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Volume 37, Issue 1
Newton’s Method and Its Hybrid with Machine Learning for Navier-Stokes Darcy Models Discretized by Mixed Element Methods

Jianguo Huang, Hui Peng & Haohao Wu

DOI: 10.4208/cicp.OA-2024-0066

Commun. Comput. Phys., 37 (2025), pp. 30-60.

Published online: 2025-01

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  • Abstract

This paper focuses on discussing Newton’s method and its hybrid with machine learning for the steady state Navier-Stokes Darcy model discretized by mixed element methods. First, a Newton iterative method is introduced for solving the relative discretized problem. It is proved technically that this method converges quadratically with the convergence rate independent of the mixed element mesh size, under certain standard conditions. Later on, a deep learning algorithm is proposed for solving this nonlinear coupled problem. Following the ideas of an earlier work by Huang, Wang and Yang (2020), an Int-Deep algorithm is constructed by combining the previous two methods so as to further improve the computational efficiency and robustness. A series of numerical examples are reported to show the numerical performance of the proposed methods.

  • Keywords

Navier-Stokes Darcy model, deep learning, Newton iterative method, Int-Deep method, convergence analysis.

  • AMS Subject Headings

65N12, 65N15, 65N22, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-37-30, author = {Huang , JianguoPeng , Hui and Wu , Haohao}, title = {Newton’s Method and Its Hybrid with Machine Learning for Navier-Stokes Darcy Models Discretized by Mixed Element Methods}, journal = {Communications in Computational Physics}, year = {2025}, volume = {37}, number = {1}, pages = {30--60}, abstract = {

This paper focuses on discussing Newton’s method and its hybrid with machine learning for the steady state Navier-Stokes Darcy model discretized by mixed element methods. First, a Newton iterative method is introduced for solving the relative discretized problem. It is proved technically that this method converges quadratically with the convergence rate independent of the mixed element mesh size, under certain standard conditions. Later on, a deep learning algorithm is proposed for solving this nonlinear coupled problem. Following the ideas of an earlier work by Huang, Wang and Yang (2020), an Int-Deep algorithm is constructed by combining the previous two methods so as to further improve the computational efficiency and robustness. A series of numerical examples are reported to show the numerical performance of the proposed methods.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2024-0066}, url = {http://global-sci.org/intro/article_detail/cicp/23780.html} }
TY - JOUR T1 - Newton’s Method and Its Hybrid with Machine Learning for Navier-Stokes Darcy Models Discretized by Mixed Element Methods AU - Huang , Jianguo AU - Peng , Hui AU - Wu , Haohao JO - Communications in Computational Physics VL - 1 SP - 30 EP - 60 PY - 2025 DA - 2025/01 SN - 37 DO - http://doi.org/10.4208/cicp.OA-2024-0066 UR - https://global-sci.org/intro/article_detail/cicp/23780.html KW - Navier-Stokes Darcy model, deep learning, Newton iterative method, Int-Deep method, convergence analysis. AB -

This paper focuses on discussing Newton’s method and its hybrid with machine learning for the steady state Navier-Stokes Darcy model discretized by mixed element methods. First, a Newton iterative method is introduced for solving the relative discretized problem. It is proved technically that this method converges quadratically with the convergence rate independent of the mixed element mesh size, under certain standard conditions. Later on, a deep learning algorithm is proposed for solving this nonlinear coupled problem. Following the ideas of an earlier work by Huang, Wang and Yang (2020), an Int-Deep algorithm is constructed by combining the previous two methods so as to further improve the computational efficiency and robustness. A series of numerical examples are reported to show the numerical performance of the proposed methods.

Huang , JianguoPeng , Hui and Wu , Haohao. (2025). Newton’s Method and Its Hybrid with Machine Learning for Navier-Stokes Darcy Models Discretized by Mixed Element Methods. Communications in Computational Physics. 37 (1). 30-60. doi:10.4208/cicp.OA-2024-0066
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