Coupled simulation-optimization model for coastal aquifer management using genetic programming-based ensemble surrogate models and multiple-realization optimization

ion in this study area. Different groundwater extraction scenarios were generated using Latin hypercube sampling. The salinity concentrations resulting from each of these pumping patterns are simulated using FEMWATER. The simulated salinity level and the corresponding pumping rates form the input-output pattern. Altogether 230 extraction patterns are used in this study. Different realizations of this input-output data set were generated using the nonparametric bootstrap method. Each of these data sets was used to build surrogate models to create the ensemble of surrogate models. Each data set was split into halves for training and testing the GP models. The input-output patterns were then used to train the genetic programmingbased surrogate models. Adaptive training [Sreekanth and Datta, 2010] was performed to reduce the number of patterns required for training. [37] Surrogates were developed for predicting salinity at three different locations. For each location 30 models in the ensemble was found to be sufficient to characterize the uncertainty. All the genetic programming surrogate models used a population size of 500, mutation frequency of 95, and crossover frequency of 50. A commercial genetic programming software Discipulus was used to develop the surrogate models. The parameters values, as per the guidelines after performing a sensitivity analysis, were used in the development of the model. The functional set in the developed GP models contained the operations addition, subtraction, multiplication, division, comparison, and data transfer. The maximum number of surrogate model parameters used was limited to 30 to prevent overfitting of the model. Squared deviation from the actual value was used as the fitness function. At the end of model training and testing source codes of the model in C language were generated using the interactive evaluator of the software and are then coupled with the multiobjective optimization algorithm NSGA II. 6. Results and Discussion 6.1. Uncertainty in Surrogate Models [38] The uncertainty in the surrogate models were quantified using the coefficient of variation of the root mean square errors of the individual surrogate models. The root mean square errors of individual surrogate model salinity predictions C1, C2, and C3 are shown in Figures 4, 5, and 6. The RMSEs are computed over the testing data set used for evaluating the genetic programming-based surrogate models. It could be observed that for different realizations of the same data set, the root mean square errors are different for different surrogate models. This is due to the predictive uncertainty of the surrogate models. The root mean square errors for the ensemble of models predicting salinity C1 are plotted against the number of surrogate models in the ensemble starting from an initial ensemble size of 10 in Figure 7. As the number of models in the ensemble increases, RMSE of the ensemble prediction decreases, at least in this example. [39] The coefficient of variation of the RMSEs, as a measure of uncertainty in prediction of salinity, is plotted against the number of surrogate models in the ensemble for each ensemble predicting C1, C2, and C3. The plots are shown in Figures 8, 9, and 10. Uncertainty of the ensemble model has a definite decreasing trend with the increasing number of models in the ensemble. For each of the salinity concentrations C1, C2, and C3 the uncertainty in the ensemble of surrogate model decreases with the number of models in the ensemble and reaches a constant value when the number of models in the ensemble is around 30. Hence the optimum number of models in the ensemble for coupled simulation optimization is chosen as 30. The optimum number of surrogate models depends on the uncertainty level in the model structure and parameters. For more complex systems the uncertainty in the model structure and parameters of surrogate models will be larger and hence more number of surrogate models will be required in the ensemble. The sensitivity of the derived Pareto-optimal solutions to the number of surrogate models in the ensemble is analyzed in section 6.4. Table 1. Parameters for Aquifer Simulation