UC Santa Barbara

The proliferation of complex interconnected systems in today’s world has necessitated the development of advanced methods for optimizing their operation and management. Bilevel Optimization (BO), a powerful mathematical framework that considers optimiza- tion problems within optimization problems, has emerged as a promising approach to tackle these challenges. This thesis delves into the realm of BO with a primary focus on its application to learning and control in complex systems, particularly addressing the critical problem of network flow estimation.The core objective of this thesis is to develop novel physics-inspired learning techniques, to provide high-performing and explainable network flow estimators. In Chapter 1, a comprehensive overview of BO is provided, emphasizing its relevance and significance in real-world scenarios. The mathematical foundations of BO problems and the challenges posed by their computational complexity are elucidated, and some numerical schemes to solve them, in exact or approximate form, are reviewed. A collection of some known BO problems in machine learning is presented, and novel connections between BO and problems in machine learning and control are established. In Chapters 2 and 3 two novel flow estimation algorithms are proposed, addressing different nuances of the flow estimation problem. Both algorithms are rooted in first principles physics, and result into two different Implicit Neural Network Layers. Our approach enables high modularity, and its effectiveness is validated both theoretically as well as empirically. Extensive experiments across different application domains are presented, namely power systems, water distribution networks and traffic systems. Finally, Chapter 4 summarizes the findings of this thesis, and highlights potential future research directions.