相关论文: Beginner's guide to Aggregation-Diffusion Equation…

The propagation of electromagnetic waves in general media is modeled by the time-dependent Maxwell's partial differential equations (PDEs), coupled with constitutive laws that describe the response of the media. In this work, we focus on…

Fractional diffusion equations (FDEs) are a mathematical tool used for describing some special diffusion phenomena arising in many different applications like porous media and computational finance. In this paper, we focus on a…

Diffusion in inhomogeneous materials can be described by both the Fick and Fokker--Planck diffusion equations. Here, we study a mixed Fick and Fokker-Planck diffusion problem with coefficients rapidly oscillating both in space and time. We…

In this paper we want to propose practical numerical methods to solve a class of initial-boundary problem of space-time fractional advection-diffusion equations. To start with, an implicit method based on two-sided Gr\"unwald formulae is…

We propose a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of parametric partial differential equations (PDEs) conditioned on partial observations, which includes,…

We consider a one-dimensional aggregation-diffusion equation, which is the gradient flow in the Wasserstein space of a functional with competing attractive-repulsive interactions. We prove that the fully deterministic particle…

In inhomogeneous environments, the correct expression of the diffusive flux is often not given by the Fick's law $\Gamma = - D \nabla n $. The most general hydrodynamic equation modelling diffusion is indeed the Fokker-Planck Equation…

An asymptotic limit of a class of Cahn-Hilliard systems is investigated to obtain a general nonlinear diffusion equation. The target diffusion equation may reproduce a number of well-known model equations: Stefan problem, porous media…

We investigate well-posedness for martingale solutions of stochastic differential equations, under low regularity assumptions on their coefficients, widely extending some results first obtained by A. Figalli. Our main results are a very…

We present a concise, self-contained derivation of diffusion-based generative models. Starting from basic properties of Gaussian distributions (densities, quadratic expectations, re-parameterisation, products, and KL divergences), we…

Although diffusion models now occupy a central place in generative modeling, introductory treatments commonly assume Euclidean data and seldom clarify their connection to discrete-state analogues. This article is a self-contained primer on…

We study diffusion-type equations supported on structures that are randomly varying in time. After settling the issue of well-posedness, we focus on the asymptotic behavior of solutions: our main result gives sufficient conditions for…

The achievement of Bose-Einstein condensation (BEC) in ultracold vapors of alkali atoms has given enormous impulse to the theoretical and experimental study of dilute atomic gases in condensed quantum states inside magnetic traps and…

Mathematical models based on probability density functions (PDF) have been extensively used in hydrology and subsurface flow problems, to describe the uncertainty in porous media properties (e.g., permeability modelled as random field).…

This survey provides a concise yet comprehensive overview on enhanced dissipation phenomena, transitioning seamlessly from the physical principles underlying the interplay between advection and diffusion to their rigorous mathematical…

The Finite Element Method (FEM) is the gold standard for spatial discretization in numerical simulations for a wide spectrum of real-world engineering problems. Prototypical areas of interest include linear heat transfer and linear…

Diffusion models provide expressive priors for forecasting trajectories of dynamical systems, but are typically unreliable in the sparse data regime. Physics-informed machine learning (PIML) improves reliability in such settings; however,…

In this article we study certain ultradiffusion equations connected with energy landscapes of exponential type. These equations are connected with the p-adic models of complex systems introduced by Avetisov et al. We show that the…

Unidirectional pulse propagation equations [UPPE, Phys. Rev. E 70, 036604 (2004)] have provided a theoretical underpinning for computer-aided investigations into dynamics of high-power ultrashort laser pulses and have been successfully…

We present an analytical closed form expression, which gives a good approximate propagator for diffusion on the sphere. Our formula is the spherical counterpart of the Gaussian propagator for diffusion on the plane. While the analytical…