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Data-Driven Deep Learning for The Multi-Hump Solitons and Parameters Discovery in NLS Equations with Generalized \({\mathcal{PT}\mathcal{}}\)-Scarf-II Potentials

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Abstract

In this paper, we investigate the data-driven forward and inverse problems of both the focusing and defocusing nonlinear Schrödinger equations (NLSEs) with generalized parity-time (\({\mathcal{PT}\mathcal{}}\))-Scarf-II potential via the physics-informed neural networks (PINNs) deep learning. The NLSE with four different initial conditions and periodic boundary condition are analyzed via the PINNs approach. And the predicted (data-driven) multi-hump solitons have been compared to the solutions, which can be obtained from the analytical or the high-accuracy numerical methods. Moreover, we explore the influences of several key factors (e.g., the depth of the neural networks, activation functions) on the performance of the PINNs algorithm. Finally, the data-driven inverse problems of the NLSE are also investigated such that the coefficients of the generalized \({\mathcal{PT}\mathcal{}}\)-Scarf-II potentials, the nonlinear and dispersion terms can be found. The results obtained in this paper can be used to further explore the NLSE with \({\mathcal{PT}\mathcal{}}\)-symmetric potentials and the applications of deep learning method in the nonlinear partial differential equations.

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References

  1. Ahmed Z (2001) Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex PT-invariant potential. Phys Lett A 282:343

    MathSciNet  MATH  Google Scholar 

  2. Alipanahi B, Delong A, Weirauch MT, Frey BJ (2015) Predicting the sequence specificities of DMA- and RNA-binding proteins by deep learning. Nat Biotechnol 33:831–838

    Google Scholar 

  3. Bao W, Cai Y (2012) Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator. SIAM J Numer Anal 50:492

    MathSciNet  MATH  Google Scholar 

  4. Bender CM (2007) Making sense of non-Hermitian Hamiltonians. Rep Prog Phys 70:947

    MathSciNet  Google Scholar 

  5. Bender CM, Boettcher S (1998) Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys Rev Lett 80:5243

    MathSciNet  MATH  Google Scholar 

  6. Bishop CM (2006) Pattern Recognition and Machine Learning. Springer, New York

    MATH  Google Scholar 

  7. Cai S, Wang Z, Lu L, Zaki TA, Karniadakis GE (2021) DeepMMnet: Inferring the electroconvection multiphysics fields based on operator approximation by neural networks. J Comput Phys 436:110296

    MATH  Google Scholar 

  8. Chen Y, Yan Z, Mihalache D, Malomed BA (2017) Families of stable solitons and excitations in the PT-symmetric nonlinear Schrödinger equations with position-dependent effective masses. Sci Rep 7:1

    Google Scholar 

  9. Chen YX, Xu FQ, Hu YL (2017) Two-dimensional Gaussian-type spatial solitons in inhomogeneous cubic-quintic-septimal nonlinear media under PT-symmetric potentials. Nonlinear Dyn 90:1115

    MathSciNet  Google Scholar 

  10. Chen Y, Yan Z, Li X (2018) One-and two-dimensional gap solitons and dynamics in the PT-symmetric lattice potential and spatially-periodic momentum modulation. Commun Nonlinear Sci Numer Simul 55:287

    MATH  Google Scholar 

  11. Chen Y, Yan Z, Mihalache D (2020) Soliton formation and stability under the interplay between parity-time-symmetric generalized Scarf-II potentials and Kerr nonlinearity. Phys Rev E 102:012216

    MathSciNet  Google Scholar 

  12. Christodoulides DN, Yang J (2018) Parity-time Symmetry and its Applications. Springer, New York

    Google Scholar 

  13. Collobert R, Weston J, Bottou G, Karlen M, Kavukcuoglu K, Kuksa P (2011) Natural language processing (almost) from scratch. J Mach Learn Res 12:2493–2537

    MATH  Google Scholar 

  14. Dag I (1999) A quadratic B-spline finite element method for solving nonlinear Schrödinger equation. Comput Method Appl M 174:247

    Google Scholar 

  15. Dehghan M, Taleei A (2010) A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients. Comput Phys Commun 181:43

    MATH  Google Scholar 

  16. Di L, Lu L, Meneveau C, Karniadakis G, Zaki TA DeepONet prediction of linear instability waves in high-speed boundary layers, arXiv:2105.08697

  17. Dorey P, Dunning C, Tateo R (2001) Spectral equivalences, Bethe ansatz equations, and reality properties in PT-symmetric quantum mechanics. J Phys A 34:5679

    MathSciNet  MATH  Google Scholar 

  18. Guo A, Salamo G, Duchesne D, Morandotti R, Volatier-Ravat M, Aimez V, Siviloglou G, Christodoulides D (2009) Observation of PT-symmetry breaking in complex optical potentials. Phys Rev Lett 103:093902

    Google Scholar 

  19. Hodaei H, Hassan AU, Wittek S, Garcia-Gracia H, ElGanainy R, Christodoulides DN, Khajavikhan M (2017) Enhanced sensitivity at higher-order exceptional points. Nature 548:187

    Google Scholar 

  20. Jagtap AD, Kharazmi E, Karniadakis GE (2020) Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems. Comput Methods Appl Mech Engrg 365:113028

    MathSciNet  MATH  Google Scholar 

  21. Jagtap AD, Kawaguchi K, Karniadakis GE (2020) Locally adaptive activation functions with slope recovery for deep and physics-informed neural networks. Proc R Soc A 476:20200334

    MathSciNet  MATH  Google Scholar 

  22. Jin X, Cai S, Li H, Karniadakis GE (2021) NSFnets(Navier-Stokes Flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. J Comput Phys 426:109951

    MathSciNet  MATH  Google Scholar 

  23. Karniadakis GE, Kevrekidis IG, Lu L, Perdikaris P, Wang S, Yang L (2021) Physics-informed machine learning. Nat Rev Phys 3:422–440

    Google Scholar 

  24. Kingma DP, Ba J Adam: A method for stochastic optimization, arXiv:1412.6980

  25. Konotop VV, Yang J, Zezyulin DA (2016) Nonlinear waves in \({{\cal{PT} }}\)-symmetric systems. Rev Mod Phys 88:035002

    Google Scholar 

  26. Krizhevsky A, Sutskever I, Hinton G (2017) Imagenet classification with deep convolutional neural networks. Commun ACM 60:84–90

    Google Scholar 

  27. Lake BM, Salakhutdinov R, Tenenbaum JB (2015) Human-level concept learning through probabilistic program induction. Science 350:1332–1338

    MathSciNet  MATH  Google Scholar 

  28. LeCun Y, Bengio Y, Hinton G (2015) Deep learning. Nature 521:436–444

    Google Scholar 

  29. Li Z, Kovachki N, Azizzadenesheli K, Liu B, Bhattacharya K, Stuart A, Anandkumar A Fourier neural operator for parametric partial differential equations arXiv:2010.08895

  30. Lin C, Maxey M, Li Z, Karniadakis GE (2021) A seamless multiscale operator neural network for inferring bubble dynamics. J Fluid Mech 929:A18

    MathSciNet  MATH  Google Scholar 

  31. Liu DC, Nocedal J (1989) On the limited memory BFGS method for large scale optimization. Math Program 45:503–28

    MathSciNet  MATH  Google Scholar 

  32. Liu X-Y, Wang J-X (2021) Physics-informed Dyna-style model-based deep reinforcement learning for dynamic control. Proc R Soc A 477:20210618

    MathSciNet  Google Scholar 

  33. Liu B, Zhang HF, Zhong RX, Zhang XL, Qin XZ, Huang C, Li YY, Malomed BA (2019) Symmetry breaking of quantum droplets in a dual-core trap. Phys Rev A 99:053602

    Google Scholar 

  34. Li Z, Zheng H, Kovachki N, Jin D, Chen H, Liu HB, Anandkumar A Physics-informed neural operator for learning partial differential equations, arXiv:2111.03794

  35. Lu L, Meng X, Mao Z, George EK (2021) DeepXDE: A deep learning library for solving differential equations. SIAM Rev 63:208–228

    MathSciNet  MATH  Google Scholar 

  36. Lu L, Jin P, Pang G, Zhang Z, Karniadakis GE (2021) Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nat Mach Intell 3:218–229

    Google Scholar 

  37. Mao Z, Jagtap AD, Karniadakis GE (2020) Physics-informed neural networks for high-speed flows. Comput Methods Appl Mech Engrg 360:112789

    MathSciNet  MATH  Google Scholar 

  38. Moiseyev N (2011) Non-Hermitian Quantum Mechanics. Cambridge University Press, New York

    MATH  Google Scholar 

  39. Musslimani Z, Makris KG, El-Ganainy R, Christodoulides DN (2008) Optical solitons in PT periodic potentials. Phys Rev Lett 100:030402

    Google Scholar 

  40. Pang G, Lu L, Karniadakis GE (2019) fPINNs: fractional physics-informed neural networks. SIAM J Sci Comput 41:A2603–A2626

    MathSciNet  MATH  Google Scholar 

  41. Park J, Sandberg IW (1991) Universal approximation using radial-basis-function networks. Neural Comput 3:246–257

    Google Scholar 

  42. Peng B, Ödemir SK, Lei F, Monifi F, Gianfreda M, Long GL, Fan S, Nori F, Bender CM, Yang L (2014) Parity-time-symmetric whispering-gallery microcavities. Nature Phys. 10:394–398

    Google Scholar 

  43. Raissi M, Wang Z, Triantafyllou MS, Karniadakis GE (2019) Deep learning of vortex-induced vibrations. J Fluid Mech 861:119

    MathSciNet  MATH  Google Scholar 

  44. Raissi M, Perdikaris P, Karniadakis GE (2019) 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

    MathSciNet  MATH  Google Scholar 

  45. Regensburger A, Bersch C, Miri MA, Onishchukov G, Christodoulides DN, Peschel U (2012) Parity-time synthetic photonic lattices. Nature 488:167

    Google Scholar 

  46. Rüter CE, Makris KG, El-Ganainy R, Christodoulides DN, Segev M, Kip D (2010) Observation of parity-time symmetry in optics. Nat Phys 6:192

    Google Scholar 

  47. Shen Y, Wen Z, Yan Z, Hang C (2018) Effect of PT symmetry on nonlinear waves for three-wave interaction models in the quadratic nonlinear media. Chaos 28:043104

    MathSciNet  MATH  Google Scholar 

  48. Shi D, Wang P, Zhao Y (2014) Superconvergence analysis of anisotropic linear triangular finite element for nonlinear Schrödinger equation. Appl Math Lett 38:129

    MathSciNet  MATH  Google Scholar 

  49. Stein ML (1987) Large sample properties of simulations using latin hypercube sampling. Technometrics 29:143–51

    MathSciNet  MATH  Google Scholar 

  50. Wang H (2005) Numerical studies on the split-step finite difference method for nonlinear Schrödinger equations. Appl Math Comput 170:17

    MathSciNet  MATH  Google Scholar 

  51. Wang L, Yan Z (2021) Data-driven rogue waves and parameter discovery in the defocusing nonlinear Schrödinger equation with a potential using the PINN deep learning. Phys Lett A 404:127408

    MATH  Google Scholar 

  52. Wang SF, Teng Y, Perdikaris P (2021) Understanding and mitigating gradient flow pathologies in physics-informed neural networks. SIAM J Sci Comput 43:A3055–A3081

    MathSciNet  MATH  Google Scholar 

  53. Wang S, Yu X, Perdikaris P (2022) When and why PINNs fail to train: A neural tangent kernel perspective. J Comput Phys 449:110768

    MathSciNet  MATH  Google Scholar 

  54. Wang S, Wang H, Perdikaris P Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets, arXiv:2103.10974

  55. Yan Z, Wen Z, Hang C (2015) Spatial solitons and stability in self-focusing and defocusing Kerr nonlinear media with generalized parity-time-symmetric Scarf-II potentials. Phys Rev E 92:022913

    MathSciNet  Google Scholar 

  56. Yang J (2010) Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM)

  57. Yang L, Zhang D, Karniadakis GE Physics-informed generative adversarial networks for stochastic differential equations arXiv:1811.02033

  58. Yu S, Zhang Z, Karniadakis GE Error estimates of residual minimization using neural networks for linear PDEs, arXiv:2010.08019

  59. Zhang D, Lu L, Guo L, Karniadakis GE (2019) Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems. J Comput Phys 397:108850

    MathSciNet  MATH  Google Scholar 

  60. Zhang D, Guo L, Karniadakis GE Learning in modal space: solving time-dependent stochastic PDEs using physics-informed neural networks arXiv: 1905.01205

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Acknowledgements

The work of Z.Y. was supported by the NSFC under Grant Nos. 11925108 and 11731014. The work of S-F.T. was supported by the NSFC under Grant No. 11975306, the NSF of Jiangsu Province of China (Grant No. BK20181351) and the the Six Talent Peaks Project in Jiangsu Province (Grant No. JY-059).

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Authors and Affiliations

  1. KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Ming Zhong, Zijian Zhou & Zhenya Yan

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Ming Zhong, Zijian Zhou & Zhenya Yan

  3. School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, China

    Jian-Guo Zhang & Shou-Fu Tian

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  1. Ming Zhong
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Zhong, M., Zhang, JG., Zhou, Z. et al. Data-Driven Deep Learning for The Multi-Hump Solitons and Parameters Discovery in NLS Equations with Generalized \({\mathcal{PT}\mathcal{}}\)-Scarf-II Potentials. Neural Process Lett 55, 2687–2705 (2023). https://doi.org/10.1007/s11063-022-10979-3

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