A Neural Network Framework for High-Dimensional Dynamic Unbalanced Optimal Transport

TL;DR

This paper presents a neural network framework for solving high-dimensional dynamic unbalanced optimal transport problems efficiently, using relaxation, discretization, and neural modeling techniques, with promising experimental results and potential for broader applications.

Contribution

It introduces a novel neural network-based approach for high-dimensional dynamic unbalanced optimal transport, incorporating relaxation, discretization, and neural modeling for velocity and source functions.

Findings

Effective in high-dimensional scenarios

Outperforms existing methods in experiments

Extensible to applications like crowd motion

Abstract

In this paper, we introduce a neural network-based method to address the high-dimensional dynamic unbalanced optimal transport (UOT) problem. Dynamic UOT focuses on the optimal transportation between two densities with unequal total mass, however, it introduces additional complexities compared to the traditional dynamic optimal transport (OT) problem. To efficiently solve the dynamic UOT problem in high-dimensional space, we first relax the original problem by using the generalized Kullback-Leibler (GKL) divergence to constrain the terminal density. Next, we adopt the Lagrangian discretization to address the unbalanced continuity equation and apply the Monte Carlo method to approximate the high-dimensional spatial integrals. Moreover, a carefully designed neural network is introduced for modeling the velocity field and source function. Numerous experiments demonstrate that the proposed framework performs excellently in high-dimensional cases. Additionally, this method can be easily extended to more general applications, such as crowd motion problem.