Abstract
Governing equations of physical problems are traditionally derived from conservation laws or physical principles. However, some complex problems still exist for which these first-principle derivations cannot be implemented. As data acquisition and storage ability have increased, data-driven methods have attracted great attention. In recent years, several works have addressed how to learn dynamical systems and partial differential equations using data-driven methods. Along this line, in this work, we investigate how to discover subsurface flow equations from data via a machine learning technique, the least absolute shrinkage and selection operator (LASSO). The learning of single-phase groundwater flow equation and contaminant transport equation are demonstrated. Considering that the parameters of subsurface formation are usually heterogeneous, we propose a procedure for learning partial differential equations with heterogeneous model parameters for the first time. Derivative calculation from discrete data is required for implementing equation learning, and we discuss how to calculate derivatives from noisy data. For a series of cases, the proposed data-driven method demonstrates satisfactory results for learning subsurface flow equations.
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References
Bear, J.: Dynamics of Fluids in Porous Media. New York: Environmental Science Series (1972)
Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imag. Sci. 2(1), 183–202 (2009). https://doi.org/10.1137/080716542
Bongard, J., Lipson, H.: Automated reverse engineering of nonlinear dynamical systems. Proc. Natl. Acad. Sci. USA 104(24), 9943–9948 (2007). https://doi.org/10.1073/pnas.0609476104
Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine Learning 3(1), 1–122 (2010). https://doi.org/10.1561/2200000016
Bruno, O., Hoch, D.: Numerical differentiation of approximated functions with limited order-of-accuracy deterioration. SIAM J. Numer. Anal. 50(3), 1581–1603 (2012). https://doi.org/10.1137/100805807
Brunton, S.L., Proctor, J.L., Kutz, J.N.: Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl. Acad. Sci. USA 113(15), 3932–3937 (2016). https://doi.org/10.1073/pnas.1517384113
Chang, H., Zhang, D.: Identification of physical processes via combined data-driven and data-assimilation methods. J. Comp. Phy. 393, 337–350 (2019). https://doi.org/10.1016/j.jcp.2019.05.008
Chartrand, R.: Numerical differentiation of noisy, nonsmooth data. ISRN Applied Mathematics 2011, 1–11 (2011). https://doi.org/10.5402/2011/164564
Cullum, J.: Numerical differentiation and regularization. SIAM J. Numer. Anal. 8(2), 254–265 (1971). https://doi.org/10.1137/0708026
Efron, B., Hastie, T., Johnstone, I., Tibshirani, R.J.: Least angle regression. Ann. Stat. 32(2), 407–451 (2004). https://doi.org/10.1214/009053604000000067
Figueiredo, M.A.T., Nowak, R.D., Wright, S.J.: Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE J. Sel. Top. Sign. Proces. 1(4), 586–597 (2007). https://doi.org/10.1109/JSTSP.2007.910281
Hastie, T., Tibshirani, R.J., Friedman, J.H.: The elements of statistical learning: data mining, inference, and prediction. New York: Springer series in statistics. https://doi.org/10.1007/978-0-387-84858-7 (2009)
Hesterberg, T., Choi, N.H., Meier, L., Fraley, C.: Least angle and l1 penalized regression: a review. Statistics Surveys 2, 61–93 (2008). https://doi.org/10.1214/08-SS035
Jauberteau, F, Jauberteau, J.L.: Numerical differentiation with noisy signal. Appl. Math. Comput. 215 (6), 2283–2297 (2009). https://doi.org/10.1016/j.amc.2009.08.042
Knowles, I., Le, T., Yan, A.: On the recovery of multiple flow parameters from transient head data. J. Comput. Appl. Math. 169(1), 1–15 (2004). https://doi.org/10.1016/j.cam.2003.10.013
Mangan, N.M., Brunton, S.L., Proctor, J.L., Kutz, J.N.: Inferring biological networks by sparse identification of nonlinear dynamics. IEEE Transactions on Molecular Biological and Multi-Scale Communications 2(1), 52–63 (2016). https://doi.org/10.1109/TMBMC.2016.2633265
Mangan, N.M., Kutz, J.N., Brunton, S.L., Proctor, J.L.: Model selection for dynamical systems via sparse regression and information criteria. Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences 473(2204), 16 (2017). https://doi.org/10.1098/rspa.2017.0009
Meng, J., Li, H.: An efficient stochastic approach for flow in porous media via sparse polynomial chaos expansion constructed by feature selection. Adv. Water Resour. 105, 13–28 (2017). https://doi.org/10.1016/j.advwatres.2017.04.019
Ramos, G., Carrera, J., Gómez, S., Minutti, C., Camacho, R.: A stable computation of log-derivatives from noisy drawdown data. Water Resour. Res. 53(9), 7904–7916 (2017). https://doi.org/10.1002/2017WR020811
Rosset, S., Zhu, J.: Piecewise linear regularized solution paths. Ann. Stat. 35(3), 1012–1030 (2007). https://doi.org/10.1214/009053606000001370
Rudy, S.H., Brunton, S.L., Proctor, J.L., Kutz, J.N.: Data-driven discovery of partial differential equations. Sci. Adv. 3(4), e1602614 (2017). https://doi.org/10.1126/sciadv.1602614
Schaeffer, H.: Learning partial differential equation via data discovery and sparse optimisation. Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences 473(2197), 20160446 (2017). https://doi.org/10.1098/rspa.2016.0446
Schmidt, M., Lipson, H.: Distilling free-form natural laws from experimental data. Science 324(5923), 81–85 (2009). https://doi.org/10.1126/science.1165893
Tibshirani, R.J.: The lasso problem and uniqueness. Electronic Journal of Statistics 7(1), 1456–1490 (2013). https://doi.org/10.1214/13-EJS815
Zou, H.: The adaptive Lasso and its oracle properties. J. Am. Stat. Assoc. 101(476), 1418–1429 (2006). https://doi.org/10.1198/016214506000000735
Acknowledgments
This work is partially funded by the National Natural Science Foundation of China (Grant No. U1663208 and 51520105005) and the National Science and Technology Major Project of China (Grant No. 2017ZX05009-005 and 2016ZX05037-003). The link for the open-source Matlab code is provided in Hesterberg et al. [13]. The other computer codes and data used are available upon request from the corresponding author.
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Appendices
Appendix: A
In this appendix, the finite difference scheme for calculating the derivatives in the candidate library is provided.
The finite difference schemes for calculating the derivatives in Eq. 10 are
where Δt and Δx are the step sizes for time and space, respectively. Here, note that, to calculate the derivatives, we need the spatially or temporally nearby data. The data near the boundaries without sufficient nearby data for calculating the derivatives are not utilized for learning PDE.
The finite difference schemes for computing the derivatives in Eq. 13 are
where Δx and Δy are the step sizes in x and y dimension in space, respectively.
The finite difference scheme for calculating ∂K/∂y in Eq. 15 is
The finite difference scheme for calculating ∂(K∂h/∂y)/∂y/∂y is
For the contaminant transport problem, the finite difference scheme for calculating the d th derivative is
where d > 0 denotes the order of the derivative, L denotes the number of data points in each side of the considered location, and al denotes the coefficient. When d is odd, a−l = −al, while when d is even, a−l = al. Tables 3, 4, and 5 show the coefficient values for the first three orders of derivatives with different choice of L.
Here, note that the derivative of u with respect to t is also calculated using Eq. A.5 by replacing x with t.
Appendix: B
In this appendix, we will discuss the reason that the coefficient K and ∂K/∂y in the PDE shown in Eq. 29 needs to be calculated using Eqs. 26 and A.3, respectively.
We take the term ∂(K∂h/∂y)/∂y for analysis. According to the equation in MODFLOW, ∂(K∂h/∂y)/∂y is calculated using the following formula:
Performing simple manipulation, we have
If ∂2u/∂y2(xi,j+ 1/4,t) and ∂2u/∂y2(xi,j− 1/4,t) are close to ∂2u/∂y2(xi,j,t), we have
where
Conductivity at the interface of nearby grid blocks is calculated using the harmonic mean in MODFLOW to ensure flux continuity.
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Chang, H., Zhang, D. Machine learning subsurface flow equations from data. Comput Geosci 23, 895–910 (2019). https://doi.org/10.1007/s10596-019-09847-2
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DOI: https://doi.org/10.1007/s10596-019-09847-2