Generative modeling for ergodic systems
Year: 2026
Type: dissertation
Abstract: Chaotic dynamical systems are characterized by strong nonlinearity, aperiodic long-term behavior, and high sensitivity to initial conditions, which make their numerical simulation extremely demanding, especially in engineering-relevant applications such as turbulent flows. Although ergodic theory provides a statistical framework for describing such systems, resolving their dynamics through fine-grid simulations of ODEs and PDEs remains computationally expensive, while reduced-order closures often introduce substantial modeling error. This thesis investigates deep generative learning as an alternative approach for the synthetic modeling of invariant state snapshots of chaotic systems, with a particular focus on generative adversarial networks (GAN). By combining principles from ergodic theory with generative learning, this work establishes for the first time that GAN training converges mathematically for ergodic systems in the limit of long observation times. This result provides a rigorous theoretical foundation for the use of GAN in modeling chaotic systems. Building on this foundation, the thesis further shows that statistics computed from GAN-generated snapshots converge to those obtained from simulations, uniformly over all uniformly bounded functions. Together, these findings establish a solid framework for evaluating generative models of chaotic dynamics through physics-based statistical measures rather than explicitly embedding such measures into the training loss. The proposed methodology is applied to a hierarchy of systems of increasing complexity, including the Lorenz attractor, the double pendulum, the Kuramoto–Sivashinsky equation, and turbulent flow configurations such as flow around a cylinder and flow under periodic wake impact. Across these cases, deep convolutional GAN are shown to reproduce snapshots of chaotic and turbulent states with high fidelity, using relatively small training datasets and without incorporating physical boundary conditions. At the same time, they offer substantial computational advantages, generating new snapshots within seconds compared with the days often required for Large Eddy Simulations. The thesis also demonstrates that conditional GAN enable controlled generation and strong generalization across unseen flow configurations, even when the available training data is heavily reduced. In addition, it presents a comparative study of GAN, variational autoencoders, and denoising diffusion probabilistic models on both simulated LES data and experimental PIV data. Among these models, GAN provide the best overall trade-off between accuracy and efficiency, while diffusion models show strong potential for future data-rich applications. Finally, the work takes a first step beyond invariant snapshots toward spatio-temporal generative modeling of turbulence, opening promising directions for future research on time-dependent chaotic flow dynamics.
Source: DepositOnce
Author: Drygala, Claudia
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214885496