Physics-informed neural network for nonlinear dynamics of self-trapped necklace beams

TL;DR

This paper employs a physics-informed neural network to model and analyze the complex nonlinear dynamics of self-trapped necklace beams governed by the (2+1)D nonlinear Schrödinger equation, revealing stable propagation and parameter discovery.

Contribution

The study introduces a PINN approach to accurately predict and analyze the evolution of necklace patterns with various angular momenta, including parameter discovery with noisy data.

Findings

PINN accurately predicts nonlinear dynamics of necklace beams.

Necklace patterns exhibit stable propagation with structure preservation.

PINN effectively discovers parameters even with noisy training data.

Abstract

A physics-informed neural network (PINN) is used to produce a variety of self-trapped necklace solutions of the (2+1)-dimensional nonlinear Schr\"{o}dinger/Gross-Pitaevskii equation. We elaborate the analysis for the existence and evolution of necklace patterns with integer, half-integer, and fractional reduced orbital angular momenta by means of PINN. The patterns exhibit phenomena similar to rotation of rigid bodies and centrifugal force. Even though the necklaces slowly expand (or shrink), they preserve their structure in the course of the quasi-stable propagation over several diffraction lengths, which is completely different from the ordinary fast diffraction-dominated dynamics. By comparing different ingredients, including the training time, loss value and $\mathbb{L}_{2}$ error, PINN accurately predicts specific nonlinear dynamical properties of the evolving necklace patterns. Furthermore, we perform the data-driven discovery of parameters for both clean and perturbed training data, adding $1\%$ random noise in the latter case. The results reveal that PINN not only effectively emulates the solution of partial differential equations, but also offers applications for predicting the nonlinear dynamics of physically relevant types of patterns.