Balasuriya, SanjeevaRobertson, WillMcGowan, Sean Peter2025-07-212025-07-212025https://hdl.handle.net/2440/146173Methods for accurately predicting and estimating the states of chaotic systems are of considerable interest to many disciplines. The interacting effects of sensitivity to initial conditions, imperfect modelling, and noisy observations cause challenges for their application. As sensors become ubiquitous, approaching this problem from a data-driven perspective may offer improved accuracy compared to traditional modelbased forecasting. It is still of great importance that these data-driven methods provide interpretable and meaningful analysis of the system of interest, as is the goal of standard modelling. Motivated by the imperfections of both models and data, this thesis explores methods of reconstructing dynamical systems from observational data to empower novel techniques for prediction, estimation, and correction of chaotic systems. The algorithms developed in this work rely on Takens embeddings for the reconstruction of attractors from partially observed chaotic data, as well as ergodic hypotheses for well-defined function approximation of dynamic maps and flows. From these premises, we develop methods to reconstruct dynamics for their analysis, prediction, and estimation. Using an optimal control approach, a technique to extract rules governing the evolution of data is presented which allows the prediction, uncertainty quantification, and Lyapunov analysis of partially observed systems in settings of high noise and limited data. We extend these techniques to a data assimilation framework, where new observations of the system may be used to update the prediction in a model-free manner which shows promise in settings where models are imperfect or not available. Finally, these ideas are used to correct the bias between observations and model output to leverage the mechanistic understandability of knowledge-based models with the predictive power of data-driven methods in a way that provides interpretability of coherent unmodelled dynamics. These approaches are validated on simulated chaotic systems as well as empirical datasets of meteorological variables. This research suggests that data-driven techniques may outperform model-based methods with tasks of prediction and estimation in chaotic, noisy, and partially observed settings particularly when available models are imperfect. They also may assist in offering new methods of extracting interpretation and mechanistic understanding to improve existing models with missing dynamics.endynamical systemschaosdata assimilationdata scienceearth systemsmeteorologyThesis