Related papers: Gradient Flow Structure of a Multidimensional Nonl…
The purpose of this work is to establish a quantitative and constructive stability result for a class of subcritical Gagliardo-Nirenberg-Sobolev inequalities which interpolates between the logarithmic Sobolev inequality and the standard…
We prove the global-in-time existence of nonnegative weak solutions to a class of fourth order partial differential equations on a convex bounded domain in arbitrary spatial dimensions. Our proof relies on the formal gradient flow structure…
We prove global-in-time existence and uniqueness of measure solutions of a nonlocal interaction system of two species in one spatial dimension. For initial data including atomic parts we provide a notion of gradient-flow solutions in terms…
We develop a gradient-flow theory for time-dependent functionals defined in abstract metric spaces. Global well-posedness and asymptotic behavior of solutions are provided. Conditions on functionals and metric spaces allow to consider the…
We study the implicit bias of gradient flow (i.e., gradient descent with infinitesimal step size) on linear neural network training. We propose a tensor formulation of neural networks that includes fully-connected, diagonal, and…
We consider a non-standard finite-volume discretization of a strongly non-linear fourth order diffusion equation on the $d$-dimensional cube, for arbitrary $d \geq 1$. The scheme preserves two important structural properties of the…
We consider a class of time dependent second order partial differential equations governed by a decaying entropy. The solution usually corresponds to a density distribution, hence positivity (non-negativity) is expected. This class of…
We study the Wasserstein gradient flow of semi-discrete energies in the space of probability measures, that is functionals depending on two measures-one being an absolutely continuous density and the other an atomic measure. These energies…
The defining equation $(\ast):\ \dot \omega\_t=-F'(\omega\_t),$ of a gradient flow is kinetic in essence. This article explores some dynamical (rather than kinetic) features of gradient flows (i) by embedding equation $(\ast)$ into the…
We study both the local and global existence of a gradient flow of the Sinai-Ruelle-Bowen entropy functional on a Hilbert manifold of expanding maps of a circle equipped with a Sobolev norm in the tangent space of the manifold. We show…
We study a class of fourth-order quasilinear degenerate parabolic equations under both time-and space-dependent and time-and space-independent forces, modeling non-Newtonian thin-film flow over a solid surface in the "complete wetting"…
We derive some fluid-dynamic models for electron transport near a Dirac point in graphene. We start from a kinetic model constituted by a set of spinorial Wigner equations, we make suitable scalings (hydrodynamic or diffusive) of the model…
In this paper we study a gradient flow approach to the problem of quantization of measures in one dimension. By embedding our problem in $L^2$, we find a continuous version of it that corresponds to the limit as the number of particles…
In this paper, we give the covariant formulation of second gradient electrodynamics, which is a generalized electrodynamics of second order including derivatives of higher order. The relativistic form of the field equations, the…
In this work, we present a new approach to analyze the gradient flow for a positive semi-definite matrix denoising problem in an extensive-rank and high-dimensional regime. We use recent linear pencil techniques of random matrix theory to…
In this paper we provide a variational characterisation for a class of non-linear evolution equations with constant non-negative Dirichlet boundary conditions on a bounded domain as gradient flows in the space of non-negative measures. The…
In this paper, we introduce a geometric flow for Lagrangian submanifolds in a K\"ahler manifold that stays in its initial Hamiltonian isotopy class and is a gradient flow for volume. The stationary solutions are the Hamiltonian stationary…
We investigate a family of gradient flows of positive and probability measures, focusing on the Hellinger-Kantorovich (HK) geometry, which unifies transport mechanism of Otto-Wasserstein, and the birth-death mechanism of Hellinger (or…
We perform numerical analysis of a nonlinear gradient flow, which can be regarded as a parabolic minimal surface problem or a regularised total variation flow, using the gradient discretisation method (GDM). GDM is a unified convergence…
Obtaining system parameters and reconstructing the full flow state from limited velocity observations using conventional fluid dynamics solvers can be prohibitively expensive. Here we employ machine learning algorithms to overcome the…