Modeling dynamical systems is essential in science and engineering, requiring accurate predictions, robustness to varying conditions, and interpretability. Traditional data-driven methods often falter in long-term forecasting, generalization to out-of-distribution (OOD) scenarios, and providing insight into system behavior. This thesis investigates the integration of supervised disentanglement into deep learning as a solution to these challenges.

We first advance the modeling of wave propagation governed by the Saint-Venant equations using deep autoencoders (U-Nets) and tailored training strategies. These models show marked improvements in prediction accuracy. Through OOD testing, we reveal the limitations of conventional deep learning models and demonstrate that embedding domain knowledge into architectures and training can enhance performance.

Next, we extend supervised disentanglement to high-dimensional data by incorporating it into Recurrent State-Space Models (RSSMs) and U-Nets. These adaptations are evaluated on tasks like pendulum dynamics from images and wave propagation. Results show that supervised disentanglement improves model robustness and is feasible in complex settings.

We analyze the latent representations using established disentanglement metrics, finding that supervised disentanglement yields more structured, interpretable latent spaces. This can facilitate better parameter inference and understanding of the system’s internal dynamics.

However, we observe trade-offs: in some high-dimensional cases, short-term prediction accuracy slightly decreases. These results emphasize the importance of context and system characteristics when applying disentanglement techniques.

Overall, this thesis shows that combining physical insights with data-driven models via supervised disentanglement leads to more accurate, generalizable, and interpretable modeling of dynamical systems. This approach has potential in fields like climate science, engineering, and healthcare, where understanding complex dynamics is vital. The work lays the groundwork for further exploration into integrating domain knowledge and learning disentangled representations in deep models for dynamic systems.