关键词: Bose-Einstein condensate, Gross-Pitaevskii equation, energy minimization, normalized residual network

Abstract: This paper introduces a novel numerical method based on an energy-minimizing normalized residual network (EM-NormResNet) to compute the ground-state solution of Bose-Einstein condensates at zero or low temperatures. Starting from the three-dimensional Gross-Pitaevskii equation (GPE), we reduce it to the 1D and 2D GPEs because of the radial symmetry and cylindrical symmetry. The ground-state solution is formulated by minimizing the energy functional under constraints, which is directly solved using the EM-NormResNet approach. The paper provides detailed solutions for the ground states in 1D, 2D (with radial symmetry), and 3D (with cylindrical symmetry). We use the Thomas-Fermi approximation as the target function to pre-train the neural network. Then, the formal network is trained using the energy minimization method. In contrast to traditional numerical methods, our neural network approach introduces two key innovations: (i) a novel normalization technique designed for high-dimensional systems within an energy-based loss function; (ii) improved training efficiency and model robustness by incorporating gradient stabilization techniques into residual networks. Extensive numerical experiments validate the method's accuracy across different spatial dimensions.

Key words: Bose-Einstein condensate, Gross-Pitaevskii equation, energy minimization, normalized residual network