Parameter Estimation and Prediction in Nonlinear Dynamics

Nonlinear dynamical systems are central to modeling complex phenomena across science and engineering, from biological networks to climate systems. A critical challenge in these systems is the accurate estimation of model parameters and the reliable prediction of future states, especially under uncertainty and limited observability. This thesis addresses this challenge by developing a proper framework that integrates classical methods, modern optimization theory, and machine learning techniques for parameter estimation and prediction in nonlinear dynamical systems. We begin by revisiting foundational approaches to parameter estimation in ordinary differential equation models, including least squares, maximum likelihood, and Bayesian inference. To bridge the gap between theory and practice, we explore advanced computational techniques such as multiple shooting, collocation methods, and robust estimation with the Huber loss function. Numerical optimization plays a central role in our methodology. The thesis presents detailed analyses of unconstrained and constrained optimization algorithms, including Newtonbased methods, trust-region strategies, and sequential quadratic programming. These methods are then applied to system identification tasks, where we contrast classical strategies with data-driven machine learning approaches. In the latter part of the thesis, we propose hybrid methods that combine traditional system identification with deep learning architectures such as feedforward neural networks and neural differential equations. We introduce a machine learning-based framework for parameter estimation, supported by theoretical analysis and extensive numerical experiments on benchmark systems including the Van der Pol oscillator, Lotka-Volterra dynamics, and the Lorenz attractor. Finally, we develop a real-time dynamic state estimation framework based on moving horizon estimation using the qpOASES solver and a reformulated Huber penalty function. This method enables robust, online estimation in noisy environments and is further extended with a neural ODE and multiple shooting-based architecture. Overall, the results underscore the critical role of accurate parameter estimation in improving the reliability of nonlinear system predictions, with implications for diverse domains including physics, biology, engineering, and finance.