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.

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).

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 Weather21 2 [2] Peregrine D H 1983 J. Aust. Math. Soc. Ser. B25 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 Nature450 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. E80 026601 [12] Ohta Y and Yang J K 2012 Proc. R. Soc. A468 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 Nature521 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 Science350 1332 [23] Krizhevsky A, Sutskever I and Hinton G 2017 Communications of the Acm60 84 [24] Mcculloch W S and Pitts W 1943 Bull. Math. Biophys.5 115 [25] Rosenblatt F 1958 Psychological Review65 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 Networks9 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. Plus134 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. B34 2050288 [42] Zhang Z, Yang X Y, Li W T and Li B 2019 Chin. Phys. B28 110201 [43] Li J and Chen Y 2020 Commun. Theor. Phys.72 105005 [44] Bongard J and Lipson H 2007 Proc. Natl. Acad. Sci. USA104 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 Technometrics29 143 [51] Choromanska A, Henaff M, Mathieu M, Arous G B and LeCun Y 2015 Proc. 18 Int. Conf. on Artificial Intelligence and Statistics, PMLR38 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. A80 033610 [55] Moslem W M 2011 Phys. Plasmas18 032301 [56] Yan Z Y 2011 Phys. Lett. A375 4274

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