## 3 Application
### Results and Discussion  
The training of the predictor lasted \(T_{\text{P}}\ =\ 3.25\) h (3:15:56), which can be subdivided according to the individual levels as follows: 16%, 7%, 42%, 35%. The training processed approximately 7.6 million randomly generated training data points. The DL based TO geometries are similar to the results obtained by 88 lines of code (top88) according to [29] for the same input data points. Table 2 contains the parameters used for the conventional SIMP-TO density filter method. An example prediction is shown in Figure 11. The left result was obtained by the PEN method, and the right result was calculated by the conventional TO (88 lines of code) method under the same boundary conditions. The training history shows the progression of the objective function (see Figure 12) and of the individual evaluator losses over the number of batches (see Figure 13). The smaller batch size at higher levels produces more oscillation of the curve and therefore, makes it difficult to identify a trend. For this reason, the curves shown in the figures are filtered, using the exponential moving average and a constant smoothing factor of 0.873 [34] for all levels. This filtering does not affect the original objective function and serves only visual purposes. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Symbol** & **Value** & **Description** \\ \hline \(\alpha\) & 2 & \(f_{\text{Q}}\) coefficient for \(c\) \\ \(\beta\) & 5 & \(f_{\text{Q}}\) coefficient for \(M\) \\ \(\gamma\) & 1 & \(f_{\text{Q}}\) coefficient for \(F\) \\ \(\delta\) & 1 & \(f_{\text{Q}}\) coefficient for \(P\) \\ \(\zeta_{\text{max}}\) & 1000 & maximal patience \\ \(\eta(\lambda)\) & \(\frac{1}{4}(\lambda-1)\) & \(\frac{1}{100}\) & learning rate for different levels \(\Lambda\) \\ \(\
u\) & 0.3 & [PERSON]’s ratio used for conventional TO \\ \(\mathbf{b}_{n}^{\text{T}}\) & \(\begin{bmatrix}2048\\ 512\\ 128\\ 8\end{bmatrix}\) & batch size of the geometry for different levels \(\Lambda\) \\ \(d_{\text{inp}}\) & 8 & size of geometry in the first level along one side \\ \(E\) & 195,000 \(\frac{\text{N}}{\text{mm}^{2}}\) & [PERSON]’s modulus \\ \(F_{\text{k}}\) & 3 & parameter for \(F\) \\ \(p\) & 3 & power penalization factor used for conventional TO \\ \(r_{\text{min}}\) & 3 & filter size \\ \(r_{\text{s,F}}\) & 100 N & magnitude of the forces of the static boundary condition \\ \(x_{\text{min}}\) & 0.001 & lower limit value for the entries of \(\mathbf{x}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter used for training and generation of validation data. Figure 11: Sample geometry (**Left**): deep learning-based topology optimization by using the predictor–evaluator network (PEN) method. (**Right**): 88 lines of code [PERSON] et al. [29]. The dashed vertical lines (labeled with the value of \(\Lambda\)) in Figures 12 and 13 indicate the change in level. It can be seen that the objective function value \(J\) improves significantly in the first level \(\Lambda\) and in the following levels only slightly. The reason for this is that learning in one level optimizes the resolution in subsequent levels as well. This is also the explanation for the fact that the objective function value hardly increases when a level is increased. A slight increase can nevertheless be seen, which is due to the increased resolution and thus, the higher number of evaluated elements and the increase in complexity. The results were validated using \(n=100\) randomly generated input data points, called validation data. These data points were not part of the training. The corresponding optimized geometries were conventionally calculated by the top88 algorithm. The results of the comparison (PEN and top88) of the 100 validation data points are summarized in the plots in Figure 14.  
On average, the ANN-based TO can deliver almost the same result as the conventional method in about 7.3 ms, while the conventional topology optimizer according to [PERSON] et al. [29] requires, on average, 1.9 s (and is, therefore, roughly 259 times slower); see Figure 14a.