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A growing body of literature has been leveraging techniques of machine learning (ML) to build novel approaches to approximating the solutions to partial differential equations. Noticeably absent from the literature is a systematic…

Wave propagation problems have many applications in physics and engineering, and the stochastic effects are important in accurately modeling them due to the uncertainty of the media. This paper considers and analyzes a fully discrete finite…

We present an ``equation-free'' multiscale approach to the simulation of unsteady diffusion in a random medium. The diffusivity of the medium is modeled as a random field with short correlation length, and the governing equations are cast…

We consider the inference problem for parameters in stochastic differential equation models from discrete time observations (e.g. experimental or simulation data). Specifically, we study the case where one does not have access to…

In the context of mathematical modeling, it is sometimes convenient to integrate models of different nature. These types of combinations, however, might entail difficulties even when individual models are well-understood, particularly in…

We propose a new classification scheme for diffusion processes for which the backward Kolmogorov equation is solvable in analytically closed form by reduction to hypergeometric equations of the Gaussian or confluent type. The construction…

Stochastic diffusion equations are crucial for modeling a range of physical phenomena influenced by uncertainties. We introduce the generalized finite difference method for solving these equations. Then, we examine its consistency,…

Temporal data such as time series can be viewed as discretized measurements of the underlying function. To build a generative model for such data we have to model the stochastic process that governs it. We propose a solution by defining the…

The goal of this paper is to create a fruitful bridge between the numerical methods for approximating partial differential equations (PDEs) in fluid dynamics and the (iterative) numerical methods for dealing with the resulting large linear…

The developments over the last five decades concerning numerical discretisations of the incompressible Navier--Stokes equations have lead to reliable tools for their approximation: those include stable methods to properly address the…

In this paper, continuous-time master equations with finite states employed in nonequilibrium statistical mechanics are formulated in the language of discrete geometry. In this formulation, chains in algebraic topology are used, and master…

Diffusion models (DMs) are a class of generative machine learning methods that sample a target distribution by transforming samples of a trivial (often Gaussian) distribution using a learned stochastic differential equation. In standard…

Generative diffusion models are extensively used in unsupervised and self-supervised machine learning with the aim to generate new samples from a probability distribution estimated with a set of known samples. They have demonstrated…

This paper is concerned with a space-time adaptive numerical method for instationary porous media flows with nonlinear interaction between porosity and pressure, with focus on problems with discontinuous initial porosities. A convergent…

We investigate the periodic and stationary solutions of distribution-dependent stochastic differential equations. While generally, the semigroups associated with the equations are nonlinear, we show that the methods of weak convergence and…

We illustrate the efficacy of a discrete wavelet based approach to characterize fluctuations in non-stationary time series. The present approach complements the multi-fractal detrended fluctuation analysis (MF-DFA) method and is quite…

This paper is focused on the convergence analysis of an adaptive stochastic collocation algorithm for the stationary diffusion equation with parametric coefficient. The algorithm employs sparse grid collocation in the parameter domain…

The discrete cosine transform is a valuable tool in analysis of data on undirected rectangular grids, like images. In this paper it is shown how one can define an analogue of the discrete cosine transform on triangles. This is done by…

We investigate linear parabolic equations in divergence form with singular coefficients and non-smooth boundary data. When the diffusion, drift, or potential terms, as well as the initial or boundary conditions, are distributions rather…

Distributed order fractional Langevin-like equations are introduced and applied to describe anomalous diffusion without unique diffusion or scaling exponent. It is shown that these fractional Langevin equations of distributed order can be…