Ukrainian Journal of Physical Optics


2024 Volume 25, Issue 5


ISSN 1609-1833 (Print)

PREDICTION OF NONDEGENERATE SOLITONS AND PARAMETERS IN NONLINEAR BIREFRINGENT OPTICAL FIBERS USING PHPINN AND DEEPONET ALGORITHMS

Su-Yong Xu, Ao-Cheng Yang and Qin Zhou

1Research Group of Nonlinear Optical Science and Technology, Research Center of Nonlinear Science, School of Mathematical and Physical Sciences, Wuhan Textile University, Wuhan 430200, China, qinzhou@whu.edu.cn
2College of Optical, Mechanical and Electrical Engineering, Zhejiang A&F University, Lin'an 311300, China
3State Key Laboratory of New Textile Materials and Advanced Processing Technologies, Wuhan Textile University, Wuhan 430200, China

ABSTRACT

This paper uses parallel hard-constraint physics-informed neural networks to investigate the prediction of nondegenerate soliton and estimate parameters for the coupled nonlinear Schrödinger’s equation. Based on our previous analytical results, three types of nondegenerate solitons have been predicted in the forward problem under the corresponding initial and boundary conditions. In the inverse problem, when employing pure data as the training set, the relative errors in predicting the system’s parameters of group velocity dispersion β2 and Kerr nonlinearity γ are both less than 1%. Moreover, upon introducing a 5% noise level to the training set, the relative errors for β2 and γ remain below 3%. Additionally, we introduce for the first time the application of Deep Operator Networks (DeepONet) to predict nondegenerate soliton, reducing relative L2 error to 10-3 and achieving a speedup of approximately 103 times higher compared to the phPINN method. This demonstrates the efficacy of operator learning methods in addressing nonlinear optical problems.

Keywords: nondegenerate solitons, coupled nonlinear Schrodinger's equation, phPINN, DeepONet

UDC: 535.32

    1. Li, N., Chen, Q., Triki, H., Liu, F., Sun, Y., Xu, S., & Zhou, Q. (2024). Bright and Dark Solitons in a (2+ 1)-Dimensional Spin-1 Bose-Einstein Condensates. Ukrainian Journal of Physical Optics, 25(5), S1060-S1074. doi:10.3116/16091833/Ukr.J.Phys.Opt.2024.S1060
    2. Zhou, Q., Triki, H., Xu, J., Zeng, Z., Liu, W., & Biswas, A. (2022). Perturbation of chirped localized waves in a dual-power law nonlinear medium. Chaos, Solitons & Fractals, 160, 112198. doi:10.1016/j.chaos.2022.112198
    3. Zhou, Q., Zhong, Y., Triki, H., Sun, Y., Xu, S., Liu, W., & Biswas, A. (2022). Chirped bright and kink solitons in nonlinear optical fibers with weak nonlocality and cubic-quantic-septic nonlinearity. Chinese Physics Letters, 39(4), 044202. doi:10.1088/0256-307X/39/4/044202
    4. Zhou, Q. (2022). Influence of parameters of optical fibers on optical soliton interactions. Chinese Physics Letters, 39(1), 010501. doi:10.1088/0256-307X/39/1/010501
    5. Zhong, Y., Triki, H., & Zhou, Q. (2023). Analytical and numerical study of chirped optical solitons in a spatially inhomogeneous polynomial law fiber with parity-time symmetry potential. Communications in Theoretical Physics, 75(2), 025003. doi:10.1088/1572-9494/aca51c
    6. Zhong, Y., Triki, H., & Zhou, Q. (2024). Bright and kink solitons of time-modulated cubic-quintic-septic-nonic nonlinear Schrödinger equation under space-time rotated PT-symmetric potentials. Nonlinear Dynamics, 112(2), 1349-1364. doi:10.1007/s11071-023-09116-z
    7. Ding, C. C., Zhou, Q., Triki, H., & Hu, Z. H. (2022). Interaction dynamics of optical dark bound solitons for a defocusing Lakshmanan-Porsezian-Daniel equation. Optics Express, 30(22), 40712-40727. doi:10.1364/OE.473024
    8. Ding, C., Zhou, Q., Xu, S., Triki, H., Mirzazadeh, M., & Liu, W. (2023). Nonautonomous breather and rogue wave in spinor Bose-Einstein condensates with space-time modulated potentials. Chinese Physics Letters, 40(4), 040501. doi:10.1088/0256-307X/40/4/040501
    9. Liu, F. Y., Triki, H., & Zhou, Q. (2024). Optical nondegenerate solitons in a birefringent fiber with a 35 degree elliptical angle. Optics Express, 32(2), 2746-2765. doi:10.1364/OE.512116
    10. Chen, S., Ye, Y., Soto-Crespo, J. M., Grelu, P., & Baronio, F. (2018). Peregrine solitons beyond the threefold limit and their two-soliton interactions. Physical Review Letters, 121(10), 104101. doi:10.1103/PhysRevLett.121.104101
    11. Kaup, D. J., & Malomed, B. A. (1993). Soliton trapping and daughter waves in the Manakov model. Physical Review A, 48(1), 599. doi:10.1103/PhysRevA.48.599
    12. Akramov, M., Sabirov, K., Matrasulov, D., Susanto, H., Usanov, S., & Karpova, O. (2022). Nonlocal nonlinear Schrödinger equation on metric graphs: A model for generation and transport of parity-time-symmetric nonlocal solitons in networks. Physical Review E, 105(5), 054205. doi:10.1103/PhysRevE.105.054205
    13. Menyuk, C. R. (2024). Solitons in birefringent optical fibers and polarization mode dispersion. Optics Communications, 550, 129841. doi:10.1016/j.optcom.2023.129841
    14. Wang, X. M., Zhang, L. L., & Hu, X. X. (2020). Various types of vector solitons for the coupled nonlinear Schrödinger equations in the asymmetric fiber couplers. Optik, 219, 164989. doi:10.1016/j.ijleo.2020.164989
    15. Zhong, H., Tian, B., Jiang, Y., Li, M., Wang, P., & Liu, W. J. (2013). All-optical soliton switching for the asymmetric fiber couplers. The European Physical Journal D, 67, 1-15. doi:10.1140/epjd/e2013-30530-y
    16. Geng, K. L., Mou, D. S., & Dai, C. Q. (2023). Nondegenerate solitons of 2-coupled mixed derivative nonlinear Schrödinger equations. Nonlinear Dynamics, 111(1), 603-617. doi:10.1007/s11071-022-07833-5
    17. Karniadakis, G. E., Kevrekidis, I. G., Lu, L., Perdikaris, P., Wang, S., & Yang, L. (2021). Physics-informed machine learning. Nature Reviews Physics, 3(6), 422-440. doi:10.1038/s42254-021-00314-5
    18. Wang, S., Yu, X., & Perdikaris, P. (2022). When and why PINNs fail to train: A neural tangent kernel perspective. Journal of Computational Physics, 449, 110768. doi:10.1016/j.jcp.2021.110768
    19. Sirignano, J., & Spiliopoulos, K. (2018). DGM: A deep learning algorithm for solving partial differential equations. Journal of Computational Physics, 375, 1339-1364. doi:10.1016/j.jcp.2018.08.029
    20. Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). 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. doi:10.1016/j.jcp.2018.10.045
    21. Wen, G., Li, Z., Azizzadenesheli, K., Anandkumar, A., & Benson, S. M. (2022). U-FNO-An enhanced Fourier neural operator-based deep-learning model for multiphase flow. Advances in Water Resources, 163, 104180. doi:10.1016/j.advwatres.2022.104180
    22. Wang, X., Wu, Z., Song, J., Han, W., & Yan, Z. (2024). Data-driven soliton solutions and parameters discovery of the coupled nonlinear wave equations via a deep learning method. Chaos, Solitons & Fractals, 180, 114509. doi:10.1016/j.chaos.2024.114509
    23. Jiang, J. H., Si, Z. Z., Dai, C. Q., & Wu, B. (2024). Prediction of multipole vector solitons and model parameters for coupled saturable nonlinear Schrödinger equations. Chaos, Solitons & Fractals, 180, 114581. doi:10.1016/j.chaos.2024.114581
    24. Miao, Z., & Chen, Y. (2023). VC-PINN: Variable coefficient physics-informed neural network for forward and inverse problems of PDEs with variable coefficient. Physica D: Nonlinear Phenomena, 456, 133945. doi:10.1016/j.physd.2023.133945
    25. Lin, S., & Chen, Y. (2022). A two-stage physics-informed neural network method based on conserved quantities and applications in localized wave solutions. Journal of Computational Physics, 457, 111053. doi:10.1016/j.jcp.2022.111053
    26. Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., & Johnson, S. G. (2021). Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing, 43(6), B1105-B1132. doi:10.1137/21M1397908
    27. Pang, G., Lu, L., & Karniadakis, G. E. (2019). fPINNs: Fractional physics-informed neural networks. SIAM Journal on Scientific Computing, 41(4), A2603-A2626. doi:10.1137/18M1229845
    28. Xu, S. Y., Zhou, Q., & Liu, W. (2023). Prediction of soliton evolution and equation parameters for NLS-MB equation based on the phPINN algorithm. Nonlinear Dynamics, 111(19), 18401-18417. doi:10.1007/s11071-023-08824-w
    29. Lu, L., Jin, P., Pang, G., Zhang, Z., & Karniadakis, G. E. (2021). Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nature Machine Intelligence, 3(3), 218-229. doi:10.1038/s42256-021-00302-5
    30. Lin, C., Li, Z., Lu, L., Cai, S., Maxey, M., & Karniadakis, G. E. (2021). Operator learning for predicting multiscale bubble growth dynamics. The Journal of Chemical Physics, 154(10). doi:10.1063/5.0041203
    31. Di Leoni, P. C., Lu, L., Meneveau, C., Karniadakis, G. E., & Zaki, T. A. (2023). Neural operator prediction of linear instability waves in high-speed boundary layers. Journal of Computational Physics, 474, 111793. doi:10.1016/j.jcp.2022.111793
    32. Osorio, J. D., Wang, Z., Karniadakis, G., Cai, S., Chryssostomidis, C., Panwar, M., & Hovsapian, R. (2022). Forecasting solar-thermal systems performance under transient operation using a data-driven machine learning approach based on the deep operator network architecture. Energy Conversion and Management, 252, 115063. doi:10.1016/j.enconman.2021.115063
    33. Mao, S., Dong, R., Lu, L., Yi, K. M., Wang, S., & Perdikaris, P. (2023). Ppdonet: Deep operator networks for fast prediction of steady-state solutions in disk-planet systems. The Astrophysical Journal Letters, 950(2), L12. doi:10.3847/2041-8213/acd77f
    34. Goswami, S., Yin, M., Yu, Y., & Karniadakis, G. E. (2022). A physics-informed variational DeepONet for predicting crack path in quasi-brittle materials. Computer Methods in Applied Mechanics and Engineering, 391, 114587. doi:10.1016/j.cma.2022.114587
    35. Zhu, M., Zhang, H., Jiao, A., Karniadakis, G. E., & Lu, L. (2023). Reliable extrapolation of deep neural operators informed by physics or sparse observations. Computer Methods in Applied Mechanics and Engineering, 412, 116064. doi:10.1016/j.cma.2023.116064
    36. Wang, S., Wang, H., & Perdikaris, P. (2021). Learning the solution operator of parametric partial differential equations with physics-informed DeepONets. Science Advances, 7(40), eabi8605. doi:10.1126/sciadv.abi8605
    37. Jin, P., Meng, S., & Lu, L. (2022). MIONet: Learning multiple-input operators via tensor product. SIAM Journal on Scientific Computing, 44(6), A3490-A3514. doi:10.1137/22M1477751
    38. Lu, L., Meng, X., Mao, Z., & Karniadakis, G. E. (2021). DeepXDE: A deep learning library for solving differential equations. SIAM Review, 63(1), 208-228. doi:10.1137/19M1274067

    У цій статті використовуються паралельні нейронні мережі з жорсткими обмеженнями, засновані на фізичних даних, для дослідження передбачення невиродженого солітону та оцінки параметрів для зв’язаного нелінійного рівняння Шредінгера. На основі наших попередніх аналітичних результатів у прямій задачі було передбачено три типи невироджених солітонів за відповідних початкових і граничних умов. У зворотній задачі, коли в якості навчального набору використовуються чисті дані, відносні похибки у передбаченні параметрів системи дисперсії групової швидкості β2 та нелінійності Керра γ становлять менше ніж 1%. Крім того, при введенні 5% рівня шуму в навчальну множину відносні похибки для β2 і γ залишаються нижчими ніж 3%. Крім того, ми вперше представляємо застосування Deep Operator Networks для прогнозування невироджених солітонів, зменшуючи відносну помилку L2 до 10-3 і досягаючи прискорення приблизно в 103 рази більшого порівняно з методом phPINN. Це демонструє ефективність методів навчання операторів у вирішенні нелінійних оптичних проблем.

    Ключові слова: невироджені солітони, зв’язане нелінійне рівняння Шредінгера, phPINN, DeepONet

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