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We develop an approach to the theory of relativistic geometric flows and emergent gravity defined by entropy functionals and related statistical thermodynamics models. Nonholonomic deformations of G. Perelman's functionals and related…
The rapid growth of graph data creates significant scalability challenges as most graph algorithms scale quadratically with size. To mitigate these issues, Graph Condensation (GC) methods have been proposed to learn a small graph from a…
Motivated by bit threads, we introduce a new prescription for computing entropy vectors outside the holographic entropy cone. By utilizing cycle flows on directed graphs, we show that the maximum cycle flow associated to any subset of…
This paper develops entropy stable (ES) adaptive moving mesh schemes for the 2D and 3D special relativistic hydrodynamic (RHD) equations. They are built on the ES finite volume approximation of the RHD equations in curvilinear coordinates,…
We study a class of variational problems for regularized conservation laws with Lax's entropy-entropy flux pairs. We first introduce a modified optimal transport space based on conservation laws with diffusion. Using this space, we…
In this paper, we propose Continuous Graph Flow, a generative continuous flow based method that aims to model complex distributions of graph-structured data. Once learned, the model can be applied to an arbitrary graph, defining a…
In this paper we consider the evolution of a graph-like hypersurface by anisotropic mean curvature flow, under some restrictions on the anisotropic area integrand. We find interior estimates (in both time and space) on the gradient of such…
Gradient descent-ascent (GDA) flows play a central role in finding saddle points of bivariate functionals, with applications in optimization, game theory, and robust control. While they are well-understood in Hilbert and Banach spaces via…
This paper proposes a framework to ensure the existence of dynamical system trajectories in the state space of labeled, weighted, and attributed graphs. The evolution of such a system exhibits hybrid behavior: discrete jumps affecting the…
We consider generalized gradient systems in Banach spaces whose evolutions are generated by the interplay between an energy functional and a dissipation potential. We focus on the case in which the dual dissipation potential is given by a…
Continuous graph neural models based on differential equations have expanded the architecture of graph neural networks (GNNs). Due to the connection between graph diffusion and message passing, diffusion-based models have been widely…
Assignment flows denote a class of dynamical models for contextual data labeling (classification) on graphs. We derive a novel parametrization of assignment flows that reveals how the underlying information geometry induces two processes…
Dynamic graphs refer to graphs whose structure dynamically changes over time. Despite the benefits of learning vertex representations (i.e., embeddings) for dynamic graphs, existing works merely view a dynamic graph as a sequence of changes…
We study the convergence of stochastic time-discretization schemes for evolution equations driven by random velocity fields, including examples like stochastic gradient descent and interacting particle systems. Using a unified framework…
We consider a system consisting of a geometric evolution equation for a hypersurface and a parabolic equation on this evolving hypersurface. More precisely, we discuss mean curvature flow scaled with a term that depends on a quantity…
We propose a new semi-discretization scheme to approximate nonlinear Fokker-Planck equations, by exploiting the gradient flow structures with respect to the 2-Wasserstein metric. We discretize the underlying state by a finite graph and…
Recent efforts have extended the flow-matching framework to discrete generative modeling. One strand of models directly works with the continuous probabilities instead of discrete tokens, which we colloquially refer to as Continuous-State…
Diffusion-driven flow is a boundary layer flow arising from the interplay of gravity and diffusion in density-stratified fluids when a gravitational field is non-parallel to an impermeable solid boundary. This study investigates…
Non-uniform grids and mesh adaptation have been a growing part of numerical simulation over the past years. It has been experimentally noted that mesh adaptation leads not only to locally improved solution but also to numerical stability of…
For discretisations of hyperbolic conservation laws, mimicking properties of operators or solutions at the continuous (differential equation) level discretely has resulted in several successful methods. While well-posedness for nonlinear…