This research develops machine learning methods for modeling, controlling, and estimating complex dynamical systems. A central theme is the integration of mathematical structure and physical knowledge into learned models, enabling them to provide formal guarantees beyond what purely data-driven approaches offer. The problems addressed span learning dynamics with guarantees from data and estimating real-world physical states from sparse observations. We leverage modern deep learning techniques such as neural ODEs, flow matching models, and Fourier neural operators. Together, this work bridges the gap between classical systems theory and modern deep learning, with applications ranging from physics-informed simulation to real-time control and inference.