相关论文: Construction of Diffusion Algebras

The design of aerodynamic shapes, such as airfoils, has traditionally required significant computational resources and relied on predefined design parameters, which limit the potential for novel shape synthesis. In this work, we introduce a…

Diffusion and flow-based models have become the state of the art for generative AI across a wide range of data modalities, including images, videos, shapes, molecules, music, and more. This tutorial provides a self-contained introduction to…

Diffusion models (DMs) have emerged as powerful tools for modeling complex data distributions and generating realistic new samples. Over the years, advanced architectures and sampling methods have been developed to make these models…

Conditional diffusion models serve as the foundation of modern image synthesis and find extensive application in fields like computational biology and reinforcement learning. In these applications, conditional diffusion models incorporate…

Diffusion models have emerged as powerful tools for molecular generation, particularly in the context of 3D molecular structures. Inspired by non-equilibrium statistical physics, these models can generate 3D molecular structures with…

We show that the relation between the Schr\"odinger equation and diffusion processes has an algebraic nature and can be revealed via the structure of "duplex numbers." This helps one to clarify that quantum mechanics cannot be reduced to…

It is the purpose of this article to outline a course that can be given to engineers looking for an understandable mathematical description of the foundations of distribution theory and the necessary functional analytic methods. Arguably,…

This article proposes an active learning method for high dimensional data, based on intrinsic data geometries learned through diffusion processes on graphs. Diffusion distances are used to parametrize low-dimensional structures on the…

We give an introduction to discrete functional analysis techniques for stationary and transient diffusion equations. We show how these techniques are used to establish the convergence of various numerical schemes without assuming…

Although diffusion models have successfully extended to function-valued data, stochastic interpolants -- which offer a flexible way to bridge arbitrary distributions -- remain limited to finite-dimensional settings. This work bridges this…

In this paper we consider a representative a priori unstable Hamiltonian system with 2+1/2 degrees of freedom, to which we apply the geometric mechanism for diffusion introduced in the paper Delshams et al., Mem. Amer. Math. Soc. 2006, and…

In this article we review classical and recent results in anomalous diffusion and provide mechanisms useful for the study of the fundamentals of certain processes, mainly in condensed matter physics, chemistry and biology. Emphasis will be…

We introduce a simplified technique for incorporating diffusive phenomena into lattice-gas molecular dynamics models. In this method, spatial interactions take place one dimension at a time, with a separate fractional timestep devoted to…

Diffusion models have emerged as powerful generative frameworks by progressively adding noise to data through a forward process and then reversing this process to generate realistic samples. While these models have achieved strong…

We consider the task of generating discrete-time realisations of a nonlinear multivariate diffusion process satisfying an It\^o stochastic differential equation conditional on an observation taken at a fixed future time-point. Such…

Diffusion models have had a profound impact on many application areas, including those where data are intrinsically infinite-dimensional, such as images or time series. The standard approach is first to discretize and then to apply…

We associate with the Grassmann algebra a topological algebra of distributions, which allows the study of processes analogous to the corresponding free stochastic processes with stationary increments, as well as their derivatives.

This work intends to present a study on relations between a Lie algebra called dispersion operators algebra, linear canonical transformation and a phase space representation of quantum mechanics that we have introduced and studied in…

We develop a linear-algebraic framework for dimensional analysis in systems with constraints, particularly when variables are numerous or related by implicit relations so that direct elimination is impractical. By expressing both…

A rigorous mathematical theory of dimensional analysis, systematically accounting for the use of physical quantities in science and engineering, perhaps surprisingly, was not developed until relatively recently. We claim that this has…