Uncovering Structure with Data-driven Reduced-Order Modeling

Abstract

In this dissertation, we apply data-driven reduced-order modeling to several example systems. In each case, we consider systems with some mathematical structure we hope to capture in the model, and we investigate what approach will best uncover or make use of that structure. First, we examine the convergence of extended dynamic mode decomposition (EDMD) in systems with continuous spectra. We test the performance of EDMD on ergodic discrete-time dynamical systems on the two-dimensional torus. We demonstrate that even in systems with some continuous spectrum, the point spectrum of the Koopman operator can be approximated using EDMD. However, the quality of this approximation depends on the observables chosen. We consider Fourier modes, delay embeddings, and radial basis functions as possible sets of observables. Second, we approximate the Koopman operator in a system with continuous symmetry. We apply linearly-recurrent autoencoder networks, EDMD using observables obtained from proper orthogonal decomposition, and a reproducing kernel Hilbert space method for approximating the Koopman generator. All of these are applied to the Kuramoto-Sivashinsky equation, in a regime with modulated traveling waves. We demonstrate that applying a template-based symmetry reduction method, where the traveling speed is modeled separately from the reduced dynamics using a deep neural network, provides a significant improvement in prediction quality for these models. We also examine the eigenvalues and eigenfunctions of these Koopman approximations. Finally, we use data-driven methods to find equations of motion for a rigidly rotating body in three dimensions, given orthogonally projected two-dimensional data. To achieve this approximation, we first apply diffusion maps to learn the manifold where the governing equations can be best represented. Next, we find the rotation matrices, and thus angular velocities, at each timestep using optimization and methods from cryogenic electron microscopy. Then, we are able to approximate the equations of motion, including approximating the principal moments of inertia which appear as coefficients. In each case, we find interpretable, low-order predictive models with data-driven methods. We also demonstrate that, in these complicated cases, careful preparation including choice of observables, preprocessing steps, or manifold learning can be beneficial or even necessary to produce useful models.