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

arXiv: https://arxiv.org/abs/2409.02339 · HTML: https://mlsanity.com/paper/2409.02339/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 57 references that are in the map.
- First posted on arXiv 4 Sep 2024.

## Builds on (most cited first, 13 of 13)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs](https://mlsanity.com/paper/1909.10145/) (2019)
- [Transfer learning enhanced physics informed neural network for phase-field modeling of fracture](https://mlsanity.com/paper/1907.02531/) (2019)
- [Parallel Physics-Informed Neural Networks via Domain Decomposition](https://mlsanity.com/paper/2104.10013/) (2021)
- [Long-time integration of parametric evolution equations with physics-informed DeepONets](https://mlsanity.com/paper/2106.05384/) (2021)
- [Evolutional Deep Neural Network](https://mlsanity.com/paper/2103.09959/) (2021)
- [Data-driven rogue waves and parameter discovery in the defocusing NLS equation with a potential using the PINN deep learning](https://mlsanity.com/paper/2012.09984/) (2020)
- [Deep learning soliton dynamics and complex potentials recognition for 1D and 2D PT-symmetric saturable nonlinear Schr"odinger equations](https://mlsanity.com/paper/2310.02276/) (2023)
- [Two-stage initial-value iterative physics-informed neural networks for simulating solitary waves of nonlinear wave equations](https://mlsanity.com/paper/2409.01124/) (2024)

## Abstract (arXiv)

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.