Computing the ground state solution of Bose–Einstein condensates by an energy-minimizing normalized residual network

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.