doi:10.1088/1674-1056/ac7554

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

doi: 10.1088/1674-1056/ac7554

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 T2N 1N4,Canada

摘要

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.
- scientific computing /
- machine learning /
- the Stokes equations /
- Darcy-Forchheimer problems /
- Beavers-Joseph-Saffman interface condition

HTML全文

1. Coupled domain with interface Γ.

下载: 全尺寸图片幻灯片

2. The structure of the CDNNs.

下载: 全尺寸图片幻灯片

3. The influence of different hidden layers and different training data on err L ² (Test 1): (a) 400 training points, (b) one hidden layer.

下载: 全尺寸图片幻灯片

4. The contrast of the exact solution and the CDNNs (Test 1).

下载: 全尺寸图片幻灯片

5. The point-wise errors (Test 1).

下载: 全尺寸图片幻灯片

6. The point-wise errors (Test 2).

下载: 全尺寸图片幻灯片

7. The value of ρ (Test 3).

下载: 全尺寸图片幻灯片

8. The point-wise errors (Test 3).

下载: 全尺寸图片幻灯片

9. The results of CDNNs (Test 4).

下载: 全尺寸图片幻灯片

10. The velocity flows of Stokes and Darcy (Test 4).

下载: 全尺寸图片幻灯片

11. The results of the CDNNs (Test 5).

下载: 全尺寸图片幻灯片

12. The velocity flows of Stokes and Darcy (Test 5).

下载: 全尺寸图片幻灯片

下载: 导出CSV

Table 1.. The relative errors of Test 1.

		400 sampled points
1 layer	U _S	P _S	U _D	P _D
err L ¹	2.49 × 10 ⁰	9.15 × 10 ⁰	8.72 × 10 ⁰	2.74 × 10 ⁻¹
err L ²	4.84 × 10 ⁰	9.42 × 10 ⁰	1.94 × 10 ⁰	2.99 × 10 ⁻¹
2 layers	U _S	P _S	U _D	P _D
err L ¹	4.85 × 10 ⁻¹	3.59 × 10 ⁰	9.62 × 10 ⁻²	4.03 × 10 ⁻²
err L ²	9.01 × 10 ⁻¹	3.21 × 10 ⁰	2.19 × 10 ⁻¹	4.18 × 10 ⁻²
3 layers	U _S	P _S	U _D	P _D
err L ¹	5.80 × 10 ⁻³	5.26 × 10 ⁻²	1.01 × 10 ⁻²	3.25 × 10 ⁻³
err L ²	1.09 × 10 ⁻²	4.66 × 10 ⁻²	2.29 × 10 ⁻²	3.38 × 10 ⁻³

下载: 导出CSV

Table 2.. The relative errors of Test 2.

		400 sampled points
1 layer	U _S	P _S	U _D	P _D
err L ¹	1.82 × 10 ⁻²	1.19 × 10 ⁻¹	1.57 × 10 ⁻²	1.08 × 10 ⁻²
err L ²	3.66 × 10 ⁻²	1.23 × 10 ⁻¹	4.62 × 10 ⁻²	1.11 × 10 ⁻²
2 layers	U _S	P _S	U _D	P _D
err L ¹	2.27 × 10 ⁻⁴	1.74 × 10 ⁻³	1.04 × 10 ⁻⁴	4.67 × 10 ⁻⁵
err L ²	4.21 × 10 ⁻⁴	2.00 × 10 ⁻³	3.18 × 10 ⁻⁴	5.55 × 10 ⁻⁵
3 layers	U _S	P _S	U _D	P _D
err L ¹	1.65 × 10 ⁻⁴	1.01 × 10 ⁻³	1.13 × 10 ⁻⁴	7.69 × 10 ⁻⁵
err L ²	3.37 × 10 ⁻⁴	1.50 × 10 ⁻³	3.44 × 10 ⁻⁴	8.32 × 10 ⁻⁵

下载: 导出CSV

Table 3.. The relative errors of Test 3.

		400 sampled points
1 layer	U _S	P _S	U _D	P _D
err L ¹	3.59 × 10 ⁻¹	5.03 × 10 ⁰	5.52 × 10 ⁻²	7.59 × 10 ⁻²
err L ²	6.79 × 10 ⁻¹	5.20 × 10 ⁰	1.73 × 10 ⁻¹	7.07 × 10 ⁻²
2 layers	U _S	P _S	U _D	P _D
err L ¹	8.42 × 10 ⁻⁴	9.41 × 10 ⁻³	1.15 × 10 ⁻³	1.32 × 10 ⁻³
err L ²	1.65 × 10 ⁻³	1.05 × 10 ⁻²	3.43 × 10 ⁻³	1.56 × 10 ⁻³
3 layers	U _S	P _S	U _D	P _D
err L ¹	1.89 × 10 ⁻⁴	3.04 × 10 ⁻³	2.97 × 10 ⁻⁴	6.70 × 10 ⁻⁵
err L ²	3.65 × 10 ⁻⁴	3.37 × 10 ⁻³	8.98 × 10 ⁻⁴	8.40 × 10 ⁻⁵

下载: 导出CSV

Table 4.. The errors in interface of Test 4 ( K = 10000).

		400 sampled points
	Condition 1	Condition 2	Condition 3
1 layer	6.49 × 10 ⁻²	9.14 × 10 ⁻²	3.03 × 10 ⁻²
2 layers	3.67 × 10 ⁻⁵	4.74 × 10 ⁻²	6.27 × 10 ⁻⁴
3 layers	3.37 × 10 ⁻⁵	7.53 × 10 ⁻³	1.44 × 10 ⁻⁵

下载: 导出CSV

参考文献(58)

[1]	Li J, Bai Y, Zhao X 2023 Modern Numerical Methods for Mathematical Physics Equations Beijing Science Press 10 in Chinese
[2]	Li J, Lin X, Chen Z 2022 Finite Volume Methods for the Incompressible Navier–Stokes Equations Berlin Springer 15
[3]	Li J 2019 Numerical Methods for the Incompressible Navier–Stokes Equations Beijing Science Press 8
[4]	Saffman P G 1971 Stud. Appl. Math. 50 93 10.1002/sapm.v50.2 doi: 10.1002/sapm.v50.2
[5]	Forchheimer P 1901 Zeitz. Ver. Duetch Ing. 45 1782 10.5917/jagh1987.45.279 doi: 10.5917/jagh1987.45.279
[6]	Park E J 1995 SIAM J. Numer. Anal. 32 865 10.1137/0732040 doi: 10.1137/0732040
[7]	Kim M Y, Park E J 1999 Comput. Math. Appl. 38 113 10.1016/S0898-1221(99)00291-6 doi: 10.1016/S0898-1221(99)00291-6
[8]	Park E J 2005 Numer. Methods Part. Differ. Equ. 21 213 10.1002/num.20035 doi: 10.1002/num.20035
[9]	Discacciati M, Miglio E, Quarteroni A 2002 Appl. Numer. Math. 43 57 10.1016/S0168-9274(02)00125-3 doi: 10.1016/S0168-9274(02)00125-3
[10]	Layton W J, Schieweck F, Yotov I 2003 SIAM J. Numer. Anal. 40 2195 10.1137/S0036142901392766 doi: 10.1137/S0036142901392766
[11]	Riviere B 2005 J. Sci. Comput. 22 479 10.1007/s10915-004-4147-3 doi: 10.1007/s10915-004-4147-3
[12]	Riviere B, Yotov I 2005 SIAM J. Numer. Anal. 42 1959 10.1137/S0036142903427640 doi: 10.1137/S0036142903427640
[13]	Burman E, Hansbo P 2007 J. Comput. Appl. Math. 198 35 10.1016/j.cam.2005.11.022 doi: 10.1016/j.cam.2005.11.022
[14]	Gatica G N, ua R, Sayas F J 2011 Math. Comput. 80 1911
[15]	Girault V, Vassilev D, Yotov I 2014 Numer. Math. 127 93 10.1007/s00211-013-0583-z doi: 10.1007/s00211-013-0583-z
[16]	Lipnikov K, Vassilev D, Yotov I 2014 Numer. Math. 126 321 10.1007/s00211-013-0563-3 doi: 10.1007/s00211-013-0563-3
[17]	Qiu C X, He X M, Li J, Lin Y P 2020 J. Comput. Phys. 411 109400 10.1016/j.jcp.2020.109400 doi: 10.1016/j.jcp.2020.109400
[18]	Li R, Gao Y L, Li J, Chen Z X 2018 J. Comput. Appl. Math. 334 111 10.1016/j.cam.2017.11.011 doi: 10.1016/j.cam.2017.11.011
[19]	He Y N, Li J 2010 Int. J. Numer. Anal. Mod. 62 647 10.1002/fld.2035 doi: 10.1002/fld.2035
[20]	Liu X, Li J, Chen Z X 2018 J. Comput. Appl. Math. 333 442 10.1016/j.cam.2017.11.010 doi: 10.1016/j.cam.2017.11.010
[21]	Li J, Mei L Q, He Y N 2006 Appl. Math. Comput. 182 24 10.1016/j.amc.2006.03.030 doi: 10.1016/j.amc.2006.03.030
[22]	Zhu L P, Li J, Chen Z X 2011 J. Comput. Appl. Math. 235 2821 10.1016/j.cam.2010.12.001 doi: 10.1016/j.cam.2010.12.001
[23]	Krizhevsky A, Sutskever I, Hinton G E 2012 Commun. ACM 64 84 10.1145/3065386 doi: 10.1145/3065386
[24]	Hinton G, Deng L, Yu D, et al. 2012 IEEE Signal Proc. Mag. 29 82 10.1109/MSP.2012.2205597 doi: 10.1109/MSP.2012.2205597
[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 770
[26]	Cotter N E 1990 IEEE Trans. Neural Networks 4 290 10.1109/72.80265 doi: 10.1109/72.80265
[27]	Hornik K, Stinchcombe M, White H 1989 Neural Networks 2 359 10.1016/0893-6080(89)90020-8 doi: 10.1016/0893-6080(89)90020-8
[28]	Hornik K, Stinchcombe M, White H 1990 Neural Networks 3 551 10.1016/0893-6080(90)90005-6 doi: 10.1016/0893-6080(90)90005-6
[29]	Hornik K 1991 Neural Networks 4 251 10.1016/0893-6080(91)90009-T doi: 10.1016/0893-6080(91)90009-T
[30]	Cybenko G 1989 Math. Control Signal. 2 303 10.1007/BF02551274 doi: 10.1007/BF02551274
[31]	Telgrasky M 2016 Proc. Mach. Learn. Res. 49 1517
[32]	Mhaskar H, Liao Q L, Poggio T 2016 arXiv:1603.00988v4 [cs.LG]
[33]	Khoo Y, Lu J F, Ying L X 2017 arXiv:1707.03351 [math.NA]
[34]	Li J, Yue J, Zhang W, et al. 2022 J. Sci. Comput. 10.1007/s10915-022-01930-8 doi: 10.1007/s10915-022-01930-8
[35]	Li J, Zhang W, Yue J 2021 Int. J. Numer. Anal. Model. 18 427
[36]	Yue J, Li J 2022 Int. J. Numer. Methods Fluids. 94 1416 10.1002/fld.5095 doi: 10.1002/fld.5095
[37]	Yue J, Li J 2023 Appl. Math. Comput. 437 127514 10.1016/j.amc.2022.127514 doi: 10.1016/j.amc.2022.127514
[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 Neurocomputing 12 327 10.1016/0925-2312(95)00070-4 doi: 10.1016/0925-2312(95)00070-4
[41]	Lagaris I E, Likas A C, Fotiadis D I 1998 IEEE Trans. Neural Network 9 987 10.1109/72.712178 doi: 10.1109/72.712178
[42]	Lagaris I E, Likas A C, Papageorgiou D G 2000 IEEE Trans. Neural Network 11 1041 10.1109/72.870037 doi: 10.1109/72.870037
[43]	McFall K S, Mahan J R 2009 IEEE Trans. Neural Network 20 1221 10.1109/TNN.2009.2020735 doi: 10.1109/TNN.2009.2020735
[44]	Raissi M, Perdikaris P, Karniadakis G E 2017 arXiv:1711.10561 [cs.AI]
[45]	Raissi M, Perdikaris P, Karniadakis G E 2017 arXiv:1711.10566 [cs.AI]
[46]	Raissi M, Perdikaris P, Karniadakis G E 2019 J. Comput. Phys. 378 686 10.1016/j.jcp.2018.10.045 doi: 10.1016/j.jcp.2018.10.045
[47]	Yang L, Meng X H, Karniadakis G E 2021 J. Comput. Phys. 425 109913 10.1016/j.jcp.2020.109913 doi: 10.1016/j.jcp.2020.109913
[48]	Rao C P, Sun H, Liu Y 2020 arXiv:2006.08472v1 [math.NA]
[49]	Olivier P, 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 10.1137/19M1274067 doi: 10.1137/19M1274067
[51]	Fang Z W, Zhan J 2020 IEEE Access 8 26328 10.1109/ACCESS.2019.2963390 doi: 10.1109/ACCESS.2019.2963390
[52]	Pang G F, Lu L, Karniadakis G E 2019 SIAM J. Sci. Comput. 41 A2603 10.1137/18M1229845 doi: 10.1137/18M1229845
[53]	Zhu Y H, Zabaras N, Koutsourelakis P S, et al. 2019 J. Comput. Phys. 394 56 10.1016/j.jcp.2019.05.024 doi: 10.1016/j.jcp.2019.05.024
[54]	Sirignano J, Spiliopoulos K 2018 J. Comput. Phys. 375 1339 10.1016/j.jcp.2018.08.029 doi: 10.1016/j.jcp.2018.08.029
[55]	Beaver G S, Joseph D D 1967 J. Fluid Mech. 30 197 10.1017/S0022112067001375 doi: 10.1017/S0022112067001375
[56]	Zhao L, Chung E T, Park E J, Zhou G 2021 SIAM J. Numer. Anal. 59 1 10.1137/19M1268525 doi: 10.1137/19M1268525
[57]	Kovasznay L I G 1948 Math. Proc. Cambridge 44 58 10.1017/S0305004100023999 doi: 10.1017/S0305004100023999
[58]	Lèon Bottou 2012 Lecture Notes in Computer Science Grègoire M, Genevieve B O, Klaus R M Berlin Springer 430 445