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

Niklas Canova, Jonas Simon Fleck

arXiv:2610.11318v1cs.AI

TL;DR

Single-cell snapshots resolve cellular states but do not uniquely reveal the dynamics connecting them, while transition kernels encode additional dynamical information without specifying continuous evolution. FFM derives directed geometry from transition-graph moments and uses it to learn continuous stochastic transport. Across synthetic and single-cell benchmarks, it improves recovery of withheld intermediate populations, particularly for strongly directional or anisotropic dynamics.

  • Problem

    Single-cell snapshots and transition kernels do not by themselves specify continuous dynamics between or beyond observed cells.

    Source: §1 · pp. 1–2

  • Method

    FFM uses first and second local transition moments to define a Freidlin–Wentzell-motivated Finsler geometry, then learns geodesic couplings and stochastic conditional paths through simulation-free score and flow matching.

    Source: §1 · p. 2, §4 · p. 4

  • Results

    FFM improves recovery of withheld intermediate populations across synthetic and single-cell benchmarks, particularly for strongly directional or anisotropic dynamics.

    Source: §1 · p. 2, §6 · p. 8

  • Takeaways & Limitations

    The results support using discrete transition probabilities to construct continuous generative dynamics that retain directional and diffusive structure.

    Source: §6 · p. 8

    Interpreting the graph-derived moments as drift and diffusion requires assuming the Markov chain discretizes an Itô diffusion.

    Source: §4 · p. 5

Abstract

from arXiv · show

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.

1 Introduction

Single-cell snapshots reveal cellular-state continua but cannot directly determine the continuous dynamics connecting states. FFM uses transition-graph moments to build directed geometry and learn continuous stochastic transport, improving intermediate-state prediction across benchmarks.

1 Introduction

  • Because sequencing is destructive, cells cannot be followed through time, so snapshots alone generally do not determine continuous state-change dynamics.Single-cell genomics resolves a continuum of states in a cross-sectional experiment.Source: §1 · p. 1
  • Markov kernels encode transition likelihoods between observed states but leave continuous evolution between or beyond those cells unspecified.They can combine dynamical information from distinct experimental and computational sources.Source: §1 · pp. 1–2
  • FFM derives a Randers geometry from first and second local transition moments, jointly representing directional preference and anisotropic accessibility.The construction is motivated by the Freidlin–Wentzell minimum-action principle.Source: §1 · p. 2
  • FFM learns neural approximations of directed geodesics, uses their costs for source–target couplings, and distills geometry-aware stochastic paths through simulation-free score and flow matching.The stochastic paths use local noise structure shaped by graph-derived diffusion.Source: §1 · p. 2
  • FFM improves intermediate-state prediction across benchmarks while incorporating transition information from RNA velocity, lineage tracing, labeling, or temporally sampled populations.The paper presents cell-cell Markov graphs as a common representation for these information sources.Source: §1 · p. 2

2 Preliminaries

The preliminaries introduce direction-dependent Finsler costs, transport between endpoint distributions, and simulation-free flow and score matching. They also connect small-noise stochastic bridges to Freidlin–Wentzell most-probable paths.

Finsler Geometry

  • A Randers structure combines a Riemannian tensor G(x) with a directional 1-form β(x), allowing a direction-dependent, asymmetric cost.Finsler geodesics minimize path length and provide locally shortest connections between endpoints.Source: §2 · pp. 2–3

Optimal Transport and Schrödinger Bridge Problem

  • Optimal transport finds a least-cost coupling between source and target distributions, while the Schrödinger bridge seeks a closest-to-reference path measure with those endpoint marginals.Using squared Finsler geodesic distance as the transport cost links path energy minimization to endpoint coupling.Source: §2 · p. 3

Schrödinger Bridge Problem

  • For non-Brownian references with anisotropic diffusion and nontrivial drift, conditional bridges are generally neither Gaussian nor available in closed form.In the small-noise regime, their most-probable paths are characterized by Freidlin–Wentzell action minimizers.Source: §2 · p. 3

Score and Flow Matching

  • Conditional flow matching trains on endpoint-pair-conditioned paths and vector fields, avoiding direct evaluation of the intractable marginal vector field.The learned flow and score can define a stochastic generative process with a matrix-valued diffusion schedule.Source: §2 · p. 3

3 Related Work

Related work spans trajectory inference from population marginals, graph-based fate inference, and geometric or stochastic flow matching. FFM distinguishes itself by deriving directed geometry and anisotropic noise structure from transition statistics.

Single-cell trajectory inference

  • FFM uses cell-cell transition statistics to extend trajectory inference beyond finite observed-cell graphs and learn continuous dynamics.Earlier approaches include multi-time-point optimal transport and manifold-aware optimal transport with neural differential equations.Source: §3 · p. 4

Stochastic generative transport

  • FFM uses transition-derived geometry to determine both interpolating paths and local anisotropic noise within stochastic generative transport.Related approaches include simulation-free Schrödinger bridge methods and closed-form or smooth Schrödinger bridges.Source: §3 · p. 4

Geometric and directional flow matching

  • Unlike Riemannian geodesic flow methods, FFM derives directed Randers geometry from local transition statistics, including first-moment directionality and second-moment anisotropy.Bamberger et al. use local kernels for anisotropic covariance, while Petrović et al. [35] use local velocity fields for directionality.Source: §3 · p. 4

Lineage-informed Finsler geometry

  • Unlike lineage-tree Finsler priors that penalize motion inconsistent with prescribed cell-type adjacency, FFM derives directional preference and accessibility from cell-cell transition probabilities.The comparison is with Zweig et al. [51].Source: §3 · p. 4

4 Finsler Flow Matching

FFM turns a row-stochastic transition graph into a directed Finsler geometry, then uses learned geodesics and geometry-aware stochastic paths to model continuous transport. Its diffusion interpretation is conditional on the graph arising from a discretized Itô diffusion.

Finsler geometry from transition moments

  • Under a short-time Itô diffusion model, local transition moments estimate drift and diffusion; the resulting Randers geometry combines directional preference with anisotropic accessibility.At finite graph scale, the second moment is interpreted as transition accessibility rather than diffusion covariance.Source: §4 · p. 4

Geodesic properties

  • A transition-supported corridor can contain every minimizing Finsler geodesic when leaving and returning costs more, and the metric locally favors the first transition moment.This gives a conditional locality property for paths consistent with the transition structure.Source: §4 · p. 5
  • When the transition graph discretizes an SDE, empirical Freidlin–Wentzell action minimizers converge to the asymptotically most probable transition path in the small-noise limit.The paper relates this result to Freidlin and Wentzell and Heymann and Vanden-Eijnden.Source: §4 · p. 5

Scope and assumptions

  • FFM accepts arbitrary row-stochastic transition matrices, but interpreting its graph-derived moments as drift and diffusion requires the discretized-diffusion assumption.This distinguishes the general method from its diffusion interpretation.Source: §4 · p. 5

Learning Geodesics

  • A neural interpolant parameterizes endpoint-constrained curves, and minimizing Finsler energy over it approximates pairwise transition-derived geodesics.The boundary correction enforces the source and target endpoints; energy minimizers are constant-speed geodesics.Source: §4 · p. 5

Finsler Optimal Transport

  • Learned Finsler geodesic costs approximate the transport cost and are used in an entropy-regularized static optimal transport problem to obtain a source–target coupling.The coupling and learned interpolants induce intermediate distributions with the source and target as endpoint marginals.Source: §4 · pp. 5–6

Stochastic conditional paths

  • Conditional paths approximate bridges with anisotropic Gaussian noise evaluated along the deterministic interpolant, injecting noise along directions accessible under the transition graph.The paths are distilled into marginal flow and score models through simulation-free conditional matching.Source: §4 · p. 6

Inference

  • At inference, the learned dynamics are integrated from source samples, with the stochastic process using graph-derived local geometry to shape its noise.The scalar noise schedule controls overall strength, while the local covariance determines anisotropic geometry.Source: §4 · p. 6

5 Experiments

Across synthetic and single-cell benchmarks, FFM recovers withheld intermediate states, with performance varying by dataset representation. The experiments evaluate midpoint recovery alongside endpoint and Finsler-cost distances.

5 Experiments

  • The primary metric is Euclidean W2 at t = 1/2, supplemented by endpoint W2 and the geometry-dependent W2,F; transition graphs are recomputed after evaluation cells are removed.Comparisons include OT-CFM, MFM/LAND, and Curly-FM under matched architectures and training budgets.Source: §5 · p. 6
  • W2 = 0.1588 and 0.1449 for deterministic and stochastic FFM, respectively, versus 0.5221 for Curly-FM on the synthetic Sheet benchmark.Ablations indicate that both directional and anisotropic components contribute to the improvement.Source: §5 · p. 7, Figure 2 · p. 7
  • In Pancreas, Curly-FM has lower midpoint W2 in two dimensions, while FFM has substantially lower midpoint W2 in 20- and 50-dimensional PCA representations.The benchmark withholds the intermediate population, enriched for Ngn3-high pre-endocrine cells.Source: §5 · p. 7, Figure 3 · p. 8
  • On iTracer, every FFM kernel variant has lower mean midpoint W2 than OT-CFM and MFM across evaluated representations, while the best kernel varies with dimensionality.Velocity performs best in two dimensions, lineage plus velocity in 20 dimensions, and expression alone in 50 dimensions.Source: §5 · p. 7, Table 1 · p. 7

6 Discussion

The discussion presents FFM as a transition-graph-derived geometry for learning continuous stochastic dynamics. Its usefulness depends on the supplied transition kernel, and its geodesic interpretation has explicit assumptions.

6 Discussion

  • FFM combines first- and second-moment information to guide coupling, geodesic interpolation, and stochastic conditional paths, with particular usefulness for strongly directional or anisotropic dynamics.Experiments indicate that additional modalities do not uniformly improve recovery.Source: §6 · p. 8
  • FFM inherits the information and limitations of its transition kernel and does not resolve the nonidentifiability of cellular dynamics from snapshots alone.Interpreting geodesics as most-probable transition curves requires diffusion-consistency and small-noise assumptions; Gaussian conditional paths are tractable approximations, not exact bridges.Source: §6 · p. 8
  • The discussion identifies uncertainty-aware transition graphs, richer conditional path families, and covariate- or perturbation-conditioned geometry as directions for future work.Source: §6 · p. 8

A Prelimiaries

The preliminaries introduce Finsler geometry and transport, then describe flow matching and its stochastic score-based extension. These pieces provide the geometric cost and conditional-path framework used to learn dynamics.

Finsler Geometry

  • A Finsler structure assigns a direction-dependent cost, so travel in v and −v can differ; geodesics minimize curve length between endpoints.A Riemannian metric is the symmetric special case.Source: A Prelimiaries · p. 14
  • The P-derived Randers metric constructs its tensor and one-form from local transition statistics, while the geometry is defined on Rd rather than an estimated data manifold.The construction makes motion along data-supported directions low-cost and weakly supported normal motion high-cost.Source: A Prelimiaries · p. 14

Optimal Transport

  • Optimal transport relaxes deterministic maps to transport plans, and its dynamic formulation connects endpoint distributions through density paths and velocity fields.The Finsler geodesic action can replace Euclidean cost in the static and dynamic formulations.Source: A Prelimiaries · p. 15

Flow Matching

  • Flow matching learns a vector field for a chosen probability path, since endpoint distributions alone do not determine the intermediate path.Conditioning on endpoint samples gives tractable conditional paths, allowing simulation-free training through a matching objective with the same gradient as its unconditional counterpart.Source: A Prelimiaries · pp. 15–16

Simulation-free Score and Flow Matching

  • Score and flow matching uses conditional paths and scores to construct a stochastic process from a deterministic flow.The process uses diffusion strength and Brownian noise.Source: A Prelimiaries · p. 16
  • The resulting ODE and SDE share the prescribed marginals, so score-based stochasticity can be added without changing the probability path.Source: A Prelimiaries · p. 16

B Theoretical Contributions

This section establishes that transition moments induce a directed, confined Finsler geometry and that, under stated assumptions, empirical geodesics converge to Freidlin–Wentzell transition curves. It also establishes a Gaussian conditional flow and a fixed-ridge consistency result.

Transition-supported geodesics

  • Large transition-weighted covariance makes directions relatively cheap, while weak transition support raises escape costs and confines minimizing geodesics near transition-supported paths.Annuli farther from the transition-supported set have larger escape cost.Source: B Theoretical Contributions · p. 17

Confinement and directional preference of geodesics

  • At fixed local speed measured by the inverse covariance, the Finsler cost is minimized when velocity aligns positively with the first transition moment.Source: B Theoretical Contributions · p. 17

Uniform Convergence of Empirical Drift and Diffusion

  • Under discrete-generator consistency, transition-kernel moments recover drift and diffusion uniformly on compact regions.For cosine RNA-velocity kernels with a first-order limit, the first moment recovers the velocity direction while covariance is interpreted as finite-scale anisotropic accessibility, not diffusion.Source: B Theoretical Contributions · pp. 18–19

Extension of empirical drift and diffusion

  • Kernel-extended empirical drift and diffusion fields are smooth, the diffusion estimate remains positive semidefinite, and both fields converge uniformly under the stated consistency and bandwidth conditions.The convergence follows from nodewise consistency and shrinking kernel bandwidth.Source: B Theoretical Contributions · p. 20

Freidlin–Wentzell Theory

  • Under the stated uniqueness, compactness, and convergence hypotheses, conditioning on a transition near its optimal time concentrates small-noise paths in arbitrarily narrow geometric tubes around the minimizing curve.The condition is reaching a small neighborhood of the target within a time frame close to the optimal transition time.Source: B Theoretical Contributions · p. 21

Empirical Finsler minimizers converge to geometric FW transition curves

  • Under compact-path, consistency, vanishing-regularization, and uniqueness assumptions, empirical Finsler minimizers converge to population geometric Freidlin–Wentzell minimizers, which are asymptotically most probable small-noise paths when globally minimizing.The small-noise interpretation also requires the hypotheses for conditional concentration around the population minimizer.Source: B Theoretical Contributions · pp. 21–22

Theoretical Contributions

  • The regularized empirical Finsler minimizer converges to the unique Freidlin–Wentzell minimizer, which under [20, Proposition 2.3] is the asymptotically most probable small-noise transition curve.The result assumes the minimizer is also the global geometric Freidlin–Wentzell minimizer and the hypotheses of [20, Proposition 2.3] hold.Source: B Theoretical Contributions · p. 23
  • For fixed positive ridge and regularization parameters, empirical regularized actions converge uniformly to the population regularized action without requiring the original diffusion tensor to be uniformly positive definite.Uniform positive definiteness is still required to pass to the unregularized limit and recover the original diffusion’s geometric Freidlin–Wentzell action.Source: B Theoretical Contributions · p. 23
  • Raw and diffusion-normalized moments produce the same metric after rescaling regularization parameters, so implementation absorbs time-step normalization into those hyperparameters.In raw-moment notation, the consistency assumptions require ρ_N/Δt_N → 0 and λ_N/√Δt_N → 0.Source: B Theoretical Contributions · p. 23
  • The Gaussian probability-path construction gives a vector field whose flow has the prescribed Gaussian marginals for every t in (0, 1).The path uses mean μ_t and covariance t(1 − t)G_t, with differentiable mean and symmetric positive-definite G_t.Source: B Theoretical Contributions · p. 24

C Experiments

Experiments assess FFM on synthetic Sheet and bifurcation benchmarks and Pancreas and Erythroid single-cell data, including withheld-intermediate recovery and robustness analyses.

Experimental protocol

  • Models share four 64-wide hidden layers, SiLU activations, and a 32-dimensional sinusoidal time embedding; results use three seeds except for Erythroid, which uses five.Hyperparameters were selected on a 15% validation split using the mean of mid- and end-time W2.Source: C Experiments · p. 24

Sheet

  • On the Sheet benchmark, directionality substantially improves intermediate recovery, and the full Finsler metric achieves the lowest midpoint errors.The benchmark uses a directed transition kernel that drives motion toward a middle corridor before crossing and away afterward.Source: Figure 4 · p. 25
  • The Sheet benchmark samples 4,000 points on a noisy saddle and uses a directed kernel to drive transitions toward and then away from the central corridor.Source and target states are defined by w ≤ −0.13 and w ≥ 0.13, respectively.Source: C Experiments · p. 25

Erythroid

  • The Erythroid evaluation uses 9,815 cells from a mouse developmental trajectory partitioned into three temporal snapshots, with an intermediate population withheld during training.The evaluation includes a two-dimensional embedding and 20- and 50-dimensional representations.Source: C Experiments · pp. 25, 27

Pancreas

  • Pancreas analyses assess recovery over an 11-point time grid, geometry-dependent Wasserstein distance across representations, and sensitivity to intermediate support for metric construction.Comparisons across support-fraction columns are descriptive because the scored population changes; methods within each column use the same scored cells.Source: C Experiments · pp. 25–26

Bifurcation

  • FFM maintains high recall on both ForkSheet fate arms, while OT-CFM and Curly-FM misassign lower-arm cells and MFM loses upper-arm recall mainly through outside outcomes.A separate allocation sweep found that performance under imbalance remained configuration-dependent.Source: C Experiments · p. 27, Table 4 · p. 28, Table 5 · p. 28

D Related Work

FFM generalizes symmetric Riemannian geodesic matching with a transition-derived directed geometry. Its connections to LAND and Curly-FM clarify how graph-derived geometry relates to existing metrics and vector-field objectives.

D Related Work

  • Metric Flow Matching (MFM) is FFM’s symmetric special case, using Riemannian geodesics without directional information.FFM is described as a generalization of MFM when a transition probability matrix supplies directionality.Source: D Related Work · pp. 28–29
  • For approximately constant Z_i, choosing ε = ρZ recovers the LAND metric [2] up to a global factor that leaves its geodesic curves unchanged.Source: D Related Work · pp. 28–29
  • Curly-FM’s Euclidean velocity residual is replaced by a graph-derived anisotropic residual, but full FFM attains lower Pancreas W2(1/2) under shared and method-specific couplings.The matched control has access to the same first and second transition moments, so the improvement is not explained solely by second-moment information.Source: D Related Work · p. 29
  • With dense, locally isotropic neighborhoods, cosine velocity kernels recover vector-field orientation; the second moment captures accessible-transition structure and can denoise noisy vector fields.Orthogonal displacement contributions cancel in pairs under local isotropy.Source: D Related Work · p. 29

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