Sum-of-Gaussians tensor neural networks for high-dimensional Schrödinger equation
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arXiv
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| Auteurs principaux: | , , , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908865777369088 |
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| author | Zhou, Qi Wu, Teng Liu, Jianghao Sun, Qingyuan Xie, Hehu Xu, Zhenli |
| author_facet | Zhou, Qi Wu, Teng Liu, Jianghao Sun, Qingyuan Xie, Hehu Xu, Zhenli |
| contents | We propose an accurate, efficient, and low-memory sum-of-Gaussians tensor neural network (SOG-TNN) algorithm for solving the high-dimensional Schrödinger equation. The SOG-TNN utilizes a low-rank tensor product representation of the solution to overcome the curse of dimensionality associated with high-dimensional integration. To handle the Coulomb interaction, we introduce an SOG decomposition to approximate the interaction kernel such that it is dimensionally separable, leading to a tensor representation with rapid convergence. We further develop a range-splitting scheme that partitions the Gaussian terms into short-, long-, and mid-range components. They are treated with the asymptotic expansion, the low-rank Chebyshev expansion, and the model reduction with singular-value decomposition, respectively, significantly reducing the number of two-dimensional integrals in computing electron-electron interactions. The SOG decomposition well resolves the computational challenge due to the singularity of the Coulomb interaction, leading to an efficient algorithm for the high-dimensional problem under the TNN framework. Numerical results demonstrate the outstanding performance of the new method, revealing that the SOG-TNN is a promising way for accurately tackling quantum systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10454 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sum-of-Gaussians tensor neural networks for high-dimensional Schrödinger equation Zhou, Qi Wu, Teng Liu, Jianghao Sun, Qingyuan Xie, Hehu Xu, Zhenli Computational Physics Numerical Analysis 35Q40, 65D40, 65N25, 68W25, 68W40 We propose an accurate, efficient, and low-memory sum-of-Gaussians tensor neural network (SOG-TNN) algorithm for solving the high-dimensional Schrödinger equation. The SOG-TNN utilizes a low-rank tensor product representation of the solution to overcome the curse of dimensionality associated with high-dimensional integration. To handle the Coulomb interaction, we introduce an SOG decomposition to approximate the interaction kernel such that it is dimensionally separable, leading to a tensor representation with rapid convergence. We further develop a range-splitting scheme that partitions the Gaussian terms into short-, long-, and mid-range components. They are treated with the asymptotic expansion, the low-rank Chebyshev expansion, and the model reduction with singular-value decomposition, respectively, significantly reducing the number of two-dimensional integrals in computing electron-electron interactions. The SOG decomposition well resolves the computational challenge due to the singularity of the Coulomb interaction, leading to an efficient algorithm for the high-dimensional problem under the TNN framework. Numerical results demonstrate the outstanding performance of the new method, revealing that the SOG-TNN is a promising way for accurately tackling quantum systems. |
| title | Sum-of-Gaussians tensor neural networks for high-dimensional Schrödinger equation |
| topic | Computational Physics Numerical Analysis 35Q40, 65D40, 65N25, 68W25, 68W40 |
| url | https://arxiv.org/abs/2508.10454 |