# Topology Optimisation under Uncertainties with Neural Networks
## 3 Neural Network Architectures
### Topology Convolutional Neural Networks (TCNN)
#### 3.1.2 Data Preparation
### Deterministic Bridge Optimisation

The motivation for the design of the TLSTM architecture was that the information contained in the time series of \(\varphi_{n}\) and \(u_{n}\) could ideally lead to an improvement in the forecast capabilities of the NN. This can be investigated as in Section 4.1.4 by calculating the optimal topology for different loading scenarios for \(g\) by Algorithm 3. Again, all results are based on the ten-fold averaged performance of the algorithms for each load \(g\). Figure 11 shows the results in the setting similar to Appendix A.1 and compares the respective metrics. Analogous to Section 4.1.3 the sequence \(\varphi_{n}\) of the optimisation by Algorithm 3 is also more resilient to porous fragments in the structures than the reference optimisation procedure. In general, the pictures of \(\varphi_{n}\) hardly differ between Algorithms 1 and 3. Hence, apart from Figure 11, no further visualisations are presented. It should also be noted that the optimisation by Algorithm 3 is much less stable than the optimisation using \(\mathcal{N}_{\text{CNN}}\) from Equation (23). This becomes apparent when the structure collapses which was the case in each of our test runs if the \(c_{m}\in\mathbb{N}\) chosen was too small in Algorithm 3. Furthermore, it could be observed that the convergence criterion of Equation (A1) was not reached after applying \(\mathcal{N}_{\text{LSTM}}\) because \(\varphi_{n}\) diverged too far from the actual minimum on \(\mathcal{T}_{m}\). In conclusion, the stability of Algorithm 3 is even more dependent on \(c_{m}\) than it is with Algorithm 1, which renders parameter calibration more difficult. However, the metrics in Figure 11 show that Algorithm 3 converges faster than Algorithm 1 and often achieves better results. One conspicuous feature is the high fluctuation in the calculation time per iteration in the applications of \(\mathcal{N}_{\mathrm{LSTM}}(\,\cdot\,;\theta)\) during the optimisation when compared to Algorithm 1. On the one hand, this is due to the comparatively high complexity of the TLSTM. In this context, high complexity means a high dimension \(d\in\mathbb{N}\) of the parameter vector \(\theta\in\mathbb{R}^{d}\). On the other hand, the main driver of the higher calculation time is the transformation given by Equation (31) since on an algorithmic level the entire input sequence \(\varphi_{n}\), \(\varphi_{n+1}\), \(\varphi_{n+2}\), \(\varphi_{n+3}\), \(\varphi_{n+4}\) has to be stored and transformed. In general, it becomes apparent at this point that the transformations in Equations (21) and (31) are the critical aspects that compromise the performance of Algorithms 1 and 3. Table 4 compares the performance of the three presented algorithms. In overall terms, Algorithm 3 achieves better compliance, whereas Algorithm 1 stands out owing to its shorter calculation time. ## 5 Discussion and Conclusions

The objective of this work is to devise neural network architectures that can be used for efficient topology optimisation problems. These tasks are computationally burdensome and typically are inevitably carried out with a large number of optimisation steps, each requiring (depending on the chosen method) the solution of state and adjoint equations to determine the gradient direction. Instead of learning a surrogate for state and adjoint equations, we present NN architectures that directly predict this gradient, leading to very efficient optimisation schemes. A noteworthy aspect of our investigation is the consideration of uncertainties of model data in a risk-averse optimisation formulation. This is a generalisation of the notion of \"loading scenarios\" that are commonly used in practice for a fixed set of parameter realisations. With our continuous presentation of uncertainties in the material and of the load acting on the considered structure, the robustness of the computed design with respect to these uncertainties can be controlled by the parameter of the CVaR used in the cost functional.