- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Dynamics creation through neural dynamical transfer learning

He Ma1,2,*, Qiyang Ge2,*, Yu Meng2, Celso Grebogi3, and Wei Lin1,2,4,5,†

  • *These authors contributed equally to this work.
  • †Contact author: wlin@fudan.edu.cn
PDF

Phys. Rev. Research 8, 033337 – Published 18 September, 2026

DOI: https://doi.org/10.1103/ccgv-hgjb

Abstract

Data-driven machine learning has established a robust foundation for reconstructing nonlinear dynamical systems from observations, primarily for the purposes of forecasting and control. However, most existing efforts focus on recovering specific observed dynamics rather than the generative synthesis of new ones. Inspired by image fusion and style transfer, we introduce a neural network framework termed neural dynamical transfer learning (NDTL) to create new systems with prescribed dynamics from pairs of parent nonlinear dynamical systems. By computing fundamental dynamical signatures, including the intrinsic dimension, the Kaplan-Yorke dimension, the invariant measure statistics, and the Lyapunov spectrum, we demonstrate that NDTL preserves key features inherited from the parent models while simultaneously generating novel dynamics. Beyond these validation examples, NDTL induces a criterion for dynamics classification, creates stable oscillatory coexistence in the Hastings-Powell food chain model, produces interpretable epidemiological models, and provides a chaotic source for image encryption.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (39)

  1. F. Takens, Detecting strange attractors in turbulence, in Dynamical Systems and Turbulence, Warwick 1980, edited by D. Rand and L.-S. Young (Springer, Berlin, 1981), pp. 366–381.
  2. J. F. Donges, Y. Zou, N. Marwan, and J. Kurths, The backbone of the climate network, Europhys. Lett. 87, 48007 (2009).
  3. G. Sugihara, R. May, H. Ye, C. Hao Hsieh, E. Deyle, M. Fogarty, and S. Munch, Detecting causality in complex ecosystems, Science 338, 496 (2012).
  4. W.-X. Wang, R. Yang, Y.-C. Lai, V. Kovanis, and C. Grebogi, Predicting catastrophes in nonlinear dynamical systems by compressive sensing, Phys. Rev. Lett. 106, 154101 (2011).
  5. W.-X. Wang, Y.-C. Lai, C. Grebogi, and J. Ye, Network reconstruction based on evolutionary-game data via compressive sensing, Phys. Rev. X 1, 021021 (2011).
  6. P. J. Schmid, Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech. 656, 5 (2010).
  7. 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).
  8. Q. Li, T. Wang, V. Roychowdhury, and M. K. Jawed, Metalearning generalizable dynamics from trajectories, Phys. Rev. Lett. 131, 067301 (2023).
  9. M. Raissi, Deep hidden physics models: Deep learning of nonlinear partial differential equations, J. Mach. Learn. Res. 19, 1 (2018).
  10. S. H. Rudy, S. L. Brunton, J. L. Proctor, and J. N. Kutz, Data-driven discovery of partial differential equations, Sci. Adv. 3, e1602614 (2017).
  11. 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).
  12. X. Li, Q. Zhu, C. Zhao, X. Duan, B. Zhao, X. Zhang, H. Ma, J. Sun, and W. Lin, Higher-order Granger reservoir computing: Simultaneously achieving scalable complex structures inference and accurate dynamics prediction, Nat. Commun. 15, 2506 (2024).
  13. Y. Yu, X. Si, C. Hu, and J. Zhang, A review of recurrent neural networks: LSTM cells and network architectures, Neural Comput. 31, 1235 (2019).
  14. A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, Attention is all you need, in Advances in Neural Information Processing Systems, edited by I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett (Curran Associates, New York, 2017), Vol. 30.
  15. A. Solera-Rico, C. Sanmiguel Vila, M. Gómez-López, Y. Wang, A. Almashjary, S. T. M. Dawson, and R. Vinuesa, β-variational autoencoders and transformers for reduced-order modelling of fluid flows, Nat. Commun. 15, 1361 (2024).
  16. G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, Physics-informed machine learning, Nat. Rev. Phys. 3, 422 (2021).
  17. M. Raissi, P. Perdikaris, and G. Karniadakis, 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).
  18. A. Choudhary, J. F. Lindner, E. G. Holliday, S. T. Miller, S. Sinha, and W. L. Ditto, Physics-enhanced neural networks learn order and chaos, Phys. Rev. E 101, 062207 (2020).
  19. J. Zhang, Q. Zhu, and W. Lin, Learning Hamiltonian neural Koopman operator and simultaneously sustaining and discovering conservation laws, Phys. Rev. Res. 6, L012031 (2024).
  20. W. Lu, H. Ma, W. Lin, and Q. Zhu, Interpretable functional Koopman learning with non-Markovian closure for spatiotemporal systems, in 43rd International Conference on Machine Learning (Microtome Publishing, Brookline, MA, 2026).
  21. R. T. Q. Chen, Y. Rubanova, J. Bettencourt, and D. K. Duvenaud, Neural ordinary differential equations, in Advances in Neural Information Processing Systems, edited by S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (Curran Associates, New York, 2018), Vol. 31.
  22. L. Böttcher, N. Antulov-Fantulin, and T. Asikis, AI Pontryagin or how artificial neural networks learn to control dynamical systems, Nat. Commun. 13, 333 (2022).
  23. Q. Zhu, Y. Guo, and W. Lin, Neural delay differential equations, in International Conference on Learning Representations (ICLR) (OpenReview.net, Vienna, Austria, 2021).
  24. W. Gilpin, Generative learning for nonlinear dynamics, Nat. Rev. Phys. 6, 194 (2024).
  25. H. Gao, S. Kaltenbach, and P. Koumoutsakos, Generative learning for forecasting the dynamics of high-dimensional complex systems, Nat. Commun. 15, 8904 (2024).
  26. D. Voina, S. Brunton, and J. N. Kutz, Deep generative modeling for identification of noisy, non-stationary dynamical systems, arXiv:2410.02079.
  27. J. Lai, A. Bao, and W. Gilpin, Panda: A pretrained forecast model for chaotic dynamics, in The Fourteenth International Conference on Learning Representations (OpenReview.net, Rio de Janeiro, Brazil, 2026).
  28. L. A. Gatys, A. S. Ecker, and M. Bethge, Image style transfer using convolutional neural networks, in Proceedings of the 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (IEEE, Piscataway, NJ, 2016), pp. 2414–2423.
  29. Y. Jing, Y. Yang, Z. Feng, J. Ye, Y. Yu, and M. Song, Neural style transfer: A review, IEEE Trans. Visual Comput. Graph. 26, 3365 (2020).
  30. C. Fefferman, S. Mitter, and H. Narayanan, Testing the manifold hypothesis, J. Am. Math. Soc. 29, 983 (2016).
  31. K. Cho, B. van Merriënboer, C. Gulcehre, D. Bahdanau, F. Bougares, H. Schwenk, and Y. Bengio, Learning phrase representations using RNN encoder–decoder for statistical machine translation, in Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), edited by A. Moschitti, B. Pang, and W. Daelemans (Association for Computational Linguistics, Doha, Qatar, 2014), pp. 1724–1734.
  32. J. Masci, U. Meier, D. Cireşan, and J. Schmidhuber, Stacked convolutional auto-encoders for hierarchical feature extraction, in Artificial Neural Networks and Machine Learning, edited by T. Honkela, W. Duch, M. Girolami, and S. Kaski (Springer, Berlin, 2011), pp. 52–59.
  33. S. Čelikovský and G. Chen, On the generalized Lorenz canonical form, Chaos, Soliton Fract. 26, 1271 (2005).
  34. W. Gilpin, Chaos as an interpretable benchmark for forecasting and data-driven modelling, in Proceedings of the Neural Information Processing Systems Track on Datasets and Benchmarks, edited by J. Vanschoren and S. Yeung (Curran Associates, Inc., Red Hook, NY, 2021), Vol. 1.
  35. A. Hastings and T. Powell, Chaos in a three-species food chain, Ecology 72, 896 (1991).
  36. M. Martcheva, An Introduction to Mathematical Epidemiology (Springer, New York, 2015), Vol. 61.
  37. N. C. Grassly and C. Fraser, Mathematical models of infectious disease transmission, Nat. Rev. Microbiol. 6, 477 (2008).
  38. H. Ma, Q. Ge, Y. Meng, C. Grebogi, and W. Lin, NDTL: Code for “Dynamics creation through neural dynamical transfer learning” (2025), https://github.com/967255/NDTL.
  39. J. W. Cooley, P. A. W. Lewis, and P. D. Welch, The fast Fourier transform and its applications, IEEE Trans. Educ. 12, 27 (1969).

Outline

Information

Phys. Rev. Research 8, 033337– Published 18 September, 2026
  • Received 24 June 2025
  • Revised 18 March 2026
  • Accepted 1 September 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation