Related papers: Gradient Flow in Morse and Analytic Settings
We propose a categorical semantics of gradient-based machine learning algorithms in terms of lenses, parametrised maps, and reverse derivative categories. This foundation provides a powerful explanatory and unifying framework: it…
Flow matching has shown state-of-the-art performance in various generative tasks, ranging from image generation to decision-making, where generation under energy guidance (abbreviated as guidance in the following) is pivotal. However, the…
This is an expository paper on the theory of gradient flows, and in particular of those PDEs which can be interpreted as gradient flows for the Wasserstein metric on the space of probability measures (a distance induced by optimal…
This article is an invitation. It is, first, an invitation to consider as a subject worthy of attention the wide range of situations where small discrete elements, either bubbles, droplets or solid particles, are embedded in turbulent…
Assuming a-priori a smooth generating vector field, we introduce a generally covariant measure of the flow geometry called the referential gradient of the flow. The main result is the explicit relation between the referential gradient and…
Finding latent structures in data is drawing increasing attention in diverse fields such as image and signal processing, fluid dynamics, and machine learning. In this work we examine the problem of finding the main modes of gradient flows.…
We introduce a framework that unifies quantum measurement dynamics, Hamiltonian dynamics, and double-bracket gradient flows. We do so by providing explicit expressions for stochastic Hamiltonians that produce state dynamics identical to…
The center of interest in this work are variational problems with integral functionals depending on special nonlocal gradients. The latter correspond to truncated versions of the Riesz fractional gradient, as introduced in [Bellido, Cueto &…
We propose a novel deep learning paradigm of differential flows that learn a stochastic differential equation transformations of inputs prior to a standard classification or regression function. The key property of differential Gaussian…
The problem of classifying turbulent environments from partial observation is key for some theoretical and applied fields, from engineering to earth observation and astrophysics, e.g. to precondition searching of optimal control policies in…
In the field of machine learning, comprehending the intricate training dynamics of neural networks poses a significant challenge. This paper explores the training dynamics of neural networks, particularly whether these dynamics can be…
We consider discrete porous medium equations of the form \partial_t \rho_t = \Delta \phi(\rho_t), where \Delta is the generator of a reversible continuous time Markov chain on a finite set X, and \phi is an increasing function. We show that…
We study a gradient flow on Sobolev diffeomorphisms for the problem of image registration. The energy functional quantifies the effect of transforming a template to a target, while also penalizing deformation of the metric tensor. The main…
In this note we consider the global convergence properties of the differential equation $\dot{\Theta}=(\Theta^{T}-\Theta)\Theta$ with $\Theta\in{SO(n)}$, which is a gradient flow of the function…
We present a convergence analysis of a finite difference scheme for the time dependent partial different equation called gradient flow associated with the Rudin-Osher-Fatemi model. We devise an iterative algorithm to compute the solution of…
Flow models are a cornerstone of modern machine learning. They are generative models that progressively transform probability distributions according to learned dynamics. Specifically, they learn a continuous-time Markov process that…
The problem of deriving a gradient flow structure for the porous medium equation which is {\em thermodynamic}, in that it arises from the large deviations of some microscopic particle system, is studied. To this end, a rescaled zero-range…
Rectified flow models have become a de facto standard in image generation due to their stable sampling trajectories and high-fidelity outputs. Despite their strong generative capabilities, they face critical limitations in image editing…
We study the sensitivity of the gradient flow coupling to sectors of different topological charge and its implications in practical situations. Furthermore, we investigate an alternative definition of the running coupling that is expected…
The metric and potential associated with the gradient property of renormalisation group flow in multiscalar models in $d=4-\varepsilon$ dimensions are studied. The metric is identified with the Zamolodchikov metric of nearly marginal…