WFR-FM: Simulation-Free Dynamic Unbalanced Optimal Transport

TL;DR

WFR-FM is a novel, simulation-free algorithm for unbalanced optimal transport that improves stability, efficiency, and accuracy in modeling dynamic systems with changing mass, with applications in biology and generative modeling.

Contribution

It introduces a unified flow matching approach for WFR geometry that simultaneously regresses displacement and growth, enabling stable and scalable unbalanced OT solutions.

Findings

WFR-FM accurately recovers WFR geodesics.

It outperforms baselines in stability and efficiency.

It effectively models biological dynamics with proliferation and apoptosis.

Abstract

The Wasserstein-Fisher-Rao (WFR) metric extends dynamic optimal transport (OT) by coupling displacement with change of mass, providing a principled geometry for modeling unbalanced snapshot dynamics. Existing WFR solvers, however, are often unstable, computationally expensive, and difficult to scale. Here we introduce WFR Flow Matching (WFR-FM), a simulation-free training algorithm that unifies flow matching with dynamic unbalanced OT. Unlike classical flow matching which regresses only a transport vector field, WFR-FM simultaneously regresses a vector field for displacement and a scalar growth rate function for birth-death dynamics, yielding continuous flows under the WFR geometry. Theoretically, we show that minimizing the WFR-FM loss exactly recovers WFR geodesics. Empirically, WFR-FM yields more accurate and robust trajectory inference in single-cell biology, reconstructing consistent dynamics with proliferation and apoptosis, estimating time-varying growth fields, and applying to generative dynamics under imbalanced data. It outperforms state-of-the-art baselines in efficiency, stability, and reconstruction accuracy. Overall, WFR-FM establishes a unified and efficient paradigm for learning dynamical systems from unbalanced snapshots, where not only states but also mass evolve over time. The Python code is available at https://github.com/QiangweiPeng/WFR-FM.