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Differential equations (DEs) are commonly used to describe dynamic systems evolving in one (ordinary differential equations or ODEs) or in more than one dimensions (partial differential equations or PDEs). In real data applications the…
A series of recent articles introduced a method to construct stochastic partial differential equations (SPDEs) which are invariant with respect to the distribution of a given conditioned diffusion. These works are restricted to the case of…
Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…
Partial differential equation (PDE) models are widely used in engineering and natural sciences to describe spatio-temporal processes. The parameters of the considered processes are often unknown and have to be estimated from experimental…
We introduce and study a new class of partial differential equations (PDEs) with hybrid fuzzy-stochastic parameters, coined fuzzy-stochastic PDEs. Compared to purely stochastic PDEs or purely fuzzy PDEs, fuzzy-stochastic PDEs offer powerful…
In this paper we discuss the analysis of a cross-diffusion PDE system for a mixture of hard spheres, which was derived by Bruna and Chapman from a stochastic system of interacting Brownian particles using the method of matched asymptotic…
We introduce and analyse a continuum model for an interacting particle system of Vicsek type. The model is given by a non-linear kinetic partial differential equation (PDE) describing the time-evolution of the density $f_t$, in the single…
We develop numerical methods for reaction-diffusion systems based on the equations of fluctuating hydrodynamics (FHD). While the FHD formulation is formally described by stochastic partial differential equations (SPDEs), it becomes similar…
The statistical mechanics of phase transitions in dense systems of polydisperse particles presents distinctive challenges to computer simulation and analytical theory alike. The core difficulty, namely dealing correctly with particle size…
Partial Differential Equations (PDEs) have long been recognized as powerful tools for image processing and analysis, providing a framework to model and exploit structural and geometric properties inherent in visual data. Over the years,…
The reaction-diffusion master equation (RDME) is a model that allows for efficient on-lattice simulation of spatially resolved stochastic chemical kinetics. Compared to off-lattice hard-sphere simulations with Brownian Dynamics (BD) or…
The simulation of large ensembles of particles is usually parallelized by partitioning the domain spatially and using message passing to communicate between the processes handling neighboring subdomains. The particles are represented as…
Many systems in physics, engineering, and biology exhibit multiscale stochastic dynamics, where low-dimensional slow variables evolve under the influence of high-dimensional fast processes. In practice, observations are often limited to a…
This work introduces a new class of cross-diffusion systems for studying overcrowding dispersal of two species. The approach, based on proximal minimization energy through a minimum flow process, offers a potential generalization of…
The dynamics of pulse solutions in a bistable reaction-diffusion system are studied analytically by reducing partial differential equations (PDEs) to finite-dimensional ordinary differential equations (ODEs). For the reduction, we apply the…
Clustering is a fundamental collective phenomenon in agent-based models (ABMs) of opinion dynamics. To study clustering in systems with co-evolving social and opinion variables, we derive stochastic partial differential equation (SPDE)…
Cross-diffusion systems arise as hydrodynamic limits of lattice multi-species interacting particle models. The objective of this work is to provide a numerical scheme for the simulation of the cross-diffusion system identified in [J.…
Diffusion-based solvers for partial differential equations (PDEs) are often bottle-necked by slow gradient-based test-time optimization routines that use PDE residuals for loss guidance. They additionally suffer from optimization…
We propose a novel framework for analyzing the dynamics of distribution shift in real-world systems that captures the feedback loop between learning algorithms and the distributions on which they are deployed. Prior work largely models…
We explore the derivation of distributed parameter system evolution laws (and in particular, partial differential operators and associated partial differential equations, PDEs) from spatiotemporal data. This is, of course, a classical…