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PINN deep learning for the Chen-Lee-Liu equation: Rogue wave on the periodic background

Weiqi Peng, Juncai Pu, Yong Chen

arXiv:2105.13027v1nlin.SInlin.PS

TL;DR

Rogue periodic waves for the Chen-Lee-Liu equation had not been studied, and PINN learning of rogue periodic waves for PDEs had not been reported. The paper derives exact periodic and rogue periodic waves using odd-order Darboux transformations and applies PINNs to four CLL wave types; it reports small errors between learned and exact solutions.

  • Problem

    Rogue periodic waves for the CLL equation had not been studied, and PINN learning of rogue periodic waves involving PDEs had not been reported.

    Source: §1 · p. 1

  • Method

    The paper uses odd-order Darboux transformations to derive exact periodic and rogue periodic CLL waves, then applies PINNs to learn periodic, rogue periodic, soliton, and breather solutions.

    Source: §2 · p. 2, §3 · p. 4, §5 · p. 12

  • Results

    The authors report that PINNs generate the four CLL wave solutions with very small errors relative to exact solutions, and describe the method as effective and stable.

    Source: §5 · pp. 12–13

  • Takeaways & Limitations

    The reported results support PINN learning of periodic, rogue periodic, soliton, and breather solutions for the CLL equation.

    Source: §5 · pp. 12–13

    The stated learning setup uses a finite rectangular domain with initial data and periodic spatial boundary conditions.

    Source: §4 · p. 6

Abstract

from arXiv · show

We consider the exact rogue periodic wave (rogue wave on the periodic background) and periodic wave solutions for the Chen-Lee-Liu equation via the odd-th order Darboux transformation. Then, the multi-layer physics-informed neural networks (PINNs) deep learning method is applied to research the data-driven rogue periodic wave, breather wave, soliton wave and periodic wave solutions of well-known Chen-Lee-Liu equation. Especially, the data-driven rogue periodic wave is learned for the first time to solve the partial differential equation. In addition, using image simulation, the relevant dynamical behaviors and error analysis for there solutions are presented. The numerical results indicate that the rogue periodic wave, breather wave, soliton wave and periodic wave solutions for Chen-Lee-Liu equation can be generated well by PINNs deep learning method.

1. Introduction

The introduction situates the Chen-Lee-Liu equation among derivative nonlinear Schrödinger models and identifies rogue periodic waves for CLL as an unstudied case. It proposes exact-wave construction and PINN learning to address gaps in both areas.

1. Introduction

  • The CLL equation is one of three derivative nonlinear Schrödinger equations related by gauge transformations, and prior work obtained its breather, rogue, and rational soliton solutions via Darboux transformations.The introduction also describes derivative nonlinear Schrödinger models in plasma astrophysics, fluid dynamics, and nonlinear optics.Source: §1 · p. 1
  • Rogue periodic waves for CLL had not been studied, motivating their construction through the equation’s odd-order Darboux transformation.The paper contrasts these waves with the more-studied rogue waves on plane-wave backgrounds.Source: §1 · p. 1
  • PINN learning had not been reported for rogue periodic waves involving PDEs, and CLL soliton, breather, and periodic solutions had not been investigated with PINNs.The paper therefore targets data-driven learning of all four CLL wave types.Source: §1 · p. 2
  • The paper derives exact periodic and rogue periodic CLL waves with odd-order Darboux transformations, then studies data-driven periodic, rogue periodic, soliton, and breather waves using PINNs.The paper outline assigns these topics to Sections 2–4.Source: §1 · p. 2

2. The PINN deep learning method

The method represents a complex-valued dispersive PDE with two real-valued neural-network outputs and physics-informed residuals. It trains these outputs against initial-boundary data and equation constraints, minimizing the combined loss with L-BFGS.

2. The PINN deep learning method

  • For the complex-valued equation qt + Nq(q, qx, qxx, qxxx, …) = 0, writing q = u + iv decomposes it into two real nonlinear dispersive equations, one for u and one for v.Source: §2 · p. 2
  • PINN residuals f_u and f_v encode the two real PDEs, while neural networks approximate latent u and v and automatic differentiation supplies the residuals.Multi-hidden-layer networks train both the latent functions and their corresponding residuals.Source: §2 · p. 2
  • The loss combines initial-boundary data errors with residual errors at collocation points, enforcing data agreement and the target dispersive equations.The paper uses L-BFGS to minimize the whole mean squared error.Source: §2 · pp. 2–3
  • The implementation uses multilayer feedforward networks with Xavier initialization and tanh activation.The reported code uses Python 3.7 and TensorFlow 1.15.Source: §2 · p. 3

3. The exact periodic wave and rogue periodic waves

The section constructs exact periodic and rogue periodic wave solutions of the Chen-Lee-Liu equation using odd-order Darboux transformations. The rogue periodic wave appears on a periodic background, and the construction also generates higher-order solutions.

3. The exact periodic wave and rogue periodic waves

  • Taking odd order N = 2n + 1 in the Darboux expression constructs periodic and rogue periodic wave solutions beyond the even-order solutions previously used for constant-background waves.The section derives a one-periodic solution with N = 1 and obtains rogue periodic waves through a Taylor expansion for N = 2n + 1.Source: §3 · p. 4
  • The exact one-periodic solution is periodic in the x direction.Figure 1 shows the solution for β = 0.5.Source: §3 · p. 4, Figure 1 · p. 4
  • The one rogue periodic wave appears on a periodic background and is distributed where the periodic wave reaches its amplitude.Figure 2 presents the solution with β = −0.2.Source: §3 · p. 5, Figure 2 · p. 5
  • Higher-order rogue periodic waves can be generated by the same expression, and the section plots a two-order solution.The plotted example takes N = 5, A = 1, a = −1, and β = −0.1.Source: §3 · p. 6, Figure 3 · p. 6

4. The data-driven periodic wave, rogue periodic wave, soliton wave, and breather wave

The paper applies PINNs to learn periodic, rogue periodic, soliton, and breather solutions of the Chen-Lee-Liu equation. Reported errors vary across solution types, while the rogue-periodic training loss has a stair-like profile.

4. The data-driven periodic wave, rogue periodic wave, soliton wave, and breather wave

  • The PINN setup represents the complex solution with real and imaginary networks and uses initial-boundary data plus collocation points to constrain the solution.The networks compute equation residuals using automatic differentiation; the study applies them under the stated initial condition and boundary conditions.Source: §4 · p. 6

4.1. The data-driven periodic wave solution.

  • 1.141566e-02 was the L2 error for the learned periodic wave, and the authors report a smooth loss curve.Figure 4 presents wave propagation at three times, density and error-density views, a three-dimensional plot, and the loss curve.Source: Figure 4 · p. 7, §4.1 · p. 8

4.2. The data-driven rogue periodic wave solution.

  • 6.103918e-03 L2 error was reported for the learned rogue periodic wave, whose loss curve had a stair-like profile.The comparison used 400 initial-boundary samples and 10,000 collocation points; using 60 neurons per layer instead produced a 1.560081e-02 L2 error.Source: §4.2 · p. 9, Figure 5 · p. 9

4.3. The data-driven soliton wave solution.

4.4. The data-driven breather wave solution.

  • 1.156422e-02 was the L2 error for the learned breather wave, whose loss curve had a gentle interregion.The training used 400 boundary samples and 20,000 collocation points; Figure 8 presents the dynamical behavior and error analysis.Source: §4.4 · p. 12, Figure 8 · p. 12

5. Conclusion

The authors derive exact periodic and rogue periodic CLL waves with odd-th order Darboux transformations, then apply PINNs to data-driven wave solutions. They report very small errors against exact solutions, while noting that rogue periodic wave training works well only over a small time range.

5. Conclusion

  • PINNs first solve the CLL equation for data-driven rogue periodic waves, extending the approach to periodic, soliton, and breather waves.The paper also derives exact periodic and rogue periodic solutions using odd-th order Darboux transformations.Source: §5 · pp. 12–13
  • Errors between exact and PINN-generated solutions are very small, supporting the method's effectiveness and stability for the studied waves.The conclusion also reports that the method has no grid-size limitation compared with traditional numerical methods.Source: §5 · p. 13
  • PINN training works well over selected time domains, but its performance declines over wider intervals, especially for rogue periodic waves.The authors identify simulating rogue periodic waves well over large spatio-temporal scales as future work.Source: §5 · p. 13

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