Data-driven two-dimensional stationary quantum droplets and wave propagations in the amended Gross–Pitaevskii equation with two potentials via deep neural networks learning

In this paper, we develop a systematic deep learning approach to solve two-dimensional stationary quantum droplets (QDs) and investigate their wave propagation in the two-dimensional amended Gross–Pitaevskii equation with Lee–Huang–Yang correction and two kinds of potentials. Firstly, we use the initial-value iterative neural network algorithm for two-dimensional stationary QDs of stationary equations. Then, the learned stationary QDs are used as the initial value conditions for physics-informed neural networks to explore their evolutions in the space-time region. Especially, we consider two types of potentials, one is the two-dimensional quadruple-well Gaussian potential and the other is the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mrow><mml:mi class="mathcal" mathvariant="script">P</mml:mi><mml:mi class="mathcal" mathvariant="script">T</mml:mi></mml:mrow></mml:mrow></mml:mstyle></mml:math> -symmetric harmonic-oscillator (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.