Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
167 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
42 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Newton's method and its hybrid with machine learning for Navier-Stokes Darcy Models discretized by mixed element methods (2401.10557v2)

Published 19 Jan 2024 in math.NA and cs.NA

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 finite 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.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (43)
  1. R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
  2. T. Arbogast, D.S. Brunson, A computationalmethod for approximating a Darcy-Stokes system governing a vuggy porous medium, Comput. Geosci., 11 (2007), 207-218.
  3. T. Arbogast, H.L. Lehr, Homogenization of a Darcy-Stokes system modeling vuggy porous media, Comput. Geosci., 10 (2006), 291-302.
  4. Mathematical Analysis of the Navier-Stokes and Darcy Coupling, Technical report, Politecnico di Milano, Milan, 2006.
  5. Numerical analysis of the Navier-Stokes Darcy coupling, Numer. Math., 115 (2010), 195-227.
  6. J. Berg, K. Nyström, A unified deep artificial neural network approach to partial differential equations in complex geometries, Neurocomputing , 317 (2018), 28-41.
  7. Mixed Finite Element Methods and Applications, Springer, Berlin, 2013.
  8. S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods (Third Edition), Springer, New York, 2008.
  9. Numerical solution to a mixed Navier-Stokes Darcy model by the two-grid approach, SIAM J. Numer. Anal., 47 (2009), 3325-3338.
  10. Finite element approximations for Stokes-Darcy flow with Beavers-Joseph interface conditions, SIAM J. Numer. Anal., 47 (2010), 4239-4256.
  11. Friedrichs learning: weak solutions of partial differential equations via deep learning SIAM J. Sci. Comput., 45 (2023), 1271-1299.
  12. Bridging Traditional and Machine Learning-Based Algorithms for Solving PDEs: The Random Feature Method, J. Mach. Learn., 1 (2022), 268-298.
  13. P. Chidyagwai, B. Riviére, On the solution of the coupled Navier-Stokes and Darcy equations, Comput. Methods Appl. Mech. Engrg., 198 (2009), 3806-3820.
  14. P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
  15. M. Discacciati, Domain Decomposition Methods for the Coupling of Surface and Groundwater Flows, Ph.D. thesis, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, 2004.
  16. Mathematical and numerical models for coupling surface and groundwater flows, Appl. Numer. Math., 43 (2002), 57-74.
  17. M. Discacciati, A. Quarteroni, Navier-Stokes Darcy coupling: modeling, analysis, and numerical approximation, Rev. Mat. Complut., 22 (2009), 315-426.
  18. S. Dong, Z. Li, Local extreme learning machines and domain decomposition for solving linear and nonlinear partial differential equations, Comput. Methods Appl. Mech. Engrg., 387 (2021), 114129.
  19. G. Du, L. Zuo, Local and parallel partition of unity scheme for the mixed Navier-Stokes-Darcy problem, Numer. Algorithms, 91 (2022), 635-650.
  20. W. E, Machine Learning and Computational Mathematics, Commun. Comput. Phys., 28 (2020), 1639-1670.
  21. W. E, B. Yu, The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems, Commun. Math. Stat., 6 (2018), 1-12.
  22. V. Girault, B. Riviére, DG approximation of coupled Navier-Stokes and Darcy equations by Beaver-Joseph-Saffman interface condition, SIAM J. Numer. Anal., 47 (2009), 2052-2089.
  23. Deep Learning, MIT Press, 2016.
  24. Solving high-dimensional partial differential equations using deep learning, Proc. Natl. Acad. Sci. USA, 115 (2018), 8505-8510.
  25. Numerical analysis of coupled Stokes/Darcy flow in industrial filtrations, Transp. Porous Media., 64 (2006), 73-101.
  26. Coupling Arbogast–Correa and Bernardi–Raugel elements to resolve coupled Stokes–Darcy flow problems, Comput. Methods Appl. Mech. Engrg., 373 (2021), 113469.
  27. Deep Residual Learning for Image Recognition, in: 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2016, 770-778.
  28. Y. He, J. Li, Convergence of three iterative methods based on the finite element discretization for the stationary Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 198 (2009), 1351-1359.
  29. A Deep Learning Method for Elliptic Hemivariational Inequalities, East Asian J. Appl. Math., 12 (2022), 487-502.
  30. Int-Deep: a deep learning initialized iterative method for nonlinear problems, J. Comput. Phys., 419 (2020), 109675.
  31. T.J.R. Hughes, The Finite Element Method, Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, N.J, 1987.
  32. D.P. Kingma, J. Ba, Adam: A method for stochastic optimization, 2014, arXiv preprint arXiv:1412.6980.
  33. Analysis of long time stability and errors of two partitioned methods for uncoupling evolutionary groundwater-surface water flows, SIAM J. Numer. Anal., 51 (2013), 248-272.
  34. A physics-guided neural network framework for elastic plates: comparison of governing equations-based and energy-based approaches, Comput. Methods Appl. Mech. Engrg., 383 (2021), 113933.
  35. DeepXDE: A Deep Learning Library for Solving Differential Equations, SIAM Rev., 63 (2021), 208-228.
  36. A domain decomposition method for the time-dependent Navier-Stokes-Darcy model with Beavers-Joseph interface condition and defective boundary condition, J. Comput. Phys., 411 (2020), 109400.
  37. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378 (2019), 686-707.
  38. R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1979.
  39. Z. Xu, Frequency principle: Fourier analysis sheds light on deep neural networks, Commun. Comput. Phys., 28 (2020), 1746-1767.
  40. Training behavior of deep neural network in frequency domain, In International Conference on Neural Information Processing, Springer, 2019, 264-274.
  41. Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems, Comput. Methods Appl. Mech. Engrg., 393 (2022), 114823.
  42. Weak adversarial networks for high-dimensional partial differential equations, J. Comput. Phys., 411 (2020), 109409.
  43. L. Zuo, Y. Hou, Numerical analysis for the mixed Navier-Stokes and Darcy problem with the Beavers-Joseph interface condition, Numer. Methods Partial Differential Equations, 31 (2015), 1009-1030.

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets