[Paper Review] Data-driven peakon and periodic peakon travelling wave solutions of some nonlinear dispersive equations via deep learning

Li Wang, Zhenya Yan|arXiv (Cornell University)|Jan 12, 2021

Nonlinear Waves and Solitons37 references4 citations

TL;DR

This paper introduces a physics-informed neural network (PINN) framework to discover data-driven peakon and periodic peakon solutions for nonlinear dispersive equations such as the Camassa-Holm, Degasperis-Procesi, and Novikov equations. By training deep neural networks on initial and boundary conditions with loss minimization, the method achieves high-accuracy approximations of peakon solutions, with L2 errors as low as 1.96e-02, demonstrating the effectiveness of PINNs in capturing discontinuous wave dynamics.

ABSTRACT

In the field of mathematical physics, there exist many physically interesting nonlinear dispersive equations with peakon solutions, which are solitary waves with discontinuous first-order derivative at the wave peak. In this paper, we apply the multi-layer physics-informed neural networks (PINNs) deep learning to successfully study the data-driven peakon and periodic peakon solutions of some well-known nonlinear dispersion equations with initial-boundary value conditions such as the Camassa-Holm (CH) equation, Degasperis-Procesi equation, modified CH equation with cubic nonlinearity, Novikov equation with cubic nonlinearity, mCH-Novikov equation, b-family equation with quartic nonlinearity, generalized modified CH equation with quintic nonlinearity, and etc. These results will be useful to further study the peakon solutions and corresponding experimental design of nonlinear dispersive equations.

Motivation & Objective

• To develop a deep learning framework for discovering peakon and periodic peakon solutions in nonlinear dispersive equations with complex nonlinearities.
• To address the challenge of modeling solitary waves with discontinuous derivatives using data-driven methods.
• To apply physics-informed neural networks (PINNs) to initial-boundary value problems of peakon equations, ensuring adherence to PDE constraints.
• To validate the accuracy of PINN-predicted solutions against numerical and exact solutions for benchmark equations like CH, DP, and Novikov.

Proposed method

• The study employs multi-layer physics-informed neural networks (PINNs) to solve initial-boundary value problems for nonlinear dispersive equations of the form (1−∂x²)ut + N(u,ux,uxx,...) = 0.
• The PINN loss function combines data fidelity from initial, boundary, and collocation points with residual minimization of the PDE over the spatio-temporal domain.
• A pseudo-spectral method generates high-fidelity training data for the PINN, using Fourier modes and time-stepping to simulate solutions over defined domains.
• The network architecture uses 5–6 hidden layers with 10–20 neurons per layer and hyperbolic tangent activation functions to model complex wave dynamics.
• Training involves minimizing a mean squared error (MSE) loss over 1000 collocation points, 10 initial, 10 boundary, and 10 end-time points to ensure solution consistency.
• The method is applied to multiple equations including the Degasperis-Procesi (DP), Camassa-Holm (CH), Novikov, and mCH-Novikov equations with varying nonlinearities and boundary conditions.

Experimental results

Research questions

• RQ1 Can physics-informed neural networks accurately recover peakon solutions with discontinuous derivatives in nonlinear dispersive equations?
• RQ2 How well can PINNs approximate periodic peakon solutions under periodic boundary conditions and complex nonlinear terms?
• RQ3 What is the accuracy of PINN-predicted solutions compared to numerical and exact solutions for benchmark peakon equations like DP and CH?
• RQ4 How do hyperparameters such as network depth, width, and training data distribution affect solution convergence and error?
• RQ5 Can PINNs generalize across different nonlinearities and parameter regimes in the b-family and generalized mCH equations?

Key findings

• The PINN framework successfully recovered the bright peakon solution of the Degasperis-Procesi equation with an L2 error of 2.74e-02 against numerical solutions.
• For the dark peakon solution of the DP equation with c = -1.05, the L2 error between the learned and numerical solutions was 3.29e-02.
• The method accurately captured periodic peakon solutions of the DP equation, achieving an L2 error of 1.96e-02 with a 6-layer PINN and 512 Fourier modes.
• The PINN model demonstrated robustness across different initial conditions and spatio-temporal domains, including [-5,5]×[0,3] and [-7,7]×[0,3] for the DP equation.
• The framework was successfully extended to multiple peakon equations, including the generalized mCH-Novikov and b-family equations with cubic and higher-order nonlinearities.
• The results confirm that PINNs can effectively model complex, non-smooth wave solutions in nonlinear dispersive PDEs with high accuracy and consistency.