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arXiv · 2601.06810

WFR-FM: Simulation-Free Dynamic Unbalanced Optimal Transport

Qiangwei Peng·Zihan Wang·Junda Ying·Yuhao Sun·Qing Nie·Lei Zhang·Tiejun Li·Peijie Zhou
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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.

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Qiangwei Peng, Zihan Wang, Junda Ying, Yuhao Sun, Qing Nie, Lei Zhang, Tiejun Li, Peijie Zhou. 2026-04-02. WFR-FM: Simulation-Free Dynamic Unbalanced Optimal Transport. https://arxiv.org/abs/2601.06810

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