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 a fundamental inverse problem in computational fluid dynamics and scientific machine learning. Classical interpolation methods fail to recover missing fine-scale structures, while existing deep learning approaches rely on convolutional architectures that lack the spectral and multiscale inductive biases necessary for physically faithful reconstruction at large upscaling factors. We introduce the Spectrally-Informed Multi-Resolution Neural Operator (SIMR-NO), a hierarchical operator learning framework that factorizes the ill-posed inverse mapping across intermediate spatial resolutions, combines deterministic interpolation priors with spectrally gated Fourier residual corrections at each stage, and incorporates local refinement modules to recover fine-scale spatial features beyond the truncated Fourier basis. The proposed method is evaluated on Kolmogorov-forced two-dimensional turbulence, where $128\times128$ vorticity fields are reconstructed from extremely coarse $8\times8$ observations representing a $16\times$ downsampling factor. Across 201 independent test realizations, SIMR-NO achieves a mean relative $\ell_2$ error of $26.04\%$ with the lowest error variance among all methods, reducing reconstruction error by $31.7\%$ over FNO, $26.0\%$ over EDSR, and $9.3\%$ over LapSRN. Beyond pointwise accuracy, SIMR-NO is the only method that faithfully reproduces the ground-truth energy and enstrophy spectra across the full resolved wavenumber range, demonstrating physically consistent super-resolution of turbulent flow fields.

Figures (9)

• Figure 1: Overview of the SIMR-NO architecture as a composition of two hierarchical stage blocks corresponding to \ref{['eq:stage1_out']}--\ref{['eq:stage2_out']}. Each stage combines a deterministic bicubic upsampling prior with a learned spectrally-informed residual correction scaled by learnable parameters $\alpha_1$ and $\alpha_2$. Stage 1 resolves large-scale coherent structures across the $32\rightarrow64$ resolution transition, while Stage 2 refines fine-scale turbulent features across the $64\rightarrow128$ transition. Both FNO and SIMR-NO receive the same pseudo high-resolution conditioning field $\widetilde{\omega}^{128}$; SIMR-NO internally constructs $a^{32} = \mathcal{U}_{128\rightarrow32}(\widetilde{\omega}^{128})$ before applying the two-stage reconstruction defined in \ref{['eq:factorization']}.
• Figure 2: Mean relative $\ell_2$ error of 201 held-out test samples by the standard deviation. SIMR-NO with the lowest mean error and standard deviation shows superiority in accuracy and consistency compared to all other baseline methods.
• Figure 3: The box plots display the distribution of relative $\ell_2$ error throughout all 201 held-out test samples. SIMR-NO demonstrates superior performance because it has the lowest median value $0.2383$ together with its most compact interquartile range, which measures between $0.1990$ and $0.3010$ and its smallest number of high-error outliers.
• Figure 4: The reconstruction results show the best results from the test sample. The first row shows input LR ($8\times8$) and pseudo HR ($\text{LR8}\rightarrow128$) and ground truth HR ($128\times128$) and model predictions. The second row displays absolute pointwise error maps. SIMR-NO achieves the lowest and most spatially uniform error while accurately reconstructing fine-scale vortical structures that all baseline reconstructions failed to detect.
• Figure 5: The best-case sample shows enstrophy spectrum $\mathcal{E}(k)$ and energy spectrum $E(k)$ and cumulative POD energy results. The SIMR-NO model tracks ground-truth spectral decay across all wavenumbers, while all baseline methods fail to accurately measure high-wavenumber energy content.