22 citations · 50 across the 14 of their papers we have counts for
14 papers
A parameterized Wasserstein Hamiltonian flow approach for solving the Schrödinger equation
Hao Wu, Shu Liu, Xiaojing Ye +1
In this paper, we propose a new method to compute the solution of time-dependent Schrödinger equation (TDSE). Using push-forward maps and Wasserstein Hamiltonian flow, we reformula…
Parameterized Wasserstein Gradient Flow
Yijie Jin, Shu Liu, Hao Wu +2
We develop a fast and scalable numerical approach to solve Wasserstein gradient flows (WGFs), particularly suitable for high-dimensional cases. Our approach is to use general reduc…
A supervised learning scheme for computing Hamilton-Jacobi equation via density coupling
Jianbo Cui, Shu Liu, Haomin Zhou
We propose a supervised learning scheme for the first order Hamilton--Jacobi PDEs in high dimensions. The scheme is designed by using the geometric structure of Wasserstein Hamilto…
Parameterized Wasserstein Hamiltonian Flow
Hao Wu, Shu Liu, Xiaojing Ye +1
In this work, we propose a numerical method to compute the Wasserstein Hamiltonian flow (WHF), which is a Hamiltonian system on the probability density manifold. Many well-known PD…
Optimal control for stochastic nonlinear Schrodinger equation on graph
Jianbo Cui, Shu Liu, Haomin Zhou
We study the optimal control formulation for stochastic nonlinear Schrodinger equation (SNLSE) on a finite graph. By viewing the SNLSE as a stochastic Wasserstein Hamiltonian flow…
Wasserstein Hamiltonian flow with common noise on graph
Jianbo Cui, Shu Liu, Haomin Zhou
We study the Wasserstein Hamiltonian flow with a common noise on the density manifold of a finite graph. Under the framework of stochastic variational principle, we first develop t…
Stochastic Wasserstein Hamiltonian Flows
Jianbo Cui, Shu Liu, Haomin Zhou
In this paper, we study the stochastic Hamiltonian flow in Wasserstein manifold, the probability density space equipped with -Wasserstein metric tensor, via the Wong--Zakai ap…
Neural Monge Map estimation and its applications
Jiaojiao Fan, Shu Liu, Shaojun Ma +2
Monge map refers to the optimal transport map between two probability distributions and provides a principled approach to transform one distribution to another. Neural network base…
Approximating the Optimal Transport Plan via Particle-Evolving Method
Shu Liu, Haodong Sun, Hongyuan Zha
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 us…
Learning High Dimensional Wasserstein Geodesics
Shu Liu, Shaojun Ma, Yongxin Chen +2
We propose a new formulation and learning strategy for computing the Wasserstein geodesic between two probability distributions in high dimensions. By applying the method of Lagran…
What is a stochastic Hamiltonian process on finite graph? An optimal transport answer
Jianbo Cui, Shu liu, Haomin Zhou
We present a definition of stochastic Hamiltonian process on finite graph via its corresponding density dynamics in Wasserstein manifold. We demonstrate the existence of stochastic…
Neural Parametric Fokker-Planck Equations
Shu Liu, Wuchen Li, Hongyuan Zha +1
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 for…
Learning Stochastic Behaviour from Aggregate Data
Shaojun Ma, Shu Liu, Hongyuan Zha +1
Learning nonlinear dynamics from aggregate data is a challenging problem because the full trajectory of each individual is not available, namely, the individual observed at one tim…
Parametric Fokker-Planck equation
Wuchen Li, Shu Liu, Hongyuan Zha +1
We derive the Fokker-Planck equation on the parametric space. It is the Wasserstein gradient flow of relative entropy on the statistical manifold. We pull back the PDE to a finite…