Feasibility of neural network metamodels for emulation and sensitivity analysis of radionuclide transport models

Abstract

In this paper we compare the outputs of neural network metamodels with numerical solutions of differential equation models in modeling cesium-137 transportation in sand. Convolutional neural networks (CNNs) were trained with differential equation simulation results. Training sets of various sizes (from 5120 to 163,840) were used. First order and total order Sobol methods were applied to both models in order to test the feasibility of neural network metamodels for sensitivity analysis of a radionuclide transport model. Convolutional neural networks were found to be capable of emulating the differential equation models with high accuracy when the training set size was 40,960 or higher. Neural network metamodels also gave similar results compared with the numerical solutions of the partial differential equation model in sensitivity analysis.

Figure 1

Liquid versus sorbed concentrations after reaching the equilibrium state. The figure is modified from. The small circles, triangles, and squares indicate the original experimental test results, the curves indicate the results of the PDE models presented in while the red and green dots indicate the 100 mM and 1000 mM NaCl solution results, respectively, obtained using the FiPy package.

Figure 2

Sorbed and liquid concentrations obtained by numerically solving the PDE using the FiPy package. The parameter values used were: $\alpha$ = 0.163, f = 0.339, K = 31.71, and n = 0.810. The simulations concern 1 mM Cs injection at the time instant 0.0 s.

Figure 3

Proposed neural network architecture to model the concentration surfaces and perform sensitivity analysis of the four parameters presented in Table 2. The first convolution layer is followed by batch normalization and the second by batch normalization and ReLU operations before the data are fed to the third convolutional layer. (The figure was generated by adapting the code from https://github.com/gwding/draw_convnet).

Figure 4

Comparison between DE and NN (trained using 163,840 realizations) concentration surfaces for both liquid (C) and sorbed (S) concentrations. The results are presented for mean parameter values $\alpha$ = 0.163, f = 0.339, K = 31.71, and n = 0.810.

Figure 5

Sobol’s first order sensitivity analysis results for narrow tolerances for DE and 40,960-trained NN concentration surfaces for liquid concentration C.

Figure 6

Sobol’s first order sensitivity analysis results for wide tolerances for DE and 40,960-trained NN concentration surfaces for liquid concentration C. The architecture was kept the same.

Figure 7

First order sensitivity analysis results along the diagonal of the time-distance plane of the model solution for the four parameters considered in the sensitivity analysis. CNN metamodel is trained using 40,960 realizations with narrow parameter tolerances. The left and right panels show the sensitivity analysis results for narrow and wide parameter tolerances, respectively.

Figure 8

Sensitivity analysis results along the diagonal of the time-distance plane of the model solution for Sobol’s first (upper row) and total (lower row) order methods. The FiPy packet solution as well as the CNN metamodel results using training sets of various sizes are presented. Sensitivity analysis is performed using 10,240 realizations with narrow parameter tolerances.

Figure 9

Sensitivity analysis results along the diagonal of the time-distance plane of the model solution for Sobol’s first and total order methods using wider tolerances. The FiPy packet solution as well as the results for the 40,960-trained CNN metamodel are presented.