How Much Is Enough? Observing Kuramoto–Sivashinsky Systems from Limited Data

Scale-resolving computational fluid dynamics tools can generate high-fidelity simulations of turbulent flows, but only when initial and boundary conditions are well specified, which is rarely the case in experimental or engineering settings. Data assimilation (DA), or state estimation, addresses this challenge by fusing sparse measurements with governing equations to reconstruct the full flow trajectory, enabling inference of real-world system behavior from limited data. For applied systems where the ground truth is unknown, inference accuracy is difficult to assess, and the number of observations needed for an accurate reconstruction is often unclear. In this work, we connect DA to state space reconstruction in dynamical systems, where measurement time series are used to recover invariants of a system's attractor. By invoking embedding theorems, we relate the number of required measurements to the dimension of the system's inertial manifold, offering a principled criterion for observability. We validate this connection using numerical experiments on the Kuramoto–Sivashinsky equation, a model chaotic system that serves as a surrogate for fluid turbulence, and we examine the optimization dynamics of adjoint–variational state estimation in this setting.