Path development, originating from rough path theory, serves as a mathematical tool to derive important properties of the path by lifting it to a desirable Lie group. The inherent analytic and geometric properties of path development render it not only theoretically significant but also promising for real-world applications when combined with appropriate learning methodologies. Consequently, the integration of path development and deep learning emerges as a natural and well-motivated approach. Path development provides a framework for analysing the functional relationships between time series and its effect on systems with rich geometric structures. On the other hand, deep learning offers powerful tools for learning these relationships from data.

The objective of this thesis is to present principled models based on path development and explore their applications in deep learning for analysing time series. The thesis encompasses two primary topics: (1) Path Development Network – a trainable network architecture based on path development that leverages representations of sequential data using finite-dimensional matrix Lie groups, and (2) Generative Model via Path Characteristic Function – a generative model that incorporates the path characteristic function as a principled representation of time series distributions to generate high-quality sequential data.

For both topics, we establish the mathematical foundations of the proposed models, providing theoretical guarantees for their feasibility and training efficiency. The models showcase promising results, demonstrating smooth training and robustness in handling irregularity and long-term dependency in time series. The versatility of the algorithm allows for integration with other deep learning techniques, enhancing performance. The proposed models achieve state-of-the-art performance in addressing general time series modelling, dynamic systems in non-Euclidean spaces, and generative tasks in various domains such as physics, finance, medicine, and more.