Data driven multi soliton solutions of the Fokas-Lenells equation using conservation laws in PINN

We propose a conserved density aided neural network to obtain data driven multi-soliton solutions of the Fokas-Lenells equation (FLE). We add the FLE and conserved densities in the loss function of the neural network. Using this modified Physics informed Neural Network (PINN) we obtain the data driven bright soliton, two-soliton interaction and dark soliton solutions of the FLE. We present a comparative result of data driven solutions generated by PINN and the modified PINN. We notice that modified PINN with conserved densities loss function achieve better accuracy compared to conventional PINN in terms of relative \(\varvec{L2}\) $\mathit{L}\mathbf{2}$ error between predicted and exact soliton solutions. We further notice that in comparison to using a single conserved density, using multiple conserved density produces better results. We believe that the present investigation with the modified PINN would be useful to study the applications of deep learning in nonlinear optics as well as other branches of nonlinear physics. Source codes are available at https://github.com/gautamksaharia/Fokas-Lenells