Chin. Phys. B, 2021, Vol. 30(6): 060202    DOI: 10.1088/1674-1056/abd7e3
 GENERAL Prev   Next

# Soliton, breather, and rogue wave solutions for solving the nonlinear Schrödinger equation using a deep learning method with physical constraints

Jun-Cai Pu(蒲俊才)1, Jun Li(李军)2, and Yong Chen(陈勇)1,3,4,†
1 School of Mathematical Sciences, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, and Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200241, China;
2 Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062, China;
3 College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China;
4 Department of Physics, Zhejiang Normal University, Jinhua 321004, China
Abstract  The nonlinear Schrödinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas. However, due to the difficulty of solving this equation, in particular in high dimensions, lots of methods are proposed to effectively obtain different kinds of solutions, such as neural networks among others. Recently, a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation's dynamical behaviors from spatiotemporal data directly. Compared with traditional neural networks, this method can obtain remarkably accurate solution with extraordinarily less data. Meanwhile, this method also provides a better physical explanation and generalization. In this paper, based on the above method, we present an improved deep learning method to recover the soliton solutions, breather solution, and rogue wave solutions of the nonlinear Schrödinger equation. In particular, the dynamical behaviors and error analysis about the one-order and two-order rogue waves of nonlinear integrable equations are revealed by the deep neural network with physical constraints for the first time. Moreover, the effects of different numbers of initial points sampled, collocation points sampled, network layers, neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions. Numerical experiments show that the dynamical behaviors of soliton solutions, breather solution, and rogue wave solutions of the integrable nonlinear Schrödinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.
Keywords:  deep learning method      neural network      soliton solutions      breather solution      rogue wave solutions
Received:  17 December 2020      Revised:  29 December 2020      Accepted manuscript online:  04 January 2021
 PACS: 02.30.Ik (Integrable systems) 05.45.Yv (Solitons) 07.05.Mh (Neural networks, fuzzy logic, artificial intelligence)
Fund: Project supported by the National Natural Science Foundation of China (Grant No. 11675054), the Fund from Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things (Grant No. ZF1213), and the Project of Science and Technology Commission of Shanghai Municipality (Grant No. 18dz2271000).
Corresponding Authors:  Yong Chen     E-mail:  ychen@sei.ecnu.edu.cn

Jun-Cai Pu(蒲俊才), Jun Li(李军), and Yong Chen(陈勇) Soliton, breather, and rogue wave solutions for solving the nonlinear Schrödinger equation using a deep learning method with physical constraints 2021 Chin. Phys. B 30 060202

 [1] Draper L 1966 Weather 21 2[2] Peregrine D H 1983 J. Aust. Math. Soc. Ser. B 25 16[3] Zabusky N J and Kruskal M D 1965 Phys. Rev. Lett. 15 240[4] Parkins A S and Walls D F 1998 Phys. Rep. 303 1[5] Ablowitz M J and Clarkson P A 1992 Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge: Cambridge University Press)[6] Schrödinger E 1926 Phys. Rev. 28 1049[7] Guo B L, Ling L M and Liu Q P 2012 Stud. Appl. Math. 130 317[8] Solli D R, Ropers C, Koonath P and Jalali B 2007 Nature 450 1054[9] Chabchoub A, Hoffmann N P and Akhmediev N 2011 Phys. Rev. Lett. 106 204502[10] Qiao Z J 1994 J. Math. Phys. 35 2971[11] Akhmediev N, Ankiewicz A and Soto-Crespo J M 2009 Phys. Rev. E 80 026601[12] Ohta Y and Yang J K 2012 Proc. R. Soc. A 468 1716[13] Hasegawa A and Tappert F 1973 Appl. Phys. Lett. 23 142[14] Kavitha L and Daniel M 2003 J. Phys. A: Math. Gen. 36 10471[15] Qiao Z J 1993 J. Math. Phys. 34 3110[16] Wang B, Zhang Z and Li B 2020 Chin. Phys. Lett. 37 030501[17] LeCun Y, Bengio Y and Hinton G 2015 Nature 521 436[18] Mitchell T M 1997 Machine Learning (McGraw-Hill Press series in computer science)[19] Bishop C M 2006 Pattern Recognition and Machine Learning (Springer)[20] Alipanahi B, Delong A, Weirauch M T and Frey B J 2015 Nat. Biotechnol. 33 831[21] Duda R O, Hart P E and Stork D G 2000 Pattern Classification (Wiley-Interscience Press)[22] Lake B M, Salakhutdinov R and Tenenbaum J B 2015 Science 350 1332[23] Krizhevsky A, Sutskever I and Hinton G 2017 Communications of the Acm 60 84[24] Mcculloch W S and Pitts W 1943 Bull. Math. Biophys. 5 115[25] Rosenblatt F 1958 Psychological Review 65 386[26] Bryson A E and Ho Y C 1975 Applied Optimal Control: Optimization, Estimation, and Control (Taylor and Francis Press)[27] Lagaris I E, Likas A and Fotiadis D I 1998 IEEE Transactions on Neural Networks 9 987[28] Hornik K, Stinchcombe M and White H 1989 Neural Netw. 2 359[29] Raissi M, Perdikaris P and Karniadakis G E 2019 J. Comput. Phys. 378 686[30] Jagtap A D, Kharazmi E and Karniadakis G E 2020 Comput. Methods Appl. Mech. Engrg. 365 113028[31] Lax P D 1968 Comm. Pure. Appl. Math. 21 467[32] Yu S J, Toda K and Fukuyama T 1998 J. Phys. A: Math. Gen. 31 10181[33] Iwao M and Hirota R 1997 J. Phys. Soc. Jpn. 66 577[34] Osman M S, Ghanbari B and Machado J A T 2019 Eur. Phys. J. Plus 134 20[35] Dong J J, Li B and Yuen M W 2020 Commun. Theor. Phys. 72 025002[36] Hirota R 2004 Direct Methods in Soliton Theory (Springer-verlag Press)[37] Geng X G and Tam H W 1999 J. Phys. Soc. Jpn. 68 1508[38] Matveev V B and Salle M A 1991 Darboux Transformation and Solitons (Springer Press)[39] Olver P J 1993 Applications of Lie Groups to Differential Equations (Springer Press)[40] Zakharov V E, Manakov S V, Novikov S P and Pitaevskii L P 1984 The Theory of Solitons: The Inverse Scattering Method (Consultants Bureau Press)[41] Pu J C and Chen Y 2020 Mod. Phys. Lett. B 34 2050288[42] Zhang Z, Yang X Y, Li W T and Li B 2019 Chin. Phys. B 28 110201[43] Li J and Chen Y 2020 Commun. Theor. Phys. 72 105005[44] Bongard J and Lipson H 2007 Proc. Natl. Acad. Sci. USA 104 9943[45] Raissi M, Perdikaris P and Karniadakis G E 2017 J. Comput. Phys. 348 683[46] Li J and Chen Y 2020 Commun. Theor. Phys. 72 115003[47] Li J and Chen Y 2020 Commun. Theor. Phys. 73 015001[48] Marcucci G, Pierangeli D and Conti C 2020 Phys. Rev. Lett. 125 093901[49] Baydin A G, Pearlmutter B A, Radul A A and Siskind J M 2018 J. Mach. Learn. Res. 18 1[50] Stein M L 1987 Technometrics 29 143[51] Choromanska A, Henaff M, Mathieu M, Arous G B and LeCun Y 2015 Proc. 18 Int. Conf. on Artificial Intelligence and Statistics, PMLR 38 192[52] Liu D C and Nocedal J 1989 Math. Program. 45 503[53] Yang J K 2010 Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, Philadelphia Press)[54] Bludov Y V, Konotop V V and Akhmediev N 2009 Phys. Rev. A 80 033610[55] Moslem W M 2011 Phys. Plasmas 18 032301[56] Yan Z Y 2011 Phys. Lett. A 375 4274
 [1] Determination of quantum toric error correction code threshold using convolutional neural network decoders Hao-Wen Wang(王浩文), Yun-Jia Xue(薛韵佳), Yu-Lin Ma(马玉林), Nan Hua(华南), and Hong-Yang Ma(马鸿洋). Chin. Phys. B, 2022, 31(1): 010303. [2] $\mathcal{H}_{\infty }$ state estimation for Markov jump neural networks with transition probabilities subject to the persistent dwell-time switching rule Hao Shen(沈浩), Jia-Cheng Wu(吴佳成), Jian-Wei Xia(夏建伟), and Zhen Wang(王震). Chin. Phys. B, 2021, 30(6): 060203. [3] Convolutional neural network for transient grating frequency-resolved optical gating trace retrieval and its algorithm optimization Siyuan Xu(许思源), Xiaoxian Zhu(朱孝先), Ji Wang(王佶), Yuanfeng Li(李远锋), Yitan Gao(高亦谈), Kun Zhao(赵昆), Jiangfeng Zhu(朱江峰), Dacheng Zhang(张大成), Yunlin Chen(陈云琳), and Zhiyi Wei(魏志义). Chin. Phys. B, 2021, 30(4): 048402. [4] General M-lumps, T-breathers, and hybrid solutions to (2+1)-dimensional generalized KDKK equation Peisen Yuan(袁培森), Jiaxin Qi(齐家馨), Ziliang Li(李子良), and Hongli An(安红利). Chin. Phys. B, 2021, 30(4): 040503. [5] Relationship between manifold smoothness and adversarial vulnerability in deep learning with local errors Zijian Jiang(蒋子健), Jianwen Zhou(周健文), and Haiping Huang(黄海平). Chin. Phys. B, 2021, 30(4): 048702. [6] Soliton interactions and asymptotic state analysis in a discrete nonlocal nonlinear self-dual network equation of reverse-space type Cui-Lian Yuan(袁翠连) and Xiao-Yong Wen(闻小永). Chin. Phys. B, 2021, 30(3): 030201. [7] Constructing reduced model for complex physical systems via interpolation and neural networks Xuefang Lai(赖学方), Xiaolong Wang(王晓龙, and Yufeng Nie(聂玉峰). Chin. Phys. B, 2021, 30(3): 030204. [8] Effective suppression of beta oscillation in Parkinsonian state via a noisy direct delayed feedback control scheme Hai-Tao Yu(于海涛), Zi-Han Meng(孟紫寒), Chen Liu(刘晨), Jiang Wang(王江), and Jing Liu(刘静). Chin. Phys. B, 2021, 30(3): 038703. [9] Discontinuous event-trigger scheme for global stabilization of state-dependent switching neural networks with communication delay Yingjie Fan(樊英杰), Zhen Wang(王震), Jianwei Xia(夏建伟), and Hao Shen(沈浩). Chin. Phys. B, 2021, 30(3): 030202. [10] Adaptive synchronization of a class of fractional-order complex-valued chaotic neural network with time-delay Mei Li(李梅), Ruo-Xun Zhang(张若洵), and Shi-Ping Yang(杨世平). Chin. Phys. B, 2021, 30(12): 120503. [11] A memristive map with coexisting chaos and hyperchaos Sixiao Kong(孔思晓), Chunbiao Li(李春彪), Shaobo He(贺少波), Serdar Çiçek, and Qiang Lai(赖强). Chin. Phys. B, 2021, 30(11): 110502. [12] Application of the edge of chaos in combinatorial optimization Yanqing Tang(唐彦卿), Nayue Zhang(张娜月), Ping Zhu(朱萍), Minghu Fang(方明虎), and Guoguang He(何国光). Chin. Phys. B, 2021, 30(10): 100505. [13] Finite-time Mittag-Leffler synchronization of fractional-order delayed memristive neural networks with parameters uncertainty and discontinuous activation functions Chong Chen(陈冲), Zhixia Ding(丁芝侠), Sai Li(李赛), Liheng Wang(王利恒). Chin. Phys. B, 2020, 29(4): 040202. [14] Multiple Lagrange stability and Lyapunov asymptotical stability of delayed fractional-order Cohen-Grossberg neural networks Yu-Jiao Huang(黄玉娇), Xiao-Yan Yuan(袁孝焰), Xu-Hua Yang(杨旭华), Hai-Xia Long(龙海霞), Jie Xiao(肖杰). Chin. Phys. B, 2020, 29(2): 020703. [15] Memristor-based vector neural network architecture Hai-Jun Liu(刘海军), Chang-Lin Chen(陈长林), Xi Zhu(朱熙), Sheng-Yang Sun(孙盛阳), Qing-Jiang Li(李清江), Zhi-Wei Li(李智炜). Chin. Phys. B, 2020, 29(2): 028502.