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.

Aviation is a complicated field that involves a wide range of operations, from commercial airline flights to Unmanned Aerial Systems (UAS). Planning and scheduling are essential components in the aviation industry that play a significant role in ensuring safe and efficient operations. Reinforcement Learning (RL) has received increasing attention in recent years due to its capability to enable autonomous decision-making. To investigate the potential advantages and effectiveness of RL in aviation planning and scheduling, three topics are explored in-depth, including obstacle avoidance, task-oriented path planning, and maintenance scheduling. A dynamic and probabilistic airspace reservation concept, called Dynamic Anisotropic (DA) bound, is first developed for UAS, which can be added around the UAS as the separation requirement. A model based on Q-leaning is proposed to integrate DA bound with path planning for obstacle avoidance. Moreover, A deep reinforcement learning algorithm based on Proximal Policy Optimization (PPO) is proposed to guide the UAS to destinations while avoiding obstacles through continuous control. Results from case studies demonstrate that the proposed model can provide accurate and robust guidance and resolve conflict with a success rate of over 99%. Next, the single-UAS path planning problem is extended to a multi-agent system where agents aim to accomplish their own complex tasks. These tasks involve non-Markovian reward functions and can be specified using reward machines. Both cooperative and competitive environments are explored. Decentralized Graph-based reinforcement learning using Reward Machines (DGRM) is proposed to improve computational efficiency for maximizing the global reward in a graph-based Markov Decision Process (MDP). Q-learning with Reward Machines for Stochastic Games (QRM-SG) is developed to learn the best-response strategy for each agent in a competitive environment. Furthermore, maintenance scheduling is investigated. The purpose is to minimize the system maintenance cost while ensuring compliance with reliability requirements. Maintenance scheduling is formulated as an MDP and determines when and what maintenance operations to conduct. A Linear Programming-enhanced RollouT (LPRT) method is developed to solve both constrained deterministic and stochastic maintenance scheduling with an infinite horizon. LPRT categorizes components according to their health condition and makes decisions for each category.

Over the past four decades, DNA nanotechnology has grown exponentially from a field focused on simple structures to one capable of synthesizing complex nano-machines capable of drug delivery, nano-robotics, digital data storage, logic gated circuitry, nano-photonics, and other applications. The construction of these nanostructures is possible because of the predictable and programmable Watson-Crick base pairing of DNA. However, there is an increasing need for the incorporation of chemical diversity and functionality into these nanostructures. To overcome this challenge, this work explored creating hybrid DNA nanostructures by making self-assembling small molecule/protein-DNA conjugates.In one direction, well studied host-guest interactions (i.e. cucurbituril[7]-adamantane) were used as the choice of self-assembling species. Binding studies using these small molecule-DNA conjugates were performed and thereafter they were used to assemble larger DNA origami nanostructures. Finally, a stimulus responsive DNA nano-box that opens and closes based on these interactions was also demonstrated. In another direction, a trimeric KDPG aldolase protein-DNA conjugate was probed as a structural building block by assembling it into a DNA origami tetrahedron with four cavities. This hybrid building block was thereafter characterized by single particle cryo-EM and the resulting electron density map was best fit by simulating origami cages with varying number of proteins (ranging from 0 to 4). Next, to increase access and for larger democratization of the field, an automation designer software tool capable of making DNA nanostructures was made. In this work, the focus was on making curved 3D DNA nanostructures. The last direction probed in this work was to make optical metamaterials based on complex 3D DNA architectures. Realization of a self-assembled 3D tetrastack geometry is still an unachieved dream in the field of DNA self-assembly. Thus, this direction was probed using DNA origami icosahedrons. Finally, the work covered in my thesis probes multiple directions for advancing DNA nanotechnology, both fundamentally and for potential applications.