Exploring data driven soliton and rogue waves in \(\mathcal{P}\mathcal{T}\) symmetric and spatio-temporal potentials using PINN and SC-PINN methods

Abstract

In this study, we present a deep learning (DL) framework for solving the nonlinear Schrödinger equation with \(\mathcal{P}\mathcal{T}\)-symmetric potentials using strongly constrained physics-informed neural networks (SC-PINNs). We focus on three types of physically compelling potentials, namely \(\mathcal{P}\mathcal{T}\)-symmetric rational, Jacobian-periodic, and spatio-temporal dependent. SC-PINNs extend the standard physics-informed neural networks (PINNs) by incorporating compound derivative information into the soft constraints, along with an adaptive weight mechanism to accelerate loss function convergence. This enhancement significantly improves the training efficiency compared to standard PINNs. We employ SC-PINNs to approximate soliton and rogue wave solutions of the system under investigation. Additionally, we uncover the impact of various factors on the neural network’s performance, including five different nonlinear activation functions: ReLU, sigmoid, sech, tanh, and sine. Our results reveal that the SC-PINNs method achieves faster convergence and lower errors compared to traditional PINNs. Notably, when using the sine activation function for the three distinct potentials mentioned above, SC-PINNs reduced errors to the order of \(10^{-7}\), \(10^{-6}\), and \(10^{-4}\), effectively capturing complex physical features for highly accurate predictions. Furthermore, we analyze the effect of \(\mathcal{P}\mathcal{T}\)-symmetric potential parameters on the obtained approximated solutions. The results demonstrate that our DL model successfully approximates soliton and rogue wave solutions of the considered system with high accuracy, outperforming traditional DL algorithms.