# Data-driven soliton mappings for integrable fractional nonlinear wave equations via deep learning with Fourier neural operator

arXiv: https://arxiv.org/abs/2209.14291 · HTML: https://mlsanity.com/paper/2209.14291/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 75 references that are in the map.
- First posted on arXiv 29 Aug 2022.

## Builds on (most cited first, 20 of 20)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
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- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fast R-CNN](https://mlsanity.com/paper/1504.08083/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)
- [The Random Feature Model for Input-Output Maps between Banach Spaces](https://mlsanity.com/paper/2005.10224/) (2020)
- [Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modeling](https://mlsanity.com/paper/2203.08205/) (2022)
- [A physics-informed operator regression framework for extracting data-driven continuum models](https://mlsanity.com/paper/2009.11992/) (2020)
- [Theory of the Frequency Principle for General Deep Neural Networks](https://mlsanity.com/paper/1906.09235/) (2019)
- [Learned coupled inversion for carbon sequestration monitoring and forecasting with Fourier neural operators](https://mlsanity.com/paper/2203.14396/) (2022)

## 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)

In this paper, we firstly extend the Fourier neural operator (FNO) to discovery the soliton mapping between two function spaces, where one is the fractional-order index space ε|ε∈ (0, 1) in the fractional integrable nonlinear wave equations while another denotes the solitonic solution function space. To be specific, the fractional nonlinear Schr"odinger (fNLS), fractional Korteweg-de Vries (fKdV), fractional modified Korteweg-de Vries (fmKdV) and fractional sine-Gordon (fsineG) equations proposed recently are studied in this paper. We present the train and evaluate progress by recording the train and test loss. To illustrate the accuracies, the data-driven solitons are also compared to the exact solutions. Moreover, we consider the influences of several critical factors (e.g., activation functions containing Relu(x), Sigmoid(x), Swish(x) and xtanh(x), depths of fully connected layer) on the performance of the FNO algorithm. We also use a new activation function, namely, xtanh(x), which is not used in the field of deep learning. The results obtained in this paper may be useful to further understand the neural networks in the fractional integrable nonlinear wave systems and the mappings between two spaces.