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The least-squares ReLU neural network (LSNN) method was introduced and studied for solving linear advection-reaction equation with discontinuous solution in \cite{Cai2021linear,cai2023least}. The method is based on an equivalent…
The reaction-diffusion equation is one of the cornerstones equations in applied science and engineering. In the present study, a deep neural network has been trained in order to predict the solution of the equation with different…
A method is developed within an adaptive framework to solve quasilinear diffusion problems with internal and possibly boundary layers starting from a coarse mesh. The solution process is assumed to start on a mesh where the problem is badly…
Approximation of scattered data is often a task in many engineering problems. The Radial Basis Function (RBF) approximation is appropriate for large scattered (unordered) datasets in d-dimensional space. This approach is useful for a higher…
Diffuse domain methods (DDMs) have gained significant attention for solving partial differential equations (PDEs) on complex geometries. These methods approximate the domain by replacing sharp boundaries with a diffuse layer of thickness…
We present two new quantum algorithms for reaction-diffusion equations that employ the truncated Chebyshev polynomial approximation. This method is employed to numerically solve the ordinary differential equation emerging from the…
Deep learning has shown great potential in accelerating diffusion tensor imaging (DTI). Nevertheless, existing methods tend to suffer from Rician noise and eddy current, leading to detail loss in reconstructing the DTI-derived parametric…
In this paper, we study the convergence behavior of the diffuse domain method (DDM) for solving a class of second-order parabolic partial differential equations with Neumann boundary condition posed on general irregular domains. The DDM…
Different relaxation approximations to partial differential equations, including conservation laws, Hamilton-Jacobi equations, convection-diffusion problems, gas dynamics problems, have been recently proposed. The present paper focuses onto…
Cataloging the complex behaviors of dynamical systems can be challenging, even when they are well-described by a simple mechanistic model. If such a system is of limited analytical tractability, brute force simulation is often the only…
The present article proposes a partitioned Dirichlet-Neumann algorithm, that allows to address unique challenges arising from a novel mixed-dimensional coupling of very slender fibers embedded in fluid flow using a regularized mortar-type…
In this work, we consider the numerical solution of an initial boundary value problem for the distributed order time fractional diffusion equation. The model arises in the mathematical modeling of ultra-slow diffusion processes observed in…
A distributed adaptive algorithm is proposed to solve a node-specific parameter estimation problem where nodes are interested in estimating parameters of local interest, parameters of common interest to a subset of nodes and parameters of…
Diffusion models have achieved state-of-the-art performance across multiple domains, with recent advancements extending their applicability to discrete data. However, aligning discrete diffusion models with task-specific preferences remains…
Rendering highly scattering participating media using brute force path tracing is a challenge. The diffusion approximation reduces the problem to solving a simple linear partial differential equation. Flux-limited diffusion introduces…
Despite the proliferation of generative models, achieving fast sampling during inference without compromising sample diversity and quality remains challenging. Existing models such as Denoising Diffusion Probabilistic Models (DDPM) deliver…
Optical Diffraction Neural Networks (DNNs), a subset of Optical Neural Networks (ONNs), show promise in mirroring the prowess of electronic networks. This study introduces the Hybrid Diffraction Neural Network (HDNN), a novel architecture…
In this work, we focus on the alignment problem of diffusion models with a continuous reward function, which represents specific objectives for downstream tasks, such as increasing darkness or improving the aesthetics of images. The central…
The gradient discretisation method (GDM) is a generic framework for designing and analysing numerical schemes for diffusion models. In this paper, we study the GDM for the porous medium equation, including fast diffusion and slow diffusion…
It is well known that Tikhonov regularization is one of the most commonly used methods for solving ill-posed problems. One of the most widely applied approaches is based on constructing a new dataset whose sample size is greater than the…