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.

翻译：复杂的时间和空间结构是湍流流体流动的固有特征，理解这些结构构成重大挑战。这种理解需要掌握湍流流体流动构型的空间。我们采用基于扩散的生成模型学习湍流涡度分布的统计规律，生成不可压缩纳维-斯托克斯方程湍流解的瞬时快照。我们考虑二维空间中的逆级串过程，生成与训练数据集不同的多样化湍流解。我们分析新湍流剖面的统计标度特性，计算其结构函数、能量功率谱、速度概率分布函数以及局部能量耗散的矩。所有学习得到的标度指数均符合预期的科尔莫戈罗夫标度，且误差低于训练数据。这种与已知湍流特征的一致性有力证明了该模型捕捉真实世界湍流本质特征的能力。