# Gaussian Process Surrogates for Modeling Uncertainties in a Use Case of Forging Superalloys
## 6 Discussion  
Evaluation of the individual GPs with 10-fold cross-validation shows promising R2-Scores (lowest: 0.8146, mean: 0.89355, highest: 0.9586), i.e., hyperparameters appear to be appropriate for further evaluations. Observation of generated manufacturing process uncertainties, i.e., \(\bar{\Sigma}_{al}(Z^{(\bar{\jmath})})\) shows a diverse data landscape, thus, we assume that further uncertainty propagation analysis is meaningful. We observe the distributions obtained from uncertainty propagation analysis by calculating Pearson kurtosis and Fisher-Pearson coefficient of skewness (A Pearson kurtosis of 3.0 and Fisher-Pearson coefficient of skewness of 0.0 describe a normal distribution). Regarding kurtosis, results shows that distributions are near to Normal distributions, where the distribution of \(h_{final}\) is slightly platykurtic (\(kurt_{Pearson}(h_{final})=2.685<3.0\)), i.e., it is less peaked than a Normal distribution and the distribution of \(d_{final}\) is little leptokurtic (\(kurt_{Pearson}(d_{final})=3.003>3.0\)), i.e., the distribution is more peaked compared to a Normal distribution. In terms of skewness, the distribution of \(d_{final}\) is more skewed compared to \(h_{final}\), however, both values are less than 0.5 so that approximate symmetry can be assumed. We evaluate the impact of data selection for model training using two metrics, R2-Score and MSE, with a 10-fold cross-validation comparing active learning with random sample  
Figure 13: Kernel density estimate plots of mean aleatoric uncertainties \(m(\sigma_{al})\) for different hyperparameter settings \(\alpha\) and \(\beta\), distributions are obtained by 10-fold cross-validation, where in each fold a target vector is randomly selected. The plots for \(\alpha=0\) show only blue lines, because the results of the different methods are the same and the lines are on top of each other. selection. With respect to mean values \(\mu\), active learning shows overall an improvement compared to random sample selection. In terms of aleatoric uncertainties \(\sigma_{al}\), random sample selection is superior to active learning up to the selection of about 20 samples, but after that active learning shows superior performance compared to random sample selection. The initially worse performance of active learning with respect to \(\sigma_{al}\) is due to a trade-off in the active learning cost function between \(\sigma(\mu)\) and \(\sigma(\sigma_{al})\) with the influence of \(\sigma(\mu)\) dominating. A possible solution for this would be the introduction of appropriate tuning parameters that regulate the influence of the respective epistemic uncertainties \(\sigma(\mu)\) and \(\sigma(\sigma_{al})\). Moreover, it should be noted that random sample selection shows only better performance at a stage where the tuning of parameters is far from complete, so the better performance is not applicable in practice. With regard to solving inverse problems, we compare our novel robust UCB based BO multi-objective optimization algorithm with two baselines: (1) combined: no distinction of uncertainties in UCB based BO and (2) epistemic: neglecting aleatoric uncertainty in UCB based BO. We show that over different values of tuning parameters \(\alpha\) and \(\beta\) there are clear tendencies of the different approaches. By disabling the influence of aleatoric uncertainty (\(\alpha=0\)), all three approaches show the same results as expected: low squared errors and neglected aleatoric uncertainty. For all approaches, slight differences can be seen over different \(\beta\) values while \(\alpha=0\), regulating the exploration vs. exploitation trade-off. Due to the fundamentals of the epistemic approach, there are no differences in the optimization result when \(\alpha\) values are changed for constant \(\beta\) values, see Figure 11. Differences in kernel density estimate plots over varying \(\alpha\) values are from random target vector selection.