Reduced-Order Discovery and Joint State-Model Tracking of High-Dimensional Dynamical Systems
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We propose a new, efficient framework to address the problem of joint state–system dynamics estimation in the case of high-dimensional nonlinear systems. Sparse Identification of Nonlinear Dynamics (SINDy) enables the discovery of dynamical systems from data, but noisy observations or incorrect modeling assumptions can lead to misspecified models. Joint estimation via an Extended Kalman Filter (EKF) allows correction of such misspecified dynamics while simultaneously tracking time-varying behavior, yet both SINDy and the EKF struggle in high-dimensional settings. We overcome this limitation by projecting the system onto a low-dimensional subspace via Proper Orthogonal Decomposition (POD), thus performing model discovery and tracking directly in the reduced space. In order to specify a prior distribution to describe model uncertainty for the EKF – which is not provided by standard SINDy – we initialize the filter with the distribution over model coefficients produced by a recently proposed variational formulation of SINDy, the so-called variational identification of nonlinear dynamics (VINDy) approach. This allows model uncertainty to be propagated through inference, improving robustness under noisy observations. We validate the proposed approach on several parameter-dependent high-dimensional dynamical systems, such as those arising from the spatial discretization of nonlinear reaction–diffusion equations, as well as elastodynamics problems, successfully recovering time-varying dynamics in multiple scenarios.
