Papers
Topics
Authors
Recent
Search
2000 character limit reached

Phase autoencoder for limit-cycle oscillators

Published 28 Feb 2024 in nlin.AO, cs.LG, and nlin.CD | (2403.06992v1)

Abstract: We present a phase autoencoder that encodes the asymptotic phase of a limit-cycle oscillator, a fundamental quantity characterizing its synchronization dynamics. This autoencoder is trained in such a way that its latent variables directly represent the asymptotic phase of the oscillator. The trained autoencoder can perform two functions without relying on the mathematical model of the oscillator: first, it can evaluate the asymptotic phase and phase sensitivity function of the oscillator; second, it can reconstruct the oscillator state on the limit cycle in the original space from the phase value as an input. Using several examples of limit-cycle oscillators, we demonstrate that the asymptotic phase and phase sensitivity function can be estimated only from time-series data by the trained autoencoder. We also present a simple method for globally synchronizing two oscillators as an application of the trained autoencoder.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (68)
  1. A. T. Winfree, The geometry of biological time, Vol. 2 (Springer, 1980).
  2. A. Pikovsky, M. Rosenblum, J. Kurths, and A. Synchronization, “A universal concept in nonlinear sciences,” Self 2, 3 (2001).
  3. Y. Kuramoto, “Chemical oscillations, waves and turbulence. mineola,”  (2003).
  4. T. Stankovski, V. Ticcinelli, P. V. McClintock, and A. Stefanovska, “Coupling functions in networks of oscillators,” New Journal of Physics 17 (2015), 10.1088/1367-2630/17/3/035002.
  5. T. Stankovski, V. Ticcinelli, P. V. McClintock, and A. Stefanovska, “Neural cross-frequency coupling functions,” Frontiers in Systems Neuroscience 11 (2017), 10.3389/fnsys.2017.00033.
  6. L. Borgius, H. Nishimaru, V. Caldeira, Y. Kunugise, P. Löw, R. Reig, S. Itohara, T. Iwasato, and O. Kiehn, “Spinal glutamatergic neurons defined by epha4 signaling are essential components of normal locomotor circuits,” Journal of Neuroscience 34 (2014), 10.1523/JNEUROSCI.4992-13.2014.
  7. J. J. Collins and I. N. Stewart, “Coupled nonlinear oscillators and the symmetries of animal gaits,” Journal of Nonlinear Science 3 (1993), 10.1007/BF02429870.
  8. R. Kobayashi, H. Nishimaru, and H. Nishijo, “Estimation of excitatory and inhibitory synaptic conductance variations in motoneurons during locomotor-like rhythmic activity,” Neuroscience 335 (2016), 10.1016/j.neuroscience.2016.08.027.
  9. T. Funato, Y. Yamamoto, S. Aoi, T. Imai, T. Aoyagi, N. Tomita, and K. Tsuchiya, “Evaluation of the phase-dependent rhythm control of human walking using phase response curves,” PLoS Computational Biology 12 (2016), 10.1371/journal.pcbi.1004950.
  10. B. Kralemann, M. Frühwirth, A. Pikovsky, M. Rosenblum, T. Kenner, J. Schaefer, and M. Moser, “In vivo cardiac phase response curve elucidates human respiratory heart rate variability,” Nature Communications 4 (2013), 10.1038/ncomms3418.
  11. M. Garcia, A. Chatterjee, A. Ruina, and M. Coleman, “The simplest walking model: Stability, complexity, and scaling,” Journal of Biomechanical Engineering 120 (1998), 10.1115/1.2798313.
  12. M. Rohden, A. Sorge, M. Timme, and D. Witthaut, “Self-organized synchronization in decentralized power grids,” Physical review letters 109, 064101 (2012).
  13. H. Nakao, “Phase reduction approach to synchronisation of nonlinear oscillators,” Contemporary Physics 57, 188–214 (2016).
  14. G. B. Ermentrout and D. H. Terman, “Mathematical foundations of neuroscience,”  (2010).
  15. B. Monga, D. Wilson, T. Matchen, and J. Moehlis, “Phase reduction and phase-based optimal control for biological systems: a tutorial,” Biological cybernetics 113, 11–46 (2019).
  16. Y. Kuramoto and H. Nakao, “On the concept of dynamical reduction: the case of coupled oscillators,” Philosophical Transactions of the Royal Society A 377, 20190041 (2019).
  17. B. Ermentrout, Y. Park, and D. Wilson, “Recent advances in coupled oscillator theory,” Philosophical Transactions of the Royal Society A 377, 20190092 (2019).
  18. L. Glass and M. C. Mackey, From clocks to chaos: The rhythms of life (Princeton University Press, 1988).
  19. S. Shirasaka, W. Kurebayashi, and H. Nakao, “Phase-amplitude reduction of limit cycling systems,” The Koopman Operator in Systems and Control: Concepts, Methodologies, and Applications , 383–417 (2020).
  20. D. Wilson, “An adaptive phase-amplitude reduction framework without o(ϵitalic-ϵ\epsilonitalic_ϵ) constraints on inputs,” SIAM Journal on Applied Dynamical Systems 21, 204–230 (2022).
  21. K. Taira and H. Nakao, “Phase-response analysis of synchronization for periodic flows,” Journal of Fluid Mechanics 846, R2 (2018).
  22. F. C. Hoppensteadt and E. M. Izhikevich, Weakly connected neural networks, Vol. 126 (Springer Science & Business Media, 1997).
  23. K. C. Wedgwood, K. K. Lin, R. Thul, and S. Coombes, “Phase-amplitude descriptions of neural oscillator models,” The Journal of Mathematical Neuroscience 3, 1–22 (2013).
  24. D. Wilson and J. Moehlis, “Isostable reduction of periodic orbits,” Physical Review E 94, 052213 (2016).
  25. A. Mauroy and I. Mezić, “Global stability analysis using the eigenfunctions of the Koopman operator,” IEEE Transactions on Automatic Control 61, 3356–3369 (2016).
  26. S. Shirasaka, W. Kurebayashi, and H. Nakao, “Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems,” Chaos: An Interdisciplinary Journal of Nonlinear Science 27 (2017).
  27. A. Mauroy and I. Mezić, “Global computation of phase-amplitude reduction for limit-cycle dynamics,” Chaos: An Interdisciplinary Journal of Nonlinear Science 28 (2018).
  28. K. Kotani, Y. Ogawa, S. Shirasaka, A. Akao, Y. Jimbo, and H. Nakao, “Nonlinear phase-amplitude reduction of delay-induced oscillations,” Physical Review Research 2, 033106 (2020).
  29. H. Nakao, “Phase and amplitude description of complex oscillatory patterns in reaction-diffusion systems,” in Physics of Biological Oscillators: New Insights into Non-Equilibrium and Non-Autonomous Systems (Springer, 2021) pp. 11–27.
  30. S. Takata, Y. Kato, and H. Nakao, “Fast optimal entrainment of limit-cycle oscillators by strong periodic inputs via phase-amplitude reduction and floquet theory,” Chaos: An Interdisciplinary Journal of Nonlinear Science 31 (2021).
  31. B. Ermentrout, “Type I membranes, phase resetting curves, and synchrony,” Neural computation 8, 979–1001 (1996).
  32. K. Ota, M. Nomura, and T. Aoyagi, “Weighted spike-triggered average of a fluctuating stimulus yielding the phase response curve,” Physical review letters 103, 024101 (2009).
  33. T. Netoff, M. A. Schwemmer, and T. J. Lewis, “Experimentally estimating phase response curves of neurons: theoretical and practical issues,” Phase Response Curves in Neuroscience: Theory, Experiment, and Analysis , 95–129 (2012).
  34. T. Imai, K. Ota, and T. Aoyagi, “Robust measurements of phase response curves realized via multicycle weighted spike-triggered averages,” Journal of the Physical Society of Japan 86, 024009 (2017).
  35. P. J. Schmid, “Dynamic mode decomposition of numerical and experimental data,” Journal of fluid mechanics 656, 5–28 (2010).
  36. M. O. Williams, I. G. Kevrekidis, and C. W. Rowley, “A data–driven approximation of the Koopman operator: Extending dynamic mode decomposition,” Journal of Nonlinear Science 25, 1307–1346 (2015).
  37. B. Lusch, J. N. Kutz, and S. L. Brunton, “Deep learning for universal linear embeddings of nonlinear dynamics,” Nature communications 9, 4950 (2018).
  38. N. Namura, S. Takata, K. Yamaguchi, R. Kobayashi, and H. Nakao, “Estimating asymptotic phase and amplitude functions of limit-cycle oscillators from time series data,” Physical Review E 106, 14204 (2022).
  39. B. Kralemann, L. Cimponeriu, M. Rosenblum, A. Pikovsky, and R. Mrowka, “Phase dynamics of coupled oscillators reconstructed from data,” Physical Review E 77, 066205 (2008).
  40. A. Mauroy, I. Mezić, and J. Moehlis, “Isostables, isochrons, and Koopman spectrum for the action–angle representation of stable fixed point dynamics,” Physica D: Nonlinear Phenomena 261, 19–30 (2013).
  41. K. Fukami, K. Taira, and H. Nakao, “Data-driven transient lift attenuation for extreme vortex gust-airfoil interactions,” in preparation .
  42. A. Mardt, L. Pasquali, H. Wu, and F. Noé, “Vampnets for deep learning of molecular kinetics,” Nature communications 9, 5 (2018).
  43. C. Wehmeyer and F. Noé, “Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics,” The Journal of chemical physics 148 (2018).
  44. S. E. Otto and C. W. Rowley, “Linearly recurrent autoencoder networks for learning dynamics,” SIAM Journal on Applied Dynamical Systems 18, 558–593 (2019).
  45. N. Takeishi, Y. Kawahara, and T. Yairi, “Learning Koopman invariant subspaces for dynamic mode decomposition,” Advances in neural information processing systems 30 (2017).
  46. E. Yeung, S. Kundu, and N. Hodas, “Learning deep neural network representations for Koopman operators of nonlinear dynamical systems,”  (2019) pp. 4832–4839.
  47. K. Champion, B. Lusch, J. N. Kutz, and S. L. Brunton, “Data-driven discovery of coordinates and governing equations,” Proceedings of the National Academy of Sciences 116, 22445–22451 (2019).
  48. O. Azencot, N. B. Erichson, V. Lin, and M. Mahoney, “Forecasting sequential data using consistent Koopman autoencoders,”  (2020) pp. 475–485.
  49. Y. Li, H. He, J. Wu, D. Katabi, and A. Torralba, “Learning compositional Koopman operators for model-based control,” arXiv preprint arXiv:1910.08264  (2019).
  50. N. Berman, I. Naiman, and O. Azencot, “Multifactor sequential disentanglement via structured Koopman autoencoders,” arXiv preprint arXiv:2303.17264  (2023).
  51. M. Han, J. Euler-Rolle, and R. K. Katzschmann, “Desko: Stability-assured robust control with a deep stochastic Koopman operator,”  (2021).
  52. G. E. Hinton and R. R. Salakhutdinov, “Reducing the dimensionality of data with neural networks,” science 313, 504–507 (2006).
  53. K. Fukami and K. Taira, “Grasping extreme aerodynamics on a low-dimensional manifold,” arXiv preprint arXiv:2305.08024  (2023).
  54. A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, “Automatic differentiation in pytorch,”   (2017).
  55. V. Nair and G. E. Hinton, “Rectified linear units improve restricted boltzmann machines,” in Proceedings of the 27th international conference on machine learning (ICML-10) (2010) pp. 807–814.
  56. S. Ioffe and C. Szegedy, “Batch normalization: Accelerating deep network training by reducing internal covariate shift,” in International conference on machine learning (pmlr, 2015) pp. 448–456.
  57. A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, et al., “Pytorch: An imperative style, high-performance deep learning library,” Advances in neural information processing systems 32 (2019).
  58. J. T. Stuart, “On the non-linear mechanics of wave disturbances in stable and unstable parallel flows part 1. the basic behaviour in plane poiseuille flow,” Journal of Fluid Mechanics 9, 353–370 (1960).
  59. L. Landau, “Comptes rendues dokl,” in Acad. Sci. USSR, Vol. 44 (1944) p. 311.
  60. A. L. Hodgkin and A. F. Huxley, “A quantitative description of membrane current and its application to conduction and excitation in nerve,” The Journal of physiology 117, 500 (1952).
  61. H. Nakao, S. Yasui, M. Ota, K. Arai, and Y. Kawamura, “Phase reduction and synchronization of a network of coupled dynamical elements exhibiting collective oscillations,” Chaos: An Interdisciplinary Journal of Nonlinear Science 28 (2018).
  62. P. Mircheski, J. Zhu, and H. Nakao, “Phase-amplitude reduction and optimal phase locking of collectively oscillating networks,” Chaos: An Interdisciplinary Journal of Nonlinear Science 33 (2023).
  63. M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,” Journal of Computational physics 378, 686–707 (2019).
  64. G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, “Physics-informed machine learning,” Nature Reviews Physics 3, 422–440 (2021).
  65. S. Greydanus, M. Dzamba, and J. Yosinski, “Hamiltonian neural networks,” Advances in neural information processing systems 32 (2019).
  66. Z. Chen, M. Feng, J. Yan, and H. Zha, “Learning neural hamiltonian dynamics: a methodological overview,” arXiv preprint arXiv:2203.00128  (2022).
  67. P. Toth, D. J. Rezende, A. Jaegle, S. Racanière, A. Botev, and I. Higgins, “Hamiltonian generative networks,” arXiv preprint arXiv:1909.13789  (2019).
  68. A. Daigavane, A. Kosmala, M. Cranmer, T. Smidt, and S. Ho, “Learning integrable dynamics with action-angle networks,” arXiv preprint arXiv:2211.15338  (2022).
Citations (3)
View on Semantic Scholar

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We found no open problems mentioned in this paper.

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Tweets

Sign up for free to view the 1 tweet with 5 likes about this paper.

Sign Up for Free