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Volume 42, Issue 6
Tensor Neural Network and Its Numerical Integration

Yifan Wang, Hehu Xie & Pengzhan Jin

DOI: 10.4208/jcm.2307-m2022-0233

J. Comp. Math., 42 (2024), pp. 1714-1742.

Published online: 2024-11

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  • Abstract

In this paper, we introduce a type of tensor neural network. For the first time, we propose its numerical integration scheme and prove the computational complexity to be the polynomial scale of the dimension. Based on the tensor product structure, we develop an efficient numerical integration method by using fixed quadrature points for the functions of the tensor neural network. The corresponding machine learning method is also introduced for solving high-dimensional problems. Some numerical examples are also provided to validate the theoretical results and the numerical algorithm.

  • Keywords

Tensor neural network, Numerical integration, Fixed quadrature points, Machine learning, High-dimensional eigenvalue problem.

  • AMS Subject Headings

65N30, 65N25, 65L15, 65B99

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-1714, author = {Wang , YifanXie , Hehu and Jin , Pengzhan}, title = {Tensor Neural Network and Its Numerical Integration}, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {42}, number = {6}, pages = {1714--1742}, abstract = {

In this paper, we introduce a type of tensor neural network. For the first time, we propose its numerical integration scheme and prove the computational complexity to be the polynomial scale of the dimension. Based on the tensor product structure, we develop an efficient numerical integration method by using fixed quadrature points for the functions of the tensor neural network. The corresponding machine learning method is also introduced for solving high-dimensional problems. Some numerical examples are also provided to validate the theoretical results and the numerical algorithm.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2307-m2022-0233}, url = {http://global-sci.org/intro/article_detail/jcm/23513.html} }
TY - JOUR T1 - Tensor Neural Network and Its Numerical Integration AU - Wang , Yifan AU - Xie , Hehu AU - Jin , Pengzhan JO - Journal of Computational Mathematics VL - 6 SP - 1714 EP - 1742 PY - 2024 DA - 2024/11 SN - 42 DO - http://doi.org/10.4208/jcm.2307-m2022-0233 UR - https://global-sci.org/intro/article_detail/jcm/23513.html KW - Tensor neural network, Numerical integration, Fixed quadrature points, Machine learning, High-dimensional eigenvalue problem. AB -

In this paper, we introduce a type of tensor neural network. For the first time, we propose its numerical integration scheme and prove the computational complexity to be the polynomial scale of the dimension. Based on the tensor product structure, we develop an efficient numerical integration method by using fixed quadrature points for the functions of the tensor neural network. The corresponding machine learning method is also introduced for solving high-dimensional problems. Some numerical examples are also provided to validate the theoretical results and the numerical algorithm.

Wang , YifanXie , Hehu and Jin , Pengzhan. (2024). Tensor Neural Network and Its Numerical Integration. Journal of Computational Mathematics. 42 (6). 1714-1742. doi:10.4208/jcm.2307-m2022-0233
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