Data-driven solitons and parameter discovery to the (2+1)-dimensional NLSE in optical fiber communications

In this paper, we investigate the (2+1)-dimensional nonlinear Schrödinger equation (NLSE) which characterizes the transmission of optical pulses through optical fibers exhibiting refractive index variations corresponding to light intensity changes. Traditional numerical methods typically require a substantial amount of data to ensure the accuracy when solving high-dimensional NLSE, resulting in high experimental costs as well as a significant demand for storage space and computing power. With physical knowledge embedded into deep neural networks, physics-informed neural network (PINN) has been widely applied to solve various complex nonlinear problems and achieved significant results with small amount of data. Setting different groups of initial conditions and boundary conditions with hyperbolic and exponential functions, we construct the corresponding loss functions which will be further applied to train PINN. All data studied here is generated on Python. Based on the predicted results, we depict different types of optical pulses. According to our data experiments, lower prediction errors can be achieved with small volume of data, which fully demonstrates the effectiveness of the PINN. In the meantime, we also perform data-driven parameter discovery to the (2+1)-dimensional NLSE to study the coefficients of the group velocity dispersion and self-phase modulation terms. It can be seen that the PINN has high accuracy and robustness for parameter discovery to the (2+1)-dimensional NLSE. In brief, the use of PINN greatly enriches the diversity of solving methods, providing a reference for research of (2+1)-dimensional solitons in the field of optical fiber communications.