# Compensating Data Shortages in Manufacturing with Monotonicity Knowledge
## 4 Results and Discussion
### Informed Machine Learning Models for Forming and Press Hardening  
As in the glass bending case, the SIAMOR method was first validated on a 1D subset of the data for the press hardening process. Namely, only those data points with \(F_{\mathrm{P}}=2250\,\mathrm{k}\mathrm{N}\), \(\Delta t_{\mathrm{h}}=4\,\mathrm{s}\) and \(t_{\mathrm{q}}=2\,\mathrm{s}\) were considered. These specifications are met by six data points, and these were used to train the models shown in Figure 5. The data are not monotonic due to experimental noise. However, they reflect the expected sigmoid-like behavior mentioned in Section 3.2, and this extends to the monotonized models. An unconstrained polynomial with \(m=3\) was chosen as the reference model to be monotonized for the comparative methods from the literature. Degrees lower than that resulted in larger deviations from the data and degrees higher than that resulted in overfitting. Thus, out of all models of the form (1), the hyperparameter choice \(m=3\) yielded the lowest RMSE values for  
Figure 4: 2D regression for laser glass bending, where the markers represent the employed training data. (**a**) Gaussian process regression (non-monotonic) with a multi-length-scale RBF kernel (noise level \(10^{-5}\)); (**b**) projection [24] of GPR; (**c**) monotonic regression of a polynomial model (\(m=7\)) using the SIAMOR method (see Sections 2.1–2.3). projection and rearrangement. For the monotonic regression with SIAMOR, \(m=6\) and five equidistant initial constraint locations in \(X^{0}\) were chosen. It converged in iteration 8 with a total of 12 monotonicity constraints. In terms of the root-mean-squared error, the SIAMOR model predicts the training data more accurately, as can be seen in Table 4. The reason is that the rearrangement- and projection-based models are dragged away from the data by the underlying reference model, especially at high \(T_{\text{f}}\). After these 1D considerations, the SIAMOR method was validated on the full 4D dataset of the press hardening process. Polynomial models with degree \(m=3\) were used for unconstrained regression and monotonic projection, and polynomials with \(m=6\) were used for the SIAMOR method. The resulting models are visualized in the surface plots in Figure 6. The unconstrained model from Figure 6a clearly shows non-monotonic predictions with respect to \(\Delta t_{\text{h}}\). Furthermore, the hardness slightly decreases with the furnace temperature at \(T_{\text{f}}\) close to 930 \({}^{\circ}\)C, which is not the behavior expected by the process expert either. Figure 6b shows the monotonic projection of the unconstrained model. It was computed according to the Appendix A on a grid \(G\) consisting of \(40^{4}\) points. The monotonic projection exhibits the kinks that are characteristic of that method, and it yields an RMSE of 28.84 HV on the entire dataset. With an overall RMSE of 10.14 HV, the model resulting from SIAMOR is more accurate for this application. A corresponding response surface is displayed in Figure 6c. In keeping with (18), monotonicity was required with respect to \(T_{\text{f}}\) (increasing), \(\Delta t_{\text{h}}\) (decreasing) and \(t_{\text{q}}\) (increasing). As \(m=6\), there are \(N_{m}=210\) model parameters and the dicretization \(X^{0}\) was initialized with a grid using four equidistant values per dimension. The algorithm converged in iteration 246 with 1372 final constraints. Our first try was with only two monotonicity requirements (namely, with respect to \(T_{\text{f}}\) and \(t_{\text{q}}\)). We observed, however, \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Monotonic Regression Type**} & **RMSE [HV]** \\ \hline projection [24] & 5.0893 \\ rearrangement [22] & 4.8346 \\ SIAMOR & 3.3583 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Root-mean-squared deviations (RMSE) of the monotonic regression models from the training data for forming and press hardening of sheet metal (1D).