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 3. The correct NLSE with ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential and the identified twos obtained by learning m and g, and the relative errors.

Item Nonlinear evolution equation Relative error [m, g](%) PDE Correct iψt = − ψxx − [V(x) + iW(x)]ψ + ∣ψ∣2ψ [0, 0] Identified (clean data) iψt = − 1.00353ψxx − [V(x) + iW(x)]ψ + 0.994 05∣ψ∣2ψ [0.353, 0.595] Identified (1% noise) iψt = − 0.98293ψxx − [V(x) + iW(x)]ψ + 1.006 51∣ψ∣2ψ [1.707, 0.651]