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Deep learning for universal linear embeddings of nonlinear dynamics

Author

Listed:
  • Bethany Lusch

    (University of Washington
    University of Washington)

  • J. Nathan Kutz

    (University of Washington)

  • Steven L. Brunton

    (University of Washington
    University of Washington)

Abstract

Identifying coordinate transformations that make strongly nonlinear dynamics approximately linear has the potential to enable nonlinear prediction, estimation, and control using linear theory. The Koopman operator is a leading data-driven embedding, and its eigenfunctions provide intrinsic coordinates that globally linearize the dynamics. However, identifying and representing these eigenfunctions has proven challenging. This work leverages deep learning to discover representations of Koopman eigenfunctions from data. Our network is parsimonious and interpretable by construction, embedding the dynamics on a low-dimensional manifold. We identify nonlinear coordinates on which the dynamics are globally linear using a modified auto-encoder. We also generalize Koopman representations to include a ubiquitous class of systems with continuous spectra. Our framework parametrizes the continuous frequency using an auxiliary network, enabling a compact and efficient embedding, while connecting our models to decades of asymptotics. Thus, we benefit from the power of deep learning, while retaining the physical interpretability of Koopman embeddings.

Suggested Citation

  • Bethany Lusch & J. Nathan Kutz & Steven L. Brunton, 2018. "Deep learning for universal linear embeddings of nonlinear dynamics," Nature Communications, Nature, vol. 9(1), pages 1-10, December.
  • Handle: RePEc:nat:natcom:v:9:y:2018:i:1:d:10.1038_s41467-018-07210-0
    DOI: 10.1038/s41467-018-07210-0
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    Cited by:

    1. Zequn Lin & Zhaofan Lu & Zengru Di & Ying Tang, 2024. "Learning noise-induced transitions by multi-scaling reservoir computing," Nature Communications, Nature, vol. 15(1), pages 1-10, December.
    2. Ding, Jiaqi & Zhao, Pu & Liu, Changjun & Wang, Xiaofang & Xie, Rong & Liu, Haitao, 2024. "From irregular to continuous: The deep Koopman model for time series forecasting of energy equipment," Applied Energy, Elsevier, vol. 364(C).
    3. Ma, Zhengjing & Mei, Gang, 2022. "A hybrid attention-based deep learning approach for wind power prediction," Applied Energy, Elsevier, vol. 323(C).
    4. Gong, Xun & Wang, Xiaozhe & Cao, Bo, 2023. "On data-driven modeling and control in modern power grids stability: Survey and perspective," Applied Energy, Elsevier, vol. 350(C).
    5. Joaquim Fernando Pinto da Costa & Manuel Cabral, 2022. "Statistical Methods with Applications in Data Mining: A Review of the Most Recent Works," Mathematics, MDPI, vol. 10(6), pages 1-22, March.
    6. Chen, Zhong & Chen, Xiaofang & Liu, Jinping & Cen, Lihui & Gui, Weihua, 2024. "Learning model predictive control of nonlinear systems with time-varying parameters using Koopman operator," Applied Mathematics and Computation, Elsevier, vol. 470(C).
    7. Cardoso, Ana Sofia & Renna, Francesco & Moreno-Llorca, Ricardo & Alcaraz-Segura, Domingo & Tabik, Siham & Ladle, Richard J. & Vaz, Ana Sofia, 2022. "Classifying the content of social media images to support cultural ecosystem service assessments using deep learning models," Ecosystem Services, Elsevier, vol. 54(C).
    8. Mandal, Ankit & Tiwari, Yash & Panigrahi, Prasanta K. & Pal, Mayukha, 2022. "Physics aware analytics for accurate state prediction of dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    9. Rijwan Khan, 2023. "Deep Learning System and It’s Automatic Testing: An Approach," Annals of Data Science, Springer, vol. 10(4), pages 1019-1033, August.
    10. Mattia Cenedese & Joar Axås & Bastian Bäuerlein & Kerstin Avila & George Haller, 2022. "Data-driven modeling and prediction of non-linearizable dynamics via spectral submanifolds," Nature Communications, Nature, vol. 13(1), pages 1-13, December.
    11. Konstantin Avchaciov & Marina P. Antoch & Ekaterina L. Andrianova & Andrei E. Tarkhov & Leonid I. Menshikov & Olga Burmistrova & Andrei V. Gudkov & Peter O. Fedichev, 2022. "Unsupervised learning of aging principles from longitudinal data," Nature Communications, Nature, vol. 13(1), pages 1-14, December.
    12. Zhao Chen & Yang Liu & Hao Sun, 2021. "Physics-informed learning of governing equations from scarce data," Nature Communications, Nature, vol. 12(1), pages 1-13, December.
    13. Gholaminejad, Tahereh & Khaki-Sedigh, Ali, 2022. "Stable deep Koopman model predictive control for solar parabolic-trough collector field," Renewable Energy, Elsevier, vol. 198(C), pages 492-504.
    14. Miao, Hua & Zhu, Wei & Dan, Yuanhong & Yu, Nanxiang, 2024. "Chaotic time series prediction based on multi-scale attention in a multi-agent environment," Chaos, Solitons & Fractals, Elsevier, vol. 183(C).

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