Related papers: Neural Galerkin Normalizing Flow for Transition Pr…
The Fokker-Planck equation describes the evolution of the probability density associated with a stochastic differential equation. As the dimension of the system grows, solving this partial differential equation (PDE) using conventional…
In this work, we propose adaptive deep learning approaches based on normalizing flows for solving fractional Fokker-Planck equations (FPEs). The solution of a FPE is a probability density function (PDF). Traditional mesh-based methods are…
In this work, we propose an adaptive learning approach based on temporal normalizing flows for solving time-dependent Fokker-Planck (TFP) equations. It is well known that solutions of such equations are probability density functions, and…
In this paper, we develop and analyze numerical methods for high dimensional Fokker-Planck equations by leveraging generative models from deep learning. Our starting point is a formulation of the Fokker-Planck equation as a system of…
We propose a high order discontinuous Galerkin (DG) method for solving nonlinear Fokker-Planck equations with a gradient flow structure. For some of these models it is known that the transient solutions converge to steady-states when time…
We investigate numerical behaviour of a convection diffusion equation with random coefficients by approximating statistical moments of the solution. Stochastic Galerkin approach, turning the original stochastic problem to a system of…
Mathematical models based on probability density functions (PDF) have been extensively used in hydrology and subsurface flow problems, to describe the uncertainty in porous media properties (e.g., permeability modelled as random field).…
Approximate weak solutions of the Fokker-Planck equation can represent a useful tool to analyze the equilibrium fluctuations of birth-death systems, as they provide a quantitative knowledge lying in between numerical simulations and exact…
In this paper, we propose a drift-diffusion process on the probability simplex to study stochastic fluctuations in probability spaces. We construct a counting process for linear detailed balanced chemical reactions with finite species such…
Normalizing flows are a popular class of models for approximating probability distributions. However, their invertible nature limits their ability to model target distributions whose support have a complex topological structure, such as…
The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability of N stochastic variables with Lochner's generalized Dirichlet distribution (R.H. Lochner, A Generalized…
This paper is concerned with structure-preserving numerical approximations for a class of nonlinear nonlocal Fokker-Planck equations, which admit a gradient flow structure and find application in diverse contexts. The solutions,…
Many systems in physics, engineering, and biology exhibit multiscale stochastic dynamics, where low-dimensional slow variables evolve under the influence of high-dimensional fast processes. In practice, observations are often limited to a…
The performance of flow matching and diffusion models can be greatly improved at inference time using reward alignment algorithms, yet efficiency remains a major limitation. While several algorithms were proposed, we demonstrate that a…
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…
The Regularised Inertial Dean-Kawasaki model (RIDK) -- introduced by the authors and J. Zimmer in earlier works -- is a nonlinear stochastic PDE capturing fluctuations around the mean-field limit for large-scale particle systems in both…
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…
In recent years, deep learning methods, exemplified by Physics-Informed Neural Networks (PINNs), have been widely applied to the numerical solution of differential equations. However, these methods may suffer from limited accuracy, high…
Diffusion models have exhibited excellent performance in various domains. The probability flow ordinary differential equation (ODE) of diffusion models (i.e., diffusion ODEs) is a particular case of continuous normalizing flows (CNFs),…
The increasing application of cardiorespiratory simulations for diagnosis and surgical planning necessitates the development of computational methods significantly faster than the current technology. To achieve this objective, we leverage…