Francesca di Mare

Claudia Drygala, Francesca di Mare, Hanno Gottschalk

Turbulent flow consists of structures with a wide range of spatial and temporal scales which are hard to resolve numerically. Classical numerical methods as the Large Eddy Simulation (LES) are able to capture fine details of turbulent structures but come at high computational cost. Applying generative adversarial networks (GAN) for the synthetic modeling of turbulence is a mathematically well-founded approach to overcome this issue. In this work, we investigate the generalization capabilites of GAN-based synthetic turbulence generators when geometrical changes occur in the flow configuration (e.g. aerodynamic geometric optimization of structures such as airfoils). As training data, we use the flow around a low-pressure turbine (LPT) stator with periodic wake impact obtained from highly resolved LES. To simulate the flow around a LPT stator, we use the conditional deep convolutional GAN framework pix2pixHD conditioned on the position of a rotating wake in front of the stator. For the generalization experiments we exclude images of wake positions located at certain regions from the training data and use the unseen data for testing. We show the abilities and limits of generalization for the conditional GAN by extending the regions of the extracted wake positions successively. Finally, we evaluate the statistical properties of the synthesized flow field by comparison with the corresponding LES results.

Claudia Drygala, Edmund Ross, Francesca di Mare et al.

Numerical simulations of turbulent flows present significant challenges in fluid dynamics due to their complexity and high computational cost. High resolution techniques such as Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES) are generally not computationally affordable, particularly for technologically relevant problems. Recent advances in machine learning, specifically in generative probabilistic models, offer promising alternatives for turbulence modeling. This paper investigates the application of three generative models - Variational Autoencoders (VAE), Deep Convolutional Generative Adversarial Networks (DCGAN), and Denoising Diffusion Probabilistic Models (DDPM) - in simulating a 2D Kármán vortex street around a fixed cylinder. Training data was obtained by means of LES. We evaluate each model's ability to capture the statistical properties and spatial structures of the turbulent flow. Our results demonstrate that DDPM and DCGAN effectively replicate the flow distribution, highlighting their potential as efficient and accurate tools for turbulence modeling. We find a strong argument for DCGAN, as although they are more difficult to train (due to problems such as mode collapse), they gave the fastest inference and training time, require less data to train compared to VAE and DDPM, and provide the results most closely aligned with the input stream. In contrast, VAE train quickly (and can generate samples quickly) but do not produce adequate results, and DDPM, whilst effective, is significantly slower at both inference and training time.

Edmund Ross, Claudia Drygala, Leonhard Schwarz et al.

In this work, we explore the use of compact latent representations with learned time dynamics ('World Models') to simulate physical systems. Drawing on concepts from control theory, we propose a theoretical framework that explains why projecting time slices into a low-dimensional space and then concatenating to form a history ('Tokenization') is so effective at learning physics datasets, and characterise when exactly the underlying dynamics admit a reconstruction mapping from the history of previous tokenized frames to the next. To validate these claims, we develop a sequence of models with increasing complexity, starting with least-squares regression and progressing through simple linear layers, shallow adversarial learners, and ultimately full-scale generative adversarial networks (GANs). We evaluate these models on a variety of datasets, including modified forms of the heat and wave equations, the chaotic regime 2D Kuramoto-Sivashinsky equation, and a challenging computational fluid dynamics (CFD) dataset of a 2D Kármán vortex street around a fixed cylinder, where our model is successfully able to recreate the flow.

Claudia Drygala, Benjamin Winhart, Francesca di Mare et al.

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.