Related papers: Gradient flows of time-dependent functionals in me…
We study the geometry of a codimension-one foliation with a time-dependent Riemannian metric. The work begins with formulae concerning deformations of geometric quantities as the Riemannian metric varies along the leaves of the foliation.…
We present a framework enabling variational data assimilation for gradient flows in general metric spaces, based on the minimizing movement (or Jordan-Kinderlehrer-Otto) approximation scheme. After discussing stability properties in the…
This is a preliminary version dated Dec. 2013. It was written independently of [Kim-Masmoudi reference] which uses a similar method. [KM] works with flows in a more general setting and does not mention CAT(K) geometry and gradient curves.…
We study the deformation and dynamics of droplets in time-dependent flows using 3D numerical simulations of two immiscible fluids based on the lattice Boltzmann model (LBM). Analytical models are available in the literature, which assume…
We provide the set of equations for non-relativistic fluid dynamics on arbitrary, possibly time-dependent spaces, in general coordinates. These equations are fully covariant under either local Galilean or local Carrollian transformations,…
We present a novel approximate inference method for diffusion processes, based on the Wasserstein gradient flow formulation of the diffusion. In this formulation, the time-dependent density of the diffusion is derived as the limit of…
We study the $L^2$-gradient flow of functionals $\mathcal F$ depending on the eigenvalues of Schr\"odinger potentials $V$ for a wide class of differential operators associated to closed, symmetric, and coercive bilinear forms, including the…
We consider a one dimensional transport model with nonlocal velocity given by the Hilbert transform and develop a global well-posedness theory of probability measure solutions. Both the viscous and non-viscous cases are analyzed. Both in…
This report is actually the in-class project of MATH489 FA2022, Dynamics and Differential Equations in University of Illinois Urbana-Champaign. The report is mainly about gradient flow of a Morse-type functions (Morse functions, Morse--Bott…
We study the gradient flow of the potential energy on the infinite-dimensional Riemannian manifold of spatial curves parametrized by the arc length, which models overdamped motion of a falling inextensible string. We prove existence of…
We study geometric properties of the gradient flow for learning deep linear convolutional networks. For linear fully connected networks, it has been shown recently that the corresponding gradient flow on parameter space can be written as a…
We prove that the standard gradient flow in parameter space that underlies many training algorithms in deep learning can be continuously deformed into an adapted gradient flow which yields (constrained) Euclidean gradient flow in output…
This paper attempts to construct monotonic entropy functionals for four-dimensional Lorentzian spacetime under physical boundary conditions, as an extension of Perelman's monotonic entropy functionals constructed for three-dimensional…
A nonlinear parabolic equation of sixth order is analyzed. The equation arises as a reduction of a model from quantum statistical mechanics, and also as the gradient flow of a second-order information functional with respect to the…
We present a Bayesian approach for modeling multivariate, dependent functional data. To account for the three dominant structural features in the data--functional, time dependent, and multivariate components--we extend hierarchical dynamic…
Conceiving a molecule as composed of smaller molecular fragments, or subunits, is one of the pillars of the chemical and physical sciences, and leads to productive methods in quantum chemistry. Using a fragmentation scheme, efficient…
Time-fractional partial differential equations are nonlocal in time and show an innate memory effect. In this work, we propose an augmented energy functional which includes the history of the solution. Further, we prove the equivalence of a…
Let $V_* : \mathbb{R}^d \to \mathbb{R}$ be some (possibly non-convex) potential function, and consider the probability measure $\pi \propto e^{-V_*}$. When $\pi$ exhibits multiple modes, it is known that sampling techniques based on…
A simple model to handle the flow of people in emergency evacuation situations is considered: at every point x, the velocity U(x) that individuals at x would like to realize is given. Yet, the incompressibility constraint prevents this…
We clarify some misunderstandings on the time-dependent current density functional theory for open quantum systems we have recently introduced [M. Di Ventra and R. D'Agosta, Phys. Rev. Lett. {\bf 98}, 226403 (2007)]. We also show that some…