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

Niklas Canova; Jonas Simon Fleck

cs.AIOct 8, 2026

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

Authors: Niklas Canova, Jonas Simon Fleck

Organizations: Institute of Human Biology (IHB), Roche, Basel, Switzerland

Abstract

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.

Figures & tables

Figure 1 : Overview of Finsler Flow Matching (FFM). A discrete Markov transition graph is converted into a continuous directed Finsler geometry, which guides score and flow matching to define a stochastic generative model with geometry-aware diffusion.

Figure 2 : $W_{2,F}$ denotes the Wasserstein distance induced by the Finsler cost $c_{F}$ . Sampling floor is computed between disjoint halves of the corresponding reference cloud.

Figure 3 : Pancreas trajectory reconstruction. Left: Two-dimensional representation. Top right: $W_{2}$ at $t=1/2$ , the average $\overline{W}_{2}$ over 11 fixed evaluation times in $[1/3,2/3]$ , endpoint $W_{2}$ , and $W_{2,F}$ Bottom right: same-information objective ablation at $d=20$ . “Shared” fixes the endpoint coupling “method-specific” constructs the coupling separately for each method. Ablation values are $W_{2}(t=1/2)$ , mean $\pm$ population standard deviation over five seeds.

Method	$d=2$ mid $W_{2}$	$d=20$ mid $W_{2}$	$d=50$ mid $W_{2}$
OT-CFM	$1.3631\pm 0.0368$	$15.9054\pm 0.0804$	$18.8807\pm 0.1172$
MFM / LAND	$1.3563\pm 0.0537$	$15.0519\pm 0.2250$	$18.2036\pm 0.2663$
FFM, expression only	$1.7162\pm 0.9919$	$12.8597\pm 0.1007$	$\mathbf{16.6979\pm 0.3020}$
FFM, + velocity	$\mathbf{1.0204\pm 0.1040}$	$12.7288\pm 0.1253$	$16.7371\pm 0.2566$
FFM, + lineage	$1.3074\pm 0.1083$	$12.7368\pm 0.1455$	$16.7415\pm 0.2426$
FFM, + lineage + velocity	$1.1773\pm 0.1404$	$\mathbf{12.7121\pm 0.1314}$	$16.9196\pm 0.2700$

Table 1 : Intermediate-distribution recovery on iTracer R2 for different transition-kernel constructions. Lower is better.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 4 — Figure 4 : Sheet benchmark and additional results. (a) Overview of the synthetic Sheet benchmark and its directed transition structure. (b) Ablation of the directional and anisotropic components of the Finsler metric. Directionality substantially improves intermediate recovery, with the full metric obtaining the lowest midpoint errors. (c) Qualitative comparison of generated trajectories.

Method	$W_{2}(1/3)$	$W_{2}(0.433)$	$W_{2}(0.567)$	$W_{2}(2/3)$	$\overline{W}_{2}^{[1/3,2/3]}$
$d=2$
FFM, deterministic	1.9878 $\pm$ 0.0631	1.8184 $\pm$ 0.0326	1.4429 $\pm$ 0.0195	1.1770 $\pm$ 0.0919	1.6175 $\pm$ 0.0348
FFM, with noise	2.0052 $\pm$ 0.0705	1.8399 $\pm$ 0.0401	1.4723 $\pm$ 0.0601	1.2148 $\pm$ 0.0851	1.6434 $\pm$ 0.0555
OT-CFM	1.8508 $\pm$ 0.0172	1.7952 $\pm$ 0.0182	1.7607 $\pm$ 0.0195	1.7682 $\pm$ 0.0201	1.7869 $\pm$ 0.0230
MFM / LAND	1.5201 $\pm$ 0.0119	1.4878 $\pm$ 0.0117	1.4438 $\pm$ 0.0108	1.4002 $\pm$ 0.0098	1.4642 $\pm$ 0.0131
Curly-FM	1.0208 $\pm$ 0.1277	0.9952 $\pm$ 0.1378	0.9480 $\pm$ 0.1457	0.9009 $\pm$ 0.1505	0.9686 $\pm$ 0.1725

Appendix

Figure 5 : Pancreas robustness analyses. (a) Intermediate-population recovery over model time. We show representative points from the fixed 11-point grid over

[1/3,2/3]

together with

\overline{W}_{2}^{[1/3,2/3]}

, the mean over all 11 points. (b) Geometry-dependent Wasserstein distance across representations. (c) Sensitivity to the fraction of intermediate-band cells made available for metric construction. Because the scored population changes with the support fraction, comparisons across columns should be interpreted descriptively. Methods within each column use the same scored cells.

Width mult.	$M_{t}=C_{\rho}(\mu_{t})$ $W_{2}(1/2)$	$M_{t}=C_{\rho}(\mu_{t})$ $W_{2}(1)$	$M_{t}=I$ $W_{2}(1/2)$	$M_{t}=I$ $W_{2}(1)$
0.003	10.4874 $\pm$ 0.0419	5.2457 $\pm$ 0.2301	10.4414 $\pm$ 0.1439	5.2980 $\pm$ 0.1595
0.010	10.4940 $\pm$ 0.1123	5.5062 $\pm$ 0.4344	10.5187 $\pm$ 0.1591	5.2420 $\pm$ 0.2684
0.030	10.4873 $\pm$ 0.0919	5.3305 $\pm$ 0.1619	10.4245 $\pm$ 0.1359	5.3180 $\pm$ 0.1351
0.100	10.3694 $\pm$ 0.0987	5.0662 $\pm$ 0.3449	10.3446 $\pm$ 0.1299	5.1521 $\pm$ 0.1702
0.300	10.4630 $\pm$ 0.2142	5.3692 $\pm$ 0.1590	10.4149 $\pm$ 0.1922	5.2433 $\pm$ 0.2451
1.000	14.0089 $\pm$ 0.0636	7.6047 $\pm$ 0.3015	13.8565 $\pm$ 0.1498	7.3363 $\pm$ 0.1651

Appendix

Figure 6 : Pancreas robustness analyses (continued). (d) Sensitivity of stochastic FFM at

d=20

to conditional-path noise width and covariance. (e) Sensitivity to conditional-path and inference covariance. (f) Sensitivity to the scalar inference-noise schedule. Values are mean

\pm

population standard deviation over three seeds unless stated otherwise. Lower is better.

Method	$d=2$	$d=10$	$d=20$	$d=50$
OT-CFM	3	3	3	3
MFM / LAND	22	61	109	254
Curly-FM	41	106	190	445
FFM (ours), deterministic	26	63	109	256
FFM (ours), with noise	15	29	48	108

Appendix

Table 2: Wall-clock time in seconds per seed on a Tesla V100-SXM2-32GB on the Pancreas dataset. The stochastic FFM arm reuses the deterministic interpolant and coupling.

$d$	Varied parameter	Fixed parameter	Values	Validation objective
2	$\rho$	$\lambda=0.1$	$0.01,\ 0.03,\ \mathbf{0.1},\ 0.3,\ 1.0$	$0.828,\ 0.749,\ \mathbf{0.535},\ 0.870,\ 1.131$
2	$\lambda$	$\rho=0.1$	$0.01,\ 0.03,\ \mathbf{0.1},\ 0.3,\ 1.0$	$1.226,\ 0.951,\ \mathbf{0.535},\ 0.826,\ 0.873$
20	$\rho$	$\lambda=1.0$	$0.003,\ 0.01,\ \mathbf{0.03},\ 0.1,\ 0.3$	$8.180,\ 8.210,\ \mathbf{8.135},\ 8.369,\ 8.288$
20	$\lambda$	$\rho=0.03$	$0.03,\ 0.1,\ 0.3,\ \mathbf{1.0},\ 3.0$	$8.990,\ 8.535,\ 8.370,\ \mathbf{8.135},\ 8.667$
50	$\rho$	$\lambda=0.03$	$3{\times}10^{-4},\ 10^{-3},\ \mathbf{3{\times}10^{-3}},\ 10^{-2},\ 3{\times}10^{-2}$	$10.455,\ 10.421,\ \mathbf{10.314},\ 10.390,\ 10.589$
50	$\lambda$	$\rho=0.003$	$0.003,\ 0.01,\ \mathbf{0.03},\ 0.1,\ 0.3$	$10.379,\ 10.314,\ \mathbf{10.314},\ 10.554,\ 10.537$

Appendix

Table 3 : One-at-a-time sensitivity of FFM on Pancreas. The reported value is the validation selection objective; lower is better. The selected value is shown in bold.

Figure 7 : Visualization of the learned erythroid trajectories in 2D and quantitative comparison across dimensions $d=2,20,50$ . “Published” denotes the values reported in Table 4 of Petrović et al. [35] , while “Reproduced” denotes our reproduction of the corresponding baselines using a common training harness at the published training budget. Due to a bug in the original Curly-FM evaluation, the Wasserstein quantity reported as $W_{2}$ in Table 4 of CurlyFM corresponds in fact to $W_{1}$ . We therefore report these published values under $W_{1}$ and additionally compute the true $W_{2}$ distance for all runs evaluated with our harness. For $d=20$ and $d=50$ , $L_{2}$ is reported in units of $10^{3}$ . Bold values indicate the best performance among methods evaluated with our common harness; published values are shown for reference and are not included when determining the best result.

Figure 8 : Learned trajectories on iTracer in the two-dimensional representation.

Panel of Figure 9 — Figure 9 : ForkSheet benchmark.

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Single-cell RNA sequencing (scRNA-seq) provides high-dimensional profiles of cellular states, enabling data-driven modeling of cellular dynamics over time. In practice, time-resolved scRNA-seq is collected at only a few discrete time points as unpaired snapshot populations, leaving substantial temporal gaps. This motivates trajectory inference at unmeasured time points. Existing methods mainly follow two directions, optimal-transport (OT) alignment provides distribution-level matching between observed snapshots, while continuous-time generative models support forecasting via learned dynamics. However, two challenges remain: (i) unpaired snapshots render local transitions between adjacent time points ambiguous, leading to unstable supervision; and (ii) long-horizon prediction relies on repeated integration, where small modeling errors compound and cause distribution drift. To address these challenges, we propose single-cell Flow Matching (scFM), a latent generative framework based on coupling-conditioned flow matching. First, we compute entropically regularized OT couplings between adjacent snapshots and use them to construct soft, weighted flow-matching targets for learning time-dependent velocity fields. Second, we learn bidirectional velocity fields and leverage their consistency to refine couplings and improve temporal coherence under sparse supervision. Third, we introduce distribution-level alignment and latent dynamic regularization to anchor long rollouts and mitigate drift. Experiments on real-world time-series scRNA-seq datasets show that scFM consistently improves distributional prediction performance for both temporal interpolation and extrapolation. Moreover, scFM yields more accurate trajectory reconstruction and temporally coherent visualizations where intermediate time points are absent, indicating a more faithful recovery of underlying temporal gene expression dynamics.

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