UC Santa Barbara

High-dimensional dynamical systems arise in many scientific and engineering applications, yet scalable probabilistic methods for estimating latent processes and forecasting future states with reliable uncertainty quantification remain limited. Latent factor models with Gaussian process (GP) priors on the latent factors offer a flexible probabilistic framework that naturally captures spatiotemporal dependence, quantifies uncertainty, and performs well in data-limited settings. However, existing approaches face two fundamental challenges. First, high-dimensional dynamical systems require estimating a large number of parameters, while GP priors on latent factors incur cubic computational complexity, making scalable inference prohibitive. Second, forecasting nonlinear dynamics with uncertainty quantification remains challenging when the governing equations are unknown.This dissertation addresses both obstacles through three contributions. First, in Chapter 2, we introduce a latent factor model with orthogonal factor loadings, where each latent factor independently follows an Ornstein-Uhlenbeck process with distinct correlation and variance parameters to capture heterogeneous underlying dynamics. To enable scalable inference, we develop a fast Expectation-Maximization (EM) algorithm that leverages the orthogonal loading structure together with the Kalman filter and Rauch-Tung-Striebel smoother to derive closed-form parameter updates. The resulting algorithm scales linearly with both the output dimension and the number of observations. Extensive simulated studies and a real-world application to slip estimation in the Cascadia region demonstrate the accuracy and scalability of the proposed approach.Second, in Chapter 3, we present three new modules in the FastGaSP R package for scalable Gaussian process and latent factor model inference. The fgasp module enables fast and exact computation for Gaussian processes with Matern kernel. The gppca module performs parameter estimation for latent factor models with orthogonal factor loadings and GP priors on the latent factors. The fmou module implements the scalable algorithm developed in Chapter 2. Together, these modules provide practical and computationally efficient tools for large-scale spatiotemporal modeling and uncertainty quantification.Finally, in Chapter 4, we develop a probabilistic forecasting framework for nonlinear dynamical systems by extending the parallel-partial Gaussian process method to estimate the unknown transition function and generate uncertainty-aware forecasts through posterior sampling. We further establish the equivalence between dynamic mode decomposition (DMD) and the maximum likelihood estimator of the linear mapping matrix in a linear state space model, providing a probabilistic interpretation of DMD that enables uncertainty quantification of prediction. Numerical examples demonstrate the role of uncertainty quantification in assessing forecast reliability and highlight the importance of correctly specifying model inputs.Together, these contributions provide probabilistic tools for scalable parameter estimation and forecasting with uncertainty quantification in high-dimensional dynamical systems.