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

Table 4. The correct NLSE with ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential and the identified twos obtained by learning V0 and W0, and the relative errors.

Item Nonlinear evolution equation Relative error [V0, W0](%) PDE Correct ${\rm{i}}{\psi }_{t}=-{\psi }_{{xx}}-[2.91{{\rm{sech}} }^{2}(x)+0.3\mathrm{isech}(x)\tanh (x)]\psi +| \psi {| }^{2}\psi $ [0, 0] Identified (clean data) ${\rm{i}}{\psi }_{t}=-{\psi }_{{xx}}-[2.91283{{\rm{sech}} }^{2}(x)+0.303\,28\mathrm{isech}(x)\tanh (x)]\psi +| \psi {| }^{2}\psi $ [0.097, 1.093] Identified (1 % noise) ${\rm{i}}{\psi }_{t}=-{\psi }_{{xx}}-[2.89035{{\rm{sech}} }^{2}(x)+0.301\,65\mathrm{isech}(x)\tanh (x)]\psi +| \psi {| }^{2}\psi $ [0.675, 0.552]