Projected Sobolev Natural Gradient Descent for Efficient Neural Network Solution of the Gross-Pitaevskii Equation
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866908795304673280 |
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| author | Bao, Chenglong Cui, Chen Jiang, Kai Shu, Shi |
| author_facet | Bao, Chenglong Cui, Chen Jiang, Kai Shu, Shi |
| contents | This paper introduces a projected Sobolev natural gradient descent (NGD) method for computing ground states of the Gross-Pitaevskii equation. By projecting a continuous Riemannian Sobolev gradient flow onto the normalized neural network tangent space, we derive a discrete NGD algorithm that preserves the normalization constraint. The numerical implementation employs variational Monte Carlo with a hybrid sampling strategy to accurately account for the normalization constant arising from nonlinear interaction terms. To enhance computational efficiency, a matrix-free Nyström-preconditioned conjugate gradient solver is adopted to approximate the NGD operator without explicit matrix assembly. Numerical experiments demonstrate that the proposed method converges significantly faster than physics-informed neural network approaches and exhibits linear scalability with respect to spatial dimensions. Moreover, the resulting neural-network solutions provide high-quality initial guesses that substantially accelerate subsequent refinement by traditional high-precision solvers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_11339 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Projected Sobolev Natural Gradient Descent for Efficient Neural Network Solution of the Gross-Pitaevskii Equation Bao, Chenglong Cui, Chen Jiang, Kai Shu, Shi Numerical Analysis This paper introduces a projected Sobolev natural gradient descent (NGD) method for computing ground states of the Gross-Pitaevskii equation. By projecting a continuous Riemannian Sobolev gradient flow onto the normalized neural network tangent space, we derive a discrete NGD algorithm that preserves the normalization constraint. The numerical implementation employs variational Monte Carlo with a hybrid sampling strategy to accurately account for the normalization constant arising from nonlinear interaction terms. To enhance computational efficiency, a matrix-free Nyström-preconditioned conjugate gradient solver is adopted to approximate the NGD operator without explicit matrix assembly. Numerical experiments demonstrate that the proposed method converges significantly faster than physics-informed neural network approaches and exhibits linear scalability with respect to spatial dimensions. Moreover, the resulting neural-network solutions provide high-quality initial guesses that substantially accelerate subsequent refinement by traditional high-precision solvers. |
| title | Projected Sobolev Natural Gradient Descent for Efficient Neural Network Solution of the Gross-Pitaevskii Equation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2512.11339 |