# Topology Optimisation under Uncertainties with Neural Networks
## 3 Neural Network Architectures
### Topology Convolutional Neural Networks (TCNN)
#### 3.1.2 Data Preparation
So instead of simply refining the previous mesh, for every refinement we start with the initial mesh \(\mathcal{T}_{0}\), interpolate the current solution and displacement onto that mesh, refine according to the associated indicators and interpolate the current solutions \(\varphi_{n}\) and \(u_{n}\) onto the finer mesh. We repeat this process until a mesh \(\mathcal{T}_{m+1}\) is obtained that is adequately finer than \(\mathcal{T}_{m}\). This refinement takes place whenever \(\varphi_{n}\) converges on the current mesh \(\mathcal{T}_{m}\). The refinement across an optimisation with Algorithm 1 is shown in Figure 14. It can be observed that the mesh is refined along the edges of \(\varphi\). This dynamic characterisation (depending on \(\varphi\) and thus on the coefficients of the associated PDE) of the domain by the mesh \(\mathcal{T}_{m}\) poses a special challenge for the presented NN architectures when solving Equation (6) or (11), respectively. The relative change \(e_{n}:=\frac{\|\varphi_{n+1}-\varphi_{n}\|_{L^{2}(D)}}{\|\varphi_{n+1}\|_{L^{ 2}(D)}}\) in combination with the step size \(\tau_{n}\) is used as convergence criterion, i.e.,

\[\frac{e_{n}}{\tau_{n}}<\varepsilon,\qquad\varepsilon>0. \tag{11}\]

This means that as soon as \(\varphi\) on a mesh \(\mathcal{T}_{m}\) does not change significantly in relation to the step size \(\tau_{n}\), \(\phi_{n}\) is considered converged. The refinement of the mesh is bounded by a chosen maximum of vertices \(|V(\mathcal{T}_{m})|\leq b\in\mathbb{N}\). We understand \(\varphi_{n}\in\mathbb{R}^{|(\mathcal{T}_{m})|}\) and \(u_{n}\in\mathbb{R}^{d\times|(\mathcal{T}_{m})|}\) as discretisation on \(\mathcal{T}_{m}\) at iteration step \(n\in\mathbb{N}\) in the sense that the value at every node \(v_{i}\in V(\mathcal{T}_{m}),\ i\leq|V(\mathcal{T}_{m})|\) is evaluated according to

\[u_{n}^{i}:=u_{n}(v_{i})\quad\text{or}\quad\varphi_{n}^{i}:=\varphi_{n}(v_{i})\]

in this node. ## Appendix C Additional Experiments

Figures 14 and 15 numerically illustrate less robust TCNN predictions for large gradient step sizes as mentioned in Section 4.1.2. In Figure 16, some iterations of a classical optimisation in comparison with the CNN assisted optimisation are depicted, showing basically identical results. However, it can also be observed that the distribution of the material converges faster when using the CNN. After 100 iterations, the first spokes for stabilising the arc can already be seen. ## References

* [PERSON] et al. (2018) [PERSON]; [PERSON]; [PERSON]; [PERSON] Risk averse stochastic structural topology optimization. _Comput. Methods Appl. Mech. Eng._**2018**, 334, 470-482. [CrossRef]
* [PERSON] et al. (2016) [PERSON]; [PERSON]; [PERSON]; [PERSON] Stochastic topology optimisation with hierarchical tensor reconstruction. _WIAS_**2016**, 2362. [CrossRef]
* [PERSON] and [PERSON] (2019) [PERSON]; [PERSON] A Novel Topology Optimization Approach using Conditional Deep Learning. _arXiv_ **2019**, arXiv:1901.04859. * [PERSON] et al. (2018) [PERSON]; [PERSON]; [PERSON] One-Shot Optimal Topology Generation through Theory-Driven Machine Learning. _arXiv_ **2018**, arXiv:1807.10787. * [PERSON] et al. (2019) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] A deep Convolutional Neural Network for topology optimization with strong generalization ability. _arXiv_ **2019**, arXiv:1901.07761. * [PERSON] (2019) [PERSON]; [PERSON] Neural networks for topology optimization. _Russ. [PERSON] Numer. Anal. Math. Model._**2019**, 34, 215-223. [CrossRef]
* [PERSON] et al. (2022) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] A deep convolutional neural network for topology optimization with perceptible generalization ability. _Eng. Optim._**2022**, 54, 973-988. [CrossRef]
* [PERSON] et al. (2019) [PERSON]; [PERSON]; [PERSON]; [PERSON] Multiscale topology optimization using neural network surrogate models. _Comput. Methods Appl. Mech. Eng._**2019**, 346, 1118-1135. [CrossRef]
* [PERSON] (2019) [PERSON] A Discussion on Solving Partial Differential Equations using Neural Networks.