相关论文: Projection based dimensionality reduction for meas…
In the present paper we discuss problems concerning evolutions of densities related to Ito diffusions in the framework of the statistical exponential manifold. We develop a rigorous approach to the problem, and we particularize it to the…
The projection filter is a technique for approximating the solutions of optimal filtering problems. In projection filters, the Kushner--Stratonovich stochastic partial differential equation that governs the propagation of the optimal…
We study optimal finite dimensional approximations of the generally infinite-dimensional Fokker-Planck-Kolmogorov (FPK) equation, finding the curve in a given finite-dimensional family that best approximates the exact solution evolution.…
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…
The Becker-D\"oring equations are an infinite dimensional system of ordinary differntial equations describing coagulation/fragmentation processes of species of integer sizes. Formal Taylor expansions motivate that its solution should be…
In this paper we study the dynamics of a fast-slow Fokker-Planck partial differential equation (PDE) viewed as the evolution equation for the density of a multiscale planar stochastic differential equation (SDE). Our key focus is on the…
In this paper we introduce a projection method for the space of probability distributions based on the differential geometric approach to statistics. This method is based on a direct L2 metric as opposed to the usual Hellinger distance and…
Dimensional reduction techniques have long been used to visualize the structure and geometry of high dimensional data. However, most widely used techniques are difficult to interpret due to nonlinearities and opaque optimization processes.…
We examine some differential geometric approaches to finding approximate solutions to the continuous time nonlinear filtering problem. Our primary focus is a new projection method for the optimal filter infinite dimensional Stochastic…
This work focuses on dimension-reduction techniques for modelling conditional extreme values. Specifically, we investigate the idea that extreme values of a response variable can be explained by nonlinear functions derived from linear…
Fourier-Motzkin elimination is a projection algorithm for solving finite linear programs. We extend Fourier-Motzkin elimination to semi-infinite linear programs which are linear programs with finitely many variables and infinitely many…
We present a new strategy to approximate the global solution of the Fokker-Planck equation efficiently in higher dimensions and show its convergence. The main ingredients are the Euler scheme to solve the associated stochastic differential…
Solving high-dimensional Fokker-Planck (FP) equations is a challenge in computational physics and stochastic dynamics, due to the curse of dimensionality (CoD) and unbounded domains. Existing deep learning approaches, such as…
We propose a novel projection-based particle method for solving the McKean-Vlasov stochastic differential equations. Our approach is based on a projection-type estimation of the marginal density of the solution in each time step. The…
Score-based diffusion models currently constitute the state of the art in continuous generative modeling. These methods are typically formulated via overdamped or underdamped Ornstein--Uhlenbeck-type stochastic differential equations, in…
We propose a novel method to solve a chemical diffusion master equation of birth and death type. This is an infinite system of Fokker-Planck equations where the different components are coupled by reaction dynamics similar in form to a…
We introduce a novel spatio-temporal discretization for nonlinear Fokker-Planck equations on the multi-dimensional unit cube. This discretization is based on two structural properties of these equations: the first is the representation as a…
For classification problems, feature extraction is a crucial process which aims to find a suitable data representation that increases the performance of the machine learning algorithm. According to the curse of dimensionality theorem, the…
In this paper, we extend the Weak Adversarial Neural Pushforward Method to the Fokker--Planck equation on compact embedded Riemannian manifolds. The method represents the solution as a probability distribution via a neural pushforward map…
We present the two new notions of projection of a stochastic differential equation (SDE) onto a submanifold, as developed in Armstrong, Brigo e Rossi Ferrucci (2019, 2018): the Ito-vector and Ito-jet projections. This allows one to…