How Many Steps Does Flow Matching Really Need? A Frequency-Domain Analysis of Discretization Error in the Base-ODE Regime

Flow matching generates high-quality images in as few as 4–50 ODE integration steps, yet existing convergence theory provides only scalar bounds that treat all frequencies equally. Under base-ODE sampling (guidance $w=1$), we derive an explicit frequency-decomposed characterization of discretization error for polynomial ODE solvers applied to the flow matching interpolant, showing that the ODE acceleration scales as $|k{|}^{\beta /2}$ for a target power spectrum $P(k)\propto |k{|}^{-\beta }$, producing a spectral ceiling $k_{\mathrm{c}}(N)\propto N^{2/\beta }$ above which high-frequency content is exponentially attenuated. We validate this prediction exactly on Gaussian targets ($R^2>0.999$) and reproduce the predicted convergence pattern on five frontier text-to-image models spanning five independent teams (SD3.5-Medium 2.5B, FLUX.1-dev 12B, AuraFlow v0.3, CogView4-6B, Lumina-Image-2.0), with monotonic high-band recovery in latent space and midpoint dominating Euler at matched NFE in every case (a $2$–$3\times$ low-NFE advantage). A single reflow iteration eliminates the ceiling entirely on a controlled synthetic setting, isolating ODE trajectory curvature as the mechanism. Distillation (FLUX.1-schnell) also eliminates the ceiling but reverses the frequency-band convergence order, providing a spectral signature of the distillation mechanism. We further report an overshoot drift at large step counts ($N\ge 128$), where solvers exceed the reference power by $60\%$+, which an exact-velocity control experiment attributes to learned velocity-field error rather than discretization. These findings yield actionable guidelines for the base-ODE regime: the midpoint solver recovers $2$–$3\times$ more high-frequency energy per function evaluation than Euler, and the optimal noise-schedule shift lies in $[1,2]$ across models.