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The existence of global weak solutions to the cross-diffusion model of Shigesada, Kawasaki, and Teramoto for an arbitrary number of species is proved. The model consists of strongly coupled parabolic equations for the population densities…
The Smoluchowski coagulation-diffusion PDE is a system of partial differential equations modelling the evolution in time of mass-bearing Brownian particles which are subject to short-range pairwise coagulation. This survey presents a fairly…
In this work we study diffusion in networks with community structure. We first replicate and extend work on networks with non-overlapping community structure. We then study diffusion on network models that have overlapping community…
In this paper we revisit the notion of grouped dispersal that have been introduced by Soubeyrand and co-authors \cite{soubeyrand2011patchy} to model the simultaneous (and hence dependent) dispersal of several propagules from a single source…
Modality translation is inherently under-constrained, as multiple cross-modal mappings may yield the same marginals. Recent work has shown that diffusion bridges are effective for this task. However, most existing approaches rely on fully…
This paper studies a stochastic model that describes the evolution of vehicle densities in a road network. It is consistent with the class of (deterministic) kinematic wave models, which describe traffic flows on the basis of conservation…
We define two algorithms for propagating information in classification problems with pairwise relationships. The algorithms are based on contraction maps and are related to non-linear diffusion and random walks on graphs. The approach is…
Various biological cells secrete diffusing chemical compounds into their environment for communication purposes. Secretion usually takes place over the cell membrane in a spatially heterogeneous manner. Mathematical models of these…
Reaction cross diffusion systems are a two species generalization of the porous media equation. These systems play an important role in the mechanical modeling of living tissues and tumor growth. Due to their mixed parabolic-hyperbolic…
Diffusion models achieved unprecedented fidelity and diversity for synthesizing image, video, 3D assets, etc. However, subject mixing is an unresolved issue for diffusion-based image synthesis, particularly for synthesizing multiple…
Coupling diffusion process of signaling molecules with nonlinear interactions of intracellular processes and cellular growth/transformation leads to a system of reaction-diffusion equations coupled with ordinary differential equations…
Diffusion models have enabled high-quality, conditional image editing capabilities. We propose to expand their arsenal, and demonstrate that off-the-shelf diffusion models can be used for a wide range of cross-domain compositing tasks.…
Diffusion, a fundamental internal mechanism emerging in many physical processes, describes the interaction among different objects. In many learning tasks with limited training samples, the diffusion connects the labeled and unlabeled data…
Heterogeneous media diffusion is often described using position-dependent diffusion coefficients and estimated indirectly through mean squared displacement in experiments. This approach may overlook other mechanisms and their interaction…
The transport of single-phase fluid mixtures in porous media is described by cross-diffusion equations for the mass densities. The equations are obtained in a thermodynamic consistent way from mass balance, Darcy's law, and the van der…
Developments in dynamical systems theory provides new support for the macroscale modelling of pdes and other microscale systems such as Lattice Boltzmann, Monte Carlo or Molecular Dynamics simulators. By systematically resolving subgrid…
Multiple time scales problems are investigated by combining geometrical and analytical approaches. More precisely, for fast-slow reaction-diffusion systems, we first prove the existence of slow manifolds for the abstract problem under the…
We study the existence, regularity and uniqueness for a general class of triangular reaction-cross-diffusion systems coming from the study of starvation driven behavior for two species in competition. This study involves an equivalent…
This paper is devoted to the study of systems of reaction-cross diffusion equations arising in population dynamics. New results of existence of weak solutions are presented, allowing to treat systems of two equations in which one of the…
Forecasting over graph-structured sensor networks demands models that capture both deterministic spatial trends and stochastic variability, while remaining efficient enough for repeated inference as new observations arrive. We propose…