Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
139 tokens/sec
GPT-4o
47 tokens/sec
Gemini 2.5 Pro Pro
43 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
47 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Prediction of discretization of online GMsFEM using deep learning for Richards equation (2403.14177v1)

Published 21 Mar 2024 in math.NA and cs.NA

Abstract: We develop a new coarse-scale approximation strategy for the nonlinear single-continuum Richards equation as an unsaturated flow over heterogeneous non-periodic media, using the online generalized multiscale finite element method (online GMsFEM) together with deep learning. A novelty of this approach is that local online multiscale basis functions are computed rapidly and frequently by utilizing deep neural networks (DNNs). More precisely, we employ the training set of stochastic permeability realizations and the computed relating online multiscale basis functions to train neural networks. The nonlinear map between such permeability fields and online multiscale basis functions is developed by our proposed deep learning algorithm. That is, in a new way, the predicted online multiscale basis functions incorporate the nonlinearity treatment of the Richards equation and refect any time-dependent changes in the problem's properties. Multiple numerical experiments in two-dimensional model problems show the good performance of this technique, in terms of predictions of the online multiscale basis functions and thus finding solutions.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (68)
  1. J. E. Aarnes and Y. Efendiev. Mixed multiscale finite element methods for stochastic porous media flows. SIAM Journal on Scientific Computing, 30(5):2319–2339, 2008.
  2. Multicontinuum homogenization for Richards’ equation: The derivation and numerical experiments. Russian Journal of Numerical Analysis and Mathematical Modelling, 38(4):207–218, 2023.
  3. Modelling fluid flow in fractured-porous rock masses by finite-element techniques. International Journal for Numerical Methods in Fluids, 4(4):337–348, 1984.
  4. A generalized multiscale finite element method for poroelasticity problems II: Nonlinear coupling. Journal of Computational and Applied Mathematics, 297:132–146, 2016.
  5. A mass conservative numerical solution for two-phase flow in porous media with application to unsaturated flow. Water Resources Research, 28(10):2819–2828, 1992.
  6. A general mass-conservative numerical solution for the unsaturated flow equation. Water Resources Research, 26(7):1483–1496, 1990.
  7. Upscaling of a class of nonlinear parabolic equations for the flow transport in heterogeneous porous media. Commun. Math. Sci., 3(4):493–515, 2005.
  8. Deep global model reduction learning in porous media flow simulation. Computational Geosciences, 24(1):261–274, 2020.
  9. François Chollet. Keras, 2015. Available online: https://keras.io/ (accessed on 6 June 2022).
  10. Multicontinuum homogenization. General theory and applications, 2023. arXiv:2309.08128.
  11. Adaptive multiscale model reduction with generalized multiscale finite element methods. Journal of Computational Physics, 320:69–95, 2016.
  12. Multiscale Model Reduction: Multiscale Finite Element Methods and Their Generalizations. Applied Mathematical Sciences. Springer Cham, first edition, 2023.
  13. Residual-driven online generalized multiscale finite element methods. Journal of Computational Physics, 302:176 – 190, 2015.
  14. Coupling of multiscale and multi-continuum approaches. GEM - International Journal on Geomathematics, 8(1):9–41, Apr 2017.
  15. Online adaptive local multiscale model reduction for heterogeneous problems in perforated domains. Applicable Analysis, 96(12):2002–2031, 2017.
  16. An adaptive GMsFEM for high-contrast flow problems. Journal of Computational Physics, 273:54–76, 2014.
  17. G. Cybenko. Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals and Systems, 2(4):303–314, 1989.
  18. Efficient uncertainty quantification techniques in inverse problems for Richards’ equation using coarse-scale simulation models. Advances in Water Resources, 32(3):329–339, 2009.
  19. Louis J. Durlofsky. Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media. Water Resources Research, 27(5):699–708, 1991.
  20. Generalized multiscale finite element methods (GMsFEM). J. Comput. Phys., 251:116–135, October 2013.
  21. Generalized multiscale finite element methods. Nonlinear elliptic equations. Communications in Computational Physics, 15(3):733–755, 003 2014.
  22. Multiscale finite element methods for high-contrast problems using local spectral basis functions. Journal of Computational Physics, 230(4):937–955, 2011.
  23. Multicontinuum homogenization and its relation to nonlocal multicontinuum theories. Journal of Computational Physics, page 111761, 2022.
  24. Hybrid explicit-implicit learning for multiscale problems with time dependent source, 2022. arXiv:2208.06790.
  25. Lawrence C. Evans. Partial Differential Equations: Second Edition, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, 2010.
  26. Numerical solution of Richards’ equation: A review of advances and challenges. Soil Science Society of America Journal, 81(6):1257–1269, 2017.
  27. Computational multiscale methods for linear poroelasticity with high contrast. Journal of Computational Physics, 395:286–297, 2019.
  28. Generalized multiscale finite element method for a strain-limiting nonlinear elasticity model. Journal of Computational and Applied Mathematics, 359:153–165, 2019.
  29. Constraint energy minimizing generalized multiscale finite element method for nonlinear poroelasticity and elasticity. Journal of Computational Physics, 417:109569, 2020.
  30. Stochastic collocation and mixed finite elements for flow in porous media. Computer methods in applied mechanics and engineering, 197(43-44):3547–3559, 2008.
  31. Deep sparse rectifier neural networks. In Geoffrey Gordon, David Dunson, and Miroslav Dudík, editors, Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, volume 15 of Proceedings of Machine Learning Research, pages 315–323, Fort Lauderdale, FL, USA, 11–13 Apr 2011. PMLR.
  32. Deep Learning. MIT Press, 2016. http://www.deeplearningbook.org.
  33. A neural network reduced order model for the elasticity problem. AIP Conference Proceedings, 2528(1):020039, 2022.
  34. Boris Hanin. Universal function approximation by deep neural nets with bounded width and ReLU activations. Mathematics, 7(10):992-1–992-9, 2019.
  35. A comparison of numerical simulation models for one-dimensional infiltration. Soil Science Society of America Journal, 41(2):285–294, 1977.
  36. Deep residual learning for image recognition. In 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 770–778, 2016.
  37. Investigation of physics-informed deep learning for the prediction of parametric, three-dimensional flow based on boundary data, 2022. arXiv:2203.09204.
  38. Deep neural networks for acoustic modeling in speech recognition: The shared views of four research groups. IEEE Signal Processing Magazine, 29(6):82–97, 2012.
  39. Kurt Hornik. Approximation capabilities of multilayer feedforward networks. Neural Networks, 4(2):251–257, 1991.
  40. Adam: A method for stochastic optimization, 2014. arXiv:1412.6980.
  41. Self-normalizing neural networks, 2017. arXiv:1706.02515.
  42. ImageNet classification with deep convolutional neural networks. Communications of the ACM, 60(6):84–90, 2017.
  43. Deep learning. Nature, 521(7553):436–444, 2015.
  44. Fully implicit time-stepping schemes and non-linear solvers for systems of reaction-diffusion equations. Applied Mathematics and Computation, 244:361–374, 2014.
  45. Theory of functional connections applied to quadratic and nonlinear programming under equality constraints. Journal of Computational and Applied Mathematics, 406:113912, 2022.
  46. Constraint energy minimizing generalized multiscale finite element method for multi-continuum Richards equations. Journal of Computational Physics, 477:111915, 2023.
  47. Learning functions: When is deep better than shallow, 2016. arXiv:1603.00988.
  48. J.R. Nimmo. Vadose Water. In Gene E. Likens, editor, Encyclopedia of Inland Waters, pages 766–777. Academic Press, Oxford, 2009.
  49. Multiscale simulations for multi-continuum Richards equations. Journal of Computational and Applied Mathematics, 397:113648, 2021.
  50. Multiscale simulations for upscaled multi-continuum flows. Journal of Computational and Applied Mathematics, 374:112782, 2020.
  51. Jun Sur Richard Park and Viet Ha Hoang. Homogenization of a multiscale multi-continuum system. Applicable Analysis, 101(4):1271–1298, 2022.
  52. Jun Sur Richard Park and Xueyu Zhu. Physics-informed neural networks for learning the homogenized coefficients of multiscale elliptic equations. Journal of Computational Physics, 467:111420, 2022.
  53. L. A. Richards. Capillary conduction of liquids through porous mediums. Physics, 1(5):318–333, 1931.
  54. Generalized multiscale finite element method for multicontinua unsaturated flow problems in fractured porous media. Journal of Computational and Applied Mathematics, 370:112594, 2020.
  55. An online generalized multiscale finite element method for unsaturated filtration problem in fractured media. Mathematics, 9(12):1382-1–1382-14, 2021.
  56. Prediction of numerical homogenization using deep learning for the Richards equation. Journal of Computational and Applied Mathematics, 424:114980, 2023.
  57. Matus Telgarsky. Benefits of depth in neural networks. In Vitaly Feldman, Alexander Rakhlin, and Ohad Shamir, editors, 29th Annual Conference on Learning Theory, volume 49 of Proceedings of Machine Learning Research, pages 1517–1539, Columbia University, New York, New York, USA, 2016. PMLR.
  58. M Vasilyeva and A Tyrylgin. Convolutional neural network for fast prediction of the effective properties of domains with random inclusions. Journal of Physics: Conference Series, 1158:042034, 2019.
  59. Learning macroscopic parameters in nonlinear multiscale simulations using nonlocal multicontinua upscaling techniques. Journal of Computational Physics, 412:109323, 2020.
  60. Preconditioning Markov Chain Monte Carlo Method for Geomechanical Subsidence using multiscale method and machine learning technique. Journal of Computational and Applied Mathematics, 392:113420, 2021.
  61. Machine learning for accelerating macroscopic parameters prediction for poroelasticity problem in stochastic media. Computers & Mathematics with Applications, 84:185–202, 2021.
  62. A method of analyzing unsteady, unsaturated flow in soils. Journal of Geophysical Research (1896-1977), 69(12):2569–2577, 1964.
  63. Prediction of discretization of GMsFEM using deep learning. Mathematics, 7(5):412-1–412-16, 2019.
  64. Reduced-order deep learning for flow dynamics. The interplay between deep learning and model reduction. Journal of Computational Physics, 401:108939, 2020.
  65. Deep multiscale model learning. Journal of Computational Physics, 406:109071, 2020.
  66. Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks. Communications in Computational Physics, 29(3):930–954, 2021.
  67. E. Wong. Stochastic Processes in Information and Dynamical Systems. McGraw-Hill, New York, 1971.
  68. A-PINN: Auxiliary physics informed neural networks for forward and inverse problems of nonlinear integro-differential equations. Journal of Computational Physics, 462:111260, 2022.
Citations (1)
View on Semantic Scholar

Summary

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

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