On examining the predictive capabilities of two variants of PINN in validating localised wave solutions in the generalized nonlinear Schr\"{o}dinger equation

TL;DR

This paper introduces a novel neural network called SCTgNN to improve the prediction of localized wave solutions in complex nonlinear Schrödinger equations, demonstrating enhanced accuracy over existing models.

Contribution

The paper presents SCTgNN, a new theory-guided neural network that combines physics-informed and theory-guided approaches to better predict localized solutions in generalized nonlinear Schrödinger equations.

Findings

Both SCTgNN and TgNN effectively predict localized wave behaviors.

SCTgNN shows improved mean square error performance.

Models handle complex nonlinear dynamics successfully.

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.