Complex spatial and temporal structures are inherent characteristics of
turbulent fluid flows and comprehending them poses a major challenge. This
comprehesion necessitates an understanding of the space of turbulent fluid flow
configurations. We employ a diffusion-based generative model to learn the
distribution of turbulent vorticity profiles and generate snapshots of
turbulent solutions to the incompressible Navier-Stokes equations. We consider
the inverse cascade in two spatial dimensions and generate diverse turbulent
solutions that differ from those in the training dataset. We analyze the
statistical scaling properties of the new turbulent profiles, calculate their
structure functions, energy power spectrum, velocity probability distribution
function and moments of local energy dissipation. All the learnt scaling
exponents are consistent with the expected Kolmogorov scaling and have lower
errors than the training ones. This agreement with established turbulence
characteristics provides strong evidence of the model's capability to capture
essential features of real-world turbulence.

利用基于扩散的生成模型学习湍流涡度轮廓的分布，生成与训练数据集不同的多样化湍流解，并分析新湍流轮廓的统计缩放特性、能量功率谱、速度概率分布函数和局部能量耗散矩。通过与已建立的湍流特性的一致性，该模型证明了其捕捉实际湍流关键特征的能力。