Skip to main content
SpringerLink
Log in

AutoKoopman: A Toolbox for Automated System Identification via Koopman Operator Linearization

  • Conference paper
  • First Online:
Automated Technology for Verification and Analysis (ATVA 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14216))

Abstract

While Koopman operator linearization has brought many advances for prediction, control, and verification of dynamical systems, its main disadvantage is that the quality of the resulting model heavily depends on the correct tuning of hyper-parameters such as the number of observables. Our toolbox is a Python package that automates learning accurate models in a Koopman linearized representation with low effort, offering several tuning strategies to optimize the hyper-parameters associated with the Koopman operator techniques automatically. supports discrete as well as continuous-time models and implements all major types of observables, which are polynomials, random Fourier features, and neural networks. As we demonstrate on several benchmarks, our toolbox is able to automatically identify very accurate dynamic models for symbolic, black-box, as well as real systems. AutoKoopman is available at https://github.com/EthanJamesLew/AutoKoopman and on PyPI as .

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 74.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alford-Lago, D.J., Curtis, C.W., Ihler, A.T., Issan, O.: Deep learning enhanced dynamic mode decomposition. Chaos: Interdisc. J. Nonlinear Sci. 32(3), 033116 (2022)

    Google Scholar 

  2. Bak, S., et al.: Reachability of black-box nonlinear systems after Koopman operator linearization. In: Proceedings of the International Conference on Analysis and Design of Hybrid Systems, pp. 253–258 (2021)

    Google Scholar 

  3. Bak, S., et al.: Reachability of Koopman linearized systems using random Fourier feature observables and polynomial zonotope refinement. In: Shoham, S., Vizel, Y. (eds.) CAV 2022. LNCS, vol. 13371, pp. 490–510. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-13185-1_24

    Chapter  Google Scholar 

  4. Bergstra, J., Bengio, Y.: Random search for hyper-parameter optimization. J. Mach. Learn. Res. 13(2), 281–305 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Bevanda, P., Sosnowski, S., Hirche, S.: Koopman operator dynamical models: learning, analysis and control. Annu. Rev. Control. 52, 197–212 (2021)

    Article  MathSciNet  Google Scholar 

  6. 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), 3932–3937 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Carleman, T.: Application de la théorie des équations intégrales linéaires aux systèmes d’équations différentielles non linéaires. Acta Math. 59, 63–87 (1932)

    Article  MathSciNet  MATH  Google Scholar 

  8. Champion, K., Lusch, B., Kutz, J.N., Brunton, S.L.: Data-driven discovery of coordinates and governing equations. Proc. Natl. Acad. Sci. 116(45), 22445–22451 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, X.: Reachability Analysis of Non-Linear Hybrid Systems Using Taylor Models. Ph.D. thesis, RWTH Aachen University (2015)

    Google Scholar 

  10. DeGennaro, A.M., Urban, N.M.: Scalable extended dynamic mode decomposition using random kernel approximation. SIAM J. Sci. Comput. 41(3), 1482–1499 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Demo, N., Tezzele, M., Rozza, G.: PyDMD: python dynamic mode decomposition. J. Open Source Softw. 3(22), 530 (2018)

    Article  Google Scholar 

  12. FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J . 1(6), 445–466 (1961)

    Article  Google Scholar 

  13. Geretti, L., et al.: ARCH-COMP20 category report: continuous and hybrid systems with nonlinear dynamics. In: Proceedings of the International Workshop on Applied Verification of Continuous and Hybrid Systems, pp. 49–75 (2020)

    Google Scholar 

  14. Geretti, L., et al.: ARCH-COMP21 category report: continuous and hybrid systems with nonlinear dynamics. In: Proceedings of the International Workshop on Applied Verification of Continuous and Hybrid Systems, pp. 32–54 (2021)

    Google Scholar 

  15. Heidlauf, P., Collins, A., Bolender, M., Bak, S.: Verification challenges in F-16 ground collision avoidance and other automated maneuvers. In: Proceedings of the International Workshop on Applied Verification of Continuous and Hybrid Systems, pp. 208–217 (2018)

    Google Scholar 

  16. Kochdumper, N., et al.: Establishing reachset conformance for the formal analysis of analog circuits. In: Proceedings of the Asia and South Pacific Design Automation Conference, pp. 199–204 (2020)

    Google Scholar 

  17. Koopman, B.O.: Hamiltonian systems and transformation in Hilbert space. Proc. Natl. Acad. Sci. 17(5), 315–318 (1931)

    Article  MATH  Google Scholar 

  18. Korda, M., Mezić, I.: Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control. Automatica 93, 149–160 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kutz, J.N., Fu, X., Brunton, S.L.: Multiresolution dynamic mode decomposition. SIAM J. Appl. Dyn. Syst. 15(2), 713–735 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, Y., et al.: Learning compositional Koopman operators for model-based control. In: Proceedings of the International Conference on Learning Representations (2020)

    Google Scholar 

  21. Liu, S.B., Althoff, M.: Reachset conformance of forward dynamic models for the formal analysis of robots. In: Proceedings of the International Conference on Intelligent Robots and Systems, pp. 370–376 (2018)

    Google Scholar 

  22. Lusch, B., Kutz, J.N., Brunton, S.L.: Deep learning for universal linear embeddings of nonlinear dynamics. Nat. Commun. 9, 4950 (2018)

    Article  Google Scholar 

  23. Maïga, M., Ramdani, N., Travé-Massuyè, L., Combastel, C.: A comprehensive method for reachability analysis of uncertain nonlinear hybrid systems. Trans. Autom. Control 61(9), 2341–2356 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Meiss, J.D.: Differential Dynamical Systems. SIAM (2007)

    Google Scholar 

  25. Mezić, I.: Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41(1), 309–325 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mezić, I.: Analysis of fluid flows via spectral properties of the Koopman operator. Annu. Rev. Fluid Mech. 45, 357–378 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Mezić, I., Banaszuk, A.: Comparison of systems with complex behavior. Physica D 197(1–2), 101–133 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Michoski, C., Milosavljević, M., Oliver, T., Hatch, D.R.: Solving differential equations using deep neural networks. Neurocomputing 399, 193–212 (2020)

    Article  Google Scholar 

  29. Morton, J., Jameson, A., Kochenderfer, M.J., Witherden, F.: Deep dynamical modeling and control of unsteady fluid flows. In: Advances in Neural Information Processing Systems, vol. 31 (2018)

    Google Scholar 

  30. O’Kelly, M., Zheng, H., Karthik, D., Mangharam, R.: F1tenth: an open-source evaluation environment for continuous control and reinforcement learning. Proc. Mach. Learn. Res. 123, 77–89 (2020)

    Google Scholar 

  31. Otto, S.E., Rowley, C.W.: Koopman operators for estimation and control of dynamical systems. Ann. Rev. Control, Robot. Auton. Syst. 4, 59–87 (2021)

    Article  Google Scholar 

  32. Pan, S., Kaiser, E., Kutz, N., Brunton, S.: PyKoopman: a python package for data-driven approximation of the Koopman operator. Bull. Am. Phys. Soc. (2022)

    Google Scholar 

  33. Proctor, J.L., Brunton, S.L., Kutz, J.N.: Dynamic mode decomposition with control. SIAM J. Appl. Dyn. Syst. 15(1), 142–161 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  34. Rahimi, A., Recht, B.: Random features for large-scale kernel machines. In: Proceedings of the International Conference on Neural Information Processing Systems, pp. 1177–1184 (2007)

    Google Scholar 

  35. Rowley, C.W., et al.: Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Sankaranarayanan, S.: Automatic abstraction of non-linear systems using change of bases transformations. In: Proceedings of the International Conference on Hybrid Systems: Computation and Control, pp. 143–152 (2011)

    Google Scholar 

  37. Schmid, M.R., Maehlisch, M., Dickmann, J., Wuensche, H.J.: Dynamic level of detail 3D occupancy grids for automotive use. In: Proceedings of the IEEE Intelligent Vehicles Symposium, pp. 269–274 (2010)

    Google Scholar 

  38. Shahriari, B., et al.: Taking the human out of the loop: a review of Bayesian optimization. Proc. IEEE 104(1), 148–175 (2015)

    Article  Google Scholar 

  39. de Silva, B., et al.: PySINDy: a python package for the sparse identification of nonlinear dynamical systems from data. J. Open Source Softw. 5(49), 2014 (2020)

    Article  Google Scholar 

  40. Stanford, A.L., Tanner, J.M.: Physics for Students of Science and Engineering. Academic Press, Cambridge (2014)

    Google Scholar 

  41. Takeishi, N., Kawahara, Y., Tabei, Y., Yairi, T.: Bayesian dynamic mode decomposition. In: Proceedings of the AAAI Conference on Artificial Intelligence, pp. 2814–2821 (2017)

    Google Scholar 

  42. Takeishi, N., Kawahara, Y., Yairi, T.: Learning Koopman invariant subspaces for dynamic mode decomposition (2017)

    Google Scholar 

  43. Williams, M.O., Kevrekidis, I.G., Rowley, C.W.: A data-driven approximation of the Koopman operator: extending dynamic mode decomposition. J. Nonlinear Sci. 25(6), 1307–1346 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  44. Williams, M.O., Rowley, C.W., Kevrekidis, I.G.: A kernel-based approach to data-driven Koopman spectral analysis. J. Comput. Dyn. 2(2), 247–265 (2014)

    Article  MATH  Google Scholar 

  45. Xiao, Y., et al.: Deep neural networks with Koopman operators for modeling and control of autonomous vehicles. Trans. Intelli. Veh. 8, 135–146 (2022). IEEE Early Access

    Google Scholar 

  46. Yeung, E., Kundu, S., Hodas, N.: Learning deep neural network representations for Koopman operators of nonlinear dynamical systems. In: Proceedings of the American Control Conference, pp. 4832–4839 (2019)

    Google Scholar 

Download references

Acknowledgements

This material is based upon work supported by the Air Force Office of Scientific Research and the Office of Naval Research under award numbers FA9550-19-1-0288, FA9550-21-1-0121, FA9550-23-1-0066 and N00014-22-1-2156. The material of Galois, Inc. is based upon work supported by the Air Force Research Laboratory (AFRL) and DARPA under Contract No. FA8750-20-C-0534. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the United States Air Force, DARPA, or the United States Navy.

Author information

Authors and Affiliations

  1. Galois Inc., Portland, OR, USA

    Ethan Lew

  2. School of Computing, Newcastle University, Newcastle Upon Tyne, UK

    Abdelrahman Hekal, Kostiantyn Potomkin & Sergiy Bogomolov

  3. Department of Computer Science, Stony Brook University, New York, NY, USA

    Niklas Kochdumper & Stanley Bak

  4. Air Force Research Laboratory, Wright-Patterson Air Force Base, Dayton, OH, USA

    Brandon Hencey

Authors
  1. Ethan Lew
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abdelrahman Hekal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kostiantyn Potomkin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Niklas Kochdumper
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Brandon Hencey
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Stanley Bak
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Sergiy Bogomolov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ethan Lew .

Editor information

Editors and Affiliations

  1. Université Sorbonne Paris Nord, Villetaneuse, France

    Étienne André

  2. Singapore Management University, Singapore, Singapore

    Jun Sun

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Lew, E. et al. (2023). AutoKoopman: A Toolbox for Automated System Identification via Koopman Operator Linearization. In: André, É., Sun, J. (eds) Automated Technology for Verification and Analysis. ATVA 2023. Lecture Notes in Computer Science, vol 14216. Springer, Cham. https://doi.org/10.1007/978-3-031-45332-8_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-45332-8_12

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-45331-1

  • Online ISBN: 978-3-031-45332-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation