  We present a mathematically well founded approach for the synthetic modeling
of turbulent flows using generative adversarial networks (GAN). Based on the
analysis of chaotic, deterministic systems in terms of ergodicity, we outline a
mathematical proof that GAN can actually learn to sample state snapshots form
the invariant measure of the chaotic system. Based on this analysis, we study a
hierarchy of chaotic systems starting with the Lorenz attractor and then carry
on to the modeling of turbulent flows with GAN. As training data, we use fields
of velocity fluctuations obtained from large eddy simulations (LES). Two
architectures are investigated in detail: we use a deep, convolutional GAN
(DCGAN) to synthesise the turbulent flow around a cylinder. We furthermore
simulate the flow around a low pressure turbine stator using the pix2pixHD
architecture for a conditional DCGAN being conditioned on the position of a
rotating wake in front of the stator. The settings of adversarial training and
the effects of using specific GAN architectures are explained. We thereby show
that GAN are efficient in simulating turbulence in technically challenging flow
problems on the basis of a moderate amount of training data. GAN training and
inference times significantly fall short when compared with classical numerical
methods, in particular LES, while still providing turbulent flows in high
resolution. We furthermore analyse the statistical properties of the
synthesized and LES flow fields, which agree excellently. We also show the
ability of the conditional GAN to generalize over changes of geometry by
generating turbulent flow fields for positions of the wake that are not
included in the training data.
