Degree-of-freedom and optimization-dynamic effects on the observability of Kuramoto-Sivashinsky systems

TL;DR

The paper addresses how many measurements are needed to reconstruct chaotic KS trajectories via adjoint state estimation by linking observability to embedding theory and the dimension $d_M$ of the inertial manifold. It shows that $m \

Abstract

Simulations of chaotic systems can only produce high-fidelity trajectories when the initial and boundary conditions are well specified. When these conditions are unknown but measurements are available, adjoint-variational state estimation can reconstruct a trajectory that is consistent with both the data and the governing equations. A key open question is how many measurements are required for accurate reconstruction, making the full system trajectory observable from sparse data. We establish observability criteria for adjoint state estimation applied to the Kuramoto-Sivashinsky equation by linking its observability to embedding theory for dissipative dynamical systems. For a system whose attractor lies on an inertial manifold of dimension $d_M$, we show that $m \geq d_M$ measurements ensures local observability from an arbitrarily good initial guess, and $m \geq 2d_M + 1$ guarantees global observability and implies the only critical point on $M$ is the global minimum. We also analyze optimization-dynamic limitations that persist even when these geometric conditions are met, including drift off the manifold, Hessian degeneracy, negative curvature, and vanishing gradients. To address these issues, we introduce a robust reconstruction strategy that combines non-convex Newton updates with a novel pseudo-projection step. Numerical simulations of the Kuramoto-Sivashinsky equation validate our analysis and show practical limits of observability for chaotic systems with low-dimensional inertial manifolds.

Figures (23)

• Figure 1: Representative trajectories of Kuramoto--Sivashinsky systems for domain lengths $L = 22$ (bottom left), $L = 44$ (bottom right), and $L = 66$ (top). All cases exhibit meandering streaks whose spatial and temporal complexity increases with $L$, reflecting the additional active degrees of freedom.
• Figure 2: Autoencoder-based estimates of $d_\mathcal{M}$ for $L \in \{22, 44, 66\}$. Singular values of the centered latent state data matrix are shown, with vertical dashed lines marking the inferred IM dimension identified by the sharp drop in eigenvalues.
• Figure 3: Exemplary measurement configurations in the $L = 22$ domain. The left panel shows a sparse layout with two spatial sensors and two observation times; the right panel shows a denser configuration with four spatial sensors and four observation times.
• Figure 4: Sample reconstructions for $L = 22$. The top row shows reconstructed trajectories with sensor locations superimposed; the bottom row shows absolute error fields. Final loss values and cosine similarities are reported above each column. From left to right: poor generalization (low loss, high error), failed optimization (high loss, high error), and successful reconstruction (low loss, low error).
• Figure 5: Optimization loss versus iteration for gradient descent, modified Newton, BFGS, and NCN applied to the same $L = 22$ case with $m_x = 4$ and $m_t = 4$. The lower axis (0--350 iterations) corresponds to Newton, BFGS, and NCN; the upper axis (0--5000 iterations) corresponds to gradient descent.