Dimensionality Reduction and Sparsity Promotion for Complex Dynamical Systems

Abstract

This thesis provides several contributions to data-driven techniques for modeling spatio-temporal data, both developing the theory of existing methods and proposing new techniques. These methods leverage underlying structures, namely low dimensional and sparse representations to extract meaningful representations for a variety of dynamical systems. We first focus on the dynamic mode decomposition (DMD), a popular dimensionality reduction technique, which assumes the data is governed by linear dynamics. We show that mean subtraction improves the performance of the method, explicitly characterizing the behavior theoretically and corroborating the results with real world datasets. This contribution is particularly significant as previous results have shown that mean subtraction has undesirable effects. We next transition to modeling nonlinear dynamics, and propose a dimensionality reduction method for time-varying linear dynamics. Our method, the spatiotemporal temporal intrinsic mode decomposition (STIMD), leverages spatial correlations to decompose data into a linear combination of intrinsic mode functions (IMFs). IMFs have the beneficial property that they have well-defined instantaneous frequencies. This method robustly models nonstationary signals, while still preserving much of the interpretibility of linear methods. We then unify results from differential geometry, time delay embeddings, and dimensionality reduction to show that nonlinear dynamical systems may be decomposed into a sparse and structured linear dynamical model plus forcing. We use this perspective to explain observed behavior in the recently developed Hankel Alternative View of Koopman (HAVOK) algorithm. We further propose modifications which not only more closely align with theory, but also improve model stability and reconstruction. Finally, we propose a new framework, uncertainty quantification for sparse identification of nonlinear dynamics (UQ-SINDy), which utilizes compressed sensing and Bayesian statistics to directly extract governing equations from data, providing explicit quantification of uncertainty in the model. This method is particularly powerful for systems which have sparse nonlinear representations and yields state of the art accuracy.