# Data-driven discoveries of B"acklund transforms and soliton evolution equations via deep neural network learning schemes

arXiv: https://arxiv.org/abs/2111.09489 · HTML: https://mlsanity.com/paper/2111.09489/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 66 references that are in the map.
- First posted on arXiv 18 Nov 2021.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [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)
- [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)
- [Data-driven peakon and periodic peakon travelling wave solutions of some nonlinear dispersive equations via deep learning](https://mlsanity.com/paper/2101.04371/) (2021)
- [Deep learning neural networks for the third-order nonlinear Schrodinger equation: Solitons, breathers, and rogue waves](https://mlsanity.com/paper/2104.14809/) (2021)

## Built on by (most cited first, 1 of 1)

- [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)

We introduce a deep neural network learning scheme to learn the B"acklund transforms (BTs) of soliton evolution equations and an enhanced deep learning scheme for data-driven soliton equation discovery based on the known BTs, respectively. The first scheme takes advantage of some solution (or soliton equation) information to study the data-driven BT of sine-Gordon equation, and complex and real Miura transforms between the defocusing (focusing) mKdV equation and KdV equation, as well as the data-driven mKdV equation discovery via the Miura transforms. The second deep learning scheme uses the explicit/implicit BTs generating the higher-order solitons to train the data-driven discovery of mKdV and sine-Gordon equations, in which the high-order solution informations are more powerful for the enhanced leaning soliton equations with higher accurates.