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.

Mechanical fatigue has been a research topic since quite a long time. It is a complex phenomenon at molecular level. The fact that fatigue failure occurs much below material’s yield point, made it much interesting area for research. So, to understand the physics behind fatigue failure became an important research topic. Fatigue failure is characterized by crack initiation and then crack propagation to finally fracture the material. If this could be modelled mathematically, then it would save lot of resources and would assure the structural integrity of given component. Many such mathematical models were published to calculate fatigue crack growth for Constant Amplitude Loading, but most of the time the applied loads are variable in nature. So, to address this problem a mathematical model which will predict fatigue life in terms of time history is needed. This research study focuses on improving previously developed subcycle fatigue crack growth model also known as small time scale model which works well in Paris regime. In the first part, focus has been given on estimating threshold point using subcycle model by applying load shedding techniques. Later subcycle model has been modified to include fatigue crack growth in threshold region. In the second part of this research study, the concept of Equivalent Initial Flaw Size (EIFS) and fracture mechanics approach has been used to compute fatigue life for Constant as well as Random Amplitude Loading. Further the model has been extended to compute the fatigue life under Mixed Mode Loading (Mode I & Mode II). Standard material properties are used to calibrate the model parameters. The fatigue life results were validated using available open literature data as well as experimental testing data. The subcycle model can be used to calculate fatigue life in case of HCF and LCF, which is suggested as a future work for this research study.

An orthotropic elasto-plastic damage material model (OEPDMM) suitable for impact analysis of composite materials has been developed through a joint research project funded by the Federal Aviation Administration (FAA) and the National Aeronautics and Space Administration (NASA). The developed material model has been implemented into LS-DYNA®, a commercial finite element program. The material model is modular comprising of deformation, damage and failure sub-models. The deformation sub-model captures the rate and/or temperature dependent elastic and inelastic behavior via a visco-elastic-plastic formulation. The damage sub-model predicts the reduction in the elastic stiffness of the material. The failure sub-model predicts when there is no more load carrying capacity in the finite element and erosion of the element from the finite element model. Most of the input parameters required to drive OEPDMM are in the form of tabulated data. The deformation sub-model is driven by a set of tabulated stress-strain data for a given strain-rate and temperature combination. The damage sub-model is driven by tabulated damage parameter-strain data. Two failure sub-models have been implemented – Puck Failure Model and Generalized Tabulated Failure Model. Puck Failure Model requires scalar parameters as input whereas, the Generalized Tabulated Failure Model is driven by a set of equivalent failure strain tabulated data. The work presented here focuses on the enhancements made to OEPDMM with emphasis on the background, development, and implementation of the failure sub-models. OEPDMM is verified and validated using a carbon/epoxy fiber reinforced composite. Two validation tests are used to evaluate the failure sub-model implementation - a stacked-ply test carried out at room temperature under quasi-static tensile and compressive loadings, and several high-speed impact tests where there is significant damage and material failure of the impacted panel. Results indicate that developed procedures provide the analyst with a reasonable and systematic approach to building predictive impact simulation models.