Benchmarking Autoregressive Conditional Diffusion Models for Turbulent Flow Simulation

TL;DR

The paper tackles the challenge of temporal stability in learned PDE solvers for turbulent flows by proposing autoregressive conditional diffusion models (ACDMs) and benchmarking them against diverse architectures across three challenging 2D flow scenarios. It demonstrates that simple diffusion-based predictors can match or surpass traditional stabilization techniques like unrolling, while uniquely providing probabilistic posterior samples that align with physical statistics. The study offers a comprehensive comparison of accuracy, stability, and spectral fidelity, highlighting tradeoffs between training versus inference costs and the value of probabilistic forecasts for turbulence. The results suggest that diffusion-based surrogates are a compelling route for robust, probabilistic turbulence prediction and motivate future exploration into sampling efficiency and three-dimensional extensions.

Abstract

Simulating turbulent flows is crucial for a wide range of applications, and machine learning-based solvers are gaining increasing relevance. However, achieving temporal stability when generalizing to longer rollout horizons remains a persistent challenge for learned PDE solvers. In this work, we analyze if fully data-driven fluid solvers that utilize an autoregressive rollout based on conditional diffusion models are a viable option to address this challenge. We investigate accuracy, posterior sampling, spectral behavior, and temporal stability, while requiring that methods generalize to flow parameters beyond the training regime. To quantitatively and qualitatively benchmark the performance of various flow prediction approaches, three challenging 2D scenarios including incompressible and transonic flows, as well as isotropic turbulence are employed. We find that even simple diffusion-based approaches can outperform multiple established flow prediction methods in terms of accuracy and temporal stability, while being on par with state-of-the-art stabilization techniques like unrolling at training time. Such traditional architectures are superior in terms of inference speed, however, the probabilistic nature of diffusion approaches allows for inferring multiple predictions that align with the statistics of the underlying physics. Overall, our benchmark contains three carefully chosen data sets that are suitable for probabilistic evaluation alongside various established flow prediction architectures.

Figures (45)

• Figure 1: Diffusion conditioning approach with the forward (black) and reverse process (red) during training and inference. White backgrounds for ${\bm{c}}$ or ${\bm{d}}$ indicate given information, i.e., inputs for each phase. In the context of the autoregressive surrogate simulator, ${\bm{c}}_0$ contains information about the simulated process like Reynolds or Mach number, as well as the initial or previous simulation state. ${\bm{d}}_0$ contains the next target simulation state during training, and is the resulting prediction of the next state during inference.
• Figure 2: Autoregressive simulation rollout during inference with diffusion models for $k=2$ input steps (left), and contents of each simulation state (right).
• Figure 3: Zoomed example trajectories from Inchigh with $\mathit{Re} = 1000$ (left, vorticity), Tralong with $\mathit{Ma} = 0.64$ (middle, pressure), and Iso with $z = 280$ (right, vorticity). Shown are trajectories from the numerical solver, and predictions by key architectures from each model class (also see https://ge.in.tum.de/publications/2023-acdm-kohl/).
• Figure 4: Accuracy visualization for the architectures from \ref{['tab: accuracy']}. Shown are MSE and LSiM errors with corresponding standard deviation.
• Figure 5: Large-scale posterior sample comparison with corresponding standard deviation. Shown are the vortices downstream of the cylinder on a trajectory from Tralong with $\mathit{Ma} = 0.64$ (pressure) at different time steps $t$ (also see https://ge.in.tum.de/publications/2023-acdm-kohl/).