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

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 $Re_{\lambda}\approx100$, decaying HIT at initial Taylor Reynolds number at $Re_{\lambda}\approx100$ and turbulent channel flow at friction Reynolds numbers $Re_{\tau}\approx395$ and $Re_{\tau}\approx590$. 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.