Generative modeling of turbulence

Abstract

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 from 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 synthesize 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 analyze 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.

Generalization capabilities of conditional GAN for turbulent flow under changes of geometry

“…Our previous work [5] showed that the application of generative learning for deterministic ergodic systems proves to be a mathematically well-founded approach since it converges in the limit of large observation time. Below, we briefly recapitulate the notion of ergodicity for better understanding and explain the mathematical foundations of conditional generative learning for ergodic systems.…”

“…We compare GAN synthesized and LES turbulence by quantities that can be cast in an abstract form E x∼µ [ψ(x)] of ( 2) and (3) given a certain evaluation function ψ. Our previous work [5] shows that this approach is reasonable since any statistic evaluated on GAN synthesized flow fields converges on average to the corresponding statistic evaluated on LES data in the limit of large data and large network capacity, which makes this convergence uniform over all uniformly bounded functions ψ.…”

“…In this work, we investigate a special type of cDCGAN called pix2pixHD [15] which allows us to generate high-resolution photo-realistic images from semantic segmentation masks by modifying the architecture of φ and D and extending the loss-function (5). The generator φ is composed of two subnetworks φ 1 and φ 2 assuming the role of a global generator and a local enhancer.…”

“…The numerical setup of the LES can be found in [5]. In total, the data set consists of 2, 250 images, corresponding to 10 bar passing periods, with a resolution of 1, 000 × 625 pixels.…”

“…In our previous work [5], we addressed the first concern and proved that the application of GAN to ergodic systems is a mathematically well-founded approach, and that we are able to synthesize high quality turbulent flows with GAN, which well match the physics based features of LES. To answer to the second concern, we also made the first successful attempt to generalize with respect to spatial changes using the deep convolutional conditional GAN framework pix2pixHD.…”