Scientific Machine Learning for Spatiotemporal Dynamics: Modeling and Control

Spatiotemporal dynamics, often governed by partial differential equations (PDEs), underpin many physical processes critical to scientific discovery and engineering innovation. Traditionally, numerical solvers and linearization-based control methods have been widely adopted. However, these first-principle approaches become challenging when (a) the underlying physics is not fully understood, making model construction difficult, or (b) the governing equations are highly nonlinear or stiff, leading to excessive computational costs or ineffective control performance. Deep learning offers a promising alternative, adept at tackling high-dimensional, nonlinear systems through data-driven approaches. However, its reliance on vast datasets, intensive training demands, and poor generalization to unseen scenarios limit its practical utility for modeling and controlling spatiotemporal dynamics. This dissertation introduces novel deep learning frameworks to address these limitations. First, it enhances deep reinforcement learning efficiency via (a) large-scale asynchronous parallel training and (b) leveraging physics-informed neural networks as surrogate environments. Second, it proposes two innovative architectures that embed physical principles into deep neural networks, improving generalizability and data efficiency for deterministic spatiotemporal dynamics modeling. Finally, it develops a generative approach based on diffusion models to capture stochastic dynamics, such as turbulence. These advancements collectively enable accurate, efficient, and scalable solutions, bridging the gap between data-driven methods and physics-based algorithms for modeling and controlling spatiotemporal dynamics.