利用湍流自身对称性，无需显式增强即可实现旋转等变性，提升超分辨率效率。
Implicit Augmentation from Distributional Symmetry in Turbulence Super-Resolution
- 利用湍流时空自有的旋转对称性，隐式实现数据增强
- 各向同性区域训练的模型等变误差更低，采样越密误差越小
- 揭示尺度相关的等变误差规律，支持柯尔莫哥洛夫局部各向同性假说
模拟湍流的计算成本极高，促使人们采用机器学习方法进行湍流超分辨率重建。核心挑战在于确保学习模型满足物理对称性，如旋转等变性。我们发现，标准卷积神经网络（CNN）可在无显式数据增强或特殊架构的情况下，部分获得该对称性，因为湍流本身在时间和空间上提供了隐式的旋转增强。通过对具有不同各向异性特征的三维通道流子域进行实验，发现以更各向同性的中平面数据训练的模型，其等变误差低于边界层数据训练的模型；且更大的时空采样可进一步降低误差。此外，无论数据集各向异性如何，均存在显著的尺度依赖性等变误差，与柯尔莫哥洛夫局部各向同性假说一致。这些结果明确了在何种情况下需显式引入对称性约束，何种情况下可直接从湍流中获取，从而实现更高效、对称感知的超分辨率。
原文摘要 · Abstract (English)
The immense computational cost of simulating turbulence has motivated the use of machine learning approaches for super-resolving turbulent flows. A central challenge is ensuring that learned models respect physical symmetries, such as rotational equivariance. We show that standard convolutional neural networks (CNNs) can partially acquire this symmetry without explicit augmentation or specialized architectures, as turbulence itself provides implicit rotational augmentation in both time and space. Using 3D channel-flow subdomains with differing anisotropy, we find that models trained on more isotropic mid-plane data achieve lower equivariance error than those trained on boundary layer data, and that greater temporal or spatial sampling further reduces this error. We show a distinct scale-dependence of equivariance error that occurs regardless of dataset anisotropy that is consistent with Kolmogorov's local isotropy hypothesis. These results clarify when rotational symmetry must be explicitly incorporated into learning algorithms and when it can be obtained directly from turbulence, enabling more efficient and symmetry-aware super-resolution.
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