Solving forward and inverse problems of the nonlinear Schrödinger equation with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential via PINN deep learning

Jiaheng Li,Biao Li

Figure 6. The activation function's influence on the learning ability of this complex-valued PINN in the self-focusing NLSE with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential and initial boundary value conditions in Case 1 given in Section 3.1: the deep learning results in four different activation functions $\{{\rm{ReLU}},{\rm{Leaky}}\ {\rm{ReLUs}},{\rm{Sigmoid}},{\rm{Tanh}}\}$ from the first row to the last row. Left column images : the magnitude of the approximate predicted solution $| \hat{\psi }(t,x)| $ with the initial and Dirichlet boundary training data and 10 000 collocation points. Right three columns : comparisons of the exact and predicted solutions at the three temporal snapshots described by the three dotted black lines in the panels in the first column corresponding to time instants t = 0.5, 1.0 and 1.5.