## 3 Application
### Results and Discussion
It can also be seen that the majority of geometries generated by PEN have a compliance that is close to the geometries generated by top88; see Figure 14b. In addition to the comparison of compliance and computing time, there are indicators that allow the comparison of geometries generated by ANN-based TO with geometries generated by conventional TO. Figure 14: Computing time and compliance comparison. Figure 12: Progression of the objective function value during training. Figure 13: Progression of the evaluator values during training. The accuracy \(\kappa\) represents the conformity of the predicted geometries with the geometries of the validation data points. To determine the deviation of a single validation geometry \(\mathbf{x}_{\
u}\) from the predicted geometry \(\mathbf{x}_{\text{p}}\), obtained from the same input data points, the following functions are used:  
\[\text{mae}(\mathbf{x}_{\text{p}},\mathbf{x}_{\
u})=\frac{1}{n}\sum_{i=1}^{n} \left|x_{\text{p}_{i}}-x_{\text{v}i}\right| \tag{33}\]  
and  
\[\text{mse}(\mathbf{x}_{\text{p}},\mathbf{x}_{\
u})=\frac{1}{n}\sum_{i=1}^{n}( x_{\text{p}_{i}}-x_{\text{v}i})^{2} \tag{34}\]  
where \(n\) represents the number of validation data points and, in this case, has the value 100. The smaller the mean squared error (mse) and mean absolute error (mae) values, the better the similarity of the results. The function \(\kappa_{0,01}\) indicates the percentage of elements with density differences of less than or equal to 0.01 with respect to the density values of the validation geometries generated by the conventional TO. For the calculation of \(\kappa_{0,01}\), the Heaviside step function \(\Theta\) is used as follows:  
\[\kappa_{0,01}(\mathbf{x}_{\text{p}},\mathbf{x}_{\
u})=\frac{1}{n}\sum_{i=1}^{ n}\Theta\Bigg{(}1-\frac{\left|x_{\text{p}_{i}}-x_{\text{v}i}\right|}{0.01} \Bigg{)}. \tag{35}\]  
With the help of the function  
\[\kappa(\mathbf{x}_{\text{p}},\mathbf{x}_{\
u})=\frac{1}{n}\sum_{j=1}^{n}\frac {1}{3} \big{[}\big{(}1-\text{mse}(x_{\text{p}_{i}},x_{\text{v}i})\big{)} \tag{36}\] Figure 11:0 Figure 11:1  
the accuracy and thus, the validity of the predictor can be determined, and predictors with different hyperparameters can be compared. The function \(\kappa\) is needed because a single indicator is not sufficient to determine the accuracy of the predictor. So, it is possible that the accuracy \(\kappa_{0,01}\) is low and the errors mae and mse are also small at the same time. Since these error indicators concentrate on different kinds of differences, the average \(\kappa\) of those is a more meaningful indicator. The examples in Figure 15 show that the predictor can deliver geometries that are similar to the conventional method, as well as some weaknesses. Unfortunately, the results are not always perfect, and some geometries lack details (see Figure 15, column four or five). This may be improved by an appropriate choice of layers or hyperparameters of the predictor and by adapting the quality function. For all sample geometries in Figure 15, the compliance is reported under the geometry diagram. For the PEN-generated geometries, the evaluator losses are summarized in Figure 11:2 It can be seen that the values for all evaluators--except the compliance evaluator--are close to zero. The compliance can only be reduced to an optimal value, which cannot be decreased without adding more material. It should also be noted that the deviation in the degree of filling \(M\) is not exactly zero, unlike what is expected by the conventional method. This is due to the fact that there is no restriction, but only a penalty for deviation. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Example** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline \(c\) & 1.044 & 0.318 & 0.152 & 0.476 & 0.391 & 0.521 \\ \(M\) & 0.010 & 0.000 & 0.004 & 0.005 & 0.006 & 0.007 \\ \(F\) & 0.002 & 0.003 & 0.001 & 0.002 & 0.002 & 0.002 \\ \(P\) & 0.014 & 0.009 & 0.005 & 0.005 & 0.007 & 0.017 \\ \hline \hline \end{tabular}
\end{table}