Whitening the Spectrum: Preconditioned Flow Matching for Turbulence Generation

Generative models of turbulence transport a spectrally flat source onto fields whose energy follows a steep power law, and the flow matching objective inherits that spectrum through the regression target and the interpolant's signal-to-noise ratio, so energy-containing scales dominate the gradient and dissipative scales are learned last. Our method, WSFM, instead solves transport in a whitened Fourier representation, an invertible, mode-wise affine transform estimated from the training ensemble that equalizes the latent's second-order statistics across wavenumbers, preserves the relative Fourier phases carrying coherent vortical structure, and leaves the modeled distribution unchanged. It therefore improves the conditioning of the optimization rather than altering the target, and we prove the whitened transport problem is uniformly conditioned. At matched backbone and parameter count, WSFM lowers relative $L_2$ and high-wavenumber spectral error against raw Fourier flow matching and retains the small-scale structure the raw coordinate smooths away. Across benchmark flows the gains concentrate in the physics, with structure-function exponents, enstrophy and dissipation statistics reproduced more faithfully than by the raw coordinate or the published surrogates, while pointwise accuracy stays comparable.