Research on pulse transmission dynamics in optical fiber based on intelligent algorithms

LI Jun; SU Jin; HAN Xiaoxiang; ZHU Weijie; YANG Ruixia; ZHANG Haiyang; YAN Xiang’an; ZHANG Yunjie; WANG Feiran

doi:10.7498/aps.74.20241473

Abstract

Nonlinear Schrödinger equation (NLSE) has important applications in quantum mechanics, nonlinear optics, plasma physics, condensed matter physics, optical fiber communication and laser system design, and its accurate solution is very important for understanding complex physical phenomena. Here, the traditional finite difference method (FDM), the split-step Fourier (SSF) method and the physics-informed neural network (PINN) method are studied, aiming to analyze in depth the solving mechanisms of various algorithms, and then realize the efficient and accurate solution of complex NLSE in optical fiber. Initially, the steps, process and results of PINN in solving the NLSE for pulse under the condition of short-distance transmission are described, and the errors of these methods are quantitatively evaluated by comparing them with the errors of PINN, FDM and SSF. On this basis, the key factors affecting the accuracy of NLSE solution for pulse under long-distance transmission are further discussed. Then, the effects of different networks, activation functions, hidden layers and the number of neurons in PINN on the accuracy of NLSE solution are discussed. It is found that selecting a suitable combination of activation functions and network types can significantly reduce the error, and the combination of FNN and tanh activation functions is particularly good. The effectiveness of ensemble learning strategy is also verified, that is, by combining the advantages of traditional numerical methods and PINN, the accuracy of NLSE solution is improved. Finally, the evolution characteristics of Airy pulse with different chirps in fiber and the solution of vector NLSE corresponding to polarization-maintaining fiber are studied by using the above algorithm. This study explores the solving mechanisms of FDM, SSF and PINN in complex NLSE, compares and analyzes the error characteristics of those methods in various transmission scenarios, proposes and verifies the ensemble learning strategy, thus providing a solid theoretical basis for studying pulse transmission dynamics and data-driven simulation.

Keywords:

Authors and contacts

1.
School of Science, Xi’an Polytechnic University, Xi’an 710048, China
2.
Engineering Research Center of Flexible Radiation Protection Technology, Universities of Shaanxi Province, Xi’an 710048, China
3.
Xi’an Key Laboratory of Nuclear Protection Textile Equipment Technology, Xi’an 710048, China

Corresponding author: HAN Xiaoxiang, hanxiaoxiang@xpu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. U2330109, 62405239, 61805212).

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References

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[6]	Xie S S, Li G X, Yi S 2009 Comput. Meth. Appl. Mech. Eng. 198 1052 Google Scholar
[7]	Wu L 2012 Numer. Meth. Partial Differ. Eq. 28 63 Google Scholar
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[33]	Lu L, Meng X H, Mao Z P, Karniadakis G E 2019 SIAM Rev. 63 208 Google Scholar
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Cited By

图 1 基于NLSE的脉冲在光纤中传输方程的FDM, SSF, PINN三种方法求解示意图

Figure 1. Schematic diagram of FDM, SSF, PINN methods for solving the propagation equation of pulses in optical fibers based on NLSE.

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图 2 基于PINN的NLSE求解示意图, 其中NN为神经网络, AD为自动微分技术, PDE为NLSE实虚部分离对应的物理信息, Loss为优化物理信息损失函数

Figure 2. Schematic diagram of NLSE solution based on PINN. NN is the neural network, AD is the automatic differentiation technology, PDE is the physical information corresponding to the real and imaginary part of the NLSE, Loss is the optimized physical information loss function.

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图 3 基于PINN的NLSE求解结果　(a) 脉冲在光纤中的传播演化图; (b) 损失函数曲线

Figure 3. Results of NLSE solving by PINN: (a) Evolution diagram of pulse propagation in fiber; (b) loss function curves.

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图 4 多种算法对复杂NLSE求解的脉冲演化过程和对比图　(a) FDM求解的脉冲在光纤中的传播演化图; (b) SSF求解的脉冲在光纤中的传播演化图; (c) PINN求解对应的脉冲在光纤中的传播演化图; (d) FDM与SSF之间的交叉对比图; (e) PINN与SSF之间的交叉对比图; (f) PINN与FDM之间的交叉对比图

Figure 4. Pulse evolution process and comparison diagram of FDM, SSF and PINN: (a) Pulse evolution process based on FDM; (b) pulse evolution process based on SSF; (c) pulse evolution process based on PINN; (d) comparison diagram between FDM and SSF; (e) comparison diagram between PINN and SSF; (f) comparison diagram between PINN and FDM.

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图 5 PINN, SSF, FDM求解基阶孤子在200 m光纤中传输的结果　(a) PINN在不同采样点数下的输入与输出对比图; (b) FDM, SSF, PINN求解的输入与输出对比图; (c) FDM, SSF, PINN求解的基阶孤子传输演化图

Figure 5. Results of PINN, SSF, and FDM solving the fundamental order soliton transmission in a 200 m fiber: (a) Results of PINN at different sampling number (SN); (b) results of FDM, SSF, PINN and input; (c) evolution of fundamental soliton transmission solved by FDM, SSF and PINN.

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图 6 不同激活函数与神经网络设置下PINN的训练误差图　(a) 神经网络为FNN, 激活函数为Tanh的训练误差图; (b) 神经网络为FNN, 激活函数为Sigmoid的训练误差图; (c) 神经网络为FNN, 激活函数为Swish的训练误差图; (d) 神经网络为PFNN, 激活函数为Tanh的训练误差图; (e) 神经网络为PFNN, 激活函数为Sigmoid的训练误差图; (f) 神经网络为PFNN, 激活函数为Swish的训练误差图

Figure 6. Training error of PINN under different activation function and NN: (a) Training error with Tanh and FNN; (b) training error with Sigmoid and FNN; (c) training error with Swish and FNN; (d) training error with Tanh and PFNN; (e) training error with Sigmoid and PFNN; (f) training error with Swish and PFNN.

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图 7 不同隐藏层和神经元设置下PINN训练误差图　(a) 隐藏层为3, 神经元为10的训练误差图; (b) 隐藏层为3, 神经元为15的训练误差图; (c) 隐藏层为3, 神经元为20的训练误差图; (d)隐藏层为4, 神经元为10的训练误差图; (e) 隐藏层为4, 神经元为15的训练误差图; (f) 隐藏层为4, 神经元为20的训练误差图; (g) 隐藏层为5, 神经元为10的训练误差图; (h) 隐藏层为5, 神经元为15的训练误差图; (i) 隐藏层为5, 神经元为20的训练误差图

Figure 7. Training error of PINN under different number of hidden layers and neurons: (a) Training error with 3 and 10; (b) training error with 3 and 15; (c) training error with 3 and 20; (d) training error with 4 and 10; (e) training error with 4 and 15; (f) training error with 4 and 20; (g) training error with 5 and 10; (h) training error with 5 and 15; (i) training error with 5 and 20.

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图 8 不同集成方式对所得结果误差的影响　(a)—(c)分别为SSF, FDM, PINN求解的脉冲传输与解析解误差; (d)—(f)分别为SSF+FDM, PINN+FDM, PINN+SSF+FDM集成求解的脉冲传输与解析解误差

Figure 8. Influence of different integration methods on the results’ error: (a)–(c) The pulse transmission and errors of SSF, FDM and PINN solutions; (d)–(f) the pulse transmission and errors of SSF+FDM, PINN+FDM and PINN+SSF+FDM solutions.

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图 9 多种NLSE求解方法对比不同啁啾艾里脉冲在光纤传输中演化　(a)—(c)分别为对应于FDM, 在C = 0.4, 0, –0.4时的脉冲演化; (d)—(f)分别为对应于SSF, 在C = 0.4, 0, –0.4时的脉冲演化; (g)—(i)分别为对应于PINN, 在C = 0.4, 0, –0.4时的脉冲演化

Figure 9. Comparison of Airy pulses’ evolution with different chirps by various NLSE solving methods in fiber transmission: (a)–(c) The pulse evolution corresponding to FDM with C values of 0.4, 0, –0.4; (d)–(f) the pulse evolution corresponding to SSF with C values of 0.4, 0, –0.4; (g)–(i) the pulse evolution corresponding to PINN with C values of 0.4, 0, –0.4.

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图 10 SSF, FDM和PINN求解VNLSE对应的脉冲演化过程的交叉对比图　(a) SSF与FDM关于$u_x^2$ 的交叉对比图; (b) SSF与FDM关于$u_y^2$的交叉对比图; (c) SSF与FDM关于$ u_x^2{+}u_y^2 $的交叉对比图; (d) PINN与FDM关于$u_x^2$的交叉对比图; (e) PINN与FDM关于$u_y^2$的交叉对比图; (f) PINN与FDM关于$ u_x^2{+}u_y^2 $的交叉对比图

Figure 10. Comparison of the pulse evolution process corresponding to VNLSE solved by SSF, FDM and PINN: (a) SSF and FDM with respect to $u_x^2$; (b) SSF and FDM with respect to $u_y^2$; (c) SSF and FDM with respect to $ u_x^2{+}u_y^2 $; (d) PINN and FDM with respect to $u_x^2$; (e) PINN and FDM with respect to $u_y^2$; (f) PINN and FDM with respect to $ u_x^2{+}u_y^2 $.

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表 1 集成学习和单一方法的误差表

Table 1. Error table of Ensemble Learning and single method.

	MAE	MSE	RMSE	MAPE/%
FDM	4.014×10^–4	4.877×10^–7	6.983×10^–4	13.159
SSF	3.396×10^–4	3.112×10^–7	5.579×10^–4	8.979
PINN	4.173×10^–4	4.744×10^–7	6.887×10^–4	11.540
SSF+FDM	2.625×10^–4	1.466×10^–7	3.829×10^–4	10.933
PINN+FDM	2.865×10^–4	2.316×10^–7	4.652×10^–4	10.048
PINN+SSF	2.793×10^–4	1.753×10^–7	4.188×10^–4	12.237
PINN+FDM+SSF	2.190×10^–4	0.978×10^–7	3.127×10^–4	11.004

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[2]	Ngo N Q 2018 Ultra-fast Fiber Lasers: Principles and Applications with MATLAB® Models (Boca Raton: CRC Press) pp139–157
[3]	Chang Q S, Jia E H, Sun W 1999 J. Comput. Phys. 148 397 Google Scholar
[4]	Wang H Q 2005 Appl. Math. Comput. 170 17 Google Scholar
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[6]	Xie S S, Li G X, Yi S 2009 Comput. Meth. Appl. Mech. Eng. 198 1052 Google Scholar
[7]	Wu L 2012 Numer. Meth. Partial Differ. Eq. 28 63 Google Scholar
[8]	Caplan R M 2013 Appl. Numer. Math. 71 24 Google Scholar
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[10]	Feng X B, Li B Y, Ma S 2021 SIAM J. Numer. Anal. 59 1566 Google Scholar
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[12]	Zhang L F, Liu K, Zhong H Z, Zhang J G, Li Y, Fan D Y 2015 Opt. Express 23 2566 Google Scholar
[13]	Yu Y, Zhang Y T, Song X X, Zhang H T, Cao M X, Che Y L, Zhang H, Yao J Q 2017 IEEE Photonics J. 9 7904107 Google Scholar
[14]	Xin W, Wang Y, Xin Z G, Li L 2021 Opt. Commun. 489 126889 Google Scholar
[15]	罗民, 张泽贤, 陈乃妙, 刘萌, 罗爱平, 徐文成, 罗智超 2023 72 204203 Google Scholar Luo M, Zhang Y Z, Chen N M, Liu M, Luo A P, Xu W C, Luo Z C 2023 Acta Phys. Sin. 72 204203 Google Scholar
[16]	Lawal Z K, Yassin H, Lai D T, Che-Idris A 2022 Big Data Cogn. Comput. 6 140 Google Scholar
[17]	Karniadakis G E, Kevrekidis I G, Lu L, Perdikaris P, Wang S, Yang L 2021 Nat. Rev. Phys. 3 422 Google Scholar
[18]	李野, 陈松灿 2022 计算机科学 49 254 Google Scholar Li Y, Chen S C 2022 Comput. Sci. 49 254 Google Scholar
[19]	Raissi M, Perdikaris P, Karniadakis G E 2019 J. Comput. Phys. 378 686 Google Scholar
[20]	Pu J C, Peng W Q, Chen Y 2021 Wave Motion. 107 102823 Google Scholar
[21]	Pu J C, Li J, Chen Y 2021 Nonlinear Dyn. 105 1723 Google Scholar
[22]	Wang L, Yan Z Y 2021 Phys. Lett. A 404 127408 Google Scholar
[23]	Zang Y B, Yu Z M, Xu K, Chen M H, Yang S G, Chen H W 2021 J. Lightwave Technol. 40 404 Google Scholar
[24]	Jiang X T, Wang D S, Chen X, Zhang M 2022 J. Lightwave Technol. 40 21 Google Scholar
[25]	Jiang X T, Wang D S, Fan Q R, Zhang M, Lu C, Lau A P 2022 Laser Photonics Rev. 16 2100483 Google Scholar
[26]	Fang Y, Bo W B, Wang R R, Wang W W, Chao Q D 2022 Chaos Soliton. Fract. 165 112908 Google Scholar
[27]	田十方, 李彪 2023 72 100202 Google Scholar Tian S F, Li B 2023 Acta Phys. Sin. 72 100202 Google Scholar
[28]	罗霄, 张民, 蒋啸天, 宋裕琛, 张希萌, 王丹石 2023 红外与激光工程 52 12 Google Scholar Luo X, Zhang M, Jiang X T, Song X M, Zhang X M, Wang D S 2023 Infrared Laser Eng. 52 12 Google Scholar
[29]	Thulasidharan K, Sinthuja N, Priya N V, Senthilvelan M 2024 Commun. Theor. Phys. 76 115801 Google Scholar
[30]	Psaros A F, Kawaguchi K, Karniadakis G E 2022 J. Comput. Phys. 458 111121 Google Scholar
[31]	Uduagbomen J, Leeson M S, Liu Z, Lakshminarayana S, Xu T H 2024 Appl. Opt. 63 3794 Google Scholar
[32]	Cuomo S, Di Cola V S, Giampaolo F, Rozza G, Raissi M, Piccialli F 2022 J. Sci. Comput. 92 88 Google Scholar
[33]	Lu L, Meng X H, Mao Z P, Karniadakis G E 2019 SIAM Rev. 63 208 Google Scholar
[34]	Yu J, Lu L, Meng X H, Karniadakis G E 2022 Comput. Methods Appl. Mech. Eng. 393 114823 Google Scholar
[35]	Wang T M, Cai L J, Xia C J, Song H, Li L B, Bai G X, Fu N Q, Xian L D, Yang R, Mu H R, Zhang G Y, Lin S H 2024 Adv. Sci. 11 2406476 Google Scholar
[36]	Gong K W, Li L B, Yu W Z 2023 Mater. Des. 228 111848 Google Scholar
[37]	Ding B B, Li L B, Gong K W 2023 Appl. Surf. Sci. 22 156944 Google Scholar
[38]	Mao D, He Z W, Zhang Y S, Du Y Q, Zeng C, Yun L, Luo Z C, Li T J, Sun Z P, Zhao J L 2022 Light Sci. Appl. 11 25 Google Scholar
[39]	Mao D, Wang H, Zhang H, Zeng C, Du Y Q, He Z W, Sun Z P, Zhao J L 2021 Nat. Commun. 12 6712 Google Scholar
[40]	Mao D, Gao Q, Li J Y, He Z W, Du Y Q, Zeng C, Sun Z P, Zhao J L 2022 Phys. Rev. Appl. 18 044044 Google Scholar
[41]	Rao C P, Ren P, Wang Q, Buyukozturk O, Sun H, Liu Y 2023 Nat. Mach. Intell. 5 765 Google Scholar
[42]	Cao R H, Su J, Feng J Q, Guo Q 2024 Electron. Res. Arch. 32 6641 Google Scholar
[43]	Ismail M S, Alamri S Z 2004 Int. J. Comput. Math. 81 333 Google Scholar
[44]	Wang T 2011 J. Comput. Appl. Math. 235 4237 Google Scholar
[45]	Wu G Z, Fang Y, Wang Y Y, Wu G C, Dai C Q 2021 Chaos Solitons Fractals 152 111393 Google Scholar

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Vol. 74, Issue 6 - 2025-03-20

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