Data-Driven Methods for Reduced Order Models, System Identification, and Feedback Control

Abstract

Data-driven modeling is an essential tool for understanding and controlling complex physical systems, particularly when first-principles models are incomplete or unavailable. We develop three distinct methodologies for learning dynamical systems from limited, noisy, and real-world data. We introduce a reduced-order modeling framework to capture the initial ejection dynamics of turbulent plumes directly from video. By combining Otsu's thresholding with a concentric-circle search to extract plume geometries, we construct interpretable low-dimensional representations of formation dynamics. Validated on unseen video data, this approach accurately recovers rapid transients that are unaddressed by classical Gaussian models and intractable for real-time computational fluid dynamics. We present an all-at-once methodology for learning systems of ordinary differential equations (ODEs) from scarce, partial, and noisy observations. This formulation utilizes a sparse recovery strategy over a function library while jointly employing reproducing kernel Hilbert space (RKHS) theory for state estimation and discretization. The approach demonstrates high sampling efficiency and robustness to noise, outperforming existing algorithms in equation discovery. We develop a stability-constrained Neural Ordinary Differential Equation (NODE) framework for learning asymptotically stable systems exhibiting multi-attractor and hysteretic dynamics. The learned model guarantees trajectory stability throughout the state space, enabling tractable feedback control policies capable of navigating nontrivial bifurcations and hysteresis loops. Collectively, these contributions advance data-efficient modeling by integrating reduced-order methods, sparse equation discovery, and stability-enforced neural networks for real-time inference and control.