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

TL;DR

The paper addresses predicting the existence and evolution of quantum droplets in binary Bose-Einstein condensates governed by a one-dimensional Gross-Pitaevskii equation with Lee-Huang-Yang corrections. It employs Physics-Informed Neural Networks (PINNs) to solve forward dynamics and perform data-driven parameter discovery by embedding the PDE residuals into the loss function, defined as $Loss = \mathrm{MSE}_{\mathrm{IC}} + \mathrm{MSE}_{\mathrm{BC}} + \mathrm{MSE}_{\mathrm{F}}$. The authors demonstrate accurate replication of monopole to sextupole QD structures and evolutions, show that shallow networks can suffice for high accuracy, and successfully recover coefficients $\lambda_1$ and $\lambda_2$ even with 1% noisy data, underscoring PINN robustness for inverse problems in nonlinear quantum fluids. This approach offers a practical, data-efficient tool for modeling complex QD dynamics in binary BECs and for extracting physically meaningful parameters from limited observations.

Abstract

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.

Figures (9)

• Figure 1: The schematic PINN architecture for data-driven prediction of the evolution of the nonlinear system. The left panel includes the input, hidden, and output layers. The middle column represents the automatic differentiation technique in Tensorflow. The right panel shows the PINN dominated by the PDE construction, which corresponds to the PDE solution produced by the split-step method. The bottom corresponds to the loss function and optimization process (the Adam and L-BFGS optimizer).
• Figure 2: The performance of the PINN for the density of the fundamental QDs. (a, b) The reference solution and the predicted output, respectively. (c) The squared error between the reference and predicted solution. (d)-(f) The comparisons between the profiles of the learned and reference fundamental QDs in three distinct snapshots, taken as $t=0.5,1.0$ and $1.5$, corresponding to the white vertical lines in (a).
• Figure 3: The performance of the PINN for the density of the flat-top fundamental QDs. (a, b) The reference solution and the predicted output, respectively. (c) The squared error between the reference and predicted solution. (d)-(f) The comparisons between the profiles of the learned and reference fundamental QDs in three distinct snapshots, taken as $t=0.5,1.0$ and $1.5$, corresponding to the white vertical lines in (a).
• Figure 4: The training results of dipole droplet. (a, b) The reference and learned solutions, respectively. (c) The squared error between the reference and learned solution. Panels (d)-(f) illustrate the comparisons between the reference solution and PINNs results at indicated value of time.
• Figure 5: The training results of flat-top dipole droplet. (a, b) The reference and learned solutions, respectively. (c) The squared error between the reference and learned solution. Panels (d)-(f) illustrate the comparisons between the reference solution and PINNs results at indicated value of time.