[Paper Review] Bayesian Conditional Diffusion Models for Versatile Spatiotemporal Turbulence Generation

Han Gao, Xu Han+5 more|arXiv (Cornell University)|Nov 14, 2023

Generative Adversarial Networks and Image Synthesis4 citations

TL;DR

This paper proposes a Bayesian conditional diffusion model for versatile spatiotemporal turbulence generation, unifying unconditional and conditional sampling within a probabilistic framework to capture chaotic, stochastic flow dynamics. The method enables long-span flow sequence generation via autoregressive gradient-based sampling without retraining, achieving high-fidelity synthesis of LES from URANS inputs, wall-bounded turbulence from diverse conditions, and super-resolution of turbulent boundary layers across input resolutions.

ABSTRACT

Turbulent flows have historically presented formidable challenges to predictive computational modeling. Traditional numerical simulations often require vast computational resources, making them infeasible for numerous engineering applications. As an alternative, deep learning-based surrogate models have emerged, offering data-drive solutions. However, these are typically constructed within deterministic settings, leading to shortfall in capturing the innate chaotic and stochastic behaviors of turbulent dynamics. We introduce a novel generative framework grounded in probabilistic diffusion models for versatile generation of spatiotemporal turbulence. Our method unifies both unconditional and conditional sampling strategies within a Bayesian framework, which can accommodate diverse conditioning scenarios, including those with a direct differentiable link between specified conditions and generated unsteady flow outcomes, and scenarios lacking such explicit correlations. A notable feature of our approach is the method proposed for long-span flow sequence generation, which is based on autoregressive gradient-based conditional sampling, eliminating the need for cumbersome retraining processes. We showcase the versatile turbulence generation capability of our framework through a suite of numerical experiments, including: 1) the synthesis of LES simulated instantaneous flow sequences from URANS inputs; 2) holistic generation of inhomogeneous, anisotropic wall-bounded turbulence, whether from given initial conditions, prescribed turbulence statistics, or entirely from scratch; 3) super-resolved generation of high-speed turbulent boundary layer flows from low-resolution data across a range of input resolutions. Collectively, our numerical experiments highlight the merit and transformative potential of the proposed methods, making a significant advance in the field of turbulence generation.

Motivation & Objective

• To address the limitations of deterministic deep learning models in capturing the chaotic and stochastic nature of turbulent flows.
• To develop a unified generative framework capable of both unconditional and conditional sampling under diverse conditioning scenarios.
• To enable long-horizon spatiotemporal turbulence generation without retraining via autoregressive gradient-based conditional sampling.
• To achieve high-fidelity synthesis of turbulent flows from low-resolution inputs, URANS data, and initial/turbulence statistics.
• To provide a robust, probabilistic surrogate model for turbulence simulation that preserves statistical and dynamical properties of real flows.

Proposed method

• The framework employs a Bayesian diffusion process to model the posterior distribution of turbulent flow fields, enabling uncertainty-aware generation.
• It integrates both unconditional and conditional sampling by defining a score-based diffusion process conditioned on diverse inputs, including differentiable and non-differentiable conditions.
• A novel autoregressive gradient-based sampling strategy enables long-span sequence generation by iteratively refining flow states using conditional score functions.
• The method uses a denoising U-Net architecture with time-conditional conditioning to model the reverse diffusion process in a latent space.
• Conditional guidance is applied via a guidance coefficient βguide, which controls the strength of conditioning during sampling.
• The model is trained end-to-end on spatiotemporal flow data using a variational lower bound objective to maximize likelihood of observed turbulence realizations.

Experimental results

Research questions

• RQ1 Can a probabilistic diffusion model effectively generate realistic, statistically consistent turbulent flow sequences across diverse conditions?
• RQ2 How well can the model reconstruct high-fidelity Large Eddy Simulation (LES) fields from low-fidelity URANS inputs?
• RQ3 To what extent can the model generate inhomogeneous, anisotropic wall-bounded turbulence from initial conditions, turbulence statistics, or from scratch?
• RQ4 Can the model perform super-resolution of turbulent boundary layers from low-resolution inputs while preserving energy spectra and mean profiles?
• RQ5 How does the autoregressive gradient-based sampling strategy compare to standard diffusion sampling in terms of long-horizon fidelity and efficiency?

Key findings

• The model successfully synthesizes LES-level flow fields from URANS inputs, with generated sequences matching DNS in statistical properties such as energy spectra and RMS velocity fluctuations.
• For wall-bounded turbulence, the model generates physically realistic flow fields from initial conditions, prescribed turbulence statistics, and from noise, with mean velocity and temperature profiles closely matching DNS.
• In super-resolution tasks, the model reconstructs 256×256 high-resolution turbulent boundary layers from inputs as coarse as 2×2, preserving key features like boundary layer thickness and energy spectra.
• The energy spectra of generated flows align closely with DNS across all tested resolutions, with relative errors in the k⁻⁵ᐟ³ inertial subrange below 10% for high-resolution outputs.
• The autoregressive sampling strategy enables stable, long-span generation (up to 300 time steps) without retraining, with inference time under 10 seconds on a single GPU.
• Wall-clock time for generating a 300-step flow sequence is ~10 seconds using the diffusion model on an RTX 4090, significantly faster than OpenFOAM (hours) and FANCFD (tens of seconds).