# Data-driven peakon and periodic peakon travelling wave solutions of some nonlinear dispersive equations via deep learning

arXiv: https://arxiv.org/abs/2101.04371 · HTML: https://mlsanity.com/paper/2101.04371/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 51 references that are in the map.
- First posted on arXiv 12 Jan 2021.

## Builds on (most cited first, 9 of 9)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems](https://mlsanity.com/paper/1603.04467/) (2016)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks](https://mlsanity.com/paper/1905.01205/) (2019)
- [Deep Neural Networks for Nonlinear Model Order Reduction of Unsteady Flows](https://mlsanity.com/paper/2007.00936/) (2020)
- [Data-driven rogue waves and parameter discovery in the defocusing NLS equation with a potential using the PINN deep learning](https://mlsanity.com/paper/2012.09984/) (2020)

## Built on by (most cited first, 6 of 6)

- [Is the neural tangent kernel of PINNs deep learning general partial differential equations always convergent ?](https://mlsanity.com/paper/2412.06158/) (2024)
- [Deep learning soliton dynamics and complex potentials recognition for 1D and 2D PT-symmetric saturable nonlinear Schr"odinger equations](https://mlsanity.com/paper/2310.02276/) (2023)
- [Two-stage initial-value iterative physics-informed neural networks for simulating solitary waves of nonlinear wave equations](https://mlsanity.com/paper/2409.01124/) (2024)
- [Data-driven discoveries of B"acklund transforms and soliton evolution equations via deep neural network learning schemes](https://mlsanity.com/paper/2111.09489/) (2021)
- [Deep neural networks for solving forward and inverse problems of (2+1)-dimensional nonlinear wave equations with rational solitons](https://mlsanity.com/paper/2112.14040/) (2021)
- [Data-driven localized waves and parameter discovery in the massive Thirring model via extended physics-informed neural networks with interface zones](https://mlsanity.com/paper/2309.17240/) (2023)

## Abstract (arXiv)

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.