# Topology Optimisation under Uncertainties with Neural Networks
## 3 Neural Network Architectures
### Topology Convolutional Neural Networks (TCNN)
#### 3.1.2 Data Preparation
However, the amount of evaluations of \(\varphi_{n}(\omega_{i})\) that Algorithm A2 can process at once is limited by the number of available processors. Since the calculation time for the evaluation of an optimisation step \(\varphi_{n+1}(\omega_{i})\) increases with finer \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Applied Load \(g\)** & **Converges after \(n_{\text{final}}\)** & **Evaluation of \(I^{*}(\varphi_{n_{\text{final}}})\)** & **Compliance \(\int_{\varepsilon}g\cdot u(\varphi)\,\text{ds}\)** \\ \hline Algorithm A1 & \((0,-5000)^{T}\) & 538 & 1475.39 & 1173.22 \\ Algorithm 1 & \((0,-5000)^{T}\) & 441 & 1206.49 & 1148.66 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of metrics between Algorithms A1 and 1. Figure 5: Comparison of metrics between Algorithms A1 and 1.

meshes, the evaluation of the approximation of all gradient steps \(\varphi_{n+1}(\omega_{1}),\ldots,\varphi_{n+1}(\omega_{S})\) at once results in a processing time advantage for the NN. It is to be expected that this time saving increases with the number of examples \(S\). As mentioned at the beginning of the experiment, we expect to achieve good results close to the mean of \(g=(0,-5000)^{T}\). In order to obtain a more general view of the quality of Algorithm 1, we have compiled a selection of extreme cases for the distribution of \(g\) (e.g., evaluations from \(g\) that deviate strongly from \((0,-5000)^{T}\)) in Figure 8 and Table 3. The figure shows the sequence of \(\varphi_{n}\) in hundreds of steps as well as the final distribution of material \((\varphi_{100},\varphi_{200},\ldots,\varphi_{n_{\text{final}}})\) for the specific loads \(g\). Table 3 indicates a noticeable saving in calculation time, but there is no guaranteed improvement in the results. In particular, when the topology \"collapses\" (i.e., the NN cannot generalise to the input data with strong deviations from the training data), the application of the CNN leads to worse results. Nevertheless, it can be seen that the NN extension gives the algorithm a greater robustness against porous fragments (see Figure 8b) in the optimisation of the topology and thus a higher stability against collapsing of the topology in the optimisation can be assumed. Finally, a critical aspect to be mentioned is the step size \(c_{m}\). The time at which Equation (22) is applied and which is controlled by \(c_{m}\in\mathbb{N}\) has a crucial impact on the viability or \"compatibility\" between the state of the optimisation procedure and the CNN. In some cases, a \(c_{m}\) that is too small or too large can lead to the collapse of the topology, i.e., the topology deteriorates and does not recover. For the reliable use of Algorithm 1, a method for controlling \(c_{m}\) would have to be devised. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Converges after \(n_{\text{final}}\)** & **Evaluation of \(I_{g}(\varphi_{n_{\text{final}}})\)** & **Compliance \(\mathbb{E}\left[f_{I_{g}}\,g\cdot u(\varphi_{n_{\text{final}}})\,\mathsf{ds}\right]\)** & **Samples** \\ \hline Algorithm A2 & 490 & 765.85 & 729.20 & 224 \\ Algorithm 2 & 517 & 744.11 & 708.61 & 224 \\ Algorithm 2 & 500 & 760.26 & 723.24 & 224 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of metrics between Algorithms A2 and 2. Figure 8: Comparison of metrics between Algorithms A1 (**top row**) and \(1\) (**bottom row**) for different loads \(g\). Figure 7: Comparison of metrics between Algorithms A2 and 2. ### TLSTM Examples

As in Section 4.1.3, randomly generated training data should be used in the following experiment with the derived LSTM transformation of Equation (31). The data tuples consist of input and output from \(\mathcal{N}_{\text{LSTM}}\) according to Example 4. #### 4.2.1 Sampling the Data

Again, we assume an expected load \(g=(0,-5000)^{T}\) and a random rotation characterised by a truncated normal distribution with bounds \([-\frac{\pi}{2},\frac{\pi}{2}]\), standard deviation \(0.3\) and mean \(0\).