We define a Yang-Mills model in 1+1 dimensions coupled to a real scalar field and we study the Yang-Mills flow equations for this simple model. Yang-Mills flows have not been thoroughly studied, especially in a physical context, but may be able to provide valuable insight into both particle physics as well as gravity. We study our model using both the Hamiltonian equations and Euler-Lagrange equations, and we calculate the flow numerically using a simple finite difference method for the case of an Abelian Lie group and static fields. We are able to find several analytic solutions to the equations of motion and the numerical calculation of the flow suggests most non-constant solutions are unstable. We also find that the flow depends upon the relative values of the coupling constant and the mass of the scalar field. The results found with this simple model provide a starting point for the study of Yang-Mills flow in the context of more complicated (but more physical) models such as the Abelian Higgs.

In this thesis, we study a linear fractional Fokker-Planck equation that models non-local (`fractional') diffusion in the presence of a potential field. The non-locality is due to the appearance of the `fractional Laplacian' in the corresponding PDE, in place of the classical Laplacian which distinguishes the case of regular (Gaussian) diffusion. Motivated by the observation that, in contrast to the classical Fokker-Planck equation (describing regular diffusion in the presence of a potential field), there is no natural gradient flow formulation for its fractional counterpart, we prove existence of weak solutions to this fractional Fokker-Planck equation by combining a splitting technique together with a Wasserstein gradient flow formulation. An explicit iterative construction is given, which we prove weakly converges to a weak solution of this PDE. / Graduate

Given an embedding of a closed k-dimensional manifold M into N-dimensional Euclidean space R^N, we aim to perform negative gradient flow of a penalty function P that acts on the space of all smooth embeddings of M into R^N to find an ideal manifold embedding. We study the computation of the gradient for a penalty function that contains both a curvature and distance term. We also find a lower bound for how long an embedding will remain in the space of embeddings when moving in a fixed, normal gradient direction. Finally, we study the distance penalty function in a special case in which we can prove short time existence of the negative gradient flow using the Cauchy-Kovalevskaya Theorem.

<p dir="ltr">This thesis delves into the development and integration of energy-dissipative methods, with applications spanning numerical analysis, optimization, and deep neural networks, primarily targeting gradient flows and porous medium equations. In the realm of optimization, we introduce the element-wise relaxed scalar auxiliary variable (E-RSAV) algorithm, showcasing its robustness and convergence through extensive numerical experiments. Complementing this, we design an Energy-Dissipative Evolutionary Deep Operator Neural Network (DeepONet) to numerically address a suite of partial differential equations. By employing a dual-subnetwork structure and utilizing the Scalar Auxiliary Variable (SAV) method, the network achieves impeccable approximations of operators while upholding the Energy Dissipation Law, even when training data comprises only the initial state. Lastly, we formulate first-order schemes tailored for Wasserstein gradient flows. Our schemes demonstrate remarkable properties, including mass conservation, unique solvability, positivity preservation, and unconditional energy dissipation. Collectively, the innovations presented here offer promising pathways for efficient and accurate numerical solutions in both gradient flows and Wasserstein gradient flows, bridging the gap between traditional optimization techniques and modern neural network methodologies.</p>

Fokker-Planck equations, along with stochastic differential equations, play vital roles in physics, population modeling, game theory and optimization (finite or infinite dimensional). In this thesis, we study three topics, both theoretically and computationally, centered around them. In part one, we consider the optimal transport for finite discrete states, which are on a finite but arbitrary graph. By defining a discrete 2-Wasserstein metric, we derive Fokker-Planck equations on finite graphs as gradient flows of free energies. By using dynamical viewpoint, we obtain an exponential convergence result to equilibrium. This derivation provides tools for many applications, including numerics for nonlinear partial differential equations and evolutionary game theory. In part two, we introduce a new stochastic differential equation based framework for optimal control with constraints. The framework can efficiently solve several real world problems in differential games and Robotics, including the path-planning problem. In part three, we introduce a new noise model for stochastic oscillators. With this model, we prove global boundedness of trajectories. In addition, we derive a pair of associated Fokker-Planck equations.

Strongly coupled systems are elusive and not suitable to be described by conventional perturbative approaches. However, they are ubiquitous in nature, especially in particle physics. The lattice formulation of quantum field theories provided a unique framework in which the physical content of these systems could be precisely determined. Combined with numerical techniques, the lattice formalism allowed to precisely determined physical quantities describing the thermodynamics, as well as the spectroscopy of strongly interacting theories. In this work, the lattice formulation has been employed to probe the effectiveness of a recently proposed method, which aims at determining the renormalized energy-momentum tensor in non perturbative regimes. The latter plays a fundamental role to quantitatively describe the thermodynamics and fluid-dynamics of hot, dense systems, or to characterize theories that enlarge the actual standard model. In all these aspects, only a non perturbative approach provides physically reliable results: hence a non perturbative determination of the energy momentum tensor is fundamental. The new method consists in defining suitable lattice Ward identities probed by observables built with the gradient flow. The new set of identities exhibits many interesting qualities, arising from the UV finiteness of such probes, and allows to define a numerical strategy for estimating the renormalization constants of the lattice energy-momentum tensor. In this work the method has been tested within two different quantum theories, with the purpose of understanding its effectiveness and reliability.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第23000号 / 理博第4677号 / 新制||理||1671(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 川合 光, 教授 田中 貴浩, 准教授 吉岡 興一 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

This thesis is devoted to the rigorous study of approximations for (multi-phase) mean curvature flow and related equations. We establish convergence towards weak solutions of the according geometric evolution equations in the BV-setting of finite perimeter sets. Our proofs are of variational nature in the sense that we use the gradient flow structure of (multi-phase) mean curvature flow. We study two classes of schemes, namely phase-field models and thresholding schemes. The starting point of our investigation is the fact that both, the Allen-Cahn Equation and the thresholding scheme, preserve this gradient flow structure. The Allen-Cahn Equation is a gradient flow itself, while the thresholding scheme is a minimizing movements scheme for an energy that Γ-converges to the total interfacial energy. In both cases we can incorporate external forces or a volume-constraint. In the spirit of the work of Luckhaus and Sturzenhecker (Calc. Var. Partial Differential Equations 3(2):253–271, 1995), our results are conditional in the sense that we assume the time-integrated energies to converge to those of the limit. Although this assumption is natural, it is not guaranteed by the a priori estimates at hand.

Nous étudions dans ce travail les mouvements de foule intervenant dans les situa- tions d’urgence. Nous proposons un modèle macroscopique (la foule est représentée par une densité de personnes) obéissant à deux principes très simples. Tout d’abord, chaque personne possède une vitesse souhaitée (typiquement celle qui la mène vers la sortie), qu’elle adopterait en l’absence des autres. Ensuite, la foule doit respecter une contrainte de congestion, et la densité de personnes doit rester inférieure à une valeur fixée. Cette contrainte impose une vitesse de déplacement différente de la vitesse souhaitée. Nous choisissons de prendre comme vitesse réelle celle qui est la plus proche, au sens des moindres carrés, de la vitesse souhaitée, parmi les champs de vitesses admissibles, au sens où ils respectent la contrainte de densité maximale. Le modèle obtenu s’écrit sous la forme d’une équation de transport impliquant une vitesse peu régulière a priori, et qui ne peut être étudiée par des méthodes classiques. Nous démontrons un résultat d’existence grâce à la théorie du transport optimal, tout d’abord dans le cas d’une vitesse donnée comme le gradient d’une fonction, puis dans le cas général. Nous mettons également en œuvre un schéma numérique de type catching-up : à chaque pas de temps, la densité est déplacée selon le champ de vitesse souhaitée, puis est projetée sur l’ensemble des densités admissibles. Les résultats obtenus fournissent des temps d’évacuation dont l’ordre de grandeur est proche de la réalité. / In this work, we aim at modelling crowd motion in emergency situations. We propose a macroscopic model (where people are represented as a density) following two basic principles. First, each individual has a spontaneous velocity (typically, the one which leads to the nearest exit) which would be fulfilled in the absence of other people. On the other hand, the crowd has to respect a congestion constraint, and its density must remain underneath a critical density. This constraint prevents people from following their desired velocity. The actual velocity we consider is the closest, in a mean square sense, to the desired one, among the velocities which respect the maximal density constraint.The mathematical formulation writes as a transport equation which cannot be studied with classical methods, since the real velocity field has no a priori regularity, even if the desired velocity is smooth. Thanks to the optimal transport theory, we prove an existence result, first in the case where the desired velocity is the gradient of a given function, and then in the general framework. We also propose a numerical scheme which follows the catching-up principle: at each time step, we move the density according to the spontaneous velocity, and then project it onto the space of admissible densities. The numerical results we obtain reproduce qualitatively the experimental observations

In this thesis, we investigate methods of finding a local minimum for unconstrained problems of non-convex functions with n variables, by following the solution curve of a system of ordinary differential equations. The motivation for this was the fact that existing methods (e.g. those based on Newton methods with line search) sometimes terminate at a non-stationary point when applied to functions f(x) that do not a have positive-definite Hessian (i.e. ∇²f → 0) for all x. Even when methods terminate at a stationary point it could be a saddle or maximum rather than a minimum. The only method which makes intuitive sense in non-convex region is the trust region approach where we seek a step which minimises a quadratic model subject to a restriction on the two-norm of the step size. This gives a well-defined search direction but at the expense of a costly evaluation. The algorithms derived in this thesis are gradient based methods which require systems of equations to be solved at each step but which do not use a line search in the usual sense. Progress along the Continuous Steepest Descent Path (CSDP) is governed both by the decrease in the function value and measures of accuracy of a local quadratic model. Numerical results on specially constructed test problems and a number of standard test problems from CUTEr [38] show that the approaches we have considered are more promising when compared with routines in the optimization tool box of MATLAB [46], namely the trust region method and the quasi-Newton method. In particular, they perform well in comparison with the, superficially similar, gradient-flow method proposed by Behrman [7].