The coupled deep neural networks for coupling of the Stokes and Darcy-Forchheimer problems

Jing Yue(岳靖)^{1}, Jian Li(李剑)^{1,†}, Wen Zhang(张文)^{1}, and Zhangxin Chen(陈掌星)^{2,3}

1 School of Electrical and Control Engineering, School of Mathematics and Data Science, Shaanxi University of Science and Technology, Xi'an 710021, China; 2 School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China; 3 Department of Chemical and Petroleum Engineering, Schulich School of Engineering, University of Calgary, 2500 University Drive N. W., Calgary, Alberta T2 N; 1 N4, Canada

Abstract We present an efficient deep learning method called coupled deep neural networks (CDNNs) for coupling of the Stokes and Darcy-Forchheimer problems. Our method compiles the interface conditions of the coupled problems into the networks properly and can be served as an efficient alternative to the complex coupled problems. To impose energy conservation constraints, the CDNNs utilize simple fully connected layers and a custom loss function to perform the model training process as well as the physical property of the exact solution. The approach can be beneficial for the following reasons: Firstly, we sample randomly and only input spatial coordinates without being restricted by the nature of samples. Secondly, our method is meshfree, which makes it more efficient than the traditional methods. Finally, the method is parallel and can solve multiple variables independently at the same time. We present the theoretical results to guarantee the convergence of the loss function and the convergence of the neural networks to the exact solution. Some numerical experiments are performed and discussed to demonstrate performance of the proposed method.

Fund: Project supported in part by the National Natural Science Foundation of China (Grant No. 11771259), the Special Support Program to Develop Innovative Talents in the Region of Shaanxi Province, the Innovation Team on Computationally Efficient Numerical Methods Based on New Energy Problems in Shaanxi Province, and the Innovative Team Project of Shaanxi Provincial Department of Education (Grant No. 21JP013).

Corresponding Authors:
Jian Li
E-mail: jianli@sust.edu.cn,jiaaanli@gmail.com

Cite this article:

Jing Yue(岳靖), Jian Li(李剑), Wen Zhang(张文), and Zhangxin Chen(陈掌星) The coupled deep neural networks for coupling of the Stokes and Darcy-Forchheimer problems 2023 Chin. Phys. B 32 010201

[1] Li J, Bai Y and Zhao X 2023 Modern Numerical Methods for Mathematical Physics Equations (Beijing: Science Press) p. 10 (in Chinese) [2] Li J, Lin X and Chen Z 2022 Finite Volume Methods for the Incompressible Navier-Stokes Equations (Berlin: Springer) p. 15 [3] Li J 2019 Numerical Methods for the Incompressible Navier-Stokes Equations (Beijing: Science Press) p. 8 [4] Saffman P G 1971 Stud. Appl. Math.50 93 [5] Forchheimer P 1901 Zeitz. Ver. Duetch Ing.45 1782 (in Japanese) [6] Park E J 1995 SIAM J. Numer. Anal.32 865 [7] Kim M Y and Park E J 1999 Comput. Math. Appl.38 113 [8] Park E J 2005 Numer. Methods Part. Differ. Equ.21 213 [9] Discacciati M, Miglio E and Quarteroni A 2002 Appl. Numer. Math.43 57 [10] Layton W J, Schieweck F and Yotov I 2003 SIAM J. Numer. Anal.40 2195 [11] Riviére B 2005 J. Sci. Comput.22 479 [12] Riviére B and Yotov I 2005 SIAM J. Numer. Anal.42 1959 [13] Burman E and Hansbo P 2007 J. Comput. Appl. Math.198 35 [14] Gatica G N, Oyarzúa R and Sayas F J 2011 Math. Comput.80 1911 [15] Girault V, Vassilev D and Yotov I 2014 Numer. Math.127 93 [16] Lipnikov K, Vassilev D and Yotov I 2014 Numer. Math.126 321 [17] Qiu C X, He X M, Li J and Lin Y P 2020 J. Comput. Phys.411 109400 [18] Li R, Gao Y L, Li J and Chen Z X 2018 J. Comput. Appl. Math.334 111 [19] He Y N and Li J 2010 Int. J. Numer. Anal. Mod.62 647 [20] Liu X, Li J and Chen Z X 2018 J. Comput. Appl. Math.333 442 [21] Li J, Mei L Q and He Y N 2006 Appl. Math. Comput.182 24 [22] Zhu L P, Li J and Chen Z X 2011 J. Comput. Appl. Math.235 2821 [23] Krizhevsky A, Sutskever I and Hinton G E 2012 Commun. ACM64 84 [24] Hinton G, Deng L, Yu D, et al. 2012 IEEE Signal Proc. Mag.29 82 [25] He K M, Zhang X Y, Ren S Q, et al. 2016 Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition June 27-30, 2016, Las Vegas, NV, USA p. 770 [26] Cotter N E 1990 IEEE Trans. Neural Networks4 290 [27] Hornik K, Stinchcombe M and White H 1989 Neural Networks2 359 [28] Hornik K, Stinchcombe M and White H 1990 Neural Networks3 551 [29] Hornik K 1991 Neural Networks4 251 [30] Cybenko G 1989 Math. Control Signal.2 303 [31] Telgrasky M 2016 Proc. Mach. Learn. Res.49 1517 [32] Mhaskar H, Liao Q L and Poggio T 2016 arXiv:1603.00988v4[cs.LG] [33] Khoo Y, Lu J F and Ying L X 2017 arXiv:1707.03351[math.NA] [34] Li J, Yue J, Zhang W, et al. 2022 J. Sci. Comput. (accepted) [35] Li J, Zhang W and Yue J 2021 Int. J. Numer. Anal. Model.18 427 [36] Yue J and Li J 2022 Int. J. Numer. Methods Fluids.94 1416 [37] Yue J and Li J 2023 Appl. Math. Comput.437 127514 [38] Fan Y W, Lin L, Ying L X, et al. 2018 arXiv:1807.01883[math.NA] [39] Wang M, Cheung S W, Chung E T, et al. 2018 arXiv:1810.12245[math.NA] [40] Li X 1996 Neurocomputing12 327 [41] Lagaris I E, Likas A C and Fotiadis D I 1998 IEEE Trans. Neural Network9 987 [42] Lagaris I E, Likas A C and Papageorgiou D G 2000 IEEE Trans. Neural Network11 1041 [43] McFall K S and Mahan J R 2009 IEEE Trans. Neural Network20 1221 [44] Raissi M, Perdikaris P and Karniadakis G E 2017 arXiv:1711.10561[cs.AI] [45] Raissi M, Perdikaris P and Karniadakis G E 2017 arXiv:1711.10566[cs.AI] [46] Raissi M, Perdikaris P and Karniadakis G E 2019 J. Comput. Phys.378 686 [47] Yang L, Meng X H and Karniadakis G E 2021 J. Comput. Phys.425 109913 [48] Rao C P, Sun H and Liu Y 2020 arXiv: 2006.08472v1[math.NA] [49] Olivier P and Fablet R 2020 arXiv:2002.01029 [physics.comp-ph] [50] Lu L, Meng X H, Mao Z P, et al. 2021 SIAM Rev.63 208 [51] Fang Z W and Zhan J 2020 IEEE Access8 26328 [52] Pang G F, Lu L and Karniadakis G E 2019 SIAM J. Sci. Comput.41 A2603 [53] Zhu Y H, Zabaras N, Koutsourelakis P S, et al. 2019 J. Comput. Phys.394 56 [54] Sirignano J and Spiliopoulos K 2018 J. Comput. Phys.375 1339 [55] Beaver G S and Joseph D D 1967 J. Fluid Mech.30 197 [56] Zhao L, Chung E T, Park E J and Zhou G 2021 SIAM J. Numer. Anal.59 1 [57] Kovasznay L I G 1948 Math. Proc. Cambridge44 58 [58] Léon Bottou 2012 Lecture Notes in Computer Science Grégoire M, Genevieve B O and Klaus R M (eds) (Berlin: Springer) pp. 430-445

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