Related papers: A Generic Approach to Solving Jump Diffusion Equat…
Using statistical physics methods, we study generative diffusion models in the regime where the dimension of space and the number of data are large, and the score function has been trained optimally. Our analysis reveals three distinct…
Mathematically modelling diffusive and advective transport of particles in heterogeneous layered media is important to many applications in computational, biological and medical physics. While deterministic continuum models of such…
In computational system biology, the mesoscopic model of reaction-diffusion kinetics is described by a continuous time, discrete space Markov process. To simulate diffusion stochastically, the jump coefficients are obtained by a…
Reaction-Diffusion systems arise in diverse areas of science and engineering. Due to the peculiar characteristics of such equations, analytic solutions are usually not available and numerical methods are the main tools for approximating the…
Density dependent Markov population processes in large populations of size $N$ were shown by Kurtz (1970, 1971) to be well approximated over finite time intervals by the solution of the differential equations that describe their average…
The use of stochastic differential equations in multi-objective optimization has been limited, in practice, by two persistent gaps: incomplete stability analyses and the absence of accessible implementations. We revisit a drift--diffusion…
In this paper we investigate deterministic diffusion in systems which are spatially extended in certain directions but are restricted in size and open in other directions, consequently particles can escape. We introduce besides the…
We propose a framework for solving evolution equations within parametric function classes, especially ones that are specified by neural networks. We call this framework the minimal neural evolution (MNE) because it is motivated by the goal…
Although real-world complex systems typically interact through sparse and heterogeneous networks, analytic solutions of their dynamics are limited to models with all-to-all interactions. Here, we solve the dynamics of a broad range of…
The problem of estimating interacting systems of multiple objects is important to a number of different fields of mathematics, physics, and engineering. Drawing from a range of disciplines, including statistical physics, variational…
We study a regulation problem for stochastic systems subject to both continuous fluctuations and rare but significant shocks, modeled as a jump-diffusion with uncertainty in both the drift and the jump intensity. Such settings arise in…
This paper presents new analytical results for a class of nonlinear parabolic systems of partial different equations with small cross-diffusion which describe the macroscopic dynamics of a variety of large systems of interacting particles.…
Human trajectory data is crucial in urban planning, traffic engineering, and public health. However, directly using real-world trajectory data often faces challenges such as privacy concerns, data acquisition costs, and data quality. A…
Jacobi diffusion is a representative diffusion process whose solution is bounded in a domain under certain drift and diffusion coefficient conditions. However, the process without such conditions has not been thoroughly investigated. We…
Stochastic reaction-diffusion equations are a popular modelling approach for studying interacting populations in a heterogeneous environment under the influence of environmental fluctuations. Although the theoretical basis of alternative…
To model discrete sequences such as DNA, proteins, and language using diffusion, practitioners must choose between three major methods: diffusion in discrete space, Gaussian diffusion in Euclidean space, or diffusion on the simplex. Despite…
The diffusion approximation (DA) is widely used in the analysis of stochastic population dynamics, from population genetics to ecology and evolution. DA is an uncontrolled approximation that assumes the smoothness of the calculated quantity…
Spatially structured neural networks driven by jump diffusion noise with monotone coefficients, fully path dependent delay and with a disorder parameter are considered. Well-posedness for the associated McKean-Vlasov equation and a…
We consider a rate control problem for an $N$-particle weakly interacting finite state Markov process. The process models the state evolution of a large collection of particles and allows for multiple particles to change state…
We extend to multi-dimensions the work of [1], where new fully explicit kinetic methods were built for the approximation of linear and non-linear convection-diffusion problems. The fundamental principles from the earlier work are retained:…