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      A Multidimensional Data-Driven Sparse Identification Technique: The Sparse Proper Generalized Decomposition

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          Abstract

          Sparse model identification by means of data is especially cumbersome if the sought dynamics live in a high dimensional space. This usually involves the need for large amount of data, unfeasible in such a high dimensional settings. This well-known phenomenon, coined as the curse of dimensionality, is here overcome by means of the use of separate representations. We present a technique based on the same principles of the Proper Generalized Decomposition that enables the identification of complex laws in the low-data limit. We provide examples on the performance of the technique in up to ten dimensions.

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          The partition of unity finite element method: Basic theory and applications

          J.M. Melenk, I. Babuška (1996)
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            THE PARTITION OF UNITY METHOD

            I. BABUŠKA, J. Melenk (1997)
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              Discovering governing equations from data: Sparse identification of nonlinear dynamical systems

              Steven L. Brunton, Joshua Proctor, J. Nathan Kutz (2015)
              The ability to discover physical laws and governing equations from data is one of humankind's greatest intellectual achievements. A quantitative understanding of dynamic constraints and balances in nature has facilitated rapid development of knowledge and enabled advanced technological achievements, including aircraft, combustion engines, satellites, and electrical power. In this work, we combine sparsity-promoting techniques and machine learning with nonlinear dynamical systems to discover governing physical equations from measurement data. The only assumption about the structure of the model is that there are only a few important terms that govern the dynamics, so that the equations are sparse in the space of possible functions; this assumption holds for many physical systems. In particular, we use sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data. The resulting models are parsimonious, balancing model complexity with descriptive ability while avoiding overfitting. We demonstrate the algorithm on a wide range of problems, from simple canonical systems, including linear and nonlinear oscillators and the chaotic Lorenz system, to the fluid vortex shedding behind an obstacle. The fluid example illustrates the ability of this method to discover the underlying dynamics of a system that took experts in the community nearly 30 years to resolve. We also show that this method generalizes to parameterized, time-varying, or externally forced systems.
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                Author and article information

                Journal
                Title: Complexity
                Abbreviated Title: Complexity
                Publisher: Hindawi Limited
                ISSN (Print): 1076-2787
                ISSN (Electronic): 1099-0526
                Publication date Created: November 01 2018
                Publication date (Print): November 01 2018
                Volume: 2018
                Pages: 1-11
                Affiliations
                [1 ]ESI Chair, ENSAM ParisTech. 151, bvd. de l'Hôpital, F-75013 Paris, France
                [2 ]LAMPA, ENSAM ParisTech. 2, bvd. de Ronceray. F-49035 Angers, France
                [3 ]Aragon Institute of Engineering Research, Universidad de Zaragoza. Maria de Luna, s.n. E-50018 Zaragoza, Spain
                [4 ]Laboratori de Càlcul Numèric, Universitat Politècnica de Catalunya. Jordi Girona 1-3, E-08034 Barcelona, Spain
                [5 ]ESI Group. Parc Icade, Immeuble le Seville, 3 bis, Saarinen, CP 50229, 94528, Rungis Cedex, France
                Article
                DOI: 10.1155/2018/5608286
                SO-VID: fdbbd724-6472-4068-8303-33637b23864d
                Copyright © © 2018
                License:

                http://creativecommons.org/licenses/by/4.0/

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