# Compensating Data Shortages in Manufacturing with Monotonicity Knowledge
## 4 Results and Discussion
### Informed Machine Learning Models for Forming and Press Hardening
Figure 5: 1D regression for forming and press hardening of sheet metal (\(F_{\text{p}}=2250\,\text{kN}\), \(\Delta t_{\text{h}}=4\,\text{s}\), \(t_{\text{q}}=2\,\text{s}\)). Dashed: (non-monotonic) polynomial of degree \(m=3\) as the reference model, dash-dotted: projection [24], dotted: rearrangement [22], solid: SIAMOR (see Sections 2.1-2.3) with degree \(m=6\). The projection and rearrangement methods were fed with the dashed polynomial curve as non-monotonic reference predictor. that the final number of iterations decreased when the third monotonicity requirement was added. Thus, the monotonicity requirements in each direction promoted each other numerically within the algorithm and for the used data. This reduction in the number of iterations was not accompanied by a decrease in total calculation time because more lower-level problems have to be solved when there are more monotonicity directions. With SIAMOR, monotonicity was achieved in all three input dimensions where it was required. See, e.g., Figure 6c, which is the monotonic counterpart of Figure 6a. A comparison of Figure 6a-c clearly shows how incorporating monotonicity expert knowledge helps compensate data shortages. Indeed, taking no monotonicity constraints into account at all (Figure 6a), we obtained an unexpected hardness minimum with respect to \(\Delta t_{\text{h}}\) at \(\Delta t_{\text{h}}\approx 2.5\,\text{s}\) and small \(T_{\text{f}}\). This also resulted in unnecessarily low predictions of the monotonic projection for small \(T_{\text{f}}\) and \(\Delta t_{\text{h}}\gtrapprox 1.5\,\text{s}\) in Figure 6b. The SIAMOR model (Figure 6c), by contrast, predicted more reasonable hardness values in this range without needing additional data, because it integrated the available monotonicity knowledge in the training phase. For the SIAMOR plots in Figure 7, \(\Delta t_{\text{h}}\) was reduced to \(0\,\text{s}\). This figure shows that monotonicity is also achieved with respect to \(t_{\text{q}}\). Without having explicitly demanded it, the hardness \(y\) shows the expected concave growth towards saturation with respect to \(t_{\text{q}}\) in Figure 7a. An additional increase in \(F_{\text{p}}\) leads to Figure 7b, where the sign of the second  
Figure 6: 4D regression for forming and press hardening of sheet metal using polynomial models (\(F_{\text{p}}=2250\,\text{kN}\), \(t_{\text{q}}=2\,\text{s}\)). The markers represent those training points matching the specification of the corresponding plane in the input space. (**a**) Unconstrained \(m=3\), (**b**) projection [24] of unconstrained \(m=3\), (**c**) SIAMOR \(m=6\). derivative of \(y\) with respect to the quenching time \(t_{\mathrm{q}}\) changes along the \(T_{\mathrm{f}}\)-axis. I.e., the model changes its convexity properties in this direction and increases convexly instead of concavely with \(t_{\mathrm{q}}\) at high \(T_{\mathrm{f}}\), \(\Delta t_{\mathrm{h}}=0\,\mathrm{s}\) and \(F_{\mathrm{p}}=2250\,\mathrm{kN}\). This contradicts the process expert's expectations. A possible way out is to measure additional data (e.g., in the rear left corner of Figure 7b), which is elaborate and costly, however. Another possible way out is to add the concavity requirement \(\partial_{x_{\mathrm{q}}}^{2}\tilde{y}_{\mathrm{w}}(\mathbf{x})\leq 0\) for all \(\mathbf{x}\in X\) with respect to the \(x_{\mathrm{q}}=t_{\mathrm{q}}\) direction to the monotonicity constraints (6) used exclusively so far. In order to solve the resulting constrained regression problem, one can use the same adaptive semi-infinite solution strategy, which was already used for the monotonicity constraints alone.