Abstract

We study unconditional generative modeling of two-dimensional Kolmogorov flow with denoising diffusion. A plain U-Net diffusion model reproduces the point-wise statistics of turbulent vorticity fields but distorts the inertial-range spectrum and the spectral fluxes that carry energy and enstrophy across scales. We add a wavenumber-resolved spectral gate: a small Fourier-domain bottleneck that learns a per-channel multiplicative correction over a coarse radial-wavenumber grid, conditioned on the diffusion noise level and a continuous log-viscosity parameter. The gate is paired with physics losses on enstrophy, modal spectral amplitude, vorticity structure functions, integral length, and spectral flux. On forced 2D Kolmogorov flow at seven viscosity regimes spanning a factor of nine in Reynolds number, and against four unconditional baselines (a plain U-Net, a squeeze and-excite bottleneck, an FNO-block bottleneck at 2.8× the parameters, and a standalone Fourier neural operator), the gated model (WRSG, 2.51M parameters) reduces the aggregate log-spectral distance to direct numerical simulation (DNS) by 59% relative to the plain U-Net, beats the larger FNO block on that metric, and raises inverse-energy and forward-enstrophy cascade recovery from 10% and 21% to 49% and 89%. Adding an FNO branch to the gate (WRSG+FNO, 7.27M) is best overall, reaching 84% and 110% cascade recovery and the lowest flux error. A factorial ablation separates the gate, which routes spectral power and corrects the large scales and cascade transport, from the losses, which correct small-scale point statistics, and shows a strong positive interaction: the gate alone barely lowers the spectral distance and the losses alone raise it, but together they cut it by more than half. A one-parameter scalar variance recalibration, fit on a regime-stratified calibration split, brings the ensemble to the 90% coverage target, raising empirical coverage from 0.72–0.85 to 0.899–0.902. The single viscosity-conditioned model generalizes to unseen and out-of-range Reynolds numbers, and the gate, though not the vorticity-tuned loss weights, transfers to a different forcing geometry and to a passive scalar, evidence that the architectural component is general rather than turbulence-specific. We report two negative results: the physics losses worsen the large-scale integral length relative to the gate alone, and no variant fully recovers the DNS cascade peaks. Code is available at https://github.com/nishattasnim15215/Wavenumber-Resolved-Spectral-Gating-for-Diffusion-Models-of-Two-Dimensional-Turbulence.