Generative AI for subgrid turbulence in large-eddy simulations

TL;DR

A denoising diffusion probabilistic model is introduced to reconstruct SGS stresses from coarse-grained velocity fields in direct numerical simulations of the atmospheric boundary layer, and consistently outperforms Smagorinsky-type models and previous deep neural networks in terms of spatial correlations and probability distributions for deviatoric stresses.

Abstract

Turbulence governs the transport of momentum, energy, and scalars in many geophysical and engineering flows. In large-eddy simulations (LES), parameterizing subgrid-scale (SGS) stresses remains a central challenge, as unresolved physical processes strongly influence turbulent transport. Traditional SGS models, such as the Smagorinsky-type models and deep neural networks (DNNs), are deterministic and cannot capture the stochastic nature of turbulence. Despite its wide application in computer vision and natural language processing, generative artificial intelligence (AI) has not previously been applied to directly compute SGS stresses in three-dimensional turbulent boundary layers at high Reynolds numbers. Here we introduce a denoising diffusion probabilistic model (DDPM) to reconstruct SGS stresses from coarse-grained velocity fields in direct numerical simulations of the atmospheric boundary layer. The DDPM consistently outperforms Smagorinsky-type models and previous deep neural networks in terms of spatial correlations and probability distributions for deviatoric stresses, and can be applied to unseen convective stability conditions and resolutions. By learning conditional distributions rather than pointwise values, this generative approach opens a new direction for SGS turbulence modeling at high Reynolds numbers.

Figures (4)

• Figure 1: Schematic of the denoising diffusion probabilistic model (DDPM) used in this study. Original DNS velocity fields $(u,v,w)$ are filtered into coarse-grained $(\overline{u},\overline{v},\overline{w})$. Gaussian noise $\epsilon \sim \mathcal{N}(0,1)$ is added to the SGS stress to generate a noisy stress $\tau_{ij,t}$, where $t$ is timestep. The input tensor consists of the filtered velocity components and $\tau_{ij,t}$ over $3\times3\times3$ grid points and is passed through a shallow 3D U-Net ronneberger2015ucciccek20163d. The network predicts the noise $\hat{\epsilon}$. The training minimizes the mean squared error (MSE) loss $L = \|\hat{\epsilon}-\epsilon \|^2$.
• Figure 2: SGS stresses of $x$--$y$ plane normalized by $U_g^2$ at $z^+ = \dfrac{z}{(\nu/u_\tau)} = 58.2$ in the log-law region cheng2021logarithmic of the moderately convective dataset Sh5. $U_g$ is geostrophic wind, $z_i$ is boundary layer height, $\nu$ is kinematic viscosity and $u_\tau$ is friction velocity. From top to bottom: SGS stresses calculated from DNS data ($\tau_{ij}^{\mathrm{DNS}}$), the Smagorinsky model ($\tau_{ij}^{\mathrm{S}}$), the Smagorinsky--Bardina mixed model ($\tau_{ij}^{\mathrm{SB}}$), the DNN model from Cheng et al. cheng2022deep ($\tau_{ij}^{\mathrm{NN}}$), and the proposed DDPM_multiSh ($\tau_{ij}^{\mathrm{DDPM}}$). Here the DDPM_multiSh is trained on the highly convective (Sh2) and weakly convective (Sh20) cases, and tested on the unseen moderately convective (Sh5) case.
• Figure 3: Comparison of $\tau_{13}^\mathrm{DNS}$ from DNS data, $\tau_{13}^\mathrm{S}$ from the Smagorinsky model, $\tau_{13}^\mathrm{SB}$ from the Smagorinsky-Bardina model, $\tau_{13}^\mathrm{NN}$ from the DNN model cheng2022deep, and $\tau_{13}^\mathrm{DDPM}$ from DDPM_multiSh. (a) Mean SGS stresses of $x$--$y$ plane at different height $z^+$. (b) Probability distribution function (pdf) of SGS stresses in the whole DNS field. (c) The correlation ($\rho$) between $\tau_{13}^{\mathrm{DNS}}$ and the predicted $\tau_{13}$ from each model in the $x$--$y$ plane at different height $z^+$. The analyzed dataset is the moderately convective Sh5, while the DDPM_multiSh is trained on the highly convective (Sh2) and weakly convective (Sh20) cases. $U_g$ is geostrophic wind, $\rho$ is correlation, and $z^+$ is the normalized distance to the wall. In (a) and (c), the height range $z^+$ between the two dotted lines denotes the log-law region cheng2021logarithmic in convective boundary layers.
• Figure 4: Comparison of the correlation coefficients ($\rho$) between $\tau_{13}^{\mathrm{DNS}}$ and $\tau_{13}^{\mathrm{DDPM}}$ from different DDPM over the whole DNS field. (a) Comparison of DDPM_multiSh (trained on Sh20 at $\Delta_z^+=21$ and Sh2 at $\Delta_z^+=6$), DDPM_multiSh_5$\times$5$\times$5 and DDPM_multiSh_7$\times$7$\times$7. The DDPM_multiSh is trained on velocity and SGS stress patches of size 3$\times$3$\times$3, while the latter two models are trained on patches of size 5$\times$5$\times$5 and 7$\times$7$\times$7, respectively. (b) Comparison of DDPM_multiSh, DDPM_multiC (trained on Sh20 at $\Delta_z^+=11$, 21 42 and 64), and DDPM_multiT (trained on Sh20 at $\Delta_z^+=21$ across multiple time steps). None of the datasets shown along the $x$-axis are used during the DDPM training. For example, DDPM_multiT (trained on Sh20 at $\Delta_z^+=21$) is evaluated on different, unseen time steps.