Uncertainty-aware Surrogate Models for Airfoil Flow Simulations with Denoising Diffusion Probabilistic Models (2312.05320v4)

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.

• The paper demonstrates that DDPMs can effectively capture turbulence uncertainty in airfoil simulations, outperforming traditional surrogate models.
• It shows that DDPMs accurately predict drag coefficient distributions and field standard deviations, offering improved precision over BNNs and heteroscedastic models.
• The study highlights that while DDPMs entail higher computational costs, their robust uncertainty quantification can significantly benefit aerodynamic optimization tasks.

A Technical Overview of Uncertainty-aware Surrogate Models for Airfoil Flow Simulations Using Denoising Diffusion Probabilistic Models

The paper, "Uncertainty-aware Surrogate Models for Airfoil Flow Simulations with Denoising Diffusion Probabilistic Models," presents a detailed exploration into modeling turbulence simulation uncertainty, employing a novel approach through denoising diffusion probabilistic models (DDPMs). This paper is particularly focused on airfoil flow simulations and compares DDPMs against traditional Bayesian neural networks (BNNs) and heteroscedastic models, revealing the potential and limitations of DDPMs in capturing the complexities of turbulence-related uncertainties.

Choice of Methodology

The authors have adeptly utilized DDPMs to enhance the surrogate modeling of airfoil flow simulations, a method sparingly explored in this domain. Typically recognized for outperforming other generative models like GANs or VAEs in generating high-quality outputs, DDPMs offer the unique advantage of providing granular access to the complete distribution of simulated solutions, thus inherently integrating uncertainty.

In this context, the paper discusses how DDPMs, unlike BNNs or heteroscedastic models, circumvent the need for a defined distribution of the given target variable by modeling it through a step-wise noise reduction process. This characteristic promotes a better representation of the uncertainty in RANS simulations, often hindered by BNNs due to their reliance on prior assumptions about network parameters or heteroscedastic models constrained by their Gaussian distribution premises.

Comparative Analysis

The authors present an extensive analysis of the learning capabilities across different methodologies:

1. Capturing Distributional Uncertainty: The key differentiator for DDPMs in this paper is their ability to generate plausible sample solutions compared to the noisy and less reliable samples generated by both BNNs and heteroscedastic models. While BNN samples are affected by the assumed prior distribution of network parameters, the heteroscedastic models assume a simplistic data distribution, which fails to capture the complexity of turbulent flow fields.
2. Numerical Results: Numerical experiments highlight that DDPMs surpass the alternative methods in both single and multi-parameter setups, particularly in accurately predicting fields’ standard deviations and handling extrapolations. The superior representation of turbulence uncertainty in airfoil simulations is evident in the DDPMs' accurate prediction of drag coefficients distributions compared to poorly modeled distributions by BNNs and heteroscedastic models.
3. Sampling Efficiency: Despite their enhanced modeling potential, the computational cost associated with DDPMs is notably higher due to the need for more iterative steps to produce viable samples.

Implications and Future Work

The implications of successfully employing DDPMs in turbulence simulations extend beyond current methodologies due to their capacity to handle uncertainty with higher accuracy. As surrogates, models built using DDPMs offer enhanced reliability, which can be highly beneficial in iterative engineering tasks, such as aerodynamic shape optimization, where understanding uncertainty is crucial.

Theoretically, the application of DDPMs can be further extended into turbulence modeling where capturing unresolved turbulent scales as probabilistic distributions has been an ongoing challenge. Future avenues for research could explore acceleration techniques for DDPM sampling and an investigation into expanding the generalization abilities beyond the current limits, potentially by varying dataset parameters more effectively.

In conclusion, the paper establishes a firm premise for adopting DDPMs as a beneficial tool in airfoil flow simulations, setting a path for their more extensive utilization in predictive modeling where uncertainty quantification is paramount.