# Topology Optimisation under Uncertainties with Neural Networks
## 1 Introduction  
Structural topology optimisation is the (engineering-oriented) process of designing a construction part using optimisation algorithms under certain constraints. The resulting designs usually have a large influence on the subsequent production costs. The starting point of the process is a design domain that represents the maximum space available for the optimised component to be developed. The outcome presents information about which parts of the design space are occupied by material and which are void. Often, the task is motivated by mechanical requirements, e.g., sufficient stiffness of the constructed part with respect to assumed forces acting on it while certain predetermined points or surfaces should connect to other parts. A typical physical model for this comes from linear elasticity, describing the displacement field given material properties and forces. For the mathematical optimisation, it is repeatedly necessary to compute the stress distribution determined by the physical model in the design domain (more precisely, in the parts of the domain with material). This potentially complex computational task usually relies on the finite element method (FEM), which is based on a discretisation of the domain into elements. Most commonly, the domain is represented as mesh consisting of disjoint simplices, i.e., triangles in 2D and tetrahedra in 3D. Since the optimisation process easily requires several hundred evaluations of the state equation to evolve the material distribution, it is of significant interest to develop techniques that reduce this computational burden. This becomes much more pronounced when uncertainties of the model data should be considered in the computations. The treatment of uncertainties has been developed extensively from a theoretical and practical point of view over the last decade in the area of Uncertainty Quantification (UQ). A common way to describe uncertainties is by means of random fields, whose (Karhunen-Loeve) expansions depend on a possibly very large number of random variables. The parameter space spanned by these random variables leads to very high-dimensional state problems for which the derived optimisation problems are very difficult to solve. This paper investigates the application of a trend in scientific computing for current topology optimisation methods, namely the use of modern machine learning techniques. More precisely, our objective is to improve the efficiency of the structural topology optimisation problem by predicting gradient steps based on generated training data. This efficiency gain directly transfers to our ability to compute much more involved risk-averse stochastic topology optimisation problems with random data. In this case, the topology is optimised with an adjusted cost functional including the CVaR (conditional value at risk), by which unlikely events can be taken into account in contrast to simply optimising with the mean value of possible load scenarios. In addition to random loads, we also include random material properties which, e.g., can enter the model in terms of material errors or impurities. We emphasise that risk-averse optimisation based on some risk measure is a timely topic, which plays a role in many application areas. Despite its relevance, this type of problem has not been covered widely in the literature yet. In fact, the authors are not aware of any other machine-learning-assisted approach for risk-averse topology optimisation. This might be due to the more involved mathematical framework and substantially higher computational complexity. To achieve performance benefits with topology optimisation in this paper, we adapt concepts from the field of deep learning to approximate multiple iterations of the optimisation process and render the overall optimisation more efficient. The goal of topology optimisation is to satisfy the technical requirements of a component (for instance, stiffness with respect to certain loading scenarios) with minimal use of material.