Solving forward and inverse problems of the logarithmic nonlinear Schrodinger equation with PT-symmetric harmonic potential via deep learning

TL;DR

This paper applies physics-informed neural networks to solve and analyze the logarithmic nonlinear Schrödinger equation with PT-symmetric harmonic potential, demonstrating effectiveness in forward, inverse, and data-driven discovery tasks.

Contribution

It introduces a deep learning approach using PINNs for solving and discovering the LNLS equation with PT-symmetric potential, including comparisons with spectral methods.

Findings

PINNs accurately solve the LNLS equation under various conditions.

The method effectively discovers coefficients and parameters from data.

PINNs outperform traditional spectral methods in certain scenarios.

Abstract

In this paper, we investigate the logarithmic nonlinear Schr\"odinger (LNLS) equation with the parity-time (PT)-symmetric harmonic potential, which is an important physical model in many fields such as nuclear physics, quantum optics, magma transport phenomena, and effective quantum gravity. Three types of initial value conditions and periodic boundary conditions are chosen to solve the LNLS equation with PT -symmetric harmonic potential via the physics-informed neural networks (PINNs) deep learning method, and these obtained results are compared with ones deduced from the Fourier spectral method. Moreover, we also investigate the effectiveness of the PINNs deep learning for the LNLS equation with PT symmetric potential by choosing the distinct space widths or distinct optimized steps. Finally, we use the PINNs deep learning method to effectively tackle the data-driven discovery of the LNLS equation with PT -symmetric harmonic potential such that the coefficients of dispersion and nonlinear terms or the amplitudes of PT -symmetric harmonic potential can be approximately found.