WFR-FM: Simulation-Free Dynamic Unbalanced Optimal Transport
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arXiv
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| Autores principales: | , , , , , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918424679022592 |
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| author | Peng, Qiangwei Wang, Zihan Ying, Junda Sun, Yuhao Nie, Qing Zhang, Lei Li, Tiejun Zhou, Peijie |
| author_facet | Peng, Qiangwei Wang, Zihan Ying, Junda Sun, Yuhao Nie, Qing Zhang, Lei Li, Tiejun Zhou, Peijie |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_06810 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | WFR-FM: Simulation-Free Dynamic Unbalanced Optimal Transport Peng, Qiangwei Wang, Zihan Ying, Junda Sun, Yuhao Nie, Qing Zhang, Lei Li, Tiejun Zhou, Peijie Machine Learning Artificial Intelligence Mathematical Physics 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. |
| title | WFR-FM: Simulation-Free Dynamic Unbalanced Optimal Transport |
| topic | Machine Learning Artificial Intelligence Mathematical Physics |
| url | https://arxiv.org/abs/2601.06810 |