Time series analysis is vital for modelling dynamic systems in applications like robotics and weather prediction, yet challenging due to the complex dynamic interactions across temporal and spatial dimensions. State space models (SSMs) are an expressive class of models well suited for time series analysis. Particle filters (PFs) provide an effective approach for approximate Bayesian inference in general non-linear and non-Gaussian SSMs. However, traditional PFs struggle with high-dimensional, non-linear, and irregular data. Differentiable particle filters (DPFs), which construct the components of PFs through neural networks and learn the parameters by minimising a loss function through gradient descent, address these gaps. This thesis advances DPFs for diverse time series, targeting data efficiency, expressiveness, irregular sampling, and spatio-temporal structure learning.

In general, the thesis makes four main contributions to advance DPFs. First, semi-supervised DPFs (SDPFs) reduce labelled data requirement by leveraging unlabelled data via pseudo-likelihood, outperforming baselines in robotics state estimation with limited ground truth data. Second, we enhance DPFs expressiveness with conditional normalising flow-based SDPFs (CNF-SDPFs), using normalising flows for the dynamic model and conditional normalising flows for the proposal distribution, improving tracking in visual disk environments. Third, continuous-discrete DPFs (CD-DPFs) extend DPFs to irregular time series using Gaussian mixture models (GMMs) and adaptive Gaussian sum particle filters, and the weights of the GMM are optimised adaptively through solving a convex optimisation problem by direct matching the Fokker-Planck-Kolmogorov equation. Evaluated on the stochastic Lorenz 63 model, CD-DPFs enhance forecasting while maintaining competitive imputation. Finally, continuous-discrete graphical DPFs (CD-GDPFs) incorporate structure learning for spatio-temporal time series, learning graph topology through neural dynamic structural model and group Lasso penalties. Tested on the Lorenz-96 model, CD-GDPFs improve graph inference with high true positive and low false discovery rates.