A Shortcut to Statistically Steady-State Turbulence with Flow Matching

Many nonlinear physical systems exhibit an initial transient phase in which perturbations grow before nonlinear interactions lead to a statistically steady state. While this saturated regime is of primary interest, direct numerical simulations must resolve the full transient dynamics to reach it, incurring significant computational cost. In Computational Fluid Dynamics, reduced-order approaches such as Large Eddy Simulation mitigate computational cost by modeling small-scale dynamics, enabling tractable approximations of turbulent flows. In contrast, for systems such as gyrokinetics, comparably effective closures for the full dynamics are not generally available, and high-fidelity simulations remain necessary. Existing surrogate modeling approaches for these systems are autoregressive, hence they suffer from accumulating error. We instead propose to bypass explicit time evolution by directly modeling the distribution of saturated states under an ergodicity assumption, stating that ensemble averages over samples are equivalent to time averages of a single long simulation. We introduce GyroFlow, a latent generative model that directly estimates steady-state statistics of gyrokinetic turbulence in 5D phase space, without resolving the transient phase. At the same time, GyroFlow generates saturated snapshots from noise, conditioned on dimensionless operating parameters and matches the strongest autoregressive and tabular baselines, without requiring a saturated solver state, while outperforming other generative approaches. GyroFlow attains a $>10000\times$ speedup compared to nonlinear gyrokinetics simulators and $>1000\times$ speedup compared to autoregressive baselines. To evaluate generation quality we propose , a distributional metric computed in the latent space of a pretrained gyrokinetic model, and show that it correlates with downstream flux accuracy and the spectral divergence of warm-started solver runs. Finally, GyroFlow can be used to warm-start the numerical code used to produce the data, which shortens its heat-flux transient by about $3\times$.