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The Deep Learning Galerkin Method for the General Stokes Equations

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Abstract

This paper considers the general stokes problems applying the Deep learning Galerkin Method (DGM) and gives the convergence of the DGM which contains two parts. First, guided by data and physical laws, depending on the \( L^{2} \) error we construct an objective function and control the performance of the approximation solution by minimizing the objective function in which the prior knowledge of PDEs and data are encoded. Then, we prove the convergence of the neural network to the exact solution. In particular, due to it is mesh free, the DGM can reduce the computational complexity and achieve the competitive results especially in face of the high dimensional problems. With this, compared with traditional numerical methods, numerical results verify the theoretical analysis and show the applicability and effectiveness of the proposed method.

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Funding

This work is supported in part by NSF of China (No. 11771259), special support program to develop innovative talents in the region of Shaanxi province, innovation team on computationally efficient numerical methods based on new energy problems in Shaanxi province, and innovative team project of Shaanxi Provincial Department of Education (No. 21JP013).

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Authors and Affiliations

  1. School of Mathematics and Data Science, Shaanxi University of Science and Technology, Xi’an, China

    Jian Li

  2. School of Electrical and Control Engineering, Shaanxi University of Science and Technology, Xi’an, China

    Jing Yue & Wen Zhang

  3. Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China

    Wansuo Duan

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  1. Jian Li
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  2. Jing Yue
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  3. Wen Zhang
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  4. Wansuo Duan
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Correspondence to Jian Li.

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Li, J., Yue, J., Zhang, W. et al. The Deep Learning Galerkin Method for the General Stokes Equations. J Sci Comput 93, 5 (2022). https://doi.org/10.1007/s10915-022-01930-8

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  • DOI: https://doi.org/10.1007/s10915-022-01930-8

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