# Topology Optimisation under Uncertainties with Neural Networks
## 3 Neural Network Architectures
### Topology Convolutional Neural Networks (TCNN)
#### 3.1.2 Data Preparation
Since computations with uncertainties require a substantial computational effort, our central goal is to extend the algorithms used in [1,2] by introducing appropriate NN predictions, reducing the iteration steps required. In contrast to other machine learning approaches, our aim is to achieve this even for adaptively adjusted finite element meshes since this has proven to be crucial for good performance in previous work. For this to function, an underlying sufficiently fine reference mesh is assumed for the training data and the prediction. Moreover, in contrast to other NN approaches for this problem, we consider the evolution of a continuous (functional) representation of a phase field determining the material distribution. Ideally, the NN architectures should hasten the deterministic topology optimisation problem and consequently the risk-averse optimisation under uncertainties. This is achieved in Section 3.1 by embedding a CNN in the optimisation for both the deterministic and the stochastic setting. The observed numerical results for a common 2D bridge benchmark are on par with the reference method presented in [1]. However, the gradient step predicted by the NN architectures allows for significantly larger iteration steps, rendering the optimisation procedure more efficient. This directly transfers to the Monte-Carlo-based risk-averse optimisation under uncertainties as defined by Algorithm 2 since the samples for the statistical estimation are obtained with minimal cost. In addition to the CNN, a second architecture is illustrated in terms of an LSTM. This generally leads to a better quality of the optimisation and is motivated by the idea that a memory of previous gradients may lead to a more accurate prediction of the next gradient step. However, it comes at the cost of longer computation times due to the transformation between the different adapted computation \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Method** & **Applied Load \(g\)** & **Computation Time** & **Converges after \(n_{\text{final}}\)** & **Evaluation of \(I^{\epsilon}(\boldsymbol{\rho_{\text{initial}}})\)** & **Compliance \(\int_{\text{f}_{\text{g}}}g\cdot u(\boldsymbol{\varphi})\,\text{ds}\)** \\ \hline Algorithm A1 & \((0,-5000)^{T}\) & 4 min 43 s & 538 & 1475.39 & 1173.22 \\ Algorithm 1 & \((0,-5000)^{T}\) & 4 min 05 s & 441 & **1206.49** & **1148.66** \\ Algorithm 3 & \((-5000)^{T}\) & 4 min 34 s & **425** & 1209.24 & 1151.27 \\ \hline Algorithm A1 & \((2632.16,-4251.09)^{T}\) & 4 min 31 s & 473 & 1487.39 & 1416.46 \\ Algorithm 1 & \((2632.16,-4251.09)^{T}\) & **4 min 14 s** & 517 & 1475.05 & 1405.39 \\ Algorithm 3 & \((2632.16,-4251.09)^{T}\) & 4 min 34 s & **436** & **1209.24** & **1151.27** \\ \hline Algorithm A1 & \((-733.23,-4945.95)^{T}\) & 4 min 50 s & **457** & 1115.66 & 1062.41 \\ Algorithm 1 & \((-733.23,-4945.95)^{T}\) & **4 min 21 s** & 497 & **1049.91** & 1042.58 \\ Algorithm 3 & \((-733.23,-4945.95)^{T}\) & 4 min 41 s & 484 & 1082.62 & **1031.62** \\ \hline Algorithm A1 & \((-1099.92,-4877.25)^{T}\) & 5 min 11 s & 470 & 1042.18 & 992.28 \\ Algorithm 1 & \((-1099.92,-4877.25)^{T}\) & **4 min 42 s** & **447** & 1048.44 & 998.34 \\ Algorithm 3 & \((-1099.92,-4877.25)^{T}\) & 4 min 43 s & 542 & **1041.28** & **992.16** \\ \hline Algorithm A1 & \((-3197.16,-3844.24)^{T}\) & 2 min 17 s & 365 & **845.93** & **805.94** \\ Algorithm 1 & \((-3197.16,-3844.24)^{T}\) & **2 min 13 s** & **346** & 852.36 & 811.15 \\ Algorithm 3 & \((-3197.16,-3844.24)^{T}\) & 2 min 59 s & 366 & 848.52 & 808.40 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of metrics between Algorithms A1, 1 and 3.

meshes (see Section 4.4). Hence, a substantial performance improvement could be achieved by reducing the complexity of the transformations of Equations (21) and (31). There are several interesting research directions from the presented approach and observed numerical results.