A Study on Data-Driven Identification and Representation of Nonlinear Dynamical Systems with a Physics-Integrated Deep Learning Approach: Koopman Operators and Nonlinear Normal Modes

Rostamijavanani, Abdolvahhab; Li, Shanwu; Yang, Yongchao

doi:10.1007/978-3-031-04086-3_30

A Study on Data-Driven Identification and Representation of Nonlinear Dynamical Systems with a Physics-Integrated Deep Learning Approach: Koopman Operators and Nonlinear Normal Modes

Abdolvahhab Rostamijavanani⁵,
Shanwu Li⁵ &
Yongchao Yang⁵

Conference paper
First Online: 29 July 2022

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Part of the Conference Proceedings of the Society for Experimental Mechanics Series book series (CPSEMS)

Abstract

In this study, we investigate the performance of data-driven Koopman operator and nonlinear normal mode (NNM) on predictive modeling of nonlinear dynamical systems using a physics-constrained deep learning approach. Two physics-constrained deep autoencoders are proposed: one to identify eigenfunction of Koopman operator and the other to identify nonlinear modal transformation function of NNMs, respectively, from the response data only. Koopman operator aims to linearize nonlinear dynamics at the cost of infinite dimensions, while NNM aims to capture invariance properties of dynamics with the same dimension as original system. We conduct numerical study on nonlinear systems with various levels of nonlinearity and observe that NNM representation has higher accuracy than Koopman autoencoder with same dimension of feature coordinates.

Keywords

Nonlinear normal modes
Koopman operators
Data-driven system identification
Deep learning
Modal analysis

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Reference

Lusch, B., Kutz, J.N., Brunton, S.L.: Deep learning for universal linear embeddings of nonlinear dynamics. Nat. Commun. 9(1), 1–10 (2018)
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Authors and Affiliations

Department of Mechanical Engineering-Engineering Mechanics, Michigan Technological University, Houghton, MI, USA
Abdolvahhab Rostamijavanani, Shanwu Li & Yongchao Yang

Authors

Abdolvahhab Rostamijavanani
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Shanwu Li
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Yongchao Yang
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Corresponding author

Correspondence to Yongchao Yang .

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Editors and Affiliations

Department of Mechanical Engineering, Rice University, Houston, TX, USA
Dr. Matthew R.W. Brake
Department of Mechanical Engineering, Imperial College London, London, UK
Ludovic Renson
Sandia National Laboratories, Albuquerque, NM, USA
Robert J. Kuether
Department of Mechanical and Process Engineering, ETH Zürich, Zürich, Switzerland
Dr. Paolo Tiso

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Rostamijavanani, A., Li, S., Yang, Y. (2023). A Study on Data-Driven Identification and Representation of Nonlinear Dynamical Systems with a Physics-Integrated Deep Learning Approach: Koopman Operators and Nonlinear Normal Modes. In: Brake, M.R., Renson, L., Kuether, R.J., Tiso, P. (eds) Nonlinear Structures & Systems, Volume 1. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-031-04086-3_30

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DOI: https://doi.org/10.1007/978-3-031-04086-3_30
Published: 29 July 2022
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-04085-6
Online ISBN: 978-3-031-04086-3
eBook Packages: EngineeringEngineering (R0)