Ensemble Driven Emulation for Correcting Coarse Resolution Trajectories in Multi-Scale Systems

High-resolution simulations of multiscale systems such as turbulent flows and climate models remain computationally prohibitive, necessitating coarse models that often fail to capture the statistics of fine-scale structures and rare events driven by transient instabilities.

We introduce a general, nonintrusive machine‑learning correction that post‑processes coarse‑resolution trajectories by combining two ideas: nudging, a weak penalization of the coarse model toward a short reference trajectory to limit chaotic divergence; and an ensemble of stochastic perturbations of the nudged state to approximate its dominant short-time-scale fluctuations with the empirical covariance. These dominant transient directions span a low‑dimensional manifold on which a conditional diffusion model maps the nudged-coarse state and its tangent basis back to the reference. Demonstrated on canonical isotropic and stratified flow benchmarks, our SDE‑ensemble driven approach faithfully recovers fine‑scale features and extreme‑event tails, far exceeding the finite training window. By informing the generative denoising with dynamically dominant instabilities, this framework offers a scalable path toward emulation of complex chaotic systems without embedding corrections into the governing equations.