Identification of physical processes and unknown parameters of 3D groundwater contaminant problems via theory-guided U-net

Abstract

Identification of unknown physical processes and parameters of groundwater contaminant problems is a challenging task due to their ill-posed and non-unique nature. Numerous works have focused on determining nonlinear physical processes through model selection methods. However, identifying corresponding nonlinear systems for different physical phenomena using numerical methods can be computationally prohibitive. With the advent of machine learning (ML) algorithms, more efficient surrogate models based on neural networks (NNs) have been developed in various disciplines. In this work, a theory-guided U-net (TgU-net) framework is proposed for surrogate modeling of three-dimensional (3D) groundwater contaminant problems in order to efficiently elucidate their involved processes and unknown parameters. In TgU-net, the underlying governing equations are embedded into the loss function of U-net as soft constraints. Herein, sorption is considered to be a potential process of an uncertain type, and three equilibrium sorption isotherm types (i.e., linear, Freundlich, and Langmuir) are considered. Different from traditional approaches in which one model corresponds to one equation (Schoeniger et al. in Water Resour Res 50(12):9484–9513, 2014; Cao et al. in Hydrogeol J 27(8):2907–2918, 2019), these three sorption types are modeled through only one TgU-net surrogate. Accurate predictions illustrate the satisfactory generalizability and extrapolability of the constructed TgU-net. Furthermore, based on the constructed TgU-net surrogate, a data assimilation method is employed to identify the physical process and parameters simultaneously. The convergence of indicators demonstrates the validity of the proposed method. The influence of sparsity-promoting techniques, data noise, and quantity of observation information is also explored. Results demonstrate the feasibility of neural network learning a cluster of equations that have similar behaviors. This work shows the possibility of governing equation discovery of physical problems that contain multiple and even uncertain processes by using deep learning and data assimilation methods.

Appendices

Appendix

Appendix A

The fitting ability, generalizability, and extrapolability of the constructed TgU-net-based surrogate models are tested with sparse data points. In addition, the corresponding traditional U-nets are trained in the same configurations as comparisons. Predicted h fields of U-net-h and TgU-net-h are presented in Figs. 22 and 23 in Appendix A, respectively. RMSEs of these points predicted by U-net and TgU-net are given in Table 12 in Appendix A.

Fig. 22

Prediction and error of U-net-h when $\stackrel{-}{lnK(\mathbf{x})}$ is set to be 1 and variance of lnK is 0.5

Fig. 23

Prediction and error of TgU-net-h when $\stackrel{-}{lnK(\mathbf{x})}$ is set to be 1 and variance of lnK is 0.5

Table 12 RMSE of points where observation data are extracted

Appendix B

In this section, the trained TgU-net-based surrogate models are employed to process forward predictions, and are combined with the IES method for inverse modeling. Firstly, observation data of hydraulic head fields are used to estimate K fields. Then, the velocity fields from the estimated K fields are inputted into TgU-net-C to estimate unknown source parameters and physical processes according to observation data of contaminant concentration. Figure 24 shows prediction and error of the hydraulic head field of the estimated lnK field in the IES method. Comparison of convergence of initialized and estimated parameters of linear and Freundlich sorption is shown in Figs. 25 and 26 in Appendix B, respectively. Convergence of contaminant parameters of linear and Freundlich sorption are presented in Tables 13 and 14 in Appendix B, respectively. Tables 15 compares the convergence results of parameters by utilizing different sparsity-promoting methods under 10% noise (Tables 16, 17 and 18).

Fig. 24

Prediction and error of the hydraulic head field of estimated lnK field in the IES method

Fig. 25

Comparison of convergence of initialized and estimated parameters (linear sorption)

Fig. 26

Comparison of convergence of initialized and estimated parameters (Freundlich sorption)

Table 13 Convergence of parameters under different noise levels (linear sorption)

Table 14 Convergence of parameters under different noise levels (Freundlich sorption)

Table 15 Comparison of parameters convergence of different methods under 10% noise (Freundlich sorption)

Table 16 Convergence of parameters versus iteration when $\mathrm{\alpha }=0$ under 10% noise

Table 17 Convergence of parameters versus iteration when $\mathrm{\alpha }=0.05$ under 10% noise

Table 18 Convergence of parameters versus iteration when $\mathrm{\alpha }=0.1$ under 10% noise

Appendix C

In this section, the influences of strong nonlinearity and information volume on data assimilation results are discussed. Estimated source parameters are displayed in Table 19.

Table 19 Parameters convergence of the IES method only using data from C fields under different noise levels (Freundlich sorption)

Appendix D

The performances of U-net-h, TgU-net-h, U-net-C, and TgU-net-C in the forward problems are summarized in Table 20. The performances of TgU-net-h and TgU-net-C in the inverse problems are summarized in Tables 21 and 22, respectively. Noise is not considered. The number of test realizations is 100. The correlation length of lnK along each side is 0.2 times the corresponding geometrical length. $\stackrel{-}{lnK(\mathbf{x})}$ is set to be 1, and the variance of lnK is 0.5. The sorption type is random. With the assistance of theory-guidance, the prediction accuracy can be significantly improved. And the estimated parameters converge well near the references, further demonstrating the validity of the proposed method.

Table 20 The performances of U-net-h, TgU-net-h, U-net-C, and TgU-net-C in the forward problems

Table 21 The RMSE of the mean of initialized and estimated lnK and the average of the variances at each point in each layer, and the ${R}^{2}$ score and RMSE of predicted h field through estimated lnK field

Table 22 The mean and variance of estimated parameters through IES and TgU-net-C