State estimation in spatiotemporal chaos via low-rank StatFEM

Trajectory reconstruction in high-dimensional spatiotemporal chaos is often difficult due to the coupled challenges of structural model misspecification and prohibitive computational costs. We introduce in this work a Low-Rank Extended Rauch-Tung-Striebel Smoother (LR-ExRTSS) for use within the Statistical Finite Element (StatFEM) framework, combining a square-root retrospective smoother with a forward looking low-rank extended Kalman filter. We apply this grid-independent, retrospective scheme to the 2D anisotropic Kuramoto-Sivashinsky equation under a severe dynamical mismatch. We force a predictive model converging to a stable 1D steady state to track a truly chaotic 2D system using sparse, noisy observations. Results demonstrate that the LR-ExRTSS acts as a bridge between misspecified physics and chaotic reality by reintroducing transverse instabilities aggressively damped by the flawed model. Spectral analysis confirms that the efficacy of the scheme is based on its ability to span the unstable-neutral subspace of the true system, regardless of the subspace predicted by the model. This work establishes low-rank StatFEM as a computationally feasible framework for robust state estimation in high-dimensional systems under severe structural error.