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Hierarchical deep learning for data-driven identification of reduced-order models of nonlinear dynamical systems

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Abstract

Identifying reduced-order models (ROMs) of nonlinear dynamical systems is difficult, especially when the system equation is unknown with only measurement data available. In such a case, not only a reduced subspace but also the associated dynamics need to be identified from data only, leading to a challenging data-driven ROM problem. In this study, we present a hierarchical deep learning approach to identify ROM from measurement only; it simultaneously identifies the nonlinear normal modal (NNM) subspace with a hierarchical order and the associated nonlinear modal dynamics. We conduct study to validate such an approach on both unforced and forced nonlinear dynamical systems, and find that the identified hierarchical NNMs-spanned subspace enables an efficient and effective dimensional truncation to achieve optimally lowest-dimensional ROM. We discuss in detail its performance and applicability.

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Data availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

  1. Strogatz, S., Friedman, M., Mallinckrodt, A.J., McKay, S.: Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Comput. Phys. 8(5), 532 (1994)

    Article  Google Scholar 

  2. Peeters, M., Kerschen, G., Golinval, J.C.: Dynamic testing of nonlinear vibrating structures using nonlinear normal modes. J. Sound Vib. 330(3), 486–509 (2011)

    Article  Google Scholar 

  3. Noël, J.P., Renson, L., Kerschen, G.: Complex dynamics of a nonlinear aerospace structure: experimental identification and modal interactions. J. Sound Vib. 333(12), 2588–2607 (2014)

    Article  Google Scholar 

  4. Rega, G., Troger, H.: Dimension reduction of dynamical systems: methods, models, applications. Nonlinear Dyn. 41(1–3), 1–15 (2005)

    Article  MathSciNet  Google Scholar 

  5. Mezić, I.: Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41(1–3), 309–325 (2005)

    Article  MathSciNet  Google Scholar 

  6. Kerschen, G., GolinvalGolinval, J.C., Vakakis, A.F., Bergman, L.A.: The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview. Nonlinear Dyn. 41(1–3), 147–169 (2005)

    Article  MathSciNet  Google Scholar 

  7. Quarteroni, A., Rozza, G., et al.: Reduced Order Methods for Modeling and Computational Reduction, vol. 9. Springer, Berlin (2014)

    MATH  Google Scholar 

  8. Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection–diffusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (1998)

    Article  MathSciNet  Google Scholar 

  9. Hughes, T.J.R., Engel, G., Mazzei, L., Larson, M.G.: The continuous Galerkin method is locally conservative. J. Comput. Phys. 163(2), 467–488 (2000)

    Article  MathSciNet  Google Scholar 

  10. Azeez, M.F.A., Vakakis, A.F.: Proper orthogonal decomposition (POD) of a class of vibroimpact oscillations. J. Sound Vib. 240(5), 859–889 (2001)

    Article  Google Scholar 

  11. Georgiou, I.: Advanced proper orthogonal decomposition tools: using reduced order models to identify normal modes of vibration and slow invariant manifolds in the dynamics of planar nonlinear rods. Nonlinear Dyn. 41(1–3), 69–110 (2005)

    Article  MathSciNet  Google Scholar 

  12. Feeny, B.F., Kappagantu, R.: On the physical interpretation of proper orthogonal modes in vibrations. J. Sound Vib. 211(4), 607–616 (1998)

    Article  Google Scholar 

  13. Feeny, B.F.: On the proper orthogonal modes and normal modes of continuous vibration systems. Trans. ASME J. Vib. Acoust. 124(1), 150–160 (2002)

    Article  Google Scholar 

  14. Lin, W.Z., Lee, K.H., Lu, P., Lim, S.P., Liang, Y.C.: The relationship between eigenfunctions of Karhunen–Loève decomposition and the modes of distributed parameter vibration system (2002)

  15. Kerschen, G., Golinval, J.C.: Physical interpretation of the proper orthogonal modes using the singular value decomposition. J. Sound Vib. 249(5), 849–865 (2003)

    Article  MathSciNet  Google Scholar 

  16. Sidhu, H.S., Narasingam, A., Siddhamshetty, P., Kwon, J.S.I.: Model order reduction of nonlinear parabolic PDE systems with moving boundaries using sparse proper orthogonal decomposition: application to hydraulic fracturing. Comput. Chem. Eng. 112, 92–100 (2018)

    Article  Google Scholar 

  17. Everson, R., Sirovich, L.: Karhunen–Loeve procedure for gappy data. J. Opt. Soc. Am. A 12(8), 1657–1664 (1995)

    Article  Google Scholar 

  18. Willcox, K.: Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition. Comput. Fluids 35(2), 208–226 (2006)

    Article  Google Scholar 

  19. Shaw, S.W., Pierre, C.: Normal modes for non-linear vibratory systems. J. Sound Vib. 164(1), 85–124 (1993)

    Article  Google Scholar 

  20. Blanc, F., Touzé, C., Mercier, J.F., Ege, K., Bonnet Ben-Dhia, A.S.: On the numerical computation of nonlinear normal modes for reduced-order modelling of conservative vibratory systems. Mech. Syst. Signal Process. 36(2), 520–539 (2013)

    Article  Google Scholar 

  21. Touzé, C., Amabili, M.: Nonlinear normal modes for damped geometrically nonlinear systems: application to reduced-order modelling of harmonically forced structures. J. Sound Vib. 298(4–5), 958–981 (2006)

    Article  Google Scholar 

  22. Srinil, N., Rega, G.: Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part II: internal resonance activation, reduced-order models and nonlinear normal modes. Nonlinear Dyn. 48(3), 253–274 (2007)

    Article  Google Scholar 

  23. Pesheck, E., Pierre, C., Shaw, S.W.: New Galerkin-based approach for accurate non-linear normal modes through invariant manifolds. J. Sound Vib. 249(5), 971–993 (2003)

    Article  MathSciNet  Google Scholar 

  24. Amabili, M., Touzé, C.: Reduced-order models for nonlinear vibrations of fluid-filled circular cylindrical shells: comparison of POD and asymptotic nonlinear normal modes methods. J. Fluids Struct. 23(6), 885–903 (2007)

    Article  Google Scholar 

  25. Rahman, S.M., Pawar, S., San, O., Rasheed, A., Iliescu, T.: Nonintrusive reduced order modeling framework for quasigeostrophic turbulence. Phys. Rev. E 100(5), 1–25 (2019)

    Article  MathSciNet  Google Scholar 

  26. Hesthaven, J.S., Ubbiali, S.: Non-intrusive reduced order modeling of nonlinear problems using neural networks. J. Comput. Phys. 363, 55–78 (2018)

    Article  MathSciNet  Google Scholar 

  27. Chua, A.J.K., Galley, C.R., Vallisneri, M.: Reduced-order modeling with artificial neurons for gravitational-wave inference. Phys. Rev. Lett. 122(21), 211101 (2019)

    Article  Google Scholar 

  28. Mou, C., Koc, B., San, O., Rebholz, L.G., Iliescu, T.: Data-driven variational multiscale reduced order models. Comput. Methods Appl. Mech. Eng. 373, 113470 (2021)

    Article  MathSciNet  Google Scholar 

  29. Gonzalez, F.J., Balajewicz, M.: Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems. arXiv preprint (2018)

  30. Maulik, R., Lusch, B., Balaprakash, P.: Reduced-order modeling of advection-dominated systems with recurrent neural networks and convolutional autoencoders. arXiv preprint (2020)

  31. Psichogios, D.C., Ungar, L.H.: A hybrid neural network-first principles approach to process modeling. AIChE J. 38(10), 1499–1511 (1992)

    Article  Google Scholar 

  32. Bangi, M.S.F., Kwon, J.S.I.: Deep hybrid modeling of chemical process: application to hydraulic fracturing. Comput. Chem. Eng. 134, 106696 (2020)

    Article  Google Scholar 

  33. Scholz, M., Vigário, R.: Nonlinear PCA: a new hierarchical approach. In: Proceedings of the 10th European Symposium on Artificial Neural Networks (ESANN), pp. 439–444 (2002)

  34. Li, S., Laima, S., Li, H.: Data-driven modeling of vortex-induced vibration of a long-span suspension bridge using decision tree learning and support vector regression. J. Wind Eng. Ind. Aerodyn. 172, 196–211 (2018)

    Article  Google Scholar 

  35. Li, S., Kaiser, E., Laima, S., Li, H., Brunton, S.L., Kutz, J.N.: Discovering time-varying aerodynamics of a prototype bridge by sparse identification of nonlinear dynamical systems. Phys. Rev. E 100(2), 022220 (2019)

    Article  MathSciNet  Google Scholar 

  36. Champion, K., Lusch, B., Kutz, J.N., Brunton, S.L.: Data-driven discovery of coordinates and governing equations. Proc. Natl. Acad. Sci. U.S.A. 116(45), 22445–22451 (2019)

    Article  MathSciNet  Google Scholar 

  37. Teng, Q., Zhang, L.: Data driven nonlinear dynamical systems identification using multi-step CLDNN. AIP Adv. 9(8), 085311 (2019)

    Article  Google Scholar 

  38. Claeskens, G., Jansen, M.: Model selection and model averaging. International Encyclopedia of the Social & Behavioral Sciences. 2nd edn., pp. 647–652 (2015)

  39. Montavon, G., Orr, G., Müller, K.-R.: Neural Networks: Tricks of the Trade, 2nd edn. Springer, Berlin (2012)

    Book  Google Scholar 

  40. Domingos, P.: The role of Occam’s Razor in knowledge discovery. Data Min. Knowl. Discov.3(4), 409–425 (1999)

  41. Kingma, D.P., Ba, J.L.: Adam: a method for stochastic optimization. In: 3rd International Conference on Learning Representations, ICLR 2015—Conference Track Proceedings. International Conference on Learning Representations, ICLR (2015)

  42. Kendall, A., Gal, Y.: What uncertainties do we need in Bayesian deep learning for computer vision? In: Proceedings of the 31st International Conference on Neural Information Processing Systems, NIPS’17, Red Hook, NY, USA, pp. 5580–5590. Curran Associates Inc (2017)

  43. Gal, Y., Ghahramani, Z.: Dropout as a Bayesian approximation: representing model uncertainty in deep learning. In: Balcan, M.F., Weinberger, K.Q. (eds.) Proceedings of The 33rd International Conference on Machine Learning, volume 48 of Proceedings of Machine Learning Research, New York, New York, USA, pp. 1050–1059 (2016)

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Acknowledgements

This research was partially funded by the Physics of Artificial Intelligence Program of U.S. Defense Advanced Research Projects Agency (DARPA) and the faculty startup fund of Michigan Tech.

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  1. Department of Mechanical Engineering-Engineering Mechanics, Michigan Technological University, Houghton, MI, 49931, USA

    Shanwu Li & Yongchao Yang

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  1. Shanwu Li
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Correspondence to Yongchao Yang.

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Li, S., Yang, Y. Hierarchical deep learning for data-driven identification of reduced-order models of nonlinear dynamical systems. Nonlinear Dyn 105, 3409–3422 (2021). https://doi.org/10.1007/s11071-021-06772-x

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