Skip to main content
SpringerLink
Log in

Continuous probabilistic solution to the transient self-oscillation under stochastic forcing: a PINN approach

  • Original Article
  • Published:
Journal of Mechanical Science and Technology Aims and scope Submit manuscript

Abstract

The dynamics of self-oscillators under stochastic excitation are of great interest for scientific and engineering purposes. In this study, we propose a method for computing continuous probabilistic solutions to stochastic oscillators developing into self-oscillation. We employ a physics-informed neural network to solve the transient Fokker-Planck equation that corresponds to the Langevin equation characterizing a stochastic self-oscillator. The proposed framework is parametrically demonstrated by varying linear and nonlinear coefficients, noise amplitude, and oscillation frequency. Furthermore, the robustness of the present method is verified with the time-marching numerical solution, which shows great agreement with the PINN-based solution.

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

Similar content being viewed by others

References

  1. C. Chen, D. H. Zanette, J. R. Guest, D. A. Czaplewski and D. López, Self-sustained micromechanical oscillator with linear feedback, Phys. Rev.Lett., 117(1) (2016) 017203.

    Article  Google Scholar 

  2. A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 12 (2001).

    Book  MATH  Google Scholar 

  3. M. Lee, K. T. Kim, V. Gupta and L. K. B. Li, System identification and early warning detection of thermoacoustic oscillations in a turbulent combustor using its noise-induced dynamics, Proc. Combust. Inst., 38(4) (2021) 6025–6033.

    Article  Google Scholar 

  4. A. Jenkins, Self-oscillation, Phys. Rep., 525(2) (2013) 167–222.

    Article  MathSciNet  MATH  Google Scholar 

  5. S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, CRC Press (2018).

  6. C. Audoin and B. Guinot, The Measurement of Time: Time, Frequency and the Atomic Clock, Cambridge University Press (2001).

  7. H. Kawaguchi, Optical bistability and chaos in a semiconductor laser with a saturable absorber, Appl. Phys. Lett., 45(12) (1984) 1264–1266.

    Article  Google Scholar 

  8. M. Lee, Y. Guan, V. Gupta and L. K. B. Li, Input-output system identification of a thermoacoustic oscillator near a Hopf bifurcation using only fixed-point data, Phys. Rev. E, 101(1) (2020) 013102.

    Article  Google Scholar 

  9. M. Lee, S. Yoon, J. Kim, Y. Wang, K. Lee, F. C. Park and C. H. Sohn, Classification of impinging jet flames using convolutional neural network with transfer learning, J. Mech. Sci. Tech., 36(3) (2022) 1547–1556.

    Article  Google Scholar 

  10. M. Lee, Y. Zhu, L. K. B. Li and V. Gupta, System identification of a low-density jet via its noise-induced dynamics, J. Fluid Mech., 862 (2019) 200–215.

    Article  MathSciNet  MATH  Google Scholar 

  11. W. Horsthemke, Noise Induced Transitions, Springer (1984).

  12. M. Lee, System identification near a Hopf bifurcation via the noise-induced dynamics in the fixed-point regime, Ph.D. Thesis, The Hong Kong University of Science and Technology (2020).

  13. R. Benzi, A. Sutera and A. Vulpiani, The mechanism of stochastic resonance, J. Phys. A, 14(11) (1981) L453.

    Article  MathSciNet  Google Scholar 

  14. L. Gammaitoni, P. Hänggi, P. Jung and F. Marchesoni, Stochastic resonance, Rev. Mod. Phys., 70 (1998) 223–287.

    Article  Google Scholar 

  15. A. S. Pikovsky and J. Kurths, Coherence resonance in a noise-driven excitable system, Phys. Rev. Lett., 78 (1997) 775–778.

    Article  MathSciNet  MATH  Google Scholar 

  16. L. Kabiraj, R. Steinert, A. Saurabh and C. O. Paschereit, Coherence resonance in a thermoacoustic system, Phys. Rev. E, 92 (2015) 042909.

    Article  Google Scholar 

  17. V. Jegadeesan and R. I. Sujith, Experimental investigation of noise induced triggering in thermoacoustic systems, P. Combust. Inst., 34(2) (2013) 3175–3183.

    Article  Google Scholar 

  18. J. M. R. Parrondo, C. van den Broeck, J. Buceta and F. J. de la Rubia, Noise-induced spatial patterns, Physica A, 224(1) (1996) 153–161.

    Article  Google Scholar 

  19. H. Risken, Fokker-Planck Equation, Springer (1984).

  20. 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 (2019) 686–707.

    Article  MathSciNet  MATH  Google Scholar 

  21. G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang and L. Yang, Physics-informed machine learning, Nat. Rev. Phys., 3(6) (2021) 422–440.

    Article  Google Scholar 

  22. Z. Hao, S. Liu, Y. Zhang, C. Ying, Y. Feng, H. Su and J. Zhu, Physics-informed machine learning: a survey on problems, methods and applications, arXiv:2211.08064 (2022).

  23. J. Han and A. Jentzen, Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Commun. Math. Stat., 5(4) (2017) 349–380.

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Sirignano and K. Spiliopoulos, DGM: a deep learning algorithm for solving partial differential equations, J. Comput. Phys., 375 (2018) 1339–1364.

    Article  MathSciNet  MATH  Google Scholar 

  25. H. Son, J. W. Jang, W. J. Han and H. J. Hwang, Sobolev training for physics informed neural networks, arXiv:2101. 08932 (2021).

  26. A. D. Jagtap, Z. Mao, N. Adams and G. E. Karniadakis, Physics-informed neural networks for inverse problems in supersonic flows, arXiv:2202.11821 (2022).

  27. L. Lu, R. Pestourie, W. Yao, Z. Wang, F. Verdugo and S. G. Johnson, Physics-informed neural networks with hard constraints for inverse design, SIAM J. Sci. Comput., 43(6) (2021) B1105–B1132.

    Article  MathSciNet  MATH  Google Scholar 

  28. Y. Chen, L. Lu, G. E. Karniadakis and L. Dal Negro, Physics-informed neural networks for inverse problems in nano-optics and metamaterials, Opt. Express, 28(8) (2020) 11618–11633.

    Article  Google Scholar 

  29. H. Jo, H. Son, H. J. Hwang and E. H. Kim, Deep neural network approach to forward-inverse problems, Netw. Heterog. Media, 15(2) (2020) 247.

    Article  MathSciNet  MATH  Google Scholar 

  30. L. Lu, X. Meng, Z. Mao and G. E. Karniadakis, DeepXDE: a deep learning library for solving differential equations, SIAM Rev., 63(1) (2021) 208–228.

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Kennedy and L. Chua, Van der Pol and chaos, IEEE Trans. Circuits Syst., 33(10) (1986) 974–980.

    Article  MathSciNet  Google Scholar 

  32. A. H. Nayfeh, Introduction to Perturbation Techniques, John Wiley, New York (1981).

    MATH  Google Scholar 

  33. J. Roberts and P. D. Spanos, Stochastic averaging: an approximate method of solving random vibration problems, Int. J. Nonlin. Mech., 21 (1986) 111–134.

    Article  MathSciNet  MATH  Google Scholar 

  34. W. Zhu and J. Yu, On the response of the van der Pol oscillator to white noise excitation, J. Sound Vib., 117(3) (1987) 421–431.

    Article  MathSciNet  MATH  Google Scholar 

  35. H. J. Hwang and H. Son, Lagrangian dual framework for conservative neural network solutions of kinetic equations, Kinet. Relat. Models, 15(4) (2022) 551.

    Article  MathSciNet  MATH  Google Scholar 

  36. H. J. Hwang, J. W. Jang, H. Jo and J. Y. Lee, Trend to equilibrium for the kinetic Fokker-Planck equation via the neural network approach, J. Comput. Phys., 419 (2020) 109665.

    Article  MathSciNet  MATH  Google Scholar 

  37. S. W. Cho, H. J. Hwang and H. Son, Traveling wave solutions of partial differential equations via neural networks, J. Sci. Comput., 89(1) (2021) 1–26.

    Article  MathSciNet  MATH  Google Scholar 

  38. X. Huang, H. Liu, B. Shi, Z. Wang, K. Yang, Y. Li, M. Wang, H. Chu, J. Zhou and F. Yu, A universal PINNs method for solving partial differential equations with a point source, Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence (IJCAI-22) (2022) 3839–3846.

  39. G. Xavier and Y. Bengio, Understanding the difficulty of training deep feedforward neural networks, Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics 2010 (2010) 249–256.

  40. D. P. Kingma and J. Ba, Adam: a method for stochastic optimization, arXiv:1412.6980 (2014).

  41. A. Paszke et al., Pytorch: an imperative style, high-performance deep learning library, Adv. Neural Inf. Process. Syst., 32 (2019).

  42. P. A. Markowich and C. Villani, On the trend to equilibrium for the fokker-planck equation: an interplay between physics and functional analysis, Mat. Contemp., 19 (2000) 1–29.

    MathSciNet  MATH  Google Scholar 

  43. M. Lee, D. Kim, J. Lee, Y. Kim and M. Yi, A data-driven approach for analyzing Hall thruster discharge instability leading to plasma blowoff, Acta Astronautica, 206 (2023) 1–8.

    Article  Google Scholar 

Download references

Acknowledgments

Hwijae Son was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. NRF-2022R1F1A1073732) and the research fund of Hanbat National University in 2022. Minwoo Lee was supported by NRF grant funded by MSIT (No. NRF-2021R1G1A1091278).

Author information

Authors and Affiliations

  1. Department of Artificial Intelligence Software, Hanbat National University, 125 Dongseodaero, Yuseong-gu, 34158, Daejeon, Korea

    Hwijae Son

  2. Department of Mechanical Engineering, Hanbat National University, 125 Dongseodaero, Yuseong-gu, 34158, Daejeon, Korea

    Minwoo Lee

Authors
  1. Hwijae Son
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Minwoo Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Minwoo Lee.

Additional information

Hwijae Son received his B.S. degree in mathematics from Hanyang University and Ph.D. degree from POSTECH in 2021. He is currently an Assistant Professor in the Department of Artificial Intelligence Software at Hanbat National University, Korea. His research interest is mainly focused on physics-informed machine learning and scientific computing.

Minwoo Lee received his B.S. and M.S. degrees in aerospace engineering from the Korea Advanced Institute of Science and Technology in 2012 and 2014, respectively, and a Ph.D. degree in mechanical engineering from the Hong Kong University of Science and Technology in 2020. He is currently an Assistant Professor in the Department of Mechanical Engineering at Hanbat National University, Korea. His major research interest is the data-driven analysis and low-order modeling of thermofluid systems.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Son, H., Lee, M. Continuous probabilistic solution to the transient self-oscillation under stochastic forcing: a PINN approach. J Mech Sci Technol 37, 3911–3918 (2023). https://doi.org/10.1007/s12206-023-0707-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12206-023-0707-z

Keywords

Search

Navigation