Cascaded PhyFormer-SRNet: A physics-informed transformer-based framework with cascaded refinement for spatiotemporal super-resolution of turbulent flows

Reconstructing high-resolution turbulent flow fields from sparse low-resolution data remains a significant challenge for conventional data-driven methods, primarily due to the added difficulty of resolving the spatiotemporally coupled multi-scale dynamics inherent in turbulence. We propose a novel framework, named as Cascaded PhyFormer-SRNet, that provides a comprehensive solution through improved model architecture design, imposition of physical constraints, and a method for enhanced fine-scale refinement. First, an end-to-end spatiotemporal transformer-based super-resolution model is developed to integrate spatial feature extraction and attention mechanism for temporal enhancement. Unlike conventional methods that separately construct spatial and temporal super-resolution models, a unified architecture enables simultaneous spatiotemporal super-resolution, allowing more effective learning of the spatiotemporal multi-scale interactions present in turbulence. Second, finite difference-based physical constraints are imposed on the reconstructed flow fields to ensure physically consistent predictions of fine-scale turbulence characteristics. Building on the proposed physics-informed transformer-based super-resolution model, a cascaded refinement strategy to improve multi-scale spectral reconstruction is further applied to complete our proposed Cascaded PhyFormer-SRNet framework. Through these three innovations, the Cascaded PhyFormer-SRNet achieves superior spectral accuracy across scales and facilitates high-fidelity turbulence reconstruction in space and time. Experiments show that our framework outperforms conventional methods in reconstructing multi-scale channel turbulence from highly compressed data, even at compression ratios over 99%. Separate experiments show effective reconstruction of Kolmogorov flow from both regularly and randomly distributed noisy observations, with a reduction in reconstruction error exceeding 32% for the former compared to conventional methods.