SIMR-NO: A Spectrally-Informed Multi-Resolution Neural Operator for Turbulent Flow Super-Resolution

Abstract Reconstructing high-resolution turbulent flow fields from severely under-resolved observations is an ill-posed inverse problem of central importance in computational fluid dynamics. Classical interpolation cannot recover the missing fine scales, and convolutional super-resolution networks lack spectral and multiscale inductive biases suited to turbulence at large upscaling factors. We introduce the \textit{Spectrally-Informed Multi-Resolution Neural Operator} (SIMR-NO), a hierarchical operator-learning framework that factorizes the reconstruction across intermediate resolutions. Each stage combines a deterministic interpolation prior with a residual correction computed by Fourier layers whose spectral multipliers are modulated by a learned radial gate, and local convolutional refinement modules recover features beyond the truncated Fourier basis. SIMR-NO is evaluated on two two-dimensional flows with distinct dynamics, statistically stationary Kolmogorov-forced turbulence and freely decaying homogeneous isotropic turbulence at Reynolds numbers of $1000$, $4000$, and $8000$, in which $128\times128$ vorticity fields are reconstructed from $8\times8$ observations, a $16\times$ upscaling factor. In comparison with bicubic interpolation, the Fourier neural operator, EDSR, and LapSRN, SIMR-NO achieves the lowest reconstruction error and the highest structural similarity for both flows and at every Reynolds number. On decaying turbulence, SIMR-NO attains a mean relative error of about $3\%$ at every Reynolds number, up to $55\%$ lower than that of the strongest baseline, and its accuracy is nearly insensitive to the Reynolds number. Spectral diagnostics, vorticity-conditioned error statistics, and localized comparisons show that these gains are accompanied by a more accurate reconstruction of intense vortex cores, fine-scale filaments, and high-wavenumber content.