# Gaussian Process Surrogates for Modeling Uncertainties in a Use Case of Forging Superalloys
## 5 Results
### Solving Inverse Problem  
We evaluate our proposed multi-objective optimization strategy by solving inverse problems, i.e., we try to find input settings that lead to an output that is as near as possible to an initially defined target vector. In addition to minimize distances between a target vector \(y_{target}\) and random mean vector \(m(\mu^{(j)})\), we try to achieve results that also keep mean aleatoric uncertainty \(m(\sigma_{al}^{(j)})\) low. We utilize a 10-fold cross-validation, where in each cross-validation step the target vector is randomly drawn from the pool dataset and the best found prediction after drawing 50 datapoints out of the pool dataset is used for evaluation. This means that for each method, a dataset of 50 datapoints is generated, and each best prediction is found by evaluating the respective acquisition function on the corresponding generated dataset. We compare our approach with two other baselines, namely: \begin{table}
\begin{tabular}{l c c} \hline \hline \multicolumn{3}{c}{**Distribution Properties**} \\ \hline \(kurt_{Pearson}\) & 3.003 & 2.685 \\ \(skew_{Fisher-Pearson}\) & 0.449 & 0.015 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Mean values of Pearson kurtosis and Fisher-Pearson coefficient of skewness calculated from uncertainty propagation analysis results. Figure 10: MSEs of 10-fold cross-validation over number of drawn training data \(N\). In each cross-validation step, models are initially trained on two randomly selected datapoints drawn from the pool dataset. Solid lines depict the mean R2-Score values and shaded areas the upper and lower confidence bounds obtained by adding and subtracting the standard deviations, calculated from the 10 obtained results. (**a**) \(m(\mu_{Diameter})\); (**b**) \(m(\sigma_{al,Diameter})\); (**c**) \(m(\mu_{Height})\); (**d**) \(m(\sigma_{al,Height})\). Figure 9: R2-Scores of 10-fold cross-validation over number of drawn training data \(N\). In each cross-validation step, models are initially trained on two randomly selected datapoints drawn from the pool dataset. Solid lines depict the mean R2-Score values and shaded areas the upper and lower confidence bounds obtained by adding and subtracting the standard deviations, calculated from the obtained results. (**a**) \(m(\mu_{Diameter})\); (**b**) \(m(\sigma_{al,Diameter})\); (**c**) \(m(\mu_{Height})\); (**d**) \(m(\sigma_{al,Height})\). 1. Combined (This baseline can be considered as an approximation to the use of heteroskedastic GP in UCB BO. ): no distinction of uncertainties in UCB based BO, i.e., simply adding aleatoric and epistemic uncertainty with loss: \[ \begin{array}{c}\mathcal{L}_{combined}(d(\mu_{target},m(\mu)),m(\sigma_{al}), \sigma_{ep}(\sigma(\sigma_{al}),\sigma(\mu)))=\\ d(\mu_{target},m(\mu))-[\alpha\cdot m(\sigma_{al})+\beta\cdot\sigma_{ep}( \sigma(\sigma_{al}),\sigma(\mu)))]\end{array} \] (18)
2. Epistemic: neglecting aleatoric uncertainty in UCB based BO with loss: \[ \begin{array}{c}\mathcal{L}_{epistemic}(d(\mu_{target},m(\mu)),m(\sigma_{al }),\sigma_{ep}(\sigma(\sigma_{al}),\sigma(\mu)))=\\ d(\mu_{target},m(\mu))-\beta\cdot\sigma_{ep}(\sigma(\sigma_{al}),\sigma(\mu) )).\end{array} \] (19)  
Figure 11 shows representative plots of optimization results for one random target vector (i.e., one cross-validation step) over 50 draws of \(x_{inv}\), where solid lines depict squared errors and dotted lines show mean aleatoric uncertainty \(m(\sigma_{al})\). Figures 12 and 13 show different distributions of optimization results obtained by 10-fold cross-validation in respect to squared errors and mean aleatoric uncertainty \(m(\sigma_{al})\). Distributions are visualized by kernel density estimation. Figure 12: Kernel density estimate plots of squared errors 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.