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A feed-forward neural network has a remarkable property which allows the network itself to be a universal approximator for any functions.Here we present a universal, machine-learning based solver for multi-variable partial differential…
In this paper the use of nonlinear cross-diffu\-sion systems to model image restoration is investigated, theoretically and numerically. In the first case, well-posedness, scale-space properties and long time behaviour are analyzed. From a…
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…
Diffusion models are a class of probabilistic generative models that have been widely used as a prior for image processing tasks like text conditional generation and inpainting. We demonstrate that these models can be adapted to make…
Stochastic simulation methods can be applied successfully to model exact spatio-temporally resolved reaction-diffusion systems. However, in many cases, these methods can quickly become extremely computationally intensive with increasing…
The nonlocal Fisher equation is a diffusion-reaction equation with a nonlocal quadratic competition, which describes the reaction between distant individuals. This equation arises in evolutionary biological systems, where the arena for the…
Over the past few decades, there has been substantial interest in evolution equations that involving a fractional-order derivative of order $\alpha\in(0,1)$ in time, due to their many successful applications in engineering, physics, biology…
We introduce and study a notion of Asymptotic Preserving schemes, related to convergence in distribution, for a class of slow-fast Stochastic Differential Equations. In some examples, crude schemes fail to capture the correct limiting…
In this work, we derive particle schemes, based on micro-macro decomposition, for linear kinetic equations in the diffusion limit. Due to the particle approximation of the micro part, a splitting between the transport and the collision part…
In this paper, a modification of the conventional approximations to the quasi-maximum likelihood method is introduced for the parameter estimation of diffusion processes from discrete observations. This is based on a convergent…
In this paper, we develop an efficient numerical solver for unsteady diffusion-type partial differential equations with random coefficients. A major computational challenge in such problems lies in repeatedly handling large-scale linear…
We propose a semiparametric model for autonomous nonlinear dynamical systems and devise an estimation procedure for model fitting. This model incorporates subject-specific effects and can be viewed as a nonlinear semiparametric mixed…
We present a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional derivative of order $\alpha\in(0,1)$ in time. The framework relies on three…
Describing particle transport at the macroscopic or mesoscopic level in non-ideal environments poses fundamental theoretical challenges in domains ranging from inter and intra-cellular transport in biology to diffusion in porous media. Yet,…
We study a nonlinear pseudodifferential equation describing the dynamics of dislocations. The long time asymptotics of solutions is described by the self-similar profiles.
In this paper we introduce a new mathematical tool to solve fractional equations representing models of fractional systems : The Ultradistributions. Ultradistributions permit us to unify the notion of integral and derivative in one only…
This work presents a probabilistic scheme for solving semilinear nonlocal diffusion equations with volume constraints and integrable kernels. The nonlocal model of interest is defined by a time-dependent semilinear partial…
In this work, we consider the coupled systems of linear unsteady partial differential equations, which arise in the modeling of poroelasticity processes. Stability estimates of weighted difference schemes for the coupled system of equations…
In this paper we introduce a numerical method for nonlinear parabolic PDEs that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational…
In this article, we are concerned with the analysis on the numerical reconstruction of the spatial component in the source term of a time-fractional diffusion equation. This ill-posed problem is solved through a stabilized nonlinear…