Wave-packet behaviors of the defocusing nonlinear Schrödinger equation based on the modified physics-informed neural networks.

Investigated in this paper is the defocusing nonlinear Schrödinger (NLS) equation, which is used for describing the wave-packet dynamics in certain weakly nonlinear media. With the physics-informed neural networks (PINNs), we modify the corresponding loss function in the existing literature and obtain two types of dark solitons, type-I and type-II solitons. It is demonstrated that the modified loss function presents higher-precision wave-packet behaviors based on fewer initial and boundary data. Taking type-I solitons into consideration, we find that when only a small fraction of initial and boundary data are given, the prediction accuracy of the wave packets will be increased one or two orders of magnitude at least if the modification term of the loss function is introduced. Furthermore, for the inverse problem, the modified loss function provides a better estimate of the nonlinear coefficient of the NLS equation based on fewer observed data of the wave packets. For type-II solitons, we compare the required data and predicted results of the PINNs with those of the conventional time-splitting finite difference (TSFD) method and reveal that achieving the same precision of the wave-packet behavior, the PINNs with the modified loss functions require only one tenth of the amount of the initial and boundary data of the TSFD method. Besides, both unmodified and modified loss functions are exploited for predicting the behaviors of Gaussian wave packets, and it is observed that the predicted result of the modified loss function agrees with the high-precision solution of the time-splitting Fourier pseudospectral method, whereas the unmodified loss function fails.