The increasing demands of air travel and the escalating complexity of air traffic management (ATM) necessitate advanced air traffic flow prediction and optimization methodologies. This dissertation delves into integrating physics-guided machine learning techniques to address these challenges. By encompassing four pivotal studies, it contributes to the ATM field, showcasing how data-driven insights and physical principles can revolutionize our understanding and management of air traffic density, state predictions, flight delays, and airspace sectorization. The first study investigates the Bayesian Ensemble Graph Attention Network (BEGAN), a novel machine learning framework designed for precise air traffic density prediction. BEGAN combines spatial-temporal analysis with domain knowledge, enabling the model to interpret complex air traffic patterns in a highly dynamic and regulated airspace environment. The second study introduces the Physics-Informed Graph Attention Transformer, a novel approach integrating graph-based spatial learning with temporal Transformers. This model excels in capturing dynamic spatial-temporal interdependencies and integrates partial differential equations from fluid mechanics, enhancing the predictive accuracy and interpretability in ATM. The third study shifts focus to predictive modeling of aircraft delays, employing Physics-Informed Neural Networks. By utilizing sparse regression for system identification, this approach adeptly deciphers the intricate partial differential equations that dictate near-terminal air traffic dynamics, providing a novel perspective in forecasting flight delays with enhanced precision. The final study focuses on dynamic airspace sectorization, deploying an attention-based deep learning model that adeptly navigates the complexities of workload dynamics. In conjunction with constrained K-means clustering and evolutionary algorithms, it facilitates a more efficient and adaptable approach to airspace management, ensuring optimal traffic flow and safety. The findings of these studies demonstrate the significant impact of physics-guided machine learning in advancing ATM's safety and efficiency. They mark a shift from traditional empirical methods to innovative, data-driven approaches for air traffic management. This research enhances current practices and charts new paths for future technological advancements in aviation, especially in autonomous systems and digital transformation.

With the advent of new mobility services and technologies, the complexity of understanding the mobility patterns has been gradually intensified. The availability of large datasets, in conjunction with the transportation revolution, has been increased and incurs high computing costs. These two critical challenges require us to methodologically handle complex transportation problems with numerical performance: fast, high-precision solutions, and reliable structure under different impact factors. That is, it is imperative to introduce a new type of modeling strategy, advancing the conventional transportation planning models. In order to do this, we leverage the backbone of the underlying algorithm behind machine learning (ML): computational graph (CG) and automatic differentiation (AD). CG is a directed acyclic graph (DAG) where each vertex represents a mathematical operation, and each edge represents data transfer. AD is an efficient algorithm to analytically compute gradients of necessary functionality. Embedding the two key algorithms into the planning models, specifically parametric-based econometric models and network optimization models, we theoretically and practically develop different types of modeling structures and reformulate mathematical formulations on basis of the graph-oriented representation. Three closely related analytical and computational frameworks are presented in this dissertation, based on a common modeling methodology of CG abstraction. First, a two-stage interpretable machine learning framework developed by a linear regression model, coupled with a neural network layered by long short-term memory (LSTM) shows the capability of capturing statistical characteristics with enhanced predictability in the context of day-to-day streaming datasets. Second, AD-based computation in estimating for discrete choice models proves more efficiency of handling complex modeling structure than the standard optimization solver relying on numerical gradients, outperforming the standard methods, Biogeme and Apollo. Lastly, CG allows modelers to take advantage of a special problem structure for the feedback loops, a new class of problem reformulation developed through Lagrangian relaxation (LR), which makes CG based model well suited for reaching a high degree of the integrated demand-supply consistency. Overall, the deep integration of the practically important planning models with the underlying computationally efficient ML algorithms can enhance behavioral understanding of interactions in real-world urban systems, and the proposed differentiable mathematical structures will enable transportation decision-makers to accurately evaluate different demand-side and supply-side scenarios with a higher degree of convergency and optimality in more complex transportation systems.