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Computing the ground state solution of Bose-Einstein condensates by an energy-minimizing normalized residual network |
| Ren-Tao Wu(吴任涛)1, Ji-Dong Gao(高济东)1, Yu-Han Wang(王宇晗)1, Zhen-Wei Deng(邓振威)1, Ming-Jun Li(李明军)2, and Rong-Pei Zhang(张荣培)1,† |
1 School of Advanced Manufacturing, Guangdong University of Technology, Jieyang 522000, China;
2 School of Mathematics and Computational Science, Xiangtan University, Xiangtang 411105, China |
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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.
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Received: 19 February 2025
Revised: 17 May 2025
Accepted manuscript online: 04 June 2025
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PACS:
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03.75.Hh
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(Static properties of condensates; thermodynamical, statistical, and structural properties)
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07.05.Mh
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(Neural networks, fuzzy logic, artificial intelligence)
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02.60.Lj
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(Ordinary and partial differential equations; boundary value problems)
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| Fund: Project supported by the National Natural Science Foundation of China (Grant No. 11971411). |
Corresponding Authors:
Rong-Pei Zhang
E-mail: rongpei_zhang@gdut.edu.cn
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Cite this article:
Ren-Tao Wu(吴任涛), Ji-Dong Gao(高济东), Yu-Han Wang(王宇晗), Zhen-Wei Deng(邓振威), Ming-Jun Li(李明军), and Rong-Pei Zhang(张荣培) Computing the ground state solution of Bose-Einstein condensates by an energy-minimizing normalized residual network 2025 Chin. Phys. B 34 100305
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