Skip to main content

Deep global model reduction learning in porous media flow simulation


In this paper, we combine deep learning concepts and some proper orthogonal decomposition (POD) model reduction methods for predicting flow in heterogeneous porous media. Nonlinear flow dynamics is studied, where the dynamics is regarded as a multi-layer network. The solution at the current time step is regarded as a multi-layer network of the solution at the initial time and input parameters. As for input, we consider various sources, which include source terms (well rates), permeability fields, and initial conditions. We consider the flow dynamics, where the solution is known at some locations and the data is integrated to the flow dynamics by modifying the reduced-order model. This approach allows modifying the reduced-order formulation of the problem. Because of the small problem size, limited observed data can be handled. We consider enriching the observed data using the computational data in deep learning networks. The basis functions of the global reduced-order model are selected such that the degrees of freedom represent the solution at observation points. This way, we can avoid learning basis functions, which can also be done using neural networks. We present numerical results, where we consider channelized permeability fields, where the network is constructed for various channel configurations. Our numerical results show that one can achieve a good approximation using forward feed maps based on multi-layer networks.

This is a preview of subscription content, access via your institution.


  1. 1.

    Alotaibi, M., Calo, V.M., Efendiev, Y., Galvis, J., Ghommem, M.: Global-local nonlinear model reduction for flows in heterogeneous porous media. Comput. Methods. Appl. Mech. Eng. 292, 122–137 (2015)

    Article  Google Scholar 

  2. 2.

    Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer, Heidelberg (2013)

    Book  Google Scholar 

  3. 3.

    Cardoso, M., Durlofsky, L.: Linearized reduced-order models for subsurface flow simulation. J. Comput. Phys. 229 (2010)

    Article  Google Scholar 

  4. 4.

    Cardoso, M.A., Durlofsky, L.J., Sarma, P.: Development and application of reduced-order modeling procedures for subsurface flow simulation. Int. J. Numer. Methods. Eng. 77(9), 1322–1350 (2009)

    Article  Google Scholar 

  5. 5.

    Celia, M.A, Bouloutas, E.T, Zarba, R.L: A general mass-conservative numerical solution for the unsaturated flow equation. Water Resources Research 26(7), 1483–1496 (1990)

    Article  Google Scholar 

  6. 6.

    Chollet, F., et al.: Keras. (2015)

  7. 7.

    Chung, E.T., Efendiev, Y., Leung, W.T.: Residual-driven online generalized multiscale finite element methods. J. Comput. Phys. 302, 176–190 (2015)

    Article  Google Scholar 

  8. 8.

    Csaji, B.C.: Approximation with artificial neural networks. Faculty of Sciences, Etvs Lornd University, 24(48) (2001)

  9. 9.

    Cybenko, G.: Approximations by superpositions of sigmoidal functions. Mathematics of Control, Signals, and Systems 2(4), 303–314 (1989)

    Article  Google Scholar 

  10. 10.

    Dostert, P., Efendiev, Y., Mohanty, B.: Efficient uncertainty quantification techniques in inverse problems for Richards’ equation using coarse-scale simulation models. Adv. Water. Resour. 32(3), 329–339 (2009)

    Article  Google Scholar 

  11. 11.

    Efendiev, Y., Datta-Gupta, A., Ginting, V., Ma, X., Mallick, B.: An efficient two-stage Markov chain Monte Carlo method for dynamic data integration. Water Resources Research, 41(12) (2005)

  12. 12.

    Efendiev, Y., Galvis, J., Gildin, E.: Local–global multiscale model reduction for flows in high-contrast heterogeneous media. J. Comput. Phys. 231(24), 8100–8113 (2012)

    Article  Google Scholar 

  13. 13.

    Efendiev, Y., Gildin, E., Yang, Y.: Online adaptive local-global model reduction for flows in heterogeneous porous media. Computation 4(2), 22 (2016)

    Article  Google Scholar 

  14. 14.

    Gardner, W.R.: Some steady-state solutions of the unsaturated moisture flow equation with application to evaporation from a water table. Soil Science 85(4), 228–232 (1958)

    Article  Google Scholar 

  15. 15.

    Ghommem, M., Calo, V.M., Efendiev, Y.: Mode decomposition methods for flows in high-contrast porous media. a global approach. J. Comput. Phys. 257, 400–413 (2014)

    Article  Google Scholar 

  16. 16.

    Ghommem, M., Presho, M., Calo, V.M., Efendiev, Y.: Mode decomposition methods for flows in high-contrast porous media. global-local approach. J. Comput. Phys. 253, 226–238 (2013)

    Article  Google Scholar 

  17. 17.

    Glorot, X., Bordes, A., Bengio, Y.: Deep sparse rectifier neural networks. In: Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, pp 315–323. PMLR (2011)

  18. 18.

    Goodfellow, I., Bengio, Y., Courville, A., Bengio, Y.: Deep Learning, vol. 1. MIT Press, Cambridge (2016)

    Google Scholar 

  19. 19.

    Hanin, B.: Universal function approximation by deep neural nets with bounded width and ReLU activations. arXiv:1708.02691 (2017)

  20. 20.

    Hinze, M., Volkwein, S.: Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: error estimates and suboptimal control. In: Benner, P., Mehrmann, V., Sorensen, D.C. (eds.) Dimension Reduction of Large-Scale Systems, volume 45 of Lecture Notes in Computational Science and Engineering, pp 261–306. Springer, Berlin (2005)

  21. 21.

    Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural. Netw. 4(2), 251–257 (1991)

    Article  Google Scholar 

  22. 22.

    Jansen, J.D., Durlofsky, L.J: Use of reduced-order models in well control optimization. Optim. Eng. 18 (1), 105–132 (2017)

    Article  Google Scholar 

  23. 23.

    Kerschen, G., Golinval, Jean-claude, Vakakis, A.F., Bergman, L.A.: The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview. Nonlinear Dynamics 41(1), 147–169 (2005)

    Article  Google Scholar 

  24. 24.

    Khoo, Y., Lu, J., Ying, L.: Solving parametric PDE problems with artificial neural networks. arXiv:1707.03351 (2017)

  25. 25.

    Kingma, D.P., Ba, J.: Adam: a method for stochastic optimization. arXiv:1412.6980(2014)

  26. 26.

    LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521(7553), 436 (2015)

    Article  Google Scholar 

  27. 27.

    Li, Z., Shi, Z.: Deep residual learning and PDEs on manifold. arXiv:1708.05115 (2017)

  28. 28.

    Liao, Q., Mhaskar, H., Poggio, T.: Learning functions: when is deep better than shallow. arXiv:1603.00988v4 (2016)

  29. 29.

    Maas, A.L., Hannun, A.Y., Ng, A.Y.: Rectifier nonlinearities improve neural network acoustic models. Proc. icml. 30(1), 3 (2013)

    Google Scholar 

  30. 30.

    Richards, L.A.: Capillary conduction of liquids through porous mediums. Physics 1(5), 318–333 (1931)

    Article  Google Scholar 

  31. 31.

    Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid. Mech. 656, 5–28 (2010)

    Article  Google Scholar 

  32. 32.

    Schmidhuber, J.: Deep learning in neural networks: an overview. Neural Netw. 61, 85–117 (2015)

    Article  Google Scholar 

  33. 33.

    Telgrasky, M.: Benefits of depth in neural nets. JMLR: Workshop and Conference Proceedings, 49(123) (2016)

  34. 34.

    Trehan, S., Durlofsky, L.J: Trajectory piecewise quadratic reduced-order model for subsurface flow, with application to pde-constrained optimization. J. Comput. Phys. 326, 446–473 (2016)

    Article  Google Scholar 

  35. 35.

    van Doren, Jorn FM, Markovinović, R., Jansen, Jan-Dirk: Reduced-order optimal control of water flooding using proper orthogonal decomposition. Comput. Geosci. 10(1), 137–158 (2006)

    Article  Google Scholar 

  36. 36.

    Van Genuchten, M T h: A closed-form equation for predicting the hydraulic conductivity of unsaturated soils 1. Soil Science Society of America Journal 44(5), 892–898 (1980)

    Article  Google Scholar 

  37. 37.

    Vo, H.X, Durlofsky, L.J: A new differentiable parameterization based on principal component analysis for the low-dimensional representation of complex geological models. Mathematical Geosciences 46(7), 775–813 (2014)

    Article  Google Scholar 

  38. 38.

    Weinan, E., Yu, B.: The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Commun. Math. Stat. 6(1), 1–12 (2018)

    Google Scholar 

  39. 39.

    Wynn, A., Pearson, D.S., Ganapathisubramani, B., Goulart, P.J.: Optimal mode decomposition for unsteady flows. J. Fluid. Mech. 733, 473–503 (2013)

    Article  Google Scholar 

  40. 40.

    Yang, Y., Ghasemi, M., Gildin, E., Efendiev, Y., Calo, V.: Fast multiscale reservoir simulations with POD-DEIM model reduction. SPE J. 21(06), 2141–2154 (2016)

    Article  Google Scholar 

Download references


YE would like to thank the partial support from NSF 1620318. YE would also like to acknowledge the support of Mega-grant of the Russian Federation Government (N 14.Y26.31.0013)”. EG would like to thank Energi Simulation for partial support of this research.

Author information



Corresponding author

Correspondence to Eduardo Gildin.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cheung, S.W., Chung, E.T., Efendiev, Y. et al. Deep global model reduction learning in porous media flow simulation. Comput Geosci 24, 261–274 (2020).

Download citation


  • Deep learning
  • Model reduction
  • POD
  • Porous media flow
  • Neural networks