# Topology Optimisation under Uncertainties with Neural Networks
## 1 Introduction
To improve this step, separately trained NNs are applied to the respective mesh in order to approximate the last steps of the optimisation on the corresponding mesh. The result is then projected to the next finer mesh and the procedure is repeated a fixed number of times. A SIMP density field topology optimisation is directly performed in [11]. The problem can be represented in terms of the NN activation function. Different beam problems comparable to our experiments are depicted. Fully connected DNNs are used in [12] to represent implicit level set function describing the topology. For optimisation, a system of parameterised ODEs is used. A two-stage NN architecture which by construction reduces the problem of structural disconnections is developed in [13]. Deep generative models for topology optimisation problems with varying design constraints and data scenarios are explored in [14]. In [15], direct predictions without any iteration scheme and also the nonlinear elastic setting are considered. Examples are only shown for a coarse mesh discretisation of the design domain. In [16], an NN-assisted design method for topology optimisation is devised, which does not require any optimised data. A predictor NN provides the designs on the basis of boundary conditions and degree of filling as input data for which no optimisation training data are required. The main goal of this paper is to devise new NN architectures that lower the computational burden of structural topology optimisation based on a continuous phase-field description of the density in the design domain. In particular, the approach should be able to cope with adaptive mesh refinements during the optimisation process, which has shown to significantly improve the performance of the optimisation. Moreover, as a consequence of an efficient computation in a deterministic setting, a goal is to transfer the developed techniques to the stochastic setting for the risk-averse topology optimisation task. The general strategy is to combine conventional topology optimisation methods and NNs in order to reduce the number of required iteration steps within the optimisation procedure, increasing the overall performance. The main achievements of this paper are two new NN architectures that are demonstrated to yield state-of-the-art numerical results with a much lower number of iterations than with a classical optimisation. Moreover, in contrast to several other works that are solely founded on the image level of topology, our architectures make use of a very versatile functional phase-field description of the material distribution, which we have not observed in the literature with regard to NNs. This also holds true for the stochastic risk-averse framework, which to our knowledge has not been considered with NN predictions yet. Another novelty is the mixture of a fine reference mesh and adaptive iteration meshes during optimisation. Inspired by the work of [5; 10], as a first new NN architecture we develop a new CNN approach and show that it can replicate the reference results of [1; 2]. In contrast to [10], we only have to train one NN for the entire optimisation despite mesh refinement being carried out in the iterative procedure. We subsequently extend this approach to the stochastic setting with risk-averse optimisation from [2]. Based on the CNN, we further extend the optimisation with a Long Short-Term Memory (LSTM) architecture as a second novel method. It encodes the change of the topology over several iteration steps, thus resulting in a more accurate prediction of the next gradient step. In the numerical experiments, it can be observed that the two presented architectures perform equally well as our reference implementation. However, fewer iteration steps are required (i.e., larger steps can be used) since the gradient step predictions seem to be better than when computed with a classical optimisation algorithm. The structure of the paper is as follows.