[Paper Review] Deep learning neural networks for the third-order nonlinear Schrodinger equation: Solitons, breathers, and rogue waves

Zijian Zhou, Zhenya Yan|arXiv (Cornell University)|Apr 30, 2021

Advanced Fiber Laser Technologies36 references4 citations

TL;DR

This paper applies physics-informed neural networks (PINNs) to discover data-driven solutions—solitons, breathers, and rogue waves—for the third-order nonlinear Schrödinger equation (Hirota equation), and to identify unknown parameters in the equation using noisy and noise-free soliton data. The method achieves high accuracy in both solution reconstruction and parameter discovery, even with 2% noise, demonstrating robustness and generalization in nonlinear wave modeling.

ABSTRACT

The third-order nonlinear Schrodinger equation (alias the Hirota equation) is investigated via deep leaning neural networks, which describes the strongly dispersive ion-acoustic wave in plasma and the wave propagation of ultrashort light pulses in optical fibers, as well as broader-banded waves on deep water. In this paper, we use the physics-informed neural networks (PINNs) deep learning method to explore the data-driven solutions (e.g., soliton, breather, and rogue waves) of the Hirota equation when the two types of the unperturbated and unperturbated (a 2% noise) training data are considered. Moreover, we use the PINNs deep learning to study the data-driven discovery of parameters appearing in the Hirota equation with the aid of solitons.

Motivation & Objective

• To develop a deep learning framework for discovering complex nonlinear wave solutions—solitons, breathers, and rogue waves—of the third-order nonlinear Schrödinger equation.
• To investigate the robustness of physics-informed neural networks (PINNs) in reconstructing solutions from noisy training data (2% noise) for the Hirota equation.
• To enable data-driven discovery of unknown physical parameters (e.g., dispersion and nonlinearity coefficients) in the Hirota equation using soliton solutions as training data.
• To validate the PINN approach on both unperturbed and perturbed data, assessing accuracy via L² norm errors and convergence speed.

Proposed method

• The PINN framework uses a deep neural network to approximate the complex-valued solution $ q(x,t) = u(x,t) + iv(x,t) $ of the Hirota equation.
• The loss function combines data fidelity from initial and boundary conditions and residual minimization of the governing PDE, enforced via automatic differentiation.
• The PDE residual is computed as $ F(x,t) = iq_t + \alpha(q_{xx} + 2|q|^2 q) + i\beta(q_{xxx} + 6|q|^2 q_x) $, with $ F_u $ and $ F_v $ derived from real and imaginary parts.
• The network is trained using Adam (20,000 steps) followed by L-BFGS (50,000 steps) to minimize the mean squared error of the PDE residual and boundary conditions.
• For parameter discovery, $ \alpha $, $ \beta $, $ \mu $, and $ \nu $ are treated as trainable parameters, with loss functions adapted to include unknown coefficients.
• Training data includes exact soliton solutions and perturbed versions with 2% noise, sampled over spatial and temporal domains.

Experimental results

Research questions

• RQ1 Can PINNs accurately reconstruct soliton, breather, and rogue wave solutions of the Hirota equation from limited or noisy data?
• RQ2 How does the presence of 2% noise in training data affect the accuracy and convergence of PINN-based solution reconstruction?
• RQ3 Can PINNs effectively discover unknown physical parameters (e.g., $ \alpha $, $ \beta $) in the Hirota equation using only soliton solution data?
• RQ4 What is the relative performance of PINNs in reconstructing the full complex field $ q(x,t) $, and its real and imaginary parts, under different noise levels?

Key findings

• The PINN method achieved a relative $ \mathbb{L}^2 $-norm error of $ 7.0371 \times 10^{-4} $ for the exact soliton solution with no noise, indicating high solution accuracy.
• With 2% noise in training data, the relative $ \mathbb{L}^2 $-norm error increased to $ 9.4420 \times 10^{-4} $, demonstrating robustness to data perturbations.
• For parameter discovery of $ \alpha $ and $ \beta $, the method achieved an error of $ 3.85 \times 10^{-5} $ and $ 7.48 \times 10^{-5} $, respectively, under noise-free conditions.
• Under 2% noise, the parameter discovery errors for $ \alpha $ and $ \beta $ were $ 3.31 \times 10^{-4} $ and $ 2.89 \times 10^{-4} $, showing stable performance despite data corruption.
• For the alternative parameterization with $ \mu $ and $ \nu $, the method achieved errors of $ 3.69 \times 10^{-3} $ and $ 3.14 \times 10^{-2} $ without noise, and $ 1.84 \times 10^{-2} $ and $ 1.13 \times 10^{-1} $ with 2% noise.
• Training times were approximately 1970–3570 seconds depending on the case, with convergence achieved within 50,000 L-BFGS steps.