Skip to main content
SpringerLink
Log in

On data-driven identification: Is automatically discovering equations of motion from data a Chimera?

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, a simple inverted pendulum is considered in order to discover, or extract, its dynamic equation from experimental data acquired during proper motion. This textbook case study achieved both in numerical simulations and with an off-the-shelf hardware reveals structural deficiencies in algorithms pretending to distill physics from data. In short, the outcome is that the obtained equations are not reliable and thus the model is practically equivalent to a black-box one. The data appropriateness is checked against a model-based identification. Is automatically discovering equations of motion from data then a Chimera?

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Data availability

The manuscript has no associated data.

References

  1. Ljung, L.: System Identification: Theory for the User. Prentice Hall PTR, Upper Saddle River, NJ (1999)

    MATH  Google Scholar 

  2. Quaranta, G., Lacarbonara, W., Masri, S.F.: A review on computational intelligence for identification of nonlinear dynamical systems. Nonlinear Dyn. 99(2), 1709 (2020)

    Article  MATH  Google Scholar 

  3. Kutz, J.N., Brunton, S.L., Brunton, B.W., Proctor, J.L.: Dynamic mode decomposition: data-driven modeling of complex systems, SIAM, (2016)

  4. Bai, Z., Kaiser, E., Proctor, J.L., Kutz, J.N., Brunton, S.L.: Dynamic mode decomposition for compressive system identification. AIAA J. 58(2), 561 (2020)

    Article  Google Scholar 

  5. Juang, J.N., Pappa, R.S.: An eigensystem realization algorithm for modal parameter identification and model reduction. J. Guid. Control Dyn. 8(5), 620 (1985)

    Article  MATH  Google Scholar 

  6. Tu, J.H., Rowley, C.W., Luchtenburg, D.M., Brunton, S.L., Kutz, J.N.: arXiv preprint arXiv:1312.0041 (2013)

  7. Berger, E., Sastuba, M., Vogt, D., Jung, B., Ben Amor, H.: Estimation of perturbations in robotic behavior using dynamic mode decomposition. Adv. Robot. 29(5), 331 (2015)

    Article  Google Scholar 

  8. Folkestad, C., Pastor, D., Mezic, I., Mohr, R., Fonoberova, M., Burdick, J.: In 2020 American Control Conference, IEEE, (2020), pp. 3906–3913

  9. Bruder, D., Fu, X., Gillespie, R.B., Remy, C.D., Vasudevan, R.: Data-driven control of soft robots using Koopman operator theory. IEEE Trans. Robot. 37(3), 948 (2021)

    Article  Google Scholar 

  10. Crutchfield, J.P., McNamara, B.: Equations of motion from a data series. Complex Syst. 1(417–452), 121 (1987)

    MathSciNet  MATH  Google Scholar 

  11. Samek, W., Montavon, G., Vedaldi, A., Hansen, L.K., Müller, K.R.: Explainable AI: interpreting, explaining and visualizing deep learning, vol. 11700, Springer Nature, (2019)

  12. Vinuesa, R., Sirmacek, B.: Interpretable deep-learning models to help achieve the sustainable development goals. Nat. Mach. Intell. 3(11), 926 (2021)

    Article  Google Scholar 

  13. Schmidt, M., Lipson, H.: Distilling free-form natural laws from experimental data. Science 324(5923), 81 (2009)

    Article  Google Scholar 

  14. AAVV. Eureqa https://www.creativemachineslab.com/eureqa.html (2021)

  15. Schmidt, M., Lipson, H.: Supporting online material for distilling free-form natural laws from experimental data. Science 324(5923), 81 (2009)

    Article  Google Scholar 

  16. Hillar, C., Sommer, F.: Comment on the article “distilling free-form natural laws from experimental data” (2012)

  17. Brunton, S.L., Proctor, J.L., Kutz, J.N.: Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl. Acad. Sci. 113(15), 3932 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Yeung, E., Kundu, S., Hodas, N.: in 2019 American Control Conference (ACC), IEEE, (2019), pp. 4832–4839

  19. Cranmer, M.D., Sanchez-Gonzalez, A., Battaglia, P.W., Xu, R., Cranmer, K., Spergel, D.N., Ho, S.: CoRR arXiv:2006.11287 (2020)

  20. Gelß, P., Klus, S., Eisert, J., Schütte, C.: Multidimensional approximation of nonlinear dynamical systems, J. Comput. Nonlinear Dyn. 14(6) (2019)

  21. Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, Cambridge (2019)

    Book  MATH  Google Scholar 

  22. Shea, D.E., Brunton, S.L., Kutz, J.N.: Sindy-bvp: sparse identification of nonlinear dynamics for boundary value problems. Phys. Rev. Res. 3(2), 023255 (2021)

    Article  Google Scholar 

  23. Billard, A., Mirrazavi, S., Figueroa, N.: Learning for Adaptive and Reactive Robot Control: A Dynamical Systems Approach. MIT Press, Cambridge (2022)

    Google Scholar 

  24. Williams, M.O., Rowley, C.W., Kevrekidis, I.G.: A kernel-based approach to data-driven Koopman spectral analysis (2015)

  25. Cortiella, A., Park, K.C., Doostan, A.: Sparse identification of nonlinear dynamical systems via reweighted l1-regularized least squares. Comput. Methods Appl. Mech. Eng. 376, 113620 (2021)

    Article  MATH  Google Scholar 

  26. Antonelli, G., Caccavale, F., Chiacchio, P.: A systematic procedure for the identification of dynamic parameters of robot manipulators. Robotica 17, 427 (1999)

    Article  Google Scholar 

  27. Sahoo, S., Lampert, C., Martius, G.: in International Conference on Machine Learning PMLR, (2018), pp. 4442–4450

  28. Im, J., Rizzo, C.B., de Barros, F.P., Masri, S.F.: Application of genetic programming for model-free identification of nonlinear multi-physics systems. Nonlinear Dyn. 104(2), 1781 (2021)

    Article  Google Scholar 

  29. Skulstad, R., Li, G., Fossen, T.I., Wang, T., Zhang, H.: A co-operative hybrid model for ship motion prediction. Model. Identif. Control 42(1), 17 (2021). https://doi.org/10.4173/mic.2021.1.2

    Article  Google Scholar 

  30. Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686 (2019)

  31. Karniadakis, G.E., Kevrekidis, I.G., Lu, L., Perdikaris, P., Wang, S., Yang, L.: Physics-informed machine learning. Nat. Rev. Phys. 3(6), 422 (2021)

    Article  Google Scholar 

  32. Golluccio, G., Gillini, G., Marino, A., Antonelli, G.: IEEE Robotics & Automation Magazine (2020). https://doi.org/10.1109/MRA.2020.3004149

  33. Klayman, J.: Varieties of confirmation bias. Psychol. Learn. Motiv. 32, 385 (1995)

    Article  Google Scholar 

  34. Popper, K.: Conjectures and refutations: the growth of scientific knowledge (1963)

Download references

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. Department of Electrical and Information Engineering, University of Cassino and Southern Lazio, Cassino, Italy

    Gianluca Antonelli, Stefano Chiaverini & Paolo Di Lillo

Authors
  1. Gianluca Antonelli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefano Chiaverini
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Paolo Di Lillo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paolo Di Lillo.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest. The authors have no relevant financial or non-financial interests to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix

Modified STLS technique

The STLS technique implemented is a version slight modified with respect to [17]:

figure a

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Antonelli, G., Chiaverini, S. & Di Lillo, P. On data-driven identification: Is automatically discovering equations of motion from data a Chimera?. Nonlinear Dyn 111, 6487–6498 (2023). https://doi.org/10.1007/s11071-022-08192-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-08192-x

Keywords

Search

Navigation