High-frequency flow field super-resolution via physics-informed hierarchical adaptive Fourier feature networks

High-frequency flow field super-resolution remains challenging in computational fluid dynamics due to spectral bias in physics-informed neural networks, which limits the accurate reconstruction of fine-scale structures. This work introduces a hierarchical adaptive Fourier feature network for physics-informed super-resolution (HAFFN–PISR) framework that systematically addresses spectral bias through innovative frequency-scale decomposition. The framework employs a three-tier hierarchical encoding strategy that decomposes spatial–temporal features across distinct frequency scales, including low-frequency fundamental structures, characteristic frequencies aligned with external forcing, and high-frequency fine-scale features via learnable adaptive Fourier basis functions. Additionally, a prior knowledge-based hierarchical sampling algorithm leverages governing equations, external forcing frequencies, and Nyquist sampling theory for frequency parameter initialization. Neural tangent kernel theoretical analysis reveals that HAFFN–PISR achieves superior eigenvalue distributions, enabling accelerated convergence for high-frequency modes while maintaining consistency with the Navier–Stokes equations. Comprehensive validation demonstrates exceptional performance in reconstructing high-resolution flow fields from sparse observations with only 0.098% data utilization. Under varying forcing frequencies and Reynolds numbers, HAFFN–PISR achieves 50%–80% performance improvements over baseline methods while exhibiting robust noise resilience and superior computational efficiency compared to state-of-the-art approaches, establishing HAFFN–PISR as a powerful tool for high-fidelity flow field reconstruction. The source code for HAFFN–PISR will be made publicly available upon acceptance of this manuscript at https://github.com/xgxgnpu/HAFFN-PISR.