# Finsler Flow Matching: Dynamics-Aware Geodesic Interpolation for Single-Snapshot Trajectory Inference

arXiv: https://arxiv.org/abs/2610.11318 · HTML: https://mlsanity.com/paper/2610.11318/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 8 of its 51 references that are in the map.
- First posted on arXiv 8 Oct 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 8 of 8)

- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Learning stochastic dynamics from snapshots through regularized unbalanced optimal transport](https://mlsanity.com/paper/2410.00844/) (2024)
- [Wasserstein Flow Matching: Generative modeling over families of distributions](https://mlsanity.com/paper/2411.00698/) (2024)
- [A Locally Adaptive Normal Distribution](https://mlsanity.com/paper/1606.02518/) (2016)
- [Curly Flow Matching for Learning Non-gradient Field Dynamics](https://mlsanity.com/paper/2510.26645/) (2025)
- [Learning Lineage-guided Geodesics with Finsler Geometry](https://mlsanity.com/paper/2603.16708/) (2026)

## Abstract (arXiv)

Single-cell snapshot data can resolve a continuum of cellular states but do not uniquely determine the dynamics governing transitions between them. However, additional dynamical information can often be encoded in a cell-cell Markov transition kernel. Existing generative approaches for single cell trajectory inference either infer transport only from population marginals, impose a symmetric geometry on the state space, or incorporate directionality through a single velocity vector at each observed state. We introduce Finsler Flow Matching (FFM), a framework for learning continuous stochastic dynamics from discrete Markov transition graphs. We use the first and second local moments to construct a Finsler structure motivated by the Freidlin--Wentzell action, where the second moment determines anisotropic accessibility and the first moment introduces a preferred direction of motion. We learn neural approximations of the resulting directed geodesics, use their Finsler cost to construct source-target couplings, and define geometry-aware stochastic conditional paths that can be distilled into a continuous generative process through simulation-free score and flow matching. Across synthetic and single-cell trajectory inference benchmarks, FFM improves recovery of withheld intermediate populations, particularly when the transition dynamics are strongly directional or anisotropic. Our results provide a principled route from discrete transition probabilities to continuous generative dynamics while retaining both directional and diffusive structure.