Multi-parallelized PINNs for the inverse problem study of NLS typed equations in optical fiber communications: Discovery on diverse high-order terms and variable coefficients

Abstract

In this paper, we focus on the inverse problem study of nonlinear Schrödinger (NLS) typed equations in optical fiber communicaitons. As an extension of the physics-informed neural network (PINN), multi-parallelized PINNs are constructed and trained for the discovery of diverse high-order terms and variable coefficients. We firstly study various constant-coefficient combinations of a generalized high-order NLS typed equation, where the Chen-Lee-Liu equation, the Gerdjikov-Ivanov equation and the Kundu-Eckhaus equation are included. With small amount of exact solutions available to us, we predict the value of multiple coefficients under different cases to deduce the undetermined terms of the generalized equation based on the multi-parallelized PINN. Different categories of NLS typed equations are then inferred. In the meantime, high accuracy numerical solutions on localized regions can be accordingly obtained.The parameter discovery of NLS typed equations with variable coefficients has also been carried out based on the extended network, including analysis on interaction behaviors and the periodic phenomenon of solutions. According to outputs of multi-parallelized PINNs, we compare the numerical solutions and the predicted variable coefficients with exact results. Error analysis are then performed to check the accuracy of prediction, where both the absolute error and the mean squared error are given. Compared with the traditional PINN, our model exerts its state-of-art power in the inverse problem study of nonlinear systems, where different high-order terms and variable-coefficient terms can be clearly predicted while deducing diverse types of localized numerical solutions with lower fitting error and less data consumption. As the self-steepening pulses without self-phase modulation and the Raman effect are closely related to the inferred high-order terms of NLS typed equations, our research will serve as experimental basis in the field of optical fiber communications.