UC Berkeley

Mathematical models are a mathematical abstraction of the physical reality which is of great importance to understand the behavior of a system, make estimations and predictions and so on. They range from models based on physical laws to models learned empirically, as measurements are collected, and referred to as data-driven models. A model is based on a series of choices which influence its complexity and realism. These choices represent trade-offs between different competing objectives including interpretability, scalability, accuracy, adequation to the available data, robustness or computational complexity. The thesis investigates the advantages and disadvantages of models based on physical laws versus data-driven models through the example of signalized queuing networks such as urban transportation networks.

The dynamics of conservation flow networks are accurately represented by a first order partial differential equation.

Using Hamilton-Jacobi theory, the thesis underlines the importance to leverage physical laws to reconstruct missing information (\emph{e.g.} signal or bottleneck characteristics) and estimate the state of the network at any time and location. Noise and uncertainty in the measurements can be integrated in the model. When measurements are sparse, the state of the network cannot be estimated at every time and location on the network. Instead, the thesis shows how to leverage other characteristics, such as periodicity. From deterministic dynamics, the thesis derives the probability distribution functions of physical entities (\emph{e.g.} waiting time, density) by marginalizing the periodic variable. Using a Dynamic Bayesian Network formulation and exploiting the convexity structure of the system, the thesis shows how this modeling leads to realistic estimations and predictions, even when little measurements are available. Finally, the thesis investigates how sparse modeling and dimensionality reduction can provide insights on the large scale behavior of the network. Large scale dynamics and patterns are hard to model accurately based on physical laws. They can be discovered through data mining algorithms and integrated into physical models.