On examining the predictive capabilities of two variants of PINN in validating localised wave solutions in the generalized nonlinear Schrödinger equation

Abstract

We introduce a novel neural network structure called Strongly Constrained Theory-Guided Neural Network (SCTgNN), to investigate the behaviours of the localized solutions of the generalized nonlinear Schr\"{o}dinger (NLS) equation. This equation comprises four physically significant nonlinear evolution equations, namely, (i) NLS equation, Hirota equation Lakshmanan-Porsezian-Daniel (LPD) equation and fifth-order NLS equation. The generalized NLS equation demonstrates nonlinear effects up to quintic order, indicating rich and complex dynamics in various fields of physics. By combining concepts from the Physics-Informed Neural Network (PINN) and Theory-Guided Neural Network (TgNN) models, SCTgNN aims to enhance our understanding of complex phenomena, particularly within nonlinear systems that defy conventional patterns. To begin, we employ the TgNN method to predict the behaviours of localized waves, including solitons, rogue waves, and breathers, within the generalized NLS equation. We then use SCTgNN to predict the aforementioned localized solutions and calculate the mean square errors in both SCTgNN and TgNN in predicting these three localized solutions. Our findings reveal that both models excel in understanding complex behaviours and provide predictions across a wide variety of situations.