Identification of nonlinear dynamical system equations using dynamic mode decomposition under invariant quantity constraints

Florian De Vuyst; Pierre Villon

doi:10.1016/j.crme.2019.11.013

Identification of nonlinear dynamical system equations using dynamic mode decomposition under invariant quantity constraints

Florian De Vuyst ¹ ; Pierre Villon ²

¹ Laboratoire de mathématiques appliquées de Compiègne, EA 2222, Université de technologie de Compiègne, Alliance Sorbonne Université, 60200 Compiègne, France
² Laboratoire Roberval, FRE UTC–CNRS, Université de technologie de Compiègne, Alliance Sorbonne Université, 60200 Compiègne, France

Comptes Rendus. Mécanique, Volume 347 (2019) no. 11, pp. 882-890.

Résumé

In this paper, an algorithm for identifying equations representing a continuous nonlinear dynamical system from a noise-free state and time-derivative state measurements is proposed. It is based on a variant of the extended dynamic mode decomposition. A particular attention is paid to guarantee that the physical invariant quantities stay constant along the integral curves. The numerical methodology is validated on a two-dimensional Lotka–Volterra system. For this case, the differential equations are perfectly retrieved from data measurements. Perspectives of extension to more complex systems are discussed.

Reçu le : 2019-07-11
Accepté le : 2019-09-09
Publié le : 2019-11-01

DOI : 10.1016/j.crme.2019.11.013

Mots clés : Dynamical system, Identification, Invariant quantity, Symplectic, Dynamic mode decomposition, Lyapunov equations, Lotka–Volterra system

Affiliations des auteurs :

Florian De Vuyst ¹ ; Pierre Villon ²

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     author = {Florian De Vuyst and Pierre Villon},
     title = {Identification of nonlinear dynamical system equations using dynamic mode decomposition under invariant quantity constraints},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {882--890},
     publisher = {Elsevier},
     volume = {347},
     number = {11},
     year = {2019},
     doi = {10.1016/j.crme.2019.11.013},
     language = {en},
}

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Florian De Vuyst; Pierre Villon. Identification of nonlinear dynamical system equations using dynamic mode decomposition under invariant quantity constraints. Comptes Rendus. Mécanique, Volume 347 (2019) no. 11, pp. 882-890. doi : 10.1016/j.crme.2019.11.013. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2019.11.013/

Bibliographie
Cité par

[1] D. Gonzales; F. Chinesta; E. Cueto Learning corrections for hyperelastic models from data, Front. Mater., Volume 6 (2019) no. 14

[2] P. Wangersky Lotka–Volterra population models, Annu. Rev. Ecol. Syst., Volume 9 (1978), pp. 189-218

[3] A. Ionescu; R. Militaru; F. Munteanu Geometrical methods and numerical computations for prey-predator systems, Br. J. Math. Comput. Sci., Volume 10 (2015) no. 5, pp. 1-15

[4] P.J. Schmid Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., Volume 656 (2010), pp. 5-28

[5] R.H. Bartels; G.W. Stewart Solution of the matrix equation $A X + X B = C$ , Commun. ACM, Volume 15 (1972) no. 9, pp. 820-826 | DOI

[6] M.O. Williams; I.G. Kevrekidis; C.W. Rowley A data-driven approximation of the Koopman operator: extending dynamic mode decomposition, J. Nonlinear Sci., Volume 25 (2015), pp. 1307-1346

[7] M. Grmela; H.C. Öttinger Dynamics and thermodynamics of complex fluids. I. Development of a general formalism, Phys. Rev. E, Volume 56 (1997), p. 6620

[8] B. Moya; D. Gonzalez; I. Alfaro; F. Chinesta; E. Cueto Learning slosh dynamics by means of data, Comput. Mech., Volume 64 (2019) no. 2, pp. 511-552

[9] D. Gonzales; F. Chinesta; E. Cueto Contin. Mech. Thermodyn., 31 (2019) no. 1, pp. 239-253

[10] M. Asch; M. Bocquet; M. Nodet Data Assimilation: Methods, Algorithms, and Applications, Fundamentals of Algorithms, SIAM, 2017

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