Fractal-constrained deep learning for super-resolution of turbulence with zero or few label data

Abstract

The super-resolution of turbulence is of paramount importance and still remains challenging due to the inefficiency of the current technologies in retaining the intrinsic physics like the multi-scale flow structures and energy cascades. To address this challenge, this work proposes a fractal-constrained deep learning super-resolution model termed SKFSR-FIC. The model is characterized by two distinctive designs: (1) a SKip-connected Feature-reuse Super-Resolution (SKFSR) network that learns and retains multi-scale flow structures and multi-frequency dynamics, achieving efficient upscaling of flow fields while reconstructing self-similar physics; (2) a fractal invariance constraint (FIC) that utilizes the self-similarities of flow properties invariant to scales to substitute label data information in super resolution, thus achieving accurate reconstruction of multi-scale dynamics and energy cascades. The SKFSR-FIC model, for the first time, leverages fractal dimensions to guide the turbulent flow reconstruction and significantly reduces the reliance on label data and, especially, achieves zero-shot (i.e., unsupervised) super-resolution that cannot be handled by existing deep learning models. The results from five self-affine fractal images and two turbulent flow cases demonstrate the enhanced efficiency (up to 2 times) and accuracy (up to 100 times) of the unsupervised SKFSR-FIC model compared to the conventional interpolation method and deep learning models. Moreover, the SKFSR network is compatible with both FIC and label data, thereby adaptively enabling unsupervised, supervised and semi-supervised learning strategies. In particular, the semi-supervised SKFSR-FIC model, even by using one snapshot, achieves the best accuracy among the three learning strategies due to the combination of physics and data.