Physics-Informed Neural Networks for the Quantum Droplets in Binary Bose-Einstein Condensates

Physics-Informed Neural Networks (PINNs), which integrate deep learning with physical prior knowledge, have proven to be a powerful tool for studying the dynamics of high-dimensional nonlinear systems. The present work utilizes PINNs to analyze the existence and evolution of quantum droplets (QDs) in a binary Bose-Einstein condensate (BEC), revealing the ability of this technique to accurately predict structural features of the QDs, their multipeak profiles, and dynamical behavior. The stable evolution of multipole QDs is thus demonstrated. Comparing different network architectures, including the training time, loss values, and $\mathbb{L_{2}}$ error, PINNs accurately predict specific dynamical characteristics of QDs. Furthermore, the PINN robustness is evaluated by the application of PINN to parameter-discovery tasks, considering both clean training data and data contaminated by $1%$ random noise. The results highlight the efficiency of PINNs in modeling complex quantum systems and extracting reliable parameters under the noisy conditions.

💡 Research Summary

This paper investigates the application of Physics‑Informed Neural Networks (PINNs) to the study of quantum droplets (QDs) in binary Bose‑Einstein condensates (BECs). Quantum droplets are self‑bound, ultra‑dilute superfluid objects that arise from a delicate balance between mean‑field (MF) cubic repulsion and the beyond‑mean‑field Lee‑Huang‑Yang (LHY) quantum correction, which provides an attractive quartic term. In a binary BEC with equal intra‑component interaction strengths, the two‑component system can be reduced to an effective single‑component one‑dimensional Gross‑Pitaevskii equation (GPE) that contains both a cubic repulsive and a quadratic attractive term. After appropriate nondimensionalization, the governing equation reads

 i ∂ψ/∂t = −½ ∂²ψ/∂x² − |ψ|² ψ + |ψ|² ψ,

where ψ(x,t) is the complex wavefunction. Stationary droplet solutions are sought in the form ψ(x,t)=φ(x) e^{−iμt}, with φ(x)→0 as |x|→∞. The total norm N and energy E are defined by integrals of φ(x). Multi‑peak (multipole) droplets consist of several such peaks arranged in a stable array, each peak retaining the stability of an isolated droplet while the overall structure is maintained by inter‑droplet interactions.

The authors construct a PINN that directly approximates the complex solution ψ=u+i v by outputting the real and imaginary parts u(t,x) and v(t,x). The network architecture consists of an input layer (t, x), several hidden layers (typically five layers with thirty neurons each), and an output layer. A hyperbolic tangent activation function is used throughout. The loss function combines three mean‑squared‑error (MSE) terms: (i) MSE_IC enforces the initial condition at t=0, (ii) MSE_BC enforces periodic (or absorbing) boundary conditions in space, and (iii) MSE_F penalizes the residual of the governing GPE evaluated via automatic differentiation. The total loss is simply the sum of these three contributions, ensuring that the network respects both data and physics.

Training data are generated by high‑resolution numerical simulations of the GPE using a Newton‑conjugate‑gradient solver for the stationary state and a split‑step Fourier method with absorbing boundaries for the time evolution. Collocation points in the space‑time domain are sampled using Latin Hypercube Sampling (LHS), providing a quasi‑uniform coverage while keeping the number of points moderate (on the order of 10⁴). Optimization proceeds in two stages: first, the Adam stochastic gradient optimizer quickly reduces the loss; then, the quasi‑Newton L‑BFGS algorithm refines the parameters to machine‑precision accuracy. The convergence criterion is based on the relative change of the loss between successive iterations.

The paper presents two main sets of results. First, the PINN accurately reproduces the dynamics of both fundamental (single‑peak) and multipole QDs. Simulations are performed on the domain t∈