## 3 Results and Discussion
### Train Support Vector Regression Model  
The objective of machine learning is to discover a function \(E(\mathbf{D})_{FEA}=E(\mathbf{D})_{SVM}\) that best predicts the value of \(E(\mathbf{D})_{FEA}\) associated with each value of \(\mathbf{D}\). At the first step, 22 vertex pairs (Learning Points, LP) were selected from \(\Omega\) according to Figure 4, then \(E(\mathbf{D})_{FEA}\) values were determined by the use of finite element analysis. Thereby, the training set was produced for the machine learning algorithm.  
\begin{table}
\begin{tabular}{c c c c} \hline  & \(\mathbf{D}_{1}\) (mm) & \(\mathbf{D}_{2}\) (mm) & \(\mathbf{E(D)_{FEA}\) (kN\({}^{2}\)) \\ \hline Optimum Shape & 108 & 33 & 0 \\ Initial Shape & 77 & 15 & 7911.2 \\ \(D_{opt}\), best from 128 DP & 120 & 55 & 3.267 \\ \hline \end{tabular}
\end{table}
Table 1: Calculated cost function value in different design points.  
Figure 3: The objective function value in design space by applying a finite element model.  
With the use of Regression Learner application built into Matlab, it is possible to perform an automated training to search for the best regression model type, including linear regression models, regression trees, Gaussian process regression models, support vector machines, and ensembles of regression trees. Manual regression model training was run without a validation data set for the support vector machine. The goodness of the prediction highly hinges on the kernel function type. According to [32], the cubic kernel function was selected because this method's root mean square error (RMSE) value was the lowest on the predicted set. For the fitting process \(\varepsilon=0.001\) and kernel scale = 1 values were selected while the rest of the hyperparameters were selected automatically by the Regression Learner application. The predicted response \(E_{SVM}\) of the cubic SVM model is plotted against the true response \(E_{FEA}\), see Figure 4b. A perfect regression model has a predicted response equal to the true response, so all the points lay on the diagonal line. The vertical distance from the line to any point is the error of the prediction for that point. The predictions are scattered near the diagonal line, which means the SVR model accurately predicts the nonlinear objective function values.  
Using the trained cubic SVM model, predictions were made for each combination of integer values of design variables. Table 2 shows the smallest predicted object function value and the associated design variables. Based on the results, the SVM-based design optimization is not feasible to find the global optimum of a valley function. Predicted objective function values are illustrated above the design space according to Figure 5. It can be pointed out that the trained SVM model overlaps with the objective function values calculated by finite element analysis in point 128. The SVM model with the cubic kernel function was trained by using 22 support vectors. As a result, it seemed suitable for approaching the values of the nonlinear objective function.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(D_{1}\) (mm) & \(D_{2}\) (mm) & \(E(D_{opt})_{FEA}\) (\(\mathbf{k}\mathbf{N}^{2}\)) & \(E(D_{opt})_{SVM}(\mathbf{k}\mathbf{N}^{2})\) \\ \hline
**Optimum Shape** & 108 & 33 & 0 & - \\
**SVM** & 95 & 10 & 390.91 & \(-\)772.83 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The predefined and the support vector machines (SVM) model-based optimum point.  
Figure 4: (**a**) The selection of learning points from the design area. (**b**) Comparison of the finite element analysis (FEA) calculated objective function values with the support vector regression (SVR) model calculated predicted responses in the learning points.