# Compensating Data Shortages in Manufacturing with Monotonicity Knowledge
## 4 Results and Discussion
### Informed Machine Learning Models for Laser Glass Bending  
To begin with, the SIAMOR method was validated on a 1D subset of the data for laser glass bending, namely, the subset of all data points for which \(n_{\text{c}}=50\). This means that, out of the 25 data points, five points remained for training. First of all, ordinary unconstrained regression techniques were tried (see Figure 3a). A polynomial model of degree \(m=3\) (solid line) and a Gaussian process regressor [31] (GPR, dashed line) did not comply with the monotonicity knowledge at high \(T_{\text{f}}\). A radial basis function (RBF) kernel was used for the GPR. This non-parametric model is always a reasonable choice for simulated data because it accurately reproduces the data themselves if the noise-level parameter is kept small. For all GPR models in this work, that parameter was set to \(10^{-5}\). Next, the polynomial model was regularized in a ridge regression (dotted line), where the squared \(\ell^{2}\)-norm \(\lambda\|\mathbf{w}\|_{2}^{2}\) with a regularization weight \(\lambda\) was added to the objective function in (4). \(\lambda=0.003\) was chosen, which was roughly the minimum necessary value to achieve monotonicity. However, the resulting model does not predict the data very well. Thus, all three models from Figure 3a were unsatisfactory. As a next step, the monotonicity requirement with respect to \(T_{\text{f}}\) was brought to bear, and monotonic regression with the SIAMOR method (\(m=5\)) was used (see Figure 3b) and compared to the rearrangement [22] and to the monotonic projection [24] of the Gaussian process regressor from Figure 3a. As mentioned before, both comparative methods are based on a non-monotonic pre-trained reference predictor. This makes them fundamentally different to the SIAMOR method, which imposes the monotonicity already in the training phase. The projection was calculated as described in the Appendix A with \(|G|=80\) grid points. For the rearrangement method, the R package monreg was invoked from Python using the package rpy2. The degree \(m\) of the polynomial ansatz (1) used in the SIAMOR method was chosen as described in Section 2.3. For the specific case considered here, the curve started to vary unreasonably (albeit still monotonically) betw  
Figure 3: 1D regression for laser glass bending (\(n_{\text{c}}=50\)). (**a**) Unconstrained regression. Solid: polynomial model (\(m=3\)), dashed: Gaussian process regression (GPR) with RBF kernel (noise level \(10^{-5}\)), dotted: polynomial ridge regression (\(m=3\), \(\lambda=0.003\)). (**b**) Monotonic regression, with the solid line resulting from the SIAMOR method (see Sections 2.1–2.3) with degree \(m=5\). The projection [24] (dash-dotted) and rearrangement [22] (dotted) methods were fed with the dashed GPR curve as a non-monotonic reference predictor. \(m\geq 6\), and therefore \(m=5\) was chosen. The SIAMOR algorithm was initialized with five equidistant constraint locations in \(X^{0}\), and it converged in iteration 5 with a total number of nine constraints. The locations of the constraints are marked in Figure 3b by the gray, vertical lines. The adaptive algorithm automatically places the non-initial constraints in the non-monotonic region at high \(T_{\text{f}}\). In terms of the root-mean-squared error  
\[\text{RMSE}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\bigl{(}\widehat{y}(\mathbf{x}_{i})-t_ {l}\bigr{)}^{2}} \tag{19}\]  
on the training data, the SIAMOR model fits the data best; see Table 3. Another advantage of the SIAMOR model is that it is continuously differentiable, whereas the rearrangement and projection models exhibit (slight) unphysical kinks, which are typical for these methods [22]. And finally, the rearrangement and the projection models, for temperatures higher than 540 \({}^{\circ}\)C, both predict bending angles larger than 90\({}^{\circ}\), which is unphysical due to the mechanical stop used in the glass bending process. In contrast, the predictions of the SIAMOR model do not (significantly) exceed 90\({}^{\circ}\).