Uncertainty-aware Surrogate Models for Airfoil Flow Simulations with Denoising Diffusion Probabilistic Models

TL;DR

This first attempt to use denoising diffusion probabilistic models (DDPMs) to train an uncertainty-aware surrogate model for turbulence simulations shows that DDPMs can successfully capture the whole distribution of solutions and, as a consequence, accurately estimate the uncertainty of the simulations.

Abstract

Leveraging neural networks as surrogate models for turbulence simulation is a topic of growing interest. At the same time, embodying the inherent uncertainty of simulations in the predictions of surrogate models remains very challenging. The present study makes a first attempt to use denoising diffusion probabilistic models (DDPMs) to train an uncertainty-aware surrogate model for turbulence simulations. Due to its prevalence, the simulation of flows around airfoils with various shapes, Reynolds numbers, and angles of attack is chosen as the learning objective. Our results show that DDPMs can successfully capture the whole distribution of solutions and, as a consequence, accurately estimate the uncertainty of the simulations. The performance of DDPMs is also compared with varying baselines in the form of Bayesian neural networks and heteroscedastic models. Experiments demonstrate that DDPMs outperform the other methods regarding a variety of accuracy metrics. Besides, it offers the advantage of providing access to the complete distributions of uncertainties rather than providing a set of parameters. As such, it can yield realistic and detailed samples from the distribution of solutions. We also evaluate an emerging generative modeling variant, flow matching, in comparison to regular diffusion models. The results demonstrate that flow matching addresses the problem of slow sampling speed typically associated with diffusion models. As such, it offers a promising new paradigm for uncertainty quantification with generative models.

Figures (27)

• Figure 1: One instance of the encoded input and output simulation data (ah21-7 airfoil, $Re=8.616\times10^3$, and $\alpha=-18.93^\circ$).
• Figure 2: The uncertainty distribution in the datasets. a) The training dataset. b) The test dataset. The shaded area shows the extrapolation region.
• Figure 3: The sketch of uncertainty prediction process using the DDPM.
• Figure 4: Uncertainty transitions of the RANS simulation for raf30 airfoil. Labeled arrows denote parameter regions as in Fig. \ref{['fig:flow_std']}.
• Figure 5: Mean streamlines and the uncertainty distribution of velocity magnitude $\sigma_{|\mathbf{u^*}|}$ (raf30 airfoil). a,b) $Re=6.5 \times 10^6$ with varying $\alpha$. c) $\alpha=20^{\circ}$, and d) $\alpha=-20^{\circ}$, both with varying $Re$.