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The solution of partial differential equations (PDEs) on complex domains often presents a significant computational challenge by requiring the generation of fitted meshes. The Diffuse Domain Method (DDM) is an alternative which reformulates…

Text-to-image generation models have achieved remarkable capabilities in synthesizing images, but often struggle to provide fine-grained control over the output. Existing guidance approaches, such as segmentation maps and depth maps,…

Effective hydrological modeling and extreme weather analysis demand precipitation data at a kilometer-scale resolution, which is significantly finer than the 10 km scale offered by standard global products like IMERG. To address this, we…

Recent advances in fast sampling methods for diffusion models have demonstrated significant potential to accelerate generation on image modalities. We apply these methods to 3-dimensional molecular conformations by building on the recently…

When applying the finite-differences method to numerically solve the one-dimensional diffusion equation, one must choose discretization steps $\Delta x$, $\Delta t$ in space and time, respectively. By applying large-deviation theory on the…

Generative diffusion models have achieved remarkable success in producing high-quality images. However, these models typically operate in continuous intensity spaces, diffusing independently across pixels and color channels. As a result,…

In this work, we study the gradient discretisation method (GDM) of the time-dependent Navier-Stokes equations coupled with the heat equation, where the viscosity depends on the temperature. We design the discrete method and prove its…

We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=f \quad\quad\text{in}\quad\quad…

This paper introduces the first gradient-based framework for prompt optimization in text-to-image diffusion models. We formulate prompt engineering as a discrete optimization problem over the language space. Two major challenges arise in…

In this paper, we present a new model to simulate the formation, evolution, and break up of a thin film of fluid flowing over a curved surface. Referred to as the discrete droplet method (DDM), the model captures the evolution of thin fluid…

Conventional diffusion models typically relies on a fixed forward process, which implicitly defines complex marginal distributions over latent variables. This can often complicate the reverse process' task in learning generative…

We present a finite element scheme for fractional diffusion problems with varying diffusivity and fractional order. We consider a symmetric integral form of these nonlocal equations defined on general geometries and in arbitrary bounded…

We present recent finite element numerical results on a model convection-diffusion problem in the singular perturbed case when the convection term dominates the problem. We compare the standard Galerkin discretization using the linear…

Diffusion Models (DMs), also referred to as score-based diffusion models, utilize neural networks to specify score functions. Unlike most other probabilistic models, DMs directly model the score functions, which makes them more flexible to…

We study the numerical approximation of advection-diffusion equations with highly oscillatory coefficients and possibly dominant advection terms by means of the Multiscale Finite Element Method. The latter method is a now classical, finite…

We propose a nonlinear Discrete Duality Finite Volume scheme to approximate the solutions of drift diffusion equations. The scheme is built to preserve at the discrete level even on severely distorted meshes the energy / energy dissipation…

A hydrogeological model for the spread of pollution in an aquifer is considered. The model consists in a convection-diffusion-reaction equation involving the dispersion tensor which depends nonlinearly of the fluid velocity. We introduce an…

Discrete diffusion models have emerged as a powerful generative modeling framework for discrete data with successful applications spanning from text generation to image synthesis. However, their deployment faces challenges due to the high…

The problem of deriving a gradient flow structure for the porous medium equation which is {\em thermodynamic}, in that it arises from the large deviations of some microscopic particle system, is studied. To this end, a rescaled zero-range…

Seismic imaging from sparsely acquired data faces challenges such as low image quality, discontinuities, and migration swing artifacts. Existing convolutional neural network (CNN)-based methods struggle with complex feature distributions…