Sparse Identification of Delay Equations and Structured Populations

Dynamical systems exhibiting memory effects, where current evolution depends on past states, present fundamental challenges for data-driven equation discovery. This thesis develops complementary methodologies for discovering governing equations of such systems from time-series data by extending the Sparse Identification of Nonlinear Dynamics framework to three principal classes of delayed dynamical systems: deterministic delay differential equations with discrete delays, stochastic delay differential equations, and distributed delay systems. For delay differential equations with discrete delays, we introduce two approaches with distinct trade-offs. Expert-SINDy combines sparse regression with external particle swarm optimisation to simultaneously discover functional forms and discrete delay parameters when the system structure is partially known, maintaining interpretability of the resulting equations. Pragmatic-SINDy reformulates the infinite-dimensional identification problem using pseudospectral collocation, reducing it to finite-dimensional optimisation over only the maximum delay, thereby achieving improved computational efficiency at the cost of reduced equation interpretability. Comprehensive validation on logistic delay equations, Mackey-Glass dynamics, and neural network models demonstrates robust delay parameter identification and accurate trajectory reconstruction across diverse nonlinear behaviours. We then extend the framework to stochastic delay differential equations, developing It\^{o}-Taylor expansion-based moment estimators that reconstruct drift and diffusion components from noisy observations. Three practical estimation strategies accommodate different data availability scenarios, with extensive computational analysis establishing that pre-regression averaging substantially outperforms alternatives. For systems with distributed memory, we develop quadrature-based kernel identification methods that recover nonautonomous delay kernels without requiring prior functional form specification. Joint optimisation of integration bounds and kernel parameters achieves systematic discovery from data alone. Applications to logistic renewal equations and structured population dynamics validate the capability of the framework to extract interpretable sparse representations of distributed delay systems. Practical utility is demonstrated through application to Severe Fever with Thrombocytopenia Syndrome transmission dynamics, where distributed delay models are discovered directly from real surveillance data. Finally, comparative analysis contextualises sparse identification methods within the broader landscape of data-driven system identification, examining trade-offs between interpretability, computational efficiency, and data requirements compared with neural network approaches.