相关论文: A parameterized Wasserstein Hamiltonian flow appro…

We consider the fully-coupled McKean-Vlasov equation with multi-time-scale potentials, and all the coefficients depend on the distributions of both the slow component and the fast motion. By studying the smoothness of the solution of the…

Wasserstein gradient flow has emerged as a promising approach to solve optimization problems over the space of probability distributions. A recent trend is to use the well-known JKO scheme in combination with input convex neural networks to…

Using a continuous unitary transformation recently proposed by Wegner \cite{Wegner} together with an approximation that neglects irrelevant contributions, we obtain flow equations for Hamiltonians. These flow equations yield a diagonal or…

The Wasserstein metric has become increasingly important in many machine learning applications such as generative modeling, image retrieval and domain adaptation. Despite its appeal, it is often too costly to compute. This has motivated…

The goal of this thesis is the development and implementation of a non-perturbative solution method for Wegner's flow equations. We show that a parameterization of the flowing Hamiltonian in terms of a scalar function allows the flow…

We introduce an ordinary differential equation (ODE) based deep generative method for learning conditional distributions, named Conditional F\"ollmer Flow. Starting from a standard Gaussian distribution, the proposed flow could approximate…

The sliced-Wasserstein flow is an evolution equation where a probability density evolves in time, advected by a velocity field computed as the average among directions in the unit sphere of the optimal transport displacements from its 1D…

We present a theoretical method to generate a highly accurate {\em time-independent} Hamiltonian governing the finite-time behavior of a time-periodic system. The method exploits infinitesimal unitary transformation steps, from which…

In this paper, we address the problem of optimizing flows on generalized graphs that feature multiple entry points and multiple populations, each with varying cost structures. We tackle this problem by considering the multi-population…

The generative paradigm has become increasingly important in machine learning and deep learning models. Among popular generative models are normalizing flows, which enable exact likelihood estimation by transforming a base distribution…

Optimal transport (OT) provides powerful tools for comparing probability measures in various types. The Wasserstein distance which arises naturally from the idea of OT is widely used in many machine learning applications. Unfortunately,…

Many numerical and learning algorithms rely on the solution of the Monge-Kantorovich problem and Wasserstein distances, which provide appropriate distributional metrics. While the natural approach is to treat the problem as an…

Hamiltonian simulation is a fundamental algorithm in quantum computing that has attracted considerable interest owing to its potential to efficiently solve the governing equations of large-scale classical systems. Exponential speedup…

We present a new pathwise approximation scheme for stochastic differential equations driven by multidimensional Brownian motion which does not require the simulation of L\'{e}vy area and has a Wasserstein convergence rate better than the…

Generative models based on flow matching have attracted significant attention for their simplicity and superior performance in high-resolution image synthesis. By leveraging the instantaneous change-of-variables formula, one can directly…

In 1966, Edward Nelson presented an interesting derivation of the Schrodinger equation using Brownian motion. Recently, this derivation is linked to the theory of optimal transport, which shows that the Schrodinger equation is a Hamiltonian…

The higher-order nonlinear Schrodinger equation (Dysthe's equation in the context of water-waves) models the time evolution of the slowly modulated amplitude of a wave-packet in dispersive partial differential equations (PDE). These…

We introduce a time discretization for Wasserstein gradient flows based on the classical Backward Differentiation Formula of order two. The main building block of the scheme is the notion of geodesic extrapolation in the Wasserstein space,…

We introduce the Wasserstein Transform (WT), a general unsupervised framework for updating distance structures on given data sets with the purpose of enhancing features and denoising. Our framework represents each data point by a…

The Wasserstein distance received a lot of attention recently in the community of machine learning, especially for its principled way of comparing distributions. It has found numerous applications in several hard problems, such as domain…