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Prediction of soliton evolution and equation parameters for NLS–MB equation based on the phPINN algorithm

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Abstract

To enhance the precision and efficiency of result prediction, we proposed a parallel hard-constraint physics-informed neural networks (phPINN) by combining the parallel fully-connected neural network structure and the residual-based adaptive refinement method. We discussed the forward and inverse problems of the nonlinear Schrödinger–Maxwell–Bloch equation via the phPINN. In the forward problem, we predict five forms of soliton solutions and rogue wave dynamics under corresponding initial and boundary conditions; In the inverse problem, we predict the equation parameter using the training data with different noise intensities, initial values, and solution forms. The predicted parameters achieve a relative error of less than 1%. These results validate the effectiveness of the phPINN algorithm in solving forward and inverse problems of three-component coupled equations.

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Data availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

  1. 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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Zhang, E., Dao, M., Karniadakis, G.E., Suresh, S.: Analyses of internal structures and defects in materials using physics-informed neural networks. Sci. Adv. 8, eabk0644 (2022)

    Article  Google Scholar 

  3. Raissi, M., Yazdani, A., Karniadakis, G.E.: Hidden fluid mechanics: learning velocity and pressure fields from flow visualizations. Science 367, 1026–1030 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  4. Zhang, R.-F., Bilige, S.: Bilinear neural network method to obtain the exact analytical solutions of nonlinear partial differential equations and its application to p-gBKP equation. Nonlinear Dyn. 95, 3041–3048 (2019)

    Article  MATH  Google Scholar 

  5. Wu, G.Z., Fang, Y., Kudryashov, N.A., Wang, Y.Y., Dai, C.Q.: Prediction of optical solitons using an improved physics-informed neural network method with the conservation law constraint. Chaos Soliton Fract. 159, 112143 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  6. Zhu, B.W., Fang, Y., Liu, W., Dai, C.Q.: Predicting the dynamic process and model parameters of vector optical solitons under coupled higher-order effects via WL-tsPINN. Chaos Soliton Fract. 162, 112441 (2022)

    Article  MathSciNet  Google Scholar 

  7. Bo, W.-B., Wang, R.-R., Fang, Y., Wang, Y.-Y., Dai, C.-Q.: Prediction and dynamical evolution of multipole soliton families in fractional Schrodinger equation with the PT-symmetric potential and saturable nonlinearity. Nonlinear Dyn. 111, 1577–1588 (2023)

    Article  Google Scholar 

  8. 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)

    Article  Google Scholar 

  9. Jiang, X., Wang, D., Fan, Q., Zhang, M., Lu, C., Lau, A.P.T.: Physics-informed neural network for nonlinear dynamics in fiber optics. Laser Photon. Rev. 16, 2100483 (2022)

    Article  Google Scholar 

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

    Article  Google Scholar 

  11. 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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fang, Y., Wu, G.Z., Wen, X.K., Wang, Y.Y., Dai, C.Q.: Predicting certain vector optical solitons via the conservation-law deep-learning method. Opt. Laser Technol. 155, 108428 (2022)

    Article  Google Scholar 

  13. Fang, Y., Wu, G.Z., Wang, Y.Y., Dai, C.Q.: Data-driven femtosecond optical soliton excitations and parameters discovery of the high-order NLSE using the PINN. Nonlinear Dyn. 105, 603–616 (2021)

    Article  Google Scholar 

  14. Fang, Y., Han, H.B., Bo, W.B., Liu, W., Wang, B.H., Wang, Y.Y., Dai, C.Q.: Deep neural network for modeling soliton dynamics in the mode-locked laser. Opt Lett. 48, 779–782 (2023)

    Article  Google Scholar 

  15. 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 Soliton Fract. 158, 112118 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  16. Maimistov, A., Manykin, E.: Propagation of ultrashort optical pulses in resonant non-linear light guides. Zh. Eksp. Teor. Fiz. 85, 1177–1181 (1983)

    Google Scholar 

  17. Yuan, F.: New exact solutions of the (2+1)-dimensional NLS–MB equations. Nonlinear Dyn. 107, 1141–1151 (2021)

    Article  Google Scholar 

  18. Yuan, F.: The dynamics of the smooth positon and b-positon solutions for the NLS–MB equations. Nonlinear Dyn. 102, 1761–1771 (2020)

    Article  MATH  Google Scholar 

  19. Marcucci, G., Pierangeli, D., Conti, C.: Theory of neuromorphic computing by waves: machine learning by rogue waves, dispersive shocks, and solitons. Phys. Rev. Lett. 125, 093901 (2020)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  22. 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)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhou, H., Juncai, P., Chen, Y.: Data-driven forward–inverse problems for the variable coefficients Hirota equation using deep learning method. Nonlinear Dyn. 111(16), 14667–14693 (2023). https://doi.org/10.1007/s11071-023-08641-1

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. O’neill, M.E.: PCG: a family of simple fast space-efficient statistically good algorithms for random number generation. ACM Trans. Math. Softw. (2014)

  26. He, J.-S., Cheng, Y., Li, Y.-S.: The Darboux transformation for NLS–MB equations. Commun. Theor. Phys. 38, 493–496 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Guan, Y.-Y., Tian, B., Zhen, H.-L., Wang, Y.-F., Chai, J.: Soliton solutions of a generalised nonlinear Schrödinger–Maxwell–Bloch system in the erbium-doped optical fibre. Z. Naturfor. A 71, 241–247 (2016)

    Article  Google Scholar 

  28. Bai, X.-D., Zhang, D.: Search for rogue waves in Bose–Einstein condensates via a theory-guided neural network. Phys. Rev. E 106, 025305 (2022)

    Article  MathSciNet  Google Scholar 

  29. Zhang, R.-F., Li, M.-C., Yin, H.-M.: Rogue wave solutions and the bright and dark solitons of the (3+1)-dimensional Jimbo-Miwa equation. Nonlinear Dyn. 103, 1071–1079 (2021)

    Article  Google Scholar 

  30. He, J.S., Xu, S.W., Porsezian, K.: New types of rogue wave in an erbium-doped fibre system. J. Phys. Soc. Jpn. 81, 033002 (2012)

    Article  Google Scholar 

  31. Hou, J., Li, Y., Ying, S.: Enhancing PINNs for solving PDEs via adaptive collocation point movement and adaptive loss weighting. Nonlinear Dyn. 111(16), 15233–15261 (2023). https://doi.org/10.1007/s11071-023-08654-w

    Article  Google Scholar 

  32. Zhang, R., Su, J., Feng, J.: Solution of the Hirota equation using a physics-informed neural network method with embedded conservation laws. Nonlinear Dyn. 111, 13399–13414 (2023)

    Article  Google Scholar 

Download references

Funding

National Natural Science Foundation of China (Grant No. 12261131495); the Scientific Research and Developed Fund of Zhejiang A&F University (Grant No. 2021FR0009).

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Authors and Affiliations

  1. College of Optical, Mechanical and Electrical Engineering, Zhejiang A&F University, Lin’an, 311300, China

    Su-Yong Xu, Qin Zhou & Wei Liu

  2. Research Center of Nonlinear Science, School of Mathematical and Physical Sciences, Wuhan Textile University, Wuhan, 430200, China

    Qin Zhou

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  1. Su-Yong Xu
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Correspondence to Su-Yong Xu, Qin Zhou or Wei Liu.

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Xu, SY., Zhou, Q. & Liu, W. Prediction of soliton evolution and equation parameters for NLS–MB equation based on the phPINN algorithm. Nonlinear Dyn 111, 18401–18417 (2023). https://doi.org/10.1007/s11071-023-08824-w

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  • DOI: https://doi.org/10.1007/s11071-023-08824-w

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