Comptes Rendus
Incremental dynamic mode decomposition: A reduced-model learner operating at the low-data limit
Comptes Rendus. Mécanique, Volume 347 (2019) no. 11, pp. 780-792.

The present work aims at proposing a new methodology for learning reduced models from a small amount of data. It is based on the fact that discrete models, or their transfer function counterparts, have a low rank and then they can be expressed very efficiently using few terms of a tensor decomposition. An efficient procedure is proposed as well as a way for extending it to nonlinear settings while keeping limited the impact of data noise. The proposed methodology is then validated by considering a nonlinear elastic problem and constructing the model relating tractions and displacements at the observation points.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crme.2019.11.003
Keywords: Machine learning, Advanced regression, Tensor formats, PGD, Mode decomposition, Nonlinear reduced modeling

Agathe Reille 1; Nicolas Hascoet 1; Chady Ghnatios 2; Amine Ammar 3; Elias Cueto 4; Jean Louis Duval 5; Francisco Chinesta 1; Roland Keunings 6

1 ESI Group Chair @ PIMM, Arts et Métiers Institute of Technology, CNRS, CNAM, HESAM University, 151, boulevard de l'Hôpital, 75013 Paris, France
2 Notre Dame University – Louaize , P.O. Box 72, Zouk Mikael, Zouk Mosbeh, Lebanon
3 ESI Group Chair @ LAMPA, Arts et Métiers ParisTech, 2, boulevard du Ronceray, BP 93525, 49035 Angers cedex 01, France
4 ESI Group Chair @ I3A, University of Zaragoza, Maria de Luna, s.n., 50018 Zaragoza, Spain
5 ESI Group, Bâtiment Seville, 3bis, rue Saarinen, 50468 Rungis, France
6 ICTEAM, Université catholique de Louvain, av. Georges Lemaître, 4, B-1348 Louvain-la-Neuve, Belgium
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     author = {Agathe Reille and Nicolas Hascoet and Chady Ghnatios and Amine Ammar and Elias Cueto and Jean Louis Duval and Francisco Chinesta and Roland Keunings},
     title = {Incremental dynamic mode decomposition: {A} reduced-model learner operating at the low-data limit},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {780--792},
     publisher = {Elsevier},
     volume = {347},
     number = {11},
     year = {2019},
     doi = {10.1016/j.crme.2019.11.003},
     language = {en},
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Agathe Reille; Nicolas Hascoet; Chady Ghnatios; Amine Ammar; Elias Cueto; Jean Louis Duval; Francisco Chinesta; Roland Keunings. Incremental dynamic mode decomposition: A reduced-model learner operating at the low-data limit. Comptes Rendus. Mécanique, Volume 347 (2019) no. 11, pp. 780-792. doi : 10.1016/j.crme.2019.11.003. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2019.11.003/

[1] K. Karhunen Über lineare methoden in der wahrscheinlichkeitsrechnung, Ann. Acad. Sci. Fennicae, Ser. Al. Math. Phys., Volume 37 (1946)

[2] M.M. Loève Probability Theory, The University Series in Higher Mathematics, Van Nostrand, Princeton, NJ, 1963

[3] M. Meyer; H.G. Matthies Efficient model reduction in non-linear dynamics using the Karhunen-Loève expansion and dual-weighted-residual methods, Comput. Mech., Volume 31 (2003) no. 1–2, pp. 179-191

[4] D. Ryckelynck; F. Chinesta; E. Cueto; A. Ammar On the a priori model reduction: overview and recent developments, Arch. Comput. Methods Eng., Volume 12 (2006) no. 1, pp. 91-128

[5] S. Chaturantabut; D.C. Sorensen Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput., Volume 32 ( September 2010 ), pp. 2737-2764

[6] D. Ryckelynck Hyper-reduction of mechanical models involving internal variables, Int. J. Numer. Methods Eng., Volume 77 (2008) no. 1, pp. 75-89

[7] P. Ladeveze Nonlinear Computational Structural Mechanics, Springer, N.Y., 1999

[8] A. Ammar; B. Mokdad; F. Chinesta; R. Keunings A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids, J. Non-Newton. Fluid Mech., Volume 139 (2006), pp. 153-176

[9] A. Ammar; B. Mokdad; F. Chinesta; R. Keunings A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. Part II: transient simulation using space-time separated representations, J. Non-Newton. Fluid Mech., Volume 144 (2007), pp. 98-121

[10] F. Chinesta; R. Keunings; A. Leygue The Proper Generalized Decomposition for Advanced Numerical Simulations, Springer International Publishing, Switzerland, 2014

[11] E. Cueto; D. González; I. Alfaro Proper Generalized Decompositions: An Introduction to Computer Implementation with Matlab, Springer Briefs in Applied Sciences and Technology, Springer International Publishing, 2016

[12] B. Bognet; A. Leygue; F. Chinesta Separated representations of 3d elastic solutions in shell geometries, Adv. Model. Simul. Eng. Sci., Volume 1 (2014) no. 4

[13] F. Chinesta; A. Ammar; E. Cueto Recent advances in the use of the proper generalized decomposition for solving multidimensional models, Arch. Comput. Methods Eng., Volume 17 (2010) no. 4, pp. 327-350

[14] F. Chinesta; A. Leygue; F. Bordeu; J.V. Aguado; E. Cueto; D. Gonzalez; I. Alfaro; A. Ammar; A. Huerta PGD-based computational vademecum for efficient design, optimization and control, Arch. Comput. Methods Eng., Volume 20 (2013) no. 1, pp. 31-59

[15] P. Díez; S. Zlotnik; A. Huerta Generalized parametric solutions in Stokes flow, Comput. Methods Appl. Mech. Eng., Volume 326 (2017), pp. 223-240

[16] F. Chinesta; A. Ammar; E. Cueto On the use of proper generalized decompositions for solving the multidimensional chemical master equation, Eur. J. Comput. Mech. (Rev. Eur. Méc. Numér.), Volume 19 (2010) no. 1–3, pp. 53-64

[17] D. Gonzalez; E. Cueto; F. Chinesta Real-time direct integration of reduced solid dynamics equations, Int. J. Numer. Methods Eng., Volume 99 (2014) no. 9, pp. 633-653

[18] R. Ibañez; D. Borzacchiello; J. Vicente Aguado; E. Abisset-Chavanne; E. Cueto; P. Ladeveze; F. Chinesta Data-driven non-linear elasticity: constitutive manifold construction and problem discretization, Comput. Mech., Volume 60 ( Nov. 2017 ) no. 5, pp. 813-826

[19] T. Kirchdoerfer; M. Ortiz Data-driven computational mechanics, Comput. Methods Appl. Mech. Eng., Volume 304 (2016), pp. 81-101

[20] R. Ibanez; E. Abisset-Chavanne; J. Vicente Aguado; D. Gonzalez; E. Cueto; F. Chinesta A manifold learning approach to data-driven computational elasticity and inelasticity, Arch. Comput. Methods Eng., Volume 25 (2018) no. 1, pp. 47-57

[21] E. Lopez; D. Gonzalez; J.V. Aguado; E. Abisset-Chavanne; E. Cueto; C. Binetruy; F. Chinesta A manifold learning approach for integrated computational materials engineering, Arch. Comput. Methods Eng., Volume 25 ( January 2018 ) no. 1, pp. 59-68 | DOI

[22] Y. Kevrekidis; G. Samaey Equation-free modeling, Scholarpedia, Volume 5 (2010) no. 9, p. 4847

[23] Ch. Heyberger; P.-A. Boucard; D. Neron A rational strategy for the resolution of parametrized problems in the {PGD} framework, Comput. Methods Appl. Mech. Eng., Volume 259 (2013) no. 1, pp. 40-49

[24] G. Rozza Fundamentals of reduced basis method for problems governed by parametrized pdes and applications (P. Ladeveze; F. Chinesta, eds.), CISM Lectures Notes “Separated Representation and PGD Based Model Reduction: Fundamentals and Applications”, Springer Verlag, 2014

[25] D. Amsallem; Ch. Farhat An interpolation method for adapting reduced-order models and application to aeroelasticity, AIAA J., Volume 46 (2008), pp. 1803-1813

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