Superresolving Nonlinear PDE Dynamics with latent-space Ensemble Filtering for Turbulence Modeling and Flow Control

High-resolution forecasting of spatiotemporal dynamics from sparse or low-resolution (LR) observations is a critical challenge in the optimal control of nonlinear PDE systems. We present a unified framework-SR-ROAD-EnKF-that couples a reduced-order latent neural operator with a differentiable Ensemble Kalman Filter (EnKF) for physics-constrained superresolution. The method learns (i) a compact surrogate for PDE dynamics in a latent space and (ii) a nonlinear spectral decoder to recover high-resolution (HR) fields. Sequential assimilation and forecasting are performed via EnKF updates in latent space, enabling uncertainty-aware reconstruction and rollout. We demonstrate the approach on three benchmark systems: the viscous Burgers equation, the chaotic Kuramoto–Sivashinsky equation, and 2D Navier–Stokes–Kolmogorov turbulence at Re=16,000. Across all cases, SR-ROAD-EnKF successfully reconstructs HR states from downsampled, noisy LR observations--preserving shock fronts, attractor dynamics, and kinetic-energy spectra--while maintaining stability up to 250 forecast steps. Our results highlight the effectiveness of latent-space data assimilation as a robust and computationally efficient tool for PDE-constrained control and turbulence regulation.