## 3 Computational Experiments
In this respect, \(RMSE\) is a global accuracy metric that represents the difference between the true value and the surrogate model value. Figure 2: Comparison of sampling methods [47]. It is used in the study to measure the success of the methods in capturing global behaviour. Using a test data-set, the formulation is expressed as in Equation (23). \[RMSE =\sqrt{\frac{1}{N}\sum_{i=1}^{N}(y_{i}-\widehat{y_{i}})^{2}} \tag{23}\] \[NRMSE =\frac{RMSE}{(y_{\max}-y_{\min})}\]  
The maximum absolute error value represents the greatest difference in the design space between the surrogate model prediction and the actual value. \(MAE\) is frequently used in the literature to measure the prediction success of the surrogate model locally, and in this study, it is used to quantify how accurately the surrogate models capture the local features. The formulation is expressed using a test data-set as follows,  
\[MAE =\max(|y_{i}-\widehat{y}_{i}|) \tag{24}\] \[NMAE =\frac{MAE}{(y_{\max}-y_{\min})}\]  
The proposed assessment metrics might show different characteristics, as global accuracy may differ from local accuracy. In order to present a comprehensive performance evaluation of the surrogates, two different type of metrics are considered. A surrogate model's ability to capture both local and global characteristics needs to be evaluated. For example, when considering a search for an optimum or any other local response of the surrogate, the \(MAE\) has crucial importance to ensure that the model has no local deficits. At the same time, to facilitate an efficient evaluation of the surrogate performance, some kind of a compromise has to be observed between local and global accuracy of the models. * Uncertainty quantification (UQ) metrics: Evaluation of the uncertainty quantification results is performed using the probability density function. In engineering problems, the terms mean, standard deviation, skewness and kurtosis, which express the behaviour of the density function, are used to numerically compare the results of the uncertainty quantification. The mean (\(\bar{y}\)) is the general tendency of the response based on the variation of the uncertain variables, and the standard deviation(\(\sigma\)) refers to the variation in the analysis program's response based on the distribution of the uncertain variable. The mean and standard deviation formulas are given in Equation (25), where \(N\) represents the number of responses. \[\bar{y}=\frac{\sum_{i=1}^{N}y_{i}}{N},\ \ \ \sigma=\sqrt{\frac{\sum_{i=1}^{N}(y_{i}-\bar{y})^{2}}{N-1}}\] (25) Skewness is a measure of the asymmetry of the probability density of response relative to the mean and is calculated as in Equation (26). Kurtosis, on the other hand, expresses a measure of whether the data contain an abundance of outliers or lack of outliers relative to a normal distribution and is calculated as in Equation (26). \[\text{Skewness}\ =\frac{\sum_{i=1}^{N}(y_{i}-\bar{y})^{3}}{(N-1)\cdot\hat{ \sigma}^{3}},\ \ \ \text{Kurtosis}\ =\frac{\sum_{i=1}^{N}(y_{i}-\bar{y})^{4}}{(N-1)\cdot\hat{ \sigma}^{4}}\] (26)
* Fidelity cost assignment criteria: Multi-fidelity surrogate assessments are usually depicted by using a constant number of low-fidelity analyses and adding high-fidelity data as necessary. Since design problems have to be solved with limited computational budget to complete the process within a reasonable amount of time, dominance of low fidelity data is more convenient. Preliminary cost assignment is determined by using single-fidelity performance results,which means the data are fixed for low-fidelity cases when an initial convergence occurs. Then, the number of low-fidelity runs are kept constant and high-fidelity runs are increased to a level when an acceptable convergence is reached by the multi-fidelity surrogate.