Applied Mathematics
http://hdl.handle.net/10012/9926
Mon, 15 Jul 2024 01:16:43 GMT2024-07-15T01:16:43ZHopf 2-Algebras: Homotopy Higher Symmetries in Physics
http://hdl.handle.net/10012/20681
Hopf 2-Algebras: Homotopy Higher Symmetries in Physics
Chen, Hank
The theory of Hopf algebras and quantum groups have led to very rich and interesting developments in both mathematics and physics. In particular, they are known to play crucial roles in the interplay between 3d topological quantum field theories, categorical algebras, and the geometry of embedded links and tangles. Moreover, the semiclassical limits of quantum group Hopf algebras, in particular, are vital for the understanding of integrable systems in statistical mechanics and Poisson-Lie dualities in string theory. The goal of this PhD thesis is to study a higher-dimensional version of these correspondences, based on the very successful categorical ladder proposal: higher-dimensional physics and geometry is described by higher-categorical strutures. This is accomplished with the definition of a {\it higher homotopy Hopf algebra}, which can be understood as a quantization of the homotopy Lie bialgebra symmetries that have recently received attention in various fields of theoretical physics. These higher-homotopy symmetries are part of the study of the recently-popular categorical symmetries, which appear in the condensed matter literature, for instance, in relation to 1-form dipole symmetries in topologically ordered phases. However, here I will provide another physical motivation arising from the gauge theoretic perspective, which is natural in the context of the Green-Schwarz anomaly cancellation mechanism in quantum field theories. In particular, I use this perspective to prove various known structural theorems about Lie 2-bialgebras and their associated 2-graded classical $R$-matrices, as well as to provide a new definition and characterization of the so-called "quadratic 2-Casimir" elements. I will apply these higher homotopy symmetries to study the 4d 2-Chern-Simons topological quantum field theory, and to develop a notion of graded classical integrability for 2+1d bulk-boundary coupled systems. By following the philosophy of deformation quantization and the theory of $A_\infty$-algbera, I then introduce the notion of a "Hopf 2-algebra" explicitly, and prove several of their structural theorems. I will in particular derive a novel definition of a universal quantum 2-$R$-matrix and the higher-Yang-Baxter equations they satisfy. The main result of this thesis is that the 2-representation 2-category of Hopf 2-algebras is cohesively braided monoidal iff it is equipped with a universal 2-$R$-matrix, and that (weak) Hopf 2-algebras admit (weak) Lie 2-bialgebras as semiclassical limits. Finally, an application of this quantization framework will be considered, in which I will explicitly compute the higher representation theory of Drinfel'd double Hopf 2-algebras of finite groups. The corresponding 2-group Dijkgraaf-Witten topological field theories are then constructed directly from these Hopf 2-algebras, and I show that they recover the known 2-categorical characterizations of 4d $\mathbb{Z}_2$ symmetry protected topological phases of matter.
Wed, 26 Jun 2024 00:00:00 GMThttp://hdl.handle.net/10012/206812024-06-26T00:00:00ZOn Enabling Layer-Parallelism for Graph Neural Networks using IMEX Integration
http://hdl.handle.net/10012/20673
On Enabling Layer-Parallelism for Graph Neural Networks using IMEX Integration
Kara, Omer Ege
Graph Neural Networks (GNNs) are a type of neural networks designed to perform machine
learning tasks with graph data. Recently, there have been several works to train
differential equation-inspired GNN architectures, which are suitable for robust training
when equipped with a relatively large number of layers. Neural networks with more layers
are potentially more expressive. However, the training time increases linearly with the
number of layers. Parallel-in-layer training is a method that was developed to overcome
the increase in training time of deeper networks and was first applied to training residual
networks. In this thesis, we first give an overview of existing works on layer-parallel training
and graph neural networks inspired by differential equations. We then discuss issues
that are encountered when these graph neural network architectures are trained parallel-in-
layer and propose solutions to address these issues. Finally, we present and evaluate
experimental results about layer-parallel GNN training using the proposed approach.
Thu, 20 Jun 2024 00:00:00 GMThttp://hdl.handle.net/10012/206732024-06-20T00:00:00ZStability Analysis and Formally Guaranteed Tracking Control of Quadrotors
http://hdl.handle.net/10012/20669
Stability Analysis and Formally Guaranteed Tracking Control of Quadrotors
Chang, Haocheng
Reach-avoid tasks are among the most common challenges in autonomous aerial vehicle (UAV) applications. Despite the significant progress made in the research of aerial vehicle control during recent decades, the task of efficiently generating feasible trajectories amidst complex surroundings while ensuring formal safety guarantees during trajectory tracking remains an ongoing challenge. In response to this challenge, we propose a comprehensive control framework specifically for quadrotor UAVs reach-avoid tasks with robust formal safety guarantees. Our approach integrates geometric control theory with advanced trajectory generation techniques, enabling the consideration of tracking errors during the trajectory planning phase.
Our framework leverages the well-established geometric tracking controller, analyzing its stability to demonstrate the local exponential stability of tracking error dynamics with any positive control gains. Additionally, we derive precise and tight uniform bounds for tracking errors, ensuring guaranteed safety of the system's behavior under certain conditions. In the trajectory generation phase, our approach incorporates these bounds into the planning process, employing sophisticated sampling-based planning algorithms and safe hyper-rectangular set computations to define robust safe tubes within the environment. These safe tubes serve as corridors within which trajectories can be constructed, with piecewise continuous Bezier curves employed to ensure smooth and continuous motion. Furthermore, to enhance the performance and adaptability of our framework, we formulate an optimization problem aimed at determining optimal control gains, thereby enabling the quadrotor UAV to navigate with optimal safety guarantees.
To demonstrate the validation of the proposed framework, we conduct comprehensive numerical simulations as well as real experiments, demonstrating its ability to successfully plan and execute reach-avoid maneuvers while maintaining a high degree of safety and precision. Through these simulations, we illustrate the practical effectiveness and versatility of our framework in addressing real-world challenges encountered in UAV navigation and trajectory planning.
Wed, 19 Jun 2024 00:00:00 GMThttp://hdl.handle.net/10012/206692024-06-19T00:00:00ZController and Observer Designs for Partial Differential-Algebraic Equations
http://hdl.handle.net/10012/20666
Controller and Observer Designs for Partial Differential-Algebraic Equations
Alalabi, Ala'
Partial differential-algebraic equations (PDAEs) arise in numerous situations, including the coupling between differential-algebraic equations (DAEs) and partial differential equations (PDEs). They also emerge from the coupling of partial differential equations where one of the equations is in equilibrium, as seen in parabolic-elliptic systems. Stabilizing PDAEs and achieving certain performance necessitate sophisticated controller designs. Although there are well-developed controllers for each of PDEs and DAEs, research into controllers for PDAEs remains limited. Discretizing PDAEs to DAEs or reducing PDAE systems to PDEs, when feasible, often results in undesirable outcomes or a loss of the physical meaning of the algebraic constraints. Consequently, this thesis concentrates on the direct design of controllers based on PDAEs, using two control techniques: linear-quadratic and boundary control.
The thesis first addresses the stabilization of coupled parabolic-elliptic systems, an important class of PDAEs with wide applications in fields such as biology, incompressible fluid dynamics, and electrochemical processes. Even when the parabolic equation is exponentially stable on its own, the coupling between the two equations can cause instability in the overall system. A backstepping approach is used to derive a boundary control input to stabilize the system, resulting in an explicit expression for the control law in a state feedback form. Since the system state is not always available, exponentially convergent observers are designed to estimate the system state using boundary measurements. The observation error system is shown to be exponentially stable, again by employing a backstepping method. This leads to the design of observer gains in closed form. By integrating these observers with state feedback boundary control, the thesis also tackles the output feedback problem.
Next, the thesis considers finite-time linear-quadratic control of PDAEs that are radial with index 0; this corresponds to a nilpotency degree of 1. The well-known results for PDEs are generalized to this class of PDAEs. Here, the existence of a unique minimizing optimal control is established. In addition, a projection is used to derive a system of differential Riccati-like equation coupled with an algebraic equation, yielding the solution of the optimization problem in a feedback form. These equations, and hence the optimal control, can be calculated without constructing the projected PDAE.
Lastly, the thesis examines the linear-quadratic (LQ) control problem for linear DAEs of arbitrary index over a finite horizon. Without index reduction or a behavioral approach, it is shown that a certain projection can lead to the derivation of a differential Riccati equation, from which the optimal control is obtained. Numerical simulations are presented to illustrate the theoretical findings for each objective of the thesis.
Wed, 19 Jun 2024 00:00:00 GMThttp://hdl.handle.net/10012/206662024-06-19T00:00:00Z