Robust and Interpretable Learning for Operator-Theoretic Modeling of Non-linear Dynamics

Abstract

Non-linear dynamical systems are of significant interest to a wide range of science and engineering communities. This dissertation is focused on the advancement of theory and algorithms for operator-theoretic modeling and decomposition of non-linear dynamical systems, with a particular emphasis on the Koopman operator. The Koopman operator represents non-linear dynamics in the form of an infinite-dimensional linear operator over the space of observables of the system state. Despite the broad appeal of the Koopman operator in modal analysis, reduced-order modeling and control, discovering accurate finite-dimensional representations presents considerable challenges. Most of the existing data-driven approaches for non-linear systems target prediction, which in effect amounts to interpolation within parameter space. In contrast, the Koopman operator presents the potential of a systematic framework for physics-consistent, stable predictions, control and modal analysis. Data-driven methods including time delay dynamic mode decomposition (TD-DMD), extended DMD (EDMD), kernel DMD (KDMD), and deep learning-based techniques have been developed for Koopman approximations. While the promise of these techniques is clear, stability, robustness, spurious modes, and uncertainty quantification continue to be a challenge. As a consequence, the above methods are often employed as qualitative postprocessing tools instead of quantitative, robust, and accurate a posteriori models. This thesis aims to address the above challenges and demonstrates applications in modal decomposition, predictive modeling, and control of non-linear dynamics. We begin by developing theoretical results on the structure of TD-DMD models for non-linear dynamics on an attractor. We demonstrate that the minimal number of time delays required for perfect reconstruction is directly related to the sparsity of the Fourier spectrum of the dynamics. Furthermore, we explain why TD-DMD can “extrapolate in time”, i.e., why a model trained on a partial period of data can perfectly predict the future. We also prove that an increase in the number of time delays benefits numerical conditioning, making the model robust to noisy data. For example, we demonstrate the numerical stabilization effect of “over-delays” on the 3D Turbulent Rayleigh-Benard convection. We also construct a ROM based on TD-DMD, and compare it to state-of-the-art methods for a chaotic single injector combustion process. Next, we develop robust and accurate mode selection algorithms for non-linear Koopman approximation methods (e.g., EDMD/KDMD). We propose a model-agnostic sparsity promoting framework based on pruning and multi-task learning for Koopman eigenfunctions. From an analytical perspective, we show a close relationship between the well-known sparsity-promoting DMD (spDMD), and an empirical criterion. The performance of the proposed framework is demonstrated in several unsteady flow cases ranging from strongly transient flows including the cylinder wake and a 3D turbulent ship air-wake. Finally, we propose a probabilistic deep learning framework for the continuous-time Koopman operator. We introduce a novel parameterization that guarantees the temporal stability of the Koopman operator. The effectiveness of the proposed framework is evaluated on several non-linear dynamical systems with varying amounts of training data and noise levels. The proposed framework shows better robustness to noise and improved accuracy compared to the vanilla LSTM model. We reveal that linear reconstruction can be still useful for systems with symmetrical attractors. Lastly, we show an application of our framework on the data-driven optimal control of a simple non-linear dynamical system, and highlight performance benefits over traditional local linearization.