This dissertation explores the application of PAC-Bayesian frameworks to learning and generalization bounds within dynamical systems, focusing on linear time-invariant (LTI) systems and special non-linear systems like recurrent neural networks (RNNs). The research develops novel PAC-Bayesian error bounds, offering theoretical guarantees crucial for assessing the performance of learning algorithms in these contexts.

The thesis is motivated by the need to understand and predict behaviours in complex dynamical systems prevalent in numerous real-world scenarios. Traditional learning algorithms typically lack robust theoretical assurances for finite time-series data, a gap addressed by the PAC-Bayesian methodology. This approach provides a strong foundation to derive error bounds that ensure the reliability of algorithmic performance.

Key contributions include establishing PAC-Bayesian bounds for autonomous LTI systems, constructing minimal error variance estimators for such systems, and extending these bounds to LTI systems with inputs. Additionally, the thesis explores error bounds derived using Rényi divergence for LTI systems with inputs, that show tighter convergence and extends these guarantees to include special non-linear systems like stable RNNs.

Overall, this work significantly enhances the theoretical understanding of learning dynamics in systems, reinforcing the reliability and efficacy of learning algorithms in handling practical time-series data. The findings pave the way for future research on extending these bounds to more complex and non-linear systems, bridging theoretical insights with practical machine learning applications in dynamical systems.

After the defence there will be a small reception at Fredrik Bajers Vej 7A 1.103