[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Data-driven solitons dynamics and parameters discovery in the generalized nonlinear dispersive mKdV-type equation via deep neural networks learning

  • Research
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, we study the dynamics of data-driven solutions and identify the unknown parameters of the nonlinear dispersive modified KdV-type (mKdV-type) equation based on physics-informed neural networks (PINNs). Specifically, we learn the soliton solution, the combination of a soliton and an anti-soliton solution, the combination of two solitons and one anti-soliton solution, and the combination of two solitons and two anti-solitons solution of the mKdV-type equation by two different transformations. Meanwhile, we learn the data-driven kink solution, peakon solution, and periodic solution using the PINNs method. By utilizing image simulations, we conduct a detailed analysis of the nonlinear dynamical behaviors of the aforementioned solutions in the spatial-temporal domain. Our findings indicate that the PINNs method solves the mKdV-type equation with relative errors of \(\mathcal {O}(10^{-3})\) or \(\mathcal {O}(10^{-4})\) for the multi-soliton and kink solutions, respectively, while relative errors for the peakon and periodic solutions reach \(\mathcal {O}(10^{-2})\). In addition, the tanh function has the best training effect by comparing eight common activation functions (e.g., \(\textrm{ReLU}(\textbf{x})\), \(\textrm{ELU}(\textbf{x})\), \(\textrm{SiLU}(\textbf{x})\), \(\textrm{sigmoid}(\textbf{x})\), \(\textrm{swish}(\textbf{x})\), \(\textrm{sin}(\textbf{x})\), \(\textrm{cos}(\textbf{x})\), and \(\textrm{tanh}(\textbf{x})\)). For the inverse problem, we invert the soliton solution and identify the unknown parameters with relative errors reaching \(\mathcal {O}(10^{-2})\) or \(\mathcal {O}(10^{-3})\). Furthermore, we discover that adding appropriate noise to the initial condition enhances the robustness of the model. Our research results are crucial for understanding phenomena such as interactions in travelling waves, aiding in the discovery of physical processes and dynamic features in nonlinear systems, which have significant implications in fields such as nonlinear optics and plasma physics.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Data availability

This work has no associated data.

References

  1. Antontsev, S.N., Díaz, J.I., Shmarev, S.: Energy methods for free boundary problems: applications to nonlinear PDEs and fluid mechanics. progress in nonlinear differential equations and their applications. Appl. Mech. Rev. 55, 74–75 (2002)

    Google Scholar 

  2. Helal, M.A.: Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics. Chaos Solitons Fractals 13, 1917–1929 (2002)

    MathSciNet  Google Scholar 

  3. Liu, G.R.: An overview on meshfree methods: for computational solid mechanics. Int. J. Comput. Methods 13, 1630001 (2016)

    MathSciNet  Google Scholar 

  4. Haghighat, E. Raissi, M., Moure, A.: A deep learning framework for solution and discovery in solid mechanics: linear elasticity. arXiv:2003.02751 (2020)

  5. Adomian, G.: A new approach to nonlinear partial differential equations. J. Math. Anal. Appl. 102, 420–434 (1984)

    MathSciNet  Google Scholar 

  6. Dunne, G.V.: Functional determinants in quantum field theory. J. Phys. A Math. Theor. 41, 304006 (2008)

    MathSciNet  Google Scholar 

  7. Abdullaev, F.K., Galimzyanov, R.M., Brtka, M., et al.: Soliton dynamics at an interface between a uniform medium and a nonlinear optical lattice. Phys. Rev. E 79, 056220 (2009)

    Google Scholar 

  8. Garmire, E.: Nonlinear optics in daily life. Opt. Exp. 21, 30532–30544 (2013)

    Google Scholar 

  9. Novikov, D.S.: Elastic scattering theory and transport in graphene. Phys. Rev. B 76, 245435 (2007)

    Google Scholar 

  10. Keeley, N., Alamanos, N., Kemper, K.W.: Elastic scattering and reactions of light exotic beams. Prog. Part. Nucl. Phys. 63, 396–447 (2009)

    Google Scholar 

  11. Allegretto, W., Papini, D., Forti, M.: Common asymptotic behavior of solutions and almost periodicity for discontinuous, delayed, and impulsive neural networks. IEEE Trans. Neural Netw. 21, 1110–1125 (2010)

    Google Scholar 

  12. Tascan, F., Bekir, A., Koparan, M.: Travelling wave solutions of nonlinear evolution equations by using the first integral method. Commun. Nonlinear Sci. Numer. Simul. 14, 1810–1815 (2009)

    Google Scholar 

  13. Magnago, F.H., Abur, A.: Fault location using wavelets. IEEE Trans. Power Delivery 13, 1475–1480 (1998)

    Google Scholar 

  14. Ye, Z.H.: Vascular tissue differentiation and pattern formation in plants. Annu. Rev. Plant Biol. 53, 183–202 (2002)

    Google Scholar 

  15. Biswas, A., Fessak, M., Johnson, S.: Optical soliton perturbation in non-Kerr law media: traveling wave solution. Opt. Laser Technol. 44, 263–268 (2012)

    Google Scholar 

  16. Li, J., Zhang, L.: Bifurcations of traveling wave solutions in generalized Pochhammer–Chree equation. Chaos Solitons Fractals 14, 581–593 (2002)

    MathSciNet  Google Scholar 

  17. Shen, J., Xu, W., Li, W.: Bifurcations of travelling wave solutions in a new integrable equation with peakon and compactons. Chaos Solitons Fractals 27, 413–425 (2006)

    MathSciNet  Google Scholar 

  18. Battye, R.A., Sutcliffe, P.M.: Knots as stable soliton solutions in a three-dimensional classical field theory. Phys. Rev. Lett. 81, 4798 (1998)

    MathSciNet  Google Scholar 

  19. Cheemaa, N., Seadawy, A.R., Chen, S.: More general families of exact solitary wave solutions of the nonlinear Schrödinger equation with their applications in nonlinear optics. J. Phys. Chem. Lett. 133, 1–9 (2018)

    Google Scholar 

  20. Ullah, M.S., Roshid, H.O., Ali, M.Z.: New wave behaviors and stability analysis for the (2+1)-dimensional Zoomeron model. Opt. Quantum Electron. 56, 240 (2024)

    Google Scholar 

  21. Ullah, M.S., Mostafa, M., Ali, M.Z.: Soliton solutions for the Zoomeron model applying three analytical techniques. PLoS ONE 18, e0283594 (2023)

    Google Scholar 

  22. Ullah, M.S.: Interaction solution to the (3+1)-D negative-order KdV first structure. Partial Differ. Equ. Appl. Math. 8, 100566 (2023)

    Google Scholar 

  23. Ullah, M.S., Ahmed, O., Mahbub, M.A.: Collision phenomena between lump and kink wave solutions to a (3+1)-dimensional Jimbo-Miwa-like model. Partial Differ. Equ. Appl. Math. 5, 100324 (2022)

    Google Scholar 

  24. Wazwaz, A.M.: New solitons and kink solutions for the Gardner equation. Commun. Nonlinear Sci. Numer. Simul. 12, 1395–1404 (2007)

    MathSciNet  Google Scholar 

  25. Wazwaz, A.M.: The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations. Appl. Math. Comput. 188, 1467–1475 (2007)

    MathSciNet  Google Scholar 

  26. Degasperis, A., Holm, D.D., Hone, A.N.W.: A new integrable equation with peakon solutions. Theor. Math. Phys. 133, 1463–1474 (2002)

    MathSciNet  Google Scholar 

  27. Geng, X., Xue, B.: A three-component generalization of Camassa–Holm equation with N-peakon solutions. Adv. Math. 226, 827–839 (2011)

    MathSciNet  Google Scholar 

  28. Daubechies, I.: The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inf. Theory 36, 961–1005 (1990)

    MathSciNet  Google Scholar 

  29. Kumar, S., Niwas, M.: Exploring lump soliton solutions and wave interactions using new inverse \((G^{^{\prime }}/G)\)-expansion approach: applications to the (2+ 1)-dimensional nonlinear Heisenberg ferromagnetic spin chain equation. Nonlinear Dyn. 111, 20257–20273 (2023)

    Google Scholar 

  30. Kumar, S., Niwas, M.: Analyzing multi-peak and lump solutions of the variable-coefficient Boiti–Leon–Manna–Pempinelli equation: a comparative study of the Lie classical method and unified method with applications. Nonlinear Dyn. 111, 22457–22475 (2023)

    Google Scholar 

  31. Niwas, M., Kumar, S.: Multi-peakons, lumps, and other solitons solutions for the (2+1)-dimensional generalized Benjamin-Ono equation: an inverse \((G^{^{\prime }}/G)\)-expansion method and real-world applications. Nonlinear Dyn. 111, 22499–22512 (2023)

    Google Scholar 

  32. Tian, Y.: Artificial intelligence image recognition method based on convolutional neural network algorithm. IEEE Access 8, 125731–125744 (2020)

    Google Scholar 

  33. Dilsizian, S.E., Siegel, E.L.: Artificial intelligence in medicine and cardiac imaging: harnessing big data and advanced computing to provide personalized medical diagnosis and treatment. Curr. Cardiol. Rep. 16, 1–8 (2014)

    Google Scholar 

  34. Boulesteix, A.L., Wright, M.: Artificial intelligence in genomics. Hum. Genet. 141, 1449–1450 (2022)

    Google Scholar 

  35. Raissi, M., Perdikaris, P., Karniadakis, G. E.: Physics informed deep learning (Part I): data-driven solutions of nonlinear partial differential equations. arXiv preprint arXiv: 1711.10561 (2017)

  36. Raissi, M., Perdikaris, P., Karniadakis, G. E.: Physics informed deep learning (Part II): data-driven discovery of nonlinear partial differential equations. arXiv: 1711.10566 (2017)

  37. Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686–707 (2019)

    MathSciNet  Google Scholar 

  38. Lu, L., Meng, X., Mao, Z.: DeepXDE: a deep learning library for solving differential equations. SIAM Rev. 63, 208–228 (2021)

    MathSciNet  Google Scholar 

  39. Yu, J., Lu, L., Meng, X.: Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems. Comput. Methods Appl. Mech. Eng. 393, 114823 (2022)

    MathSciNet  Google Scholar 

  40. Lu, L., Pestourie, R., Yao, W.: Physics-informed neural networks with hard constraints for inverse design. SIAM J. Sci. Comput. 43, 1105–1132 (2021)

    MathSciNet  Google Scholar 

  41. Lu, L., Jin, P., Pang, G.: Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nat. Mach. Intell. 3, 218–229 (2021)

    Google Scholar 

  42. Li, J., Chen, Y.: Solving second-order nonlinear evolution partial differential equations using deep learning. Commun. Theor. Phys. 72, 105005 (2020)

    MathSciNet  Google Scholar 

  43. Li, J., Chen, Y.: A deep learning method for solving third-order nonlinear evolution equations. Commun. Theor. Phys. 72, 115003 (2020)

    MathSciNet  Google Scholar 

  44. Pu, J.C., Li, J., Chen, Y.: Soliton, breather, and rogue wave solutions for solving the nonlinear Schrödinger equation using a deep learning method with physical constraints. Chin. Phys. B 30, 060202 (2021)

    Google Scholar 

  45. Peng, W.Q., Pu, J.C., Chen, Y.: PINN deep learning method for the Chen–Lee–Liu equation: Rogue wave on the periodic background. Commun. Nonlinear Sci. Numer. Simul. 105, 106067 (2022)

    MathSciNet  Google Scholar 

  46. Pu, J., Li, J., Chen, Y.: Solving localized wave solutions of the derivative nonlinear Schrödinger equation using an improved PINN method. Nonlinear Dyn. 105, 1723–1739 (2021)

    Google Scholar 

  47. Lin, S., Chen, Y.: A two-stage physics-informed neural network method based on conserved quantities and applications in localized wave solutions. J. Comput. Phys. 457, 111053 (2022)

    MathSciNet  Google Scholar 

  48. Miao, Z.W., Chen, Y.: Physics-informed neural networks method in high-dimensional integrable systems. Mod. Phys. Lett. B 36, 2150531 (2022)

    MathSciNet  Google Scholar 

  49. Wang, L., Yan, Z.: Data-driven peakon and periodic peakon solutions and parameter discovery of some nonlinear dispersive equations via deep learning. Physica D 428, 133037 (2021)

    MathSciNet  Google Scholar 

  50. Wang, X., Wu, Z., Han, W., Yan, Z.: Deep learning data-driven multi-soliton dynamics and parameters discovery for the fifth-order Kaup–Kuperschmidt equation. Physica D 454, 133862 (2023)

    MathSciNet  Google Scholar 

  51. Zhong, M., Gong, S., Tian, S.F., Yan, Z.: Data-driven rogue waves and parameters discovery in nearly integrable PT-symmetric Gross–Pitaevskii equations via PINNs deep learning. Physica D 439, 133430 (2022)

    MathSciNet  Google Scholar 

  52. Wang, L., Yan, Z.: Data-driven rogue waves and parameter discovery in the defocusing nonlinear Schrödinger equation with a potential using the PINN deep learning. Phys. Lett. A 404, 127408 (2021)

    Google Scholar 

  53. Zhou, Z., Yan, Z.: Solving forward and inverse problems of the logarithmic nonlinear Schrödinger equation with PT-symmetric harmonic potential via deep learning. Phys. Lett. A 387, 127010 (2021)

    Google Scholar 

  54. Song, J., Yan, Z.: Deep learning soliton dynamics and complex potentials recognition for 1D and 2D PT-symmetric saturable nonlinear Schrödinger equations. Physica D 448, 133729 (2023)

    Google Scholar 

  55. Li, J., Chen, J., Li, B.: Gradient-optimized physics-informed neural networks (GOPINNs): a deep learning method for solving the complex modified KdV equation. Nonlinear Dyn. 107, 781–792 (2022)

    Google Scholar 

  56. Li, J., Li, B.: Mix-training physics-informed neural networks for the rogue waves of nonlinear Schrödinger equation. Chaos Solitons Fractals 164, 112712 (2022)

    Google Scholar 

  57. Wu, G.Z., Fang, Y., Wang, Y.Y., Wu, G.C., Dai, C.Q.: Predicting the dynamic process and model parameters of the vector optical solitons in birefringent fibers via the modified PINN. Chaos Solitons Fractals 152, 111393 (2021)

    MathSciNet  Google Scholar 

  58. Fang, Y., Wu, G.Z., Kudryashov, N.A., Wang, Y.Y., Dai, C.Q.: Data-driven soliton solutions and model parameters of nonlinear wave models via the conservation-law constrained neural network method. Chaos Solitons Fractals 158, 112118 (2022)

    MathSciNet  Google Scholar 

  59. Wen, X.K., Wu, G.Z., Liu, W., Dai, C.Q.: Dynamics of diverse data-driven solitons for the three-component coupled nonlinear Schrödinger model by the MPS-PINN method. Nonlinear Dyn. 109, 3041–3050 (2022)

  60. Cui, S., Wang, Z., Han, J.: A deep learning method for solving high-order nonlinear soliton equations. Commun. Theor. Phys. 74, 075007 (2022)

    MathSciNet  Google Scholar 

  61. Yang, X., Wang, Z.: Solving Benjamin-Ono equation via gradient balanced PINNs approach. Eur. Phys. J. Plus 137, 864 (2022)

    Google Scholar 

  62. Mishra, S., Molinaro, R.: Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs. IMA J. Numer. Anal. 42, 981–1022 (2022)

    MathSciNet  Google Scholar 

  63. Wight, C.L., Zhao, J.: Solving Allen–Cahn and Cahn–Hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv: 2007.04542 (2020)

  64. Geng, X., Xue, B.: Soliton solutions and quasi-periodicsolutions of modified Korteweg–de Vries type equations. J. Math. Phys. 51, 063516 (2010)

    MathSciNet  Google Scholar 

  65. Wazwaz, A.M.: A modified KdV-type equation that admits a variety of travelling wave solutions: kinks, solitons, peakons and cuspons. Phys. Scr. 86, 045501 (2012)

    Google Scholar 

  66. Ullah, M.S., Roshid, H.O., Ali, M.Z.: New wave behaviors of the Fokas–Lenells model using three integration techniques. PLoS ONE 18, e0291071 (2023)

    Google Scholar 

  67. Ullah, M.S., Baleanu, D., Ali, M.Z.: Novel dynamics of the Zoomeron model via different analytical methods. Chaos Solitons Fractals 174, 113856 (2023)

    MathSciNet  Google Scholar 

Download references

Funding

The work was supported by the National Natural Science Foundation of China (Nos. 11925108, 12226322, and 12275017).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan, 250353, China

    Xiaoli Wang, Wenjing Han & Zekang Wu

  2. Shandong Artificial Intelligence Institute, Qilu University of Technology (Shandong Academy of Sciences), Jinan, 250353, China

    Xiaoli Wang, Wenjing Han & Zekang Wu

  3. KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Zhenya Yan

  4. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Zhenya Yan

Authors
  1. Xiaoli Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wenjing Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zekang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhenya Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

X.W., W.H. and Z.W. wrote the main manuscript text. All authors reviewed the manuscript.

Corresponding author

Correspondence to Zhenya Yan.

Ethics declarations

Conflict of interest

The authors declare no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, X., Han, W., Wu, Z. et al. Data-driven solitons dynamics and parameters discovery in the generalized nonlinear dispersive mKdV-type equation via deep neural networks learning. Nonlinear Dyn 112, 7433–7458 (2024). https://doi.org/10.1007/s11071-024-09454-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-024-09454-6

Keywords

Search

Navigation