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Data-driven 2D stationary quantum droplets and wave propagations in the amended GP equation with two potentials via deep neural networks learning

Jin Song, Zhenya Yan

Abstract

In this paper, we develop a systematic deep learning approach to solve two-dimensional (2D) stationary quantum droplets (QDs) and investigate their wave propagation in the 2D amended Gross-Pitaevskii equation with Lee-Huang-Yang correction and two kinds of potentials. Firstly, we use the initial-value iterative neural network (IINN) algorithm for 2D stationary quantum droplets of stationary equations. Then the learned stationary QDs are used as the initial value conditions for physics-informed neural networks (PINNs) to explore their evolutions in the some space-time region. Especially, we consider two types of potentials, one is the 2D quadruple-well Gaussian potential and the other is the PT-symmetric HO-Gaussian potential, which lead to spontaneous symmetry breaking and the generation of multi-component QDs. The used deep learning method can also be applied to study wave propagations of other nonlinear physical models.

Data-driven 2D stationary quantum droplets and wave propagations in the amended GP equation with two potentials via deep neural networks learning

Abstract

In this paper, we develop a systematic deep learning approach to solve two-dimensional (2D) stationary quantum droplets (QDs) and investigate their wave propagation in the 2D amended Gross-Pitaevskii equation with Lee-Huang-Yang correction and two kinds of potentials. Firstly, we use the initial-value iterative neural network (IINN) algorithm for 2D stationary quantum droplets of stationary equations. Then the learned stationary QDs are used as the initial value conditions for physics-informed neural networks (PINNs) to explore their evolutions in the some space-time region. Especially, we consider two types of potentials, one is the 2D quadruple-well Gaussian potential and the other is the PT-symmetric HO-Gaussian potential, which lead to spontaneous symmetry breaking and the generation of multi-component QDs. The used deep learning method can also be applied to study wave propagations of other nonlinear physical models.
Paper Structure (8 sections, 20 equations, 12 figures)

This paper contains 8 sections, 20 equations, 12 figures.

Table of Contents

  1. Introduction
  2. The framework of deep learning method
  3. The IINN framework for stationary QDs
  4. The PINNs framework for the evolution of QDs
  5. Data-driven 2D QDs in amended GP equation with potentials
  6. Data-driven QDs in amended GP equation with quadruple-well Gaussian potential
  7. Data-driven QDs in amended GP equation with ${\cal PT}$-symmetric HOG potential
  8. Conclusions and discussions

Figures (12)

  • Figure 1: (a1) Norm $N$ vs the chemical potential $\mu$ for 2D QDs in quadruple-well Gaussian potential (\ref{['potential1']}) starting from the ground state in the linear regime (dotted: unstable; solid: stable). (a2) Local diagrams of relevant bifurcations in (a1). (a3) Norm $N$ (big) vs the chemical potential $\mu$ for 2D QDs starting from the ground state in the linear regime including lower and upper branches fourwell.
  • Figure 2: The QDs in branch A0 of 2D amended GP equation with quadruple-well Gaussian potential. (a1) The intensity diagram $|\phi({\bf r})|$ of learned solution by IINN method. (a2) The 3D profile of the learned solution. (a3) The module of absolute error between the exact and learned solutions. (b1, b2, b3) The intensity diagram of the learned solutions at different time $t = 0,\, 2.5$, and $5.0$, respectively. (c1, c2) The initial state of the learned solution by IINN method and PINNs method. (c3) Isosurface of learned QDs at values 0.1, 0.5 and 0.9.
  • Figure 3: The loss-iteration plots. (a1) The QDs in branch A0 for IINN. (a2) The QDs in branch A0 for PINNs. (b1) The QDs in branch A1 for IINN. (b2) The QDs in branch A1 for PINNs.
  • Figure 4: The QDs in branch A1 of 2D amended GP equation with quadruple-well Gaussian potential. (a1) The intensity diagram $|\phi({\bf r})|$ of learned solution by IINN method. (a2) The 3D profile of the learned solution. (a3) The module of absolute error between the exact and learned solutions. (b1, b2, b3) The intensity diagram of the learned solutions at different time $t = 0,\, 2.5$, and $5.0$, respectively. (c1, c2) The initial state of the learned solution by IINN method and PINNs method. (c3) Isosurface of learned QDs at values 0.1, 0.5 and 0.9.
  • Figure 5: The QDs in branch A3 of 2D amended GP equation with quadruple-well Gaussian potential. (a1) The intensity diagram $|\phi({\bf r})|$ of learned solution by IINN method. (a2) The 3D profile of the learned solution. (a3) The module of absolute error between the exact and learned solutions. (b1, b2, b3) The intensity diagram of the learned solutions at different time $t = 0,\, 2.5$, and $5.0$, respectively. (c1, c2) The initial state of the learned solution by IINN method and PINNs method. (c3) Isosurface of learned QDs at values 0.1, 0.5 and 0.9.
  • ...and 7 more figures