Integrating Fourier Neural Operator with Diffusion Model for Autoregressive Predictions of Three-dimensional Turbulence

TL;DR

The paper addresses the challenge of autoregressive forecasting for three-dimensional turbulence by introducing DiAFNO, a diffusion-model–driven framework that embeds an implicit adaptive Fourier neural operator (IAFNO) within an elucidated diffusion model (EDM) to achieve long-term, continuous predictions. By combining IAFNO's global frequency sensitivity with EDM's denoising capabilities, DiAFNO delivers higher accuracy in velocity spectra, RMS statistics, and Reynolds stresses across forced HIT, decaying HIT, and turbulent channel flow, at Reynolds numbers up to $Re_{\tau}\approx590$. Empirical results show DiAFNO outperforms the EDM and the conventional DSM in most metrics, while offering competitive or superior inference speed, suggesting practical value for efficient LES-like forecasting. The work highlights the potential of diffusion-based operators for turbulent flow prediction and points to future enhancements via physics-informed constraints and extensions to more complex geometries. Overall, DiAFNO advances scalable, autoregressive 3D turbulence forecasting with improved accuracy and efficiency over state-of-the-art diffusion-augmented operators and traditional LES closures.

Abstract

Accurately autoregressive prediction of three-dimensional (3D) turbulence has been one of the most challenging problems for machine learning approaches. Diffusion models have demonstrated high accuracy in predicting two-dimensional (2D) turbulence, but their applications in 3D turbulence are relatively limited. To achieve reliable autoregressive predictions of 3D turbulence, we propose the DiAFNO model which integrates the implicit adaptive Fourier neural operator (IAFNO) with diffusion model. IAFNO can effectively capture the global frequency and structural features, which is crucial for global consistent reconstructions of the denoising process in diffusion models. Furthermore, based on conditional generation from diffusion models, we design an autoregressive framework in DiAFNO to achieve long-term stable predictions of 3D turbulence. The proposed DiAFNO model is systematically tested with fixed hyperparameters in several types of 3D turbulence, including forced homogeneous isotropic turbulence (HIT) at Taylor Reynolds number $Re_λ\approx100$, decaying HIT at initial Taylor Reynolds number at $Re_λ\approx100$ and turbulent channel flow at friction Reynolds numbers $Re_τ\approx395$ and $Re_τ\approx590$. The results in the a posteriori tests demonstrate that DiAFNO exhibits a significantly higher accuracy in terms of the velocity spectra, the root-mean-square (RMS) values of both velocity and vorticity, and Reynolds stresses, as compared to the elucidated diffusion model (EDM) and the traditional large-eddy simulation (LES) using dynamic Smagorinsky model (DSM). Meanwhile, the well-trained DiAFNO is faster than LES with the DSM.

Figures (19)

• Figure 1: The autoregressive prediction architecture of DiAFNO: (a) the training process; (b) the sampling process. Dataset $x_m$ contains flow fields from $U_0$ to $U_{N-1}$ and dataset $y_m$ contains flow fields from $U_1$ to $U_{N}$.
• Figure 2: The architecture of IAFNO: (a) the macro architecture of IAFNO as $F_{\theta}$ in DiAFNO; (b) the architecture of $\mathcal{L}^{\textrm{IAFNO}}$.
• Figure 3: The velocity spectra of various models in the forced HIT at different time instants: (a) $t/\tau\approx 4.0$; (b) $t/\tau\approx 6.0$; (c) $t/\tau\approx 8.0$; (d) $t/\tau\approx 50.0$.
• Figure 4: The PDFs of the normalized vorticity $\bar{\omega} / \bar{\omega}^{\textrm{rms}}_{\textrm{fDNS}}$ of various models in the forced HIT at different time instants: (a) $t/\tau\approx 4.0$; (b) $t/\tau\approx 6.0$; (c) $t/\tau\approx 8.0$; (d) $t/\tau\approx 50.0$.
• Figure 5: Temporal evolutions of (a) the velocity rms value and (b) vorticity rms value of various models in the forced HIT.