# Topology Optimisation under Uncertainties with Neural Networks
## 1 Introduction
There are different approaches to describe the topology in a flexible way such that substantial changes are possible. We follow our previous work in [1] and use a phase field model which describes the density of material with a value in \([0,1]\). The starting point is the definition of a physical design space available for the component under consideration. This space is completely filled with a material in the sense of an initial solution. Furthermore, all points at which loads act on the component, as well as the type of the respective load, are prescribed. The optimisation aims to achieve a homogeneous, minimum possible deformation at all optimised points of the component under the imposed (possibly continuous and thus infinitely many) loading scenarios. Here, a minimum compliance corresponds to a maximum stiffness. In general, even solving the underlying partial differential equation (PDE) of this problem for deterministic coefficients of the PDE presents a complex task. Furthermore, PDE coefficients which describe material and the loads have a strong influence on the resulting topology, i.e., even small changes in these coefficients can lead to large differences in the resulting topology. This results in considerable computational effort in the stochastic settings, since the solution has to be calculated for a sufficient number of data realisations to become reliable. Hence, the modelling of these stochastic settings for example (with the most obvious approach) by a Monte Carlo (MC) simulation increases the required iteration steps linearly in the number of simulations. A method to numerically tackle topology optimisation uncertainty was presented in [2]. In this paper, we extend the previous work by introducing Deep Neuronal Networks (DNN) that are designed to provide a prediction of the next gradient step. Since topologies discretised with finite elements can be represented as images, Convolutional Neural Networks (CNN) seem natural candidate architectures for this task and there has already been some research on this approach for the deterministic setting. An introduction is presented in [3] where the conventional topology optimisation algorithms are replicated in a computationally inexpensive way. Furthermore, a CNN is used in [4] to approximate the last iteration steps of a gradient method of a topology optimisation after a fixed number of steps to refine a \"fuzzy\" solution. A CNN architecture is also used in [5] to solve a topology optimisation problem and trained with large amounts of data. The resulting NNs were able to solve problems with boundary conditions different to their training data. In [6], the problem is stated as an image segmentation task and a deep NN with encoder-decoder architecture is leveraged for pixel-wise labeling of the predicted topology. Another encoding-decoding U-Net CNN architecture is presented in [7], providing up- and down-sampling operators based on training with large datasets. In [8] a multilevel topology optimisation is considered where the macroscale elastic structure is optimised subject to spatially varying microscale metamaterials. Instead of density, the parameters of the micromaterial are optimised in the iteration, using a single-layer feedforward Gaussian basis function network as surrogate model for the elastic response of the microscale material. A discussion on solving PDEs with the help of Neural Networks (NN) for instance of the Poisson equation and the steady Navier-Stokes equations is provided in [9]. In a relatively new approach, through a combination of Deep Learning and conventional topology optimisation, the Solid Isotropic Material with Penalisation (SIMP) was presented in [10], which could reduce the computational time compared to the classical approach. The authors use a similar method as [4] except that the underlying optimisation algorithm performs a mesh refinement after a fixed number of iterations.