Constrained sparse Galerkin regression. The sparse identification of nonlinear dynamics (SINDy) is a recently proposed data-driven modelling framework that uses sparse regression techniques to identify nonlinear low-order models. With the goal of low-order models of a fluid flow, we combine this approach with dimensionality reduction techniques (e.g. proper orthogonal decomposition) and extend it to enforce physical constraints in the regression, e.g. energy-preserving quadratic nonlinearities. The resulting models, hereafter referred to as Galerkin regression models, incorporate many beneficial aspects of Galerkin projection, but without the need for a high-fidelity solver to project the Navier-Stokes equations. Instead, the most parsimonious nonlinear model is determined that is consistent with observed measurement data and satisfies necessary constraints. Galerkin regression models also readily generalize to include higher-order nonlinear terms that model the effect of truncated modes. The effectiveness of such an approach is demonstrated on two canonical flow configurations: the two-dimensional flow past a circular cylinder and the shear-driven cavity flow. For both cases, the accuracy of the identified models compare favourably against reduced-order models obtained from a standard Galerkin projection procedure. Finally, the entire code base for our constrained sparse Galerkin regression algorithm is freely available online.

References in zbMATH (referenced in 29 articles , 1 standard article )

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  1. Bramburger, Jason J.; Brunton, Steven L.; Nathan Kutz, J.: Deep learning of conjugate mappings (2021)
  2. Flaschel, Moritz; Kumar, Siddhant; De Lorenzis, Laura: Unsupervised discovery of interpretable hyperelastic constitutive laws (2021)
  3. Fukami, Kai; Murata, Takaaki; Zhang, Kai; Fukagata, Koji: Sparse identification of nonlinear dynamics with low-dimensionalized flow representations (2021)
  4. Kang, Sung Ha; Liao, Wenjing; Liu, Yingjie: IDENT: identifying differential equations with numerical time evolution (2021)
  5. Loiseau, Jean-Christophe; Brunton, Steven L.; Noack, Bernd R.: From the POD-Galerkin method to sparse manifold models (2021)
  6. Mou, Changhong; Koc, Birgul; San, Omer; Rebholz, Leo G.; Iliescu, Traian: Data-driven variational multiscale reduced order models (2021)
  7. Benner, Peter; Goyal, Pawan; Kramer, Boris; Peherstorfer, Benjamin; Willcox, Karen: Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms (2020)
  8. Bramburger, Jason J.; Kutz, J. Nathan: Poincaré maps for multiscale physics discovery and nonlinear Floquet theory (2020)
  9. Cheng, M.; Fang, F.; Pain, C. C.; Navon, I. M.: An advanced hybrid deep adversarial autoencoder for parameterized nonlinear fluid flow modelling (2020)
  10. Erichson, N. Benjamin; Zheng, Peng; Manohar, Krithika; Brunton, Steven L.; Kutz, J. Nathan; Aravkin, Aleksandr Y.: Sparse principal component analysis via variable projection (2020)
  11. Hijazi, Saddam; Stabile, Giovanni; Mola, Andrea; Rozza, Gianluigi: Data-driven POD-Galerkin reduced order model for turbulent flows (2020)
  12. Mou, Changhong; Liu, Honghu; Wells, David R.; Iliescu, Traian: Data-driven correction reduced order models for the quasi-geostrophic equations: a numerical investigation (2020)
  13. Schaeffer, Hayden; Tran, Giang; Ward, Rachel; Zhang, Linan: Extracting structured dynamical systems using sparse optimization with very few samples (2020)
  14. Sipp, Denis; Fosas de Pando, Miguel; Schmid, Peter J.: Nonlinear model reduction: a comparison between POD-Galerkin and POD-DEIM methods (2020)
  15. Bengana, Y.; Loiseau, J.-Ch.; Robinet, J.-Ch.; Tuckerman, L. S.: Bifurcation analysis and frequency prediction in shear-driven cavity flow (2019)
  16. Koc, Birgul; Mohebujjaman, Muhammad; Mou, Changhong; Iliescu, Traian: Commutation error in reduced order modeling of fluid flows (2019)
  17. Leclercq, Colin; Demourant, Fabrice; Poussot-Vassal, Charles; Sipp, Denis: Linear iterative method for closed-loop control of quasiperiodic flows (2019)
  18. Mangan, N. M.; Askham, T.; Brunton, S. L.; Kutz, J. N.; Proctor, J. L.: Model selection for hybrid dynamical systems via sparse regression (2019)
  19. Rudy, Samuel; Alla, Alessandro; Brunton, Steven L.; Kutz, J. Nathan: Data-driven identification of parametric partial differential equations (2019)
  20. Rudy, Samuel H.; Nathan Kutz, J.; Brunton, Steven L.: Deep learning of dynamics and signal-noise decomposition with time-stepping constraints (2019)

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