Data-driven soliton mappings for integrable fractional nonlinear wave equations via deep learning with Fourier neural operator
This work addresses the challenge of modeling complex wave systems in physics and engineering, but it is incremental as it applies an existing deep learning method to new fractional equations.
The paper tackled the problem of discovering soliton mappings for fractional integrable nonlinear wave equations by extending the Fourier neural operator (FNO) to map between fractional-order index spaces and solitonic solution spaces, achieving results compared to exact solutions with recorded train and test losses.
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 $\{ε|ε\in (0, 1)\}$ in the fractional integrable nonlinear wave equations while another denotes the solitonic solution function space. To be specific, the fractional nonlinear Schrödinger (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.