Chaotic deterministic systems are characterized by a high degree of nonlinearity and unsteady, aperiodic behavior that is sensitive to initial conditions [1]. Numerical simulations of these systems often require the solution of partial differential equations on very fine computational meshes with small time steps. For engineering problems (e.g., high Reynolds numbers in turbulent flow simulations), the computational effort is unfeasible in a reasonable time despite rapidly increasing computer power [2]. To overcome this problem, we apply generative adversarial networks (GAN) as a mathematically well-founded approach [3] for the synthetic modeling of sample state snapshots from the invariant measure of chaotic deterministic systems at different hierarchy levels. With the increase of the complexity of the chaotic systems, the requirements for the GAN architecture also increase. Starting with a Vanilla GAN, we synthesize data points of the three-dimensional trajectory of the Lorenz attractor and proceed to the four-dimensional problem of the double pendulum. This is followed by modeling the turbulent flow around a cylinder using a deep convolutional GAN (DCGAN). We end with the synthesis of the flow around a low-pressure turbine stator using the conditional DCGAN pix2pixHD conditioned on the position of a rotating wake in front of the stator as the most complex example of a chaotic deterministic system. Furthermore, the ability of the conditional GAN to generalize over changes in geometry is demonstrated by generating turbulent flow fields for wake positions not included in the training data [4]. We compare the statistical properties of the synthesized state snapshots with those obtained by classical numerical methods. For the generative modeling of turbulence, we use fields of velocity fluctuations obtained from large-eddy simulations (LES) as training data. We show that GAN are efficient for simulating turbulence with a moderate amount of training data. The GAN training and inference times are significantly reduced compared to LES, while still providing turbulent flows with high resolution.

[1] S.H. Strogatz (2018). "Nonlinear dynamics and chaos with student solutions manual: With applications to physics, biology, chemistry, and engineering". CRC press.

[2] B.Winhart, M. Sinkwitz, A. Schramm, P. Post and F. di Mare, "Large Eddy Simulation of Periodic Wake Impact on Boundary Layer Transition Mechanisms on a Highly Loaded Low-Pressure Turbine Blade." Proceedings of the ASME Turbo Expo 2020: Turbomachinery Technical Conference and Exposition. Volume 2E: Turbomachinery. Virtual, Online. September 21–25, 2020. V02ET41A013. ASME. https://doi.org/10.1115/GT2020-14555.

[3] C. Drygala, B. Winhart, F. di Mare, and H. Gottschalk, "Generative modeling of turbulence", Physics of Fluids 34, 035114 (2022) https://doi.org/10.1063/5.0082562.

[4] C. Drygala, F. di Mare, and H. Gottschalk (2023). "Generalization capabilities of conditional GAN for turbulent flow under changes of geometry". arXiv preprint arXiv:2302.09945 - accepted at EUROGEN (International Conference on Evolutionary and Deterministic Methods for Design, Optimization and Control) 2023.