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

TL;DR

This paper extends Fourier neural operators to discover soliton mappings in fractional nonlinear wave equations, demonstrating high accuracy and exploring the effects of various neural network parameters, including a novel activation function.

Contribution

The paper introduces a new application of Fourier neural operators for fractional wave equations and proposes a novel activation function, enhancing the understanding of neural network performance in this context.

Findings

FNO effectively models soliton mappings in fractional wave equations.

The new activation function $x\tanh(x)$ improves neural network performance.

Comparison with exact solutions confirms high accuracy of the data-driven approach.

Abstract

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 $\{\epsilon|\epsilon\in (0, 1)\}$ in the fractional integrable nonlinear wave equations while another denotes the solitonic solution function space. To be specific, the fractional nonlinear Schr\"{o}dinger (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 $x\tanh(x)$, depths of fully connected layer) on the performance of the FNO algorithm. We also use a new activation function, namely, $x\tanh(x)$, 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.