# Gaussian Process Surrogates for Modeling Uncertainties in a Use Case of Forging Superalloys
## 5 Results  
### GPs  
Before utilizing optimization methods, we evaluate each individual GP, see Table 5. The screwpress GP is trained with data that is generated by using inputs from Table 1 with (15) and tested on data generated by using inputs from Table 2 with (15). As covariance function, \(k\) we found out that an RBF kernel is appropriate for this GP. The GPs of our proposed surrogate model are both designed with a Matern kernel with \(\
u=2.5\) and are independently evaluated by 10-fold cross-validation with inputs from Table 2 and \(Z^{(j)}\) obtained from the screwpress GP. Outputs are obtained from FEM simulations, see Section 4.2. In each cross-validation step, we split the respective data randomly such that 10 percent are in the test dataset and the remaining 90 percent are used for model training.  
In addition, we calculate mean Pearson kurtosis  
\[kurt_{Pearson}=\frac{1}{N}\sum_{j=1}^{N}\frac{1}{M}\sum_{i=1}^{M}\left(\frac{y ^{(i)(j)}-\mu^{(j)}}{\sigma_{al}^{(j)}}\right)^{4} \tag{16}\]  
\begin{table}
\begin{tabular}{l l} \hline \hline \multicolumn{2}{c}{**Abaqus FEM Simulation Settings**} \\ \hline Simulation type & Static, General \\ Time period & 1 \\ Nlgeom & On \\ Max number of increments & 1000 \\ Initial increment size & 0.001 \\ Min increment size & \(1\times 10^{-5}\) \\ Max increment size & 1 \\ Equation solver method & Direct \\ Solution technique & Full Newton \\ \hline \hline \end{tabular}
\end{table}
Table 4: Abaqus FEM simulation control parameters for our use case.  
Figure 8: Preforming an Inconel 625 superalloy billet: (**a**) initial billet and randomly drawn velocity profile \(z^{(i)(j)}\), (**b**) FEM simulation result with graphical presentation of the horizontal displacement \(U,U1\) and selected output variables \(y^{(i)(j)}\), i.e., final diameter of 288 mm and final height of 92.83 mm.  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multicolumn{5}{c}{**Individual GP Evaluations**} \\ \hline \multicolumn{2}{c}{Screwpress} & \multicolumn{2}{c}{Aleatoric Uncertainty} & \multicolumn{2}{c}{Mean Result} \\ \multicolumn{2}{c}{\(m(Z)\)} & \(m(\sigma_{al,d_{final}})\) & \(m(\sigma_{al,h_{final}})\) & \(m(\mu_{d_{final}})\) & \(m(\mu_{h_{final}})\) \\ R2-Score & 0.9923 & 0.8146 & 0.8455 & 0.9586 & 0.9555 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Evaluation of individual GPs by average R2-Scores over 10 folds.  
and mean Fisher-Pearson coefficient of skewness  
\[skew_{Fisher-Pearson}=\frac{1}{N}\sum_{j=1}^{N}\frac{1}{M}\sum_{i=1}^{M}\left(\frac{ y^{(i)(j)}-\mu^{(j)}}{\sigma_{al}^{(j)}}\right)^{3} \tag{17}\]  
to describe the distribution shapes obtained from uncertainty propagation analysis, see Table 6.  
GP models were implemented with the GPflow library version 2.2.1 and Python 3.8.10. Inferences were run on a machine with 16 GB RAM, 8 CPUs and Intel(R) i7-8565 2.0 GHz processor. We utilized a L-BFGS-B algorithm to train the models. Training our surrogate model on all available data took an average of 1.36 s based on 10 measurements. For one prediction our model needs on average 0.046 s. A FEM simulation lasted on average 149.78 s.