SIMR-NO rebuilds 128×128 turbulent vorticity from 8×8 observations, holding about 3% relative error on decaying turbulence
Synopsis
Rebuilding a 128×128 vorticity field from 8×8 observations is a 16× upscaling in which the coarse grid resolves only wavenumbers up to 4, so fine vortical structure must be inferred rather than interpolated. SIMR-NO splits the reconstruction into 32→64 and 64→128 stages, each adding a Fourier-layer residual correction gated by radial wavenumber on top of a bicubic base, with local convolutions recovering features beyond the truncated Fourier basis. Its mean relative error is 0.2506 on Kolmogorov-forced turbulence, below LapSRN, FNO and EDSR, and about 3% on decaying homogeneous isotropic turbulence at Re = 1000, 4000 and 8000, with little change across Reynolds number.
Interpretation
Turbulence super-resolution no longer relies on a single mapping that fills in every missing scale at once; it reconstructs progressively across intermediate resolutions, and each stage only doubles the resolution. Earlier single-stage methods learned a direct coarse-to-high-resolution output, and convolutional super-resolution networks lacked the spectral and multiscale inductive biases suited to turbulence, so the task became harder as the resolution gap grew. On 201 test snapshots of Kolmogorov-forced turbulence, SIMR-NO reaches a mean relative ℓ2 error of 0.2506±0.0983, 17.7%, 21.5% and 30.8% lower than LapSRN's 0.3045, FNO's 0.3194 and EDSR's 0.3623, with SSIM 0.8075 and PSNR 24.77 dB.
Each stage starts from a deterministic bicubic base, adds a learned spectral residual correction to restore high-frequency content, and finishes with local convolutional refinement. A standard Fourier neural operator keeps only mode-wise multipliers without a shared weighting by scale, while convolutional baselines filter locally with a fixed receptive field. The spectral gate multiplies each retained mode's Fourier multiplier by the output of a two-layer perceptron of normalized radial wavenumber, with 16 modes in the first stage and 32 in the second; on decaying turbulence at Re = 1000, 4000 and 8000 the mean relative errors are 0.0277, 0.0316 and 0.0318, 7.7%, 45.6% and 54.8% below FNO.
Reconstruction accuracy is markedly less sensitive to Reynolds number than in the compared models, with mean error growing only 1.15-fold from Re = 1000 to 8000. Over the same range FNO's error grows 2.35-fold, EDSR's 1.59-fold and LapSRN's 1.77-fold. On decaying turbulence SIMR-NO's PSNR stays between 45.23 and 46.16 dB and its SSIM at or above 0.998; inside 20×20 local regions its relative error at Re = 4000 and 8000 is about one third that of the strongest baseline.
Perspective
Validation covers vorticity fields on the 2D periodic domain (0,2π)²: statistically stationary Kolmogorov-forced turbulence and freely decaying homogeneous isotropic turbulence at Re = 1000, 4000 and 8000, all reconstructed at 128×128 from 8×8 observations. The decaying-turbulence results come from a single trajectory per Reynolds number sampled every 0.01, so test snapshots are temporally adjacent to training snapshots.
The contribution of each architectural component has not been separated by ablation, which the authors leave to future work. The decaying-turbulence results come from a single trajectory, and evaluation on independent realizations and generalization across Reynolds numbers with one trained model remain open. On Kolmogorov-forced turbulence every learned model, SIMR-NO included, retains more small-scale content than the reference beyond k≈20.