxime; Maday, Yvon; Nguyen, Ngoc Cuong
  • Comptes Rendus Mathematique, Vol. 339, Issue 9
  • DOI: 10.1016/j.crma.2004.08.006

Optimal control of vortex shedding using low-order models. Part II?model-based control
journal, March 1999

Data-Driven Time Parallelism via Forecasting
journal, January 2019

  • Carlberg, Kevin; Brencher, Lukas; Haasdonk, Bernard
  • SIAM Journal on Scientific Computing, Vol. 41, Issue 3
  • DOI: 10.1137/18m1174362

Non-ergodicity of inviscid two-dimensional flow on a beta-plane and on the surface of a rotating sphere
journal, November 1987

Reduced-order modeling: new approaches for computational physics
journal, February 2004

Reduced-order modeling of time-varying systems
journal, January 1999

  • Roychowdhury, J.
  • IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, Vol. 46, Issue 10
  • DOI: 10.1109/82.799678

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
  • DOI: 10.1007/s11075-004-2866-z

Ergodic results for stochastic navier-stokes equation
journal, April 1997

Non-Intrusive Inference Reduced Order Model for Fluids Using Deep Multistep Neural Network
journal, August 2019

  • Xie, Xuping; Zhang, Guannan; Webster, Clayton G.
  • Mathematics, Vol. 7, Issue 8
  • DOI: 10.3390/math7080757

An intrinsic stabilization scheme for proper orthogonal decomposition based low-dimensional models
journal, May 2007

  • Kalb, Virginia L.; Deane, Anil E.
  • Physics of Fluids, Vol. 19, Issue 5
  • DOI: 10.1063/1.2723149

A Hybrid Approach for Model Order Reduction of Barotropic Quasi-Geostrophic Turbulence
journal, October 2018

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
  • DOI: 10.1007/s00332-012-9130-9

Merger and cancellation of strained vortices
journal, August 1989

Enablers for robust POD models
journal, February 2009

  • Bergmann, M.; Bruneau, C. -H.; Iollo, A.
  • Journal of Computational Physics, Vol. 228, Issue 2
  • DOI: 10.1016/

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
  • DOI: 10.1007/s001620050119

Optimal large-eddy simulation of forced Burgers equation
journal, December 2002

  • Das, Arup; Moser, Robert D.
  • Physics of Fluids, Vol. 14, Issue 12
  • DOI: 10.1063/1.1516212

Online Adaptive Model Reduction for Nonlinear Systems via Low-Rank Updates
journal, January 2015

  • Peherstorfer, Benjamin; Willcox, Karen
  • SIAM Journal on Scientific Computing, Vol. 37, Issue 4
  • DOI: 10.1137/140989169

Deep learning in fluid dynamics
journal, January 2017

A POD reduced order unstructured mesh ocean modelling method for moderate Reynolds number flows
journal, January 2009