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Diffusion models have emerged as powerful generative frameworks by progressively adding noise to data through a forward process and then reversing this process to generate realistic samples. While these models have achieved strong…

Complex network theory provides an elegant and powerful framework to statistically investigate different types of systems such as society, brain or the structure of local and long-range dynamical interrelationships in the climate system.…

We propose certain approach of solving two-dimensional non-stationary and stationary advection-diffusion-reaction boundary value problems through their reduction to the set of corresponding one-dimensional problems. This method leverages…

This study examines anomalous diffusion and dynamical phase transitions in a nonlinear bouncer model with short-range interactions leading to velocity-dependent (adiabatic) collisions. By varying a control parameter, transitions between…

Diffusion models, which leverage stochastic processes to capture complex data distributions effectively, have shown their performance as generative models, achieving notable success in image-related tasks through iterative denoising…

A thermodynamic approach to rapid phase transformations within a diffuse interface in a binary system is developed. Assuming an extended set of independent thermodynamic variables formed by the union of the classic set of slow variables and…

This paper is devoted to a nonlocal reaction-diffusion-advection model that describes the spatial dynamics of freshwater organisms in a river with a directional motion. Our goal is to investigate how the advection rate affects the dynamic…

Significant progress has been made on the model development for simulating turbulent reacting flows. As a consequence, we are currently in a position where key-physical aspects of fairly complex combustion processes are well understood at a…

Wildfires are among the most severe disturbances affecting forest ecosystems, with over 50,000 hectares burned in Patagonia, Argentina, during 2025 alone. This study implements a Reaction-Diffusion-Convection (RDC) model to simulate…

This paper proposes a physical-statistical modeling approach for spatio-temporal data arising from a class of stochastic convection-diffusion processes. Such processes are widely found in scientific and engineering applications where…

Many systems in life sciences have been modeled by reaction-diffusion equations. However, under some circumstances, these biological systems may experience instantaneous and periodic perturbations (e.g. harvest, birth, release, fire events,…

The review is devoted to analysis of mathematical models used for describing epidemic processes. A main focus is done on the models that are based on partial differential equations (PDEs), especially those that were developed and used for…

Wildfire forecasting problems usually rely on complex grid-based mathematical models, mostly involving Computational fluid dynamics(CFD) and Celluar Automata, but these methods have always been computationally expensive and difficult to…

Stochastic models of diffusion with excluded-volume effects are used to model many biological and physical systems at a discrete level. The average properties of the population may be described by a continuum model based on partial…

Reaction-diffusion equations are one of the most common mathematical models in the natural sciences and are used to model systems that combine reactions with diffusive motion. However, rather than normal diffusion, anomalous subdiffusion is…

Most avalanching systems in nature should involve diffusive processes as well which can change the behavior of such systems and should be taken into account. We examine the effects of diffusion on the model of a dissipative bi-directional…

In this paper, we examine the long-time dynamics of an epidemic model whose diffusion and reaction terms involve nonlocal effects described by suitable convolution operators.The spreading front of the disease is represented by the free…

Operational weather forecasting models have advanced for decades on both the explicit numerical solvers and the empirical physical parameterization schemes. However, the involved high computational costs and uncertainties in these existing…

Understanding and simulating how a quantum system interacts and exchanges information or energy with its surroundings is a ubiquitous problem, one which must be carefully addressed in order to establish a coherent framework to describe the…

One of the main challenges in diffusion-based molecular communication is dealing with the non-linearity of reaction-diffusion chemical equations. While numerical methods can be used to solve these equations, a change in the input signals or…