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.

This dissertation focuses on a comprehensive exploration of machine learning (ML) and topological data analysis (TDA) with applications for engineering and clinical diagnostics and prognostics. The interface of TDA and ML is called topological machine learning (TML). The key focus and benefit of the proposed TML are on the automated, consistent, and robust handling of high-dimensional data, specifically for the complexities inherent in spatial-temporal datasets. TML's unique ability to capture and quantify high-dimensional geometric and topological features (such as homology) facilitates a deep understanding of the underlying structures of data. The associated dimension reduction capabilities significantly enhance diagnostics and prognostics accuracy and interpretability. TML is first demonstrated using an unsupervised learning setting, where the label information is not required for machine learning. Spatial-temporal data from resting-state functional magnetic resonance imaging (rs-fMRI) are collected and analyzed for Parkinson's disease. Fractal analysis is used to extract topological characteristics of the signal, and extracted features are used in a manifold embedding and projection model for low-dimensional space visualization. The low-dimensional data is integrated with a neural network-based classifier for disease diagnosis. A similar methodology is extended to structural health monitoring problems in engineering. Following this, the TML is developed for a supervised learning setting, where the major application is regression and prediction. Euler characteristics using filtration are used as the topological feature extraction method and extracted features are used in Gaussian Process (GP) modeling for regression analysis. The methodology is first demonstrated with a toy random field problem where a time-dependent field is characterized by varying topological features. The developed method is then demonstrated with crack growth problems with numerical and experimental data. Finally, the topological data analysis is Reflecting on the significant strides made in pushing the envelope of theoretical knowledge while showcasing tangible applications, this work not only charts a course for future progress in the field but also enriches our understanding of machine learning, structural health monitoring, predictive modeling, and beyond. The exploration initiated in this dissertation is just the beginning, with each chapter paving the way for new realms of exploration, innovation, and discovery.

Material behavior under high strain rate deformation has always been an interesting topic. Under this extreme impact, possible structure changes such as phase transformation, chemical reaction, and densification occur in materials. It is helpful to develop a fundamental understanding of structure-property relationship, which helps to build a theoretical model and speed up the material design process. Although shock experiment techniques have been widely developed, numerical approaches such as first principle calculations and molecular dynamics simulations have demonstrated their power in predicting shock behavior and revealing structure-property relationship in an economic and feasible manner. In this dissertation, the mechanical properties and shock responses of three materials, polyurea, silicate glass, and erythritol were investigated, among which polyurea and silicate glass are proposed to be protective materials, while erythritol is proposedto be a surrogate of the explosive material pentaerythritol tetranitrate. First principle calculations and classical molecular dynamics were carried out to predict the shock Hugoniot, and other thermomechanical properties. The simulations also explored potential shock-induced phase transformations in these three materials and seek to draw connections between shock-driven transformations and the underlying chemical composition and material structure. composition and material structure.