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

In this paper, we firstly extend the Fourier neural operator (FNO) to discovery the mapping between two infinite-dimensional 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 soliton solution in the spatio-temporal function space. In other words, once the FNO network is trained, for any given ϵ∈(0,1), the corresponding soliton solution can be quickly obtained. To be specific, the soliton solutions are learned for the fractional nonlinear Schrödinger (fNLS), fractional Korteweg–de Vries (fKdV), fractional modified Korteweg–de Vries (fmKdV) and fractional sine-Gordon (fsineG) equations. The FNO architecture is utilized to learn the soliton mappings of the above four equations. The data-driven solitons are also compared with exact solutions to illustrate the powerful approximation capability of the FNO. Moreover, we study the influences of several critical factors (e.g., activation functions containing Relu(x), Sigmoid(x), Swish(x) and the new one xtanh(x), channels of fully connected layer) on the performance of the FNO algorithm. As a result, we find that the xtanh(x) and Swish(x) functions perform better than the Relu(x) and Sigmoid(x) functions in the FNO, and the FNO network with a more-channel fully-connected layer performs better as we expect. These results obtained in this paper will be useful to further understand the neural networks for fractional integrable nonlinear wave equations and the mappings between two infinite-dimensional function spaces.