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

# Finsler Flow Matching:Dynamics-Aware Geodesic Interpolation for Single-Snapshot Trajectory Inference
Source: [https://arxiv.org/html/2610.11318](https://arxiv.org/html/2610.11318)
Jonas Simon FleckAffiliation:Institute of Human Biology \(IHB\), Roche, Basel, SwitzerlandEmail:[niklas\.canova@roche\.com](mailto:)

###### 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\.

††footnotetext:Code:[https://github\.com/mlbelab/FinslerFlowMatching](https://github.com/mlbelab/FinslerFlowMatching)![Refer to caption](https://arxiv.org/html/2610.11318v1/FFM_schematic.png)

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\.## 1 Introduction

Understanding how cells change their state in response to intrinsic programs and extrinsic perturbations is a central challenge in biology\. Such transitions underlie development and differentiation, but also cellular reprogramming, disease progression, and responses to therapeutic intervention\. Single\-cell genomics methods have made it possible to probe these processes at high resolution by resolving a continuum of cellular states in a single cross\-sectional experiment\[[44](https://arxiv.org/html/2610.11318#bib.bib35)\]\. However, because single\-cell sequencing methods are destructive, individual cells cannot be followed through time and transitions between observed states are not directly measured\. As a result, the continuous dynamics underlying cell state changes can generally not be determined from these measurements alone\[[49](https://arxiv.org/html/2610.11318#bib.bib34)\]\.

Several experimental and computational strategies can provide additional information about developmental directionality and dynamics\. RNA velocity infers the direction of local transcriptional change\[[25](https://arxiv.org/html/2610.11318#bib.bib21),[7](https://arxiv.org/html/2610.11318#bib.bib46)\], lineage tracing tracks ancestry\[[48](https://arxiv.org/html/2610.11318#bib.bib33),[46](https://arxiv.org/html/2610.11318#bib.bib31),[31](https://arxiv.org/html/2610.11318#bib.bib32)\]and metabolic or fluorescent labelling can introduce an absolute timescale\[[6](https://arxiv.org/html/2610.11318#bib.bib29),[11](https://arxiv.org/html/2610.11318#bib.bib30),[17](https://arxiv.org/html/2610.11318#bib.bib28)\]\. Measurements collected across multiple experimental time points also carry temporal information, although the correspondence between cells at successive time points is lost upon sampling and must itself be inferred\[[38](https://arxiv.org/html/2610.11318#bib.bib27)\]\. Despite their different origins, these sources of dynamical information can often be translated into a common representation, a cell\-cell Markov transition kernel, whose entries quantify the likelihood of transitions between observed cellular states\[[26](https://arxiv.org/html/2610.11318#bib.bib49),[47](https://arxiv.org/html/2610.11318#bib.bib26)\]\. Such kernels have proven highly effective for inferring terminal states, lineage relationships, and fate probabilities\[[14](https://arxiv.org/html/2610.11318#bib.bib25),[27](https://arxiv.org/html/2610.11318#bib.bib42)\]\. However, the resulting dynamics remain defined on the finite graph of observed cells\. A Markov kernel encodes transition preferences between observed states, but this does not specify how the system evolves continuously through state space, particularly between or beyond the observed cells\.

Generative modeling of population dynamics provides a complementary approach\[[32](https://arxiv.org/html/2610.11318#bib.bib2)\]\. Given samples at multiple time points, optimal transport formulations recover a coupling between the observed marginals\[[38](https://arxiv.org/html/2610.11318#bib.bib27),[13](https://arxiv.org/html/2610.11318#bib.bib37)\], while flow and Schrödinger Bridge matching provide simulation\-free frameworks for learning continuous dynamics\[[43](https://arxiv.org/html/2610.11318#bib.bib38),[30](https://arxiv.org/html/2610.11318#bib.bib11),[37](https://arxiv.org/html/2610.11318#bib.bib9),[41](https://arxiv.org/html/2610.11318#bib.bib8)\]\. Some variants additionally incorporate the geometry of the state or distribution space\[[12](https://arxiv.org/html/2610.11318#bib.bib15),[18](https://arxiv.org/html/2610.11318#bib.bib3)\]\. Metric Flow Matching replaces Euclidean interpolation with geodesics under a learned Riemannian metric\[[24](https://arxiv.org/html/2610.11318#bib.bib12)\], capturing the anisotropic geometry of the data manifold but not a preferred direction of motion\. Curly Flow Matching incorporates RNA velocity through a nonzero\-drift reference process and can therefore capture directed and non\-gradient dynamics\[[35](https://arxiv.org/html/2610.11318#bib.bib10)\], but takes as input a single drift vector at each state, and therefore cannot exploit the second\-order structure of a local transition distribution\.

In this paper, we use a discrete Markov transition graph to define a continuous dynamical prior for endpoint\-constrained generative transport\. We introduce Finsler Flow Matching \(FFM\), which converts the directional and diffusive structure of a Markov transition graph into a smooth Finsler geometry and uses this geometry to learn continuous stochastic transport through score and flow matching\. Under the additional assumption that the Markov chain discretizes an Itô diffusion, the graph\-derived local moments recover its drift and diffusion\. Freidlin–Wentzell\[[16](https://arxiv.org/html/2610.11318#bib.bib39),[20](https://arxiv.org/html/2610.11318#bib.bib36)\]theory then motivates a Randers metric\[[4](https://arxiv.org/html/2610.11318#bib.bib50)\], in which local diffusion determines anisotropic accessibility and drift introduces directional preference\. We then learn a neural path interpolant to approximate directed Randers geodesics and use the resulting path cost to construct source–target couplings consistent with the transition dynamics\. Finally, we use these couplings and geodesic interpolants to construct tractable conditional probability paths and learn an endpoint\-constrained stochastic generative process through simulation\-free score and flow matching\[[43](https://arxiv.org/html/2610.11318#bib.bib38)\], whose local noise geometry is shaped by the graph\-derived second moment\.

To summarize, our main contributions are:

- •A principled construction of Finsler geometry from Markov graphs\.Under a short\-time Itô diffusion model, we show that the first and second local moments of a cell\-cell Markov kernel recover the drift and diffusion of an underlying stochastic process\. We use these estimates to derive a Randers geometry from the Freidlin\-Wentzell minimum\-action principle\[[16](https://arxiv.org/html/2610.11318#bib.bib39)\], jointly encoding anisotropic accessibility and directional dynamics\.
- •A framework for learning continuous stochastic dynamics from discrete Markov graphs\.We introduce Finsler Flow Matching, which learns a neural path interpolant under the induced Finsler geometry, uses directed geodesic costs to construct source\-target couplings, and defines stochastic conditional paths whose local noise structure is shaped by the graph\-derived diffusion\. These paths are then distilled into a continuous generative process through simulation\-free score and flow matching\.
- •Flexible inference of single\-cell trajectory dynamics\.By using cell\-cell Markov graphs as a common representation of transition dynamics, FFM can incorporate information derived from various experimental and computational sources, including RNA velocity, lineage tracing, labeling experiments, or temporally sampled populations within a unified generative framework\. We show that our model outperforms previous methods in intermediate state prediction on various benchmarks\.

## 2 Preliminaries

In this section, we establish the background on Finsler geometry, optimal transport, and continuous generative modeling via flow and score matching\. More information can be found in[AppendixA](https://arxiv.org/html/2610.11318#A1.SS0.SSS0.Px1)

#### Finsler Geometry:

Riemannian geometry measures local distance via a symmetric metric tensorG⁡\(x\)G\(x\)\. Finsler geometry generalizes this to a direction\-dependent, asymmetric cost\. We introduce a special class of Finsler structures called a*Randers metric*:

F⁡\(x,v\)=v⊤​G​\(x\)​v\+β​\(x\)⊤​v,F\(x,v\)=\\sqrt\{v^\{\\top\}G\(x\)v\}\+\\beta\(x\)^\{\\top\}v,whereG⁡\(x\)∈SPD⁡\(d\)G\(x\)\\in\\SPD\(d\)is a Riemannian metric tensor andβ⁡\(x\)\\beta\(x\)is a11\-form giving directionality\. For an absolutely continuous pathγ:\[0,1\]→ℝd\\gamma:\[0,1\]\\to\\mathbb\{R\}^\{d\}, the Finsler lengthLF​\(γ\)L\_\{F\}\(\\gamma\)and energyEF​\(γ\)E\_\{F\}\(\\gamma\)functionals are defined as:

LF​\(γ\)≔∫01F⁡\(γ⁡\(t\),γ˙​\(t\)\)​𝑑t,EF​\(γ\)≔∫01F​\(γ⁡\(t\),γ˙​\(t\)\)2​𝑑t\.L\_\{F\}\(\\gamma\)\\coloneqq\\int\_\{0\}^\{1\}F\(\\gamma\(t\),\\dot\{\\gamma\}\(t\)\)\\mathrm\{d\}t,\\qquad E\_\{F\}\(\\gamma\)\\coloneqq\\int\_\{0\}^\{1\}F\(\\gamma\(t\),\\dot\{\\gamma\}\(t\)\)^\{2\}\\mathrm\{d\}t\.Curves minimizingLFL\_\{F\}are called*\(Finsler\) geodesics*and are locally shortest paths connecting two points\. The*Finsler geodesic distance*cF​\(x0,x1\)c\_\{F\}\(x\_\{0\},x\_\{1\}\)is defined as:

cF​\(x0,x1\)≔infγ⁡\(0\)=x0​γ​\(1\)=x1∫01F⁡\(γ⁡\(t\),γ˙​\(t\)\)​𝑑t\.c\_\{F\}\(x\_\{0\},x\_\{1\}\)\\coloneqq\\inf\_\{\\begin\{subarray\}\{c\}\\gamma\(0\)=x\_\{0\}\\ \\gamma\(1\)=x\_\{1\}\\end\{subarray\}\}\\int\_\{0\}^\{1\}F\(\\gamma\(t\),\\dot\{\\gamma\}\(t\)\)\\mathrm\{d\}t\.\(1\)

#### Optimal Transport and Schrödinger Bridge Problem:

Optimal Transport \(OT\) aims to transport a probability distribution into another one in the most cost efficient way\[[45](https://arxiv.org/html/2610.11318#bib.bib41)\]\. Given source and target densitiesp0,p1∈𝒫⁡\(ℝd\)p\_\{0\},p\_\{1\}\\in\\mathcal\{P\}\(\\mathbb\{R\}^\{d\}\)and a cost functionc:ℝd×ℝd→ℝ≥0c:\\mathbb\{R\}^\{d\}\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}\_\{\\geq 0\}, the*Kantorovich problem*seeks an optimal transport planπ∈Π⁡\(p0,p1\)\\pi\\in\\Pi\(p\_\{0\},p\_\{1\}\)minimizing total cost:

π⋆=arginfπ∈Π⁡\(p0,p1\)∫ℝd×ℝdc\(x0,x1\)dπ\(x0,x1\),\\pi^\{\\star\}=\\arg\\inf\_\{\\pi\\in\\Pi\(p\_\{0\},p\_\{1\}\)\}\\int\_\{\\mathbb\{R\}^\{d\}\\times\\mathbb\{R\}^\{d\}\}c\(x\_\{0\},x\_\{1\}\)\\mathrm\{d\}\\pi\(x\_\{0\},x\_\{1\}\),whereΠ⁡\(p0,p1\)\\Pi\(p\_\{0\},p\_\{1\}\)is the set of joint distributions with marginalsp0p\_\{0\}andp1p\_\{1\}\. Using the squared Finsler geodesic distance as cost function and the elementary factcF\(x0,x1\)2=infγ:γ⁡\(0\)=x0,γ⁡\(1\)=x​1EF\(γ\)c\_\{F\}\(x\_\{0\},x\_\{1\}\)^\{2\}=\\inf\_\{\\gamma:\\gamma\(0\)=x\_\{0\},\\gamma\(1\)=x1\}E\_\{F\}\(\\gamma\)111It follows from the Cauchy\-Schwarz inequality and the fact that minimizers of the Energy are constant speed geodesics\.implies that

W2,F2\(p0,p1\)=infπ∈Π⁡\(p0,p1\)infγ:γ⁡\(0\)=x0,γ⁡\(1\)=x1∫ℝd×ℝd∫01F2\(γ\(t\),γ˙\(t\)\)dtdπ\(x0,x1\)\.\\displaystyle W\_\{2,F\}^\{2\}\(p\_\{0\},p\_\{1\}\)=\\inf\_\{\\pi\\in\\Pi\(p\_\{0\},p\_\{1\}\)\}\\inf\_\{\\gamma:\\gamma\(0\)=x\_\{0\},\\gamma\(1\)=x\_\{1\}\}\\int\_\{\\mathbb\{R\}^\{d\}\\times\\mathbb\{R\}^\{d\}\}\\int\_\{0\}^\{1\}F^\{2\}\(\\gamma\(t\),\\dot\{\\gamma\}\(t\)\)\\text\{ \}\\mathrm\{d\}t\\text\{ \}\\mathrm\{d\}\\pi\(x\_\{0\},x\_\{1\}\)\.\(2\)A stochastic counterpart to the OT problem is given by the Schrödinger Bridge Problem \(SBP\)\[[28](https://arxiv.org/html/2610.11318#bib.bib51)\]\. Given a reference measure onΩ=C⁡\(\[0,1\],ℝd\)\\Omega=C\(\[0,1\],\\mathbb\{R\}^\{d\}\)of a stochastic processRR, the SBP searches for the closest path measure toRRthat has endpoint marginals given byp0p\_\{0\}andp1p\_\{1\}\. More precisely

Q∗=arg​minQ:Q0=p0,;Q1=p1KL\(Q∥R\)\.\\displaystyle Q^\{\*\}=\\argmin\_\{Q:Q\_\{0\}=p\_\{0\},;Q\_\{1\}=p\_\{1\}\}\\mathrm\{KL\}\(Q\\\|R\)\.\(3\)A classical result due to[Föllmer \[15\]](https://arxiv.org/html/2610.11318#bib.bib40)tells us that the solution can be decomposed as

Q∗=∫Rx0,x1dπ∗\(x0,x1\), whereπ∗=arg​minπ∈Π⁡\(p0,p1\)KL\(π∥R01\),\\displaystyle Q^\{\*\}=\\int R^\{x\_\{0\},x\_\{1\}\}\\mathrm\{d\}\\pi^\{\*\}\(x\_\{0\},x\_\{1\}\)\\text\{, where \}\\pi^\{\*\}=\\argmin\_\{\\pi\\in\\Pi\(p\_\{0\},p\_\{1\}\)\}\\mathrm\{KL\}\\left\(\\pi\\\|R\_\{01\}\\right\),\(4\)whereRx0,x1=R\(⋅∣Y0=x0,Y1=x1\)R^\{x\_\{0\},x\_\{1\}\}=R\(\\cdot\\mid Y\_\{0\}=x\_\{0\},Y\_\{1\}=x\_\{1\}\)denotes the reference process conditioned on its endpoints andR01R\_\{01\}its endpoint law\. For a Brownian reference processd​Yt=σB​d​WtdY\_\{t\}=\\sigma\_\{B\}dW\_\{t\}, one can see thatRtx0,x1=𝒩⁡\(\(1−t\)​x0\+t​x1,σ2​t​\(1−t\)​I\)\.R\_\{t\}^\{x\_\{0\},x\_\{1\}\}=\\mathcal\{N\}\\left\(\(1\-t\)x\_\{0\}\+tx\_\{1\},\\sigma^\{2\}t\(1\-t\)I\\right\)\.For non\-Brownian reference processes, such as SDEs with anisotropic diffusion and non\-trivial drift

d​Yt=b0​\(Yt\),d​t\+Σ⁡\(Yt\)​d​Wt,\\displaystyle dY\_\{t\}=b\_\{0\}\(Y\_\{t\}\),dt\+\\Sigma\(Y\_\{t\}\)dW\_\{t\},the conditional bridges are in general neither Gaussian nor available in closed form\. In the small noise regime, their most\-probable transition paths are instead characterized by minimizers of the Freidlin\-Wentzell action, a connection that we use in[AppendixB](https://arxiv.org/html/2610.11318#A2)\.

#### Score and Flow Matching:

Flow Matching \(FM\) is a simulation\-free method for training continuous\-time generative models\. Given a target probability pathpt​\(x\)p\_\{t\}\(x\)pushed forward by an unknown vector fieldut​\(x\)u\_\{t\}\(x\), FM fits a neural vector fieldvθ​\(t,x\)v\_\{\\theta\}\(t,x\)via a regression objective:

ℒFM​\(θ\)=∫01𝔼x∼pt​\(x\)​\[\|vθ​\(t,x\)−ut​\(x\)\|2\]​𝑑t\.\\mathcal\{L\}\_\{\\mathrm\{FM\}\}\(\\theta\)=\\int\_\{0\}^\{1\}\\mathbb\{E\}\_\{x\\sim p\_\{t\}\(x\)\}\\left\[\|v\_\{\\theta\}\(t,x\)\-u\_\{t\}\(x\)\|^\{2\}\\right\]\\mathrm\{d\}t\.\(5\)Conditional Flow Matching \(CFM\) circumvents the intractability of the marginal vector fieldut​\(x\)u\_\{t\}\(x\)by conditioning on endpoint pairs\(x0,x1\)∼q⁡\(x0,x1\)\(x\_\{0\},x\_\{1\}\)\\sim q\(x\_\{0\},x\_\{1\}\)\[[30](https://arxiv.org/html/2610.11318#bib.bib11),[1](https://arxiv.org/html/2610.11318#bib.bib17)\]\. The network is trained on tractable conditional vector fieldsvt​\(x∣x0,x1\)v\_\{t\}\(x\\mid x\_\{0\},x\_\{1\}\)pushing probability paths along conditional trajectoriesμt​\(x0,x1\)\\mu\_\{t\}\(x\_\{0\},x\_\{1\}\)\. FM is often combined with minibatch optimal transport to pair source and target samples, enabling it to approximate dynamic optimal transport\[[41](https://arxiv.org/html/2610.11318#bib.bib8)\]\. Following the approach of[Tong et al\. \[43\]](https://arxiv.org/html/2610.11318#bib.bib38)one may jointly training the flow and score through the combined objective

ℒ\[SF\]2​M​\(θ\)=ℒCFM​\(θ\)\+λ​ℒCSM​\(θ\)\.\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{\[SF\]\}^\{2\}\\mathrm\{M\}\}\(\\theta\)=\\mathcal\{L\}\_\{\\mathrm\{CFM\}\}\(\\theta\)\+\\lambda\\mathcal\{L\}\_\{\\mathrm\{CSM\}\}\(\\theta\)\.The learned flowvθv\_\{\\theta\}and scoresψs\_\{\\psi\}can be used with an arbitrary matrix\-valued diffusion schedule𝐠:\[0,1\]→ℝd×d\\mathbf\{g\}:\[0,1\]\\to\\mathbb\{R\}^\{d\\times d\}to define the stochastic generative process

d​Xt=\[vθ​\(t,Xt\)\+12​𝐠​\(t\)​𝐠​\(t\)⊤​sψ​\(t,Xt\)\]​d​t\+𝐠⁡\(t\)​d​Wt,\\displaystyle dX\_\{t\}=\\left\[v\_\{\\theta\}\(t,X\_\{t\}\)\+\\frac\{1\}\{2\}\\mathbf\{g\}\(t\)\\mathbf\{g\}\(t\)^\{\\top\}s\_\{\\psi\}\(t,X\_\{t\}\)\\right\]dt\+\\mathbf\{g\}\(t\)\\,dW\_\{t\},\(6\)which satisfiesXt∼ptX\_\{t\}\\sim p\_\{t\}for allt∈\[0,1\]\.t\\in\[0,1\]\.

## 3 Related Work

#### Single\-cell trajectory inference\.

OT methods infer developmental trajectories from populations observed at multiple time points by coupling successive snapshot distributions\[[38](https://arxiv.org/html/2610.11318#bib.bib27),[42](https://arxiv.org/html/2610.11318#bib.bib22),[10](https://arxiv.org/html/2610.11318#bib.bib7),[50](https://arxiv.org/html/2610.11318#bib.bib6)\]\. Relatedly,[Huguet et al\. \[23\]](https://arxiv.org/html/2610.11318#bib.bib13)combine manifold\-aware OT with neural differential equations to learn continuous cellular dynamics from snapshot data\. Complementary approaches such as CellRank construct cell\-cell Markov transition kernels from RNA velocity, lineage information, and other dynamical measurements to infer fate probabilities and developmental relationships\[[26](https://arxiv.org/html/2610.11318#bib.bib49),[47](https://arxiv.org/html/2610.11318#bib.bib26)\]\. FFM builds on these representations by using local transition statistics to construct a smooth directed geometry and continuous generative dynamics beyond the observed cells\.

#### Stochastic generative transport\.

Simulation\-free Schrödinger bridge methods combine score and flow matching to learn stochastic dynamics between endpoint distributions\[[43](https://arxiv.org/html/2610.11318#bib.bib38)\]\. Related work studies closed\-form Gaussian Schrödinger bridges and smooth Schrödinger bridges for trajectory inference\[[9](https://arxiv.org/html/2610.11318#bib.bib5),[22](https://arxiv.org/html/2610.11318#bib.bib4)\]\. FFM builds on this framework by using the transition\-derived geometry to determine both the interpolating paths and the local anisotropic noise structure\.

#### Geometric and directional flow matching\.

[Kapusniak et al\. \[24\]](https://arxiv.org/html/2610.11318#bib.bib12)replace Euclidean conditional paths with neural approximations of geodesics under a data\-dependent Riemannian metric\. FFM extends this idea to a directed Randers geometry derived from local transition statistics\. Relatedly,[Bamberger et al\. \[3\]](https://arxiv.org/html/2610.11318#bib.bib1)uses local Markov kernels to construct anisotropic covariance for geometry\-aware flow matching, whereas FFM additionally incorporates first\-moment directionality through a Randers geometry and geodesic coupling\.[Petrović et al\. \[35\]](https://arxiv.org/html/2610.11318#bib.bib10)uses a nonzero\-drift reference process to model directed and non\-gradient dynamics from local velocity fields\. FFM instead derives directionality from the first moment of a transition kernel and uses its second moment to capture anisotropic transition structure\.

#### Lineage\-informed Finsler geometry\.

[Zweig et al\. \[51\]](https://arxiv.org/html/2610.11318#bib.bib44)recently use Finsler geometry to incorporate directed lineage\-tree priors into trajectory inference, penalizing motion inconsistent with a prescribed adjacency between cell types\. FFM instead derives its Finsler geometry directly from cell\-cell transition probabilities, using the probabilistic transition graph to encode both directional preference and local transition accessibility rather than relying on a categorical lineage prior\.

## 4 Finsler Flow Matching

#### Problem formulation\.

Let

D=\{xi\}i=1N⊆ℝdD=\\\{x\_\{i\}\\\}\_\{i=1\}^\{N\}\\subseteq\\mathbb\{R\}^\{d\}be a collection of observations sampled from an unknown distribution

pp\. Let

D0=\{xi0\}i=1N0D\_\{0\}=\\\{x\_\{i\}^\{0\}\\\}\_\{i=1\}^\{N\_\{0\}\}and

D1=\{xi1\}i=1N1D\_\{1\}=\\\{x\_\{i\}^\{1\}\\\}\_\{i=1\}^\{N\_\{1\}\}denote two subpopulations from source and target distributions

p0p\_\{0\}and

p1p\_\{1\}\. In addition, we are given a row\-stochastic transition matrix

P∈ℝN×NP\\in\\mathbb\{R\}^\{N\\times N\}, whose entries represent transition likelihoods between observed states\.

Our goal is to infer a transport mechanism from

p0p\_\{0\}to

p1p\_\{1\}and a family of intermediate distributions

\(pt\)t∈\[0,1\]\(p\_\{t\}\)\_\{t\\in\[0,1\]\}that are consistent with the dynamics encoded by

PP\. More formally we can define this as an OT problem of the form[2](https://arxiv.org/html/2610.11318#S2.E2)

\(π∗,γ∗\)=arg​min\(π,γ\)∫ℝd×ℝd∫01FP2\(γ\(t\),γ˙\(t\)\)dtdπ\(x0,x1\)where​π∈Π⁡\(p0,p1\)​and​γ​\(0\)=x0,γ⁡\(1\)=x1\\displaystyle\\begin\{aligned\} &\(\\pi^\{\*\},\\gamma^\{\*\}\)=\\argmin\_\{\(\\pi,\\gamma\)\}\\int\_\{\\mathbb\{R\}^\{d\}\\times\\mathbb\{R\}^\{d\}\}\\int\_\{0\}^\{1\}F\_\{P\}^\{2\}\(\\gamma\(t\),\\dot\{\\gamma\}\(t\)\)\\text\{ \}\\mathrm\{d\}t\\text\{ \}\\mathrm\{d\}\\pi\(x\_\{0\},x\_\{1\}\)\\\\ &\\text\{ where \}\\pi\\in\\Pi\(p\_\{0\},p\_\{1\}\)\\text\{ and \}\\gamma\(0\)=x\_\{0\},\\gamma\(1\)=x\_\{1\}\\end\{aligned\}\(7\)and

FP​\(x,v\)=v⊤​GP​\(x\)​v\+βP​\(x\)⊤​vF\_\{P\}\(x,v\)=\\sqrt\{v^\{\\top\}G\_\{P\}\(x\)v\}\+\\beta\_\{P\}\(x\)^\{\\top\}vis a

PP\-derived Randers metric \(Definition[7](https://arxiv.org/html/2610.11318#Thmtheorem7)\)\. The corresponding intermediate distributions are induced by transporting the optimal coupling along the optimal paths\.

From now on we consider the special case where the

PP\-derived Randers metric is constructed from spatial moments of

PP\. More precisely we define at each observed state

xix\_\{i\}moments as:

m⁡\(xi\)≔∑j=1NPi​j​\(xj−xi\),and​C​\(xi\)≔∑j=1NPi​j​\(xj−xi\)​\(xj−xi\)⊤\.\\displaystyle m\(x\_\{i\}\)\\coloneqq\\sum\_\{j=1\}^\{N\}P\_\{ij\}\(x\_\{j\}\-x\_\{i\}\),\\text\{ and \}C\(x\_\{i\}\)\\coloneqq\\sum\_\{j=1\}^\{N\}P\_\{ij\}\(x\_\{j\}\-x\_\{i\}\)\(x\_\{j\}\-x\_\{i\}\)^\{\\top\}\.\(8\)222At finite graph scaleC=CovP⁡\(Δ​x\)\+m​m⊤C=\\operatorname\{Cov\}\_\{P\}\(\\Delta x\)\+mm^\{\\top\}, we interpretCCas an anisotropic transition\-accessibility rather then diffusion covariance\.

m⁡\(xi\)m\(x\_\{i\}\)captures the net preferred displacement and therefore provides directionality whereasC⁡\(xi\)C\(x\_\{i\}\)captures the direction on the data manifold with high transition probabilities\. To evaluate the metric along continuous paths, we construct smooth extensions ofmmandCC, for example using kernel smoothing, neural networks, or support vector regression\[[39](https://arxiv.org/html/2610.11318#bib.bib19)\], while preserving positive semidefiniteness ofCC\. We use kernel smoothing for simplicity details are provided in[C](https://arxiv.org/html/2610.11318#A3.SS0.SSS0.Px2)\. To obtain a non\-degenerate Finsler structure, we define the regularized Freidlin–Wentzell \(FW\) action\[[16](https://arxiv.org/html/2610.11318#bib.bib39)\]\. Firstly, setCρ​\(x\)≔C⁡\(x\)\+ρ​IC\_\{\\rho\}\(x\)\\coloneqq C\(x\)\+\\rho Iwithρ\>0\\rho\>0and define

FP​\(x,v\)≔v⊤​Cρ​\(x\)−1​v⏟Anisotropic accessibility​m​\(x\)⊤​Cρ​\(x\)−1​m​\(x\)\+λ2⏟first\-moment calibration−v⊤​Cρ​\(x\)−1​m​\(x\)⏟directional alignment,F\_\{P\}\(x,v\)\\coloneqq\\underbrace\{\\sqrt\{v^\{\\top\}C\_\{\\rho\}\(x\)^\{\-1\}v\}\}\_\{\\text\{Anisotropic accessibility\}\}\\,\\underbrace\{\\sqrt\{m\(x\)^\{\\top\}C\_\{\\rho\}\(x\)^\{\-1\}m\(x\)\+\\lambda^\{2\}\}\}\_\{\\text\{first\-moment calibration\}\}\-\\underbrace\{v^\{\\top\}C\_\{\\rho\}\(x\)^\{\-1\}m\(x\)\}\_\{\\text\{directional alignment\}\},\(9\)whereλ\>0\\lambda\>0ensures Randers admissibility[5](https://arxiv.org/html/2610.11318#Thmtheorem5)\. The anisotropic accessibility term encourages geodesics to remain close to regions and directions that are well supported by the transition probabilities, directional alignment ensures that geodesics follow the preferred directionality and the first\-moment calibration ensures that both penalty terms are on the same scale\. The corresponding action is

S⁡\(γ\)=∫01FP​\(γ⁡\(t\),γ˙​\(t\)\)​𝑑t\.S\(\\gamma\)=\\int\_\{0\}^\{1\}F\_\{P\}\\\!\\left\(\\gamma\(t\),\\dot\{\\gamma\}\(t\)\\right\)\\,\\mathrm\{d\}t\.\(10\)The following proposition formalizes how geodesics underFPF\_\{P\}align with the transition structure encoded byPP:

###### Proposition 1\(Informal[8](https://arxiv.org/html/2610.11318#Thmtheorem8)\)\. Suppose the endpoints can be connected along a transition\-supported corridor at a Finsler cost lower than that required to leave and return to a prescribed surrounding neighborhood\. Then every minimizingFPF\_\{P\}\-geodesic remains within that neighborhood\. Moreover, locally the metric favors alignment with the first transition moment\.

In the special case wherePPis the discrete observation of an SDE[14](https://arxiv.org/html/2610.11318#Thmtheorem14), we can conclude even more about geodesics\. By[Freidlin and Wentzell \[16\]](https://arxiv.org/html/2610.11318#bib.bib39)and[Heymann and Vanden‐Eijnden \[20\]](https://arxiv.org/html/2610.11318#bib.bib36)we can show that geodesics under the resulting metricFPF\_\{P\}follow the maximum likelihood paths of the stochastic process\.

###### Theorem 2\(Informal[18](https://arxiv.org/html/2610.11318#Thmtheorem18)\)\. IfPPis a discretization of a SDE, then minimizers of the empirical FW Finsler action converge to the asymptotically most probable transition path in the small\-noise limit\.

FFM is defined for arbitrary row\-stochastic transition matrices, whereas its diffusion interpretation requires Assumption[11](https://arxiv.org/html/2610.11318#Thmtheorem11)\. Connections to the limits of local graph kernels, and the scope of these results for velocity\-based constructions, are discussed in[B](https://arxiv.org/html/2610.11318#A2.SS0.SSS0.Px3)\.

#### Learning Geodesics:

For endpoints\(x0,x1\)∼p0⊗p1\(x\_\{0\},x\_\{1\}\)\\sim p\_\{0\}\\otimes p\_\{1\}andt∈\[0,1\]t\\in\[0,1\], parameterize a candidate curve by

xt,η​\(x0,x1\)=\(1−t\)​x0\+t​x1\+t⁡\(1−t\)​φη​\(t,x0,x1\),x\_\{t,\\eta\}\(x\_\{0\},x\_\{1\}\)\\;=\\;\(1\-t\)\\,x\_\{0\}\+t\\,x\_\{1\}\+t\(1\-t\)\\,\\varphi\_\{\\eta\}\(t,x\_\{0\},x\_\{1\}\),\(11\)whereφη:\[0,1\]×ℝd×ℝd→ℝd\\varphi\_\{\\eta\}:\[0,1\]\\times\\mathbb\{R\}^\{d\}\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}is some neural network with parametersη\\eta\. The boundary correctiont⁡\(1−t\)t\(1\-t\)enforcesx0,η=x0x\_\{0,\\eta\}=x\_\{0\}andx1,η=x1x\_\{1,\\eta\}=x\_\{1\}\. The time derivative is given by,

x˙t,η=x1−x0\+\(1−2​t\)​φη​\(t,x0,x1\)\+t⁡\(1−t\)​∂tφη​\(t,x0,x1\)\.\\displaystyle\\dot\{x\}\_\{t,\\eta\}=x\_\{1\}\-x\_\{0\}\+\(1\-2t\)\\varphi\_\{\\eta\}\(t,x\_\{0\},x\_\{1\}\)\+t\(1\-t\)\\partial\_\{t\}\\varphi\_\{\\eta\}\(t,x\_\{0\},x\_\{1\}\)\.This interpolant was previously proposed in[Kapusniak et al\. \[24\]](https://arxiv.org/html/2610.11318#bib.bib12), and similarly to their approach, we minimize the Finsler Energy functional,

ℒgeo​\(η\)=𝔼\(x0,x1\)∼p0⊗p1t∼𝒰⁡\[0,1\]​\[FP​\(xt,η​\(x0,x1\),x˙t,η​\(x0,x1\)\)2\]\.\\mathcal\{L\}\_\{\\mathrm\{geo\}\}\(\\eta\)=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}\(x\_\{0\},x\_\{1\}\)\\sim p\_\{0\}\\otimes p\_\{1\}\\\\ t\\sim\\mathcal\{U\}\[0,1\]\\end\{subarray\}\}\\left\[F\_\{P\}\\\!\\left\(x\_\{t,\\eta\}\(x\_\{0\},x\_\{1\}\),\\dot\{x\}\_\{t,\\eta\}\(x\_\{0\},x\_\{1\}\)\\right\)^\{2\}\\right\]\.\(12\)Since energy minimizers are constant\-speed reparameterizations of geodesics, minimizing \([12](https://arxiv.org/html/2610.11318#S4.E12)\) overη\\etayields an approximation of the pairwisePP\-derived Finsler geodesics\. We denote the resulting parameters byη∗\\eta^\{\*\}\.

Algorithm 1Training the Finsler Geodesic Model1:Source and target distributions

p0,p1p\_\{0\},p\_\{1\}, Finsler metric

FPF\_\{P\}, interpolant network

φη\\varphi\_\{\\eta\}, LR

γ\\gamma\.

2:whilenot convergeddo

3:Sample

x0∼p0x\_\{0\}\\sim p\_\{0\},

x1∼p1x\_\{1\}\\sim p\_\{1\},

t∼𝒰⁡\[0,1\]t\\sim\\mathcal\{U\}\[0,1\]and compute

xt,η​\(x0,x1\)x\_\{t,\\eta\}\(x\_\{0\},x\_\{1\}\)and

x˙t,η​\(x0,x1\)\\dot\{x\}\_\{t,\\eta\}\(x\_\{0\},x\_\{1\}\)
4:

ℒgeo←FP​\(xt,η,x˙t,η\)2\\mathcal\{L\}\_\{\\mathrm\{geo\}\}\\leftarrow F\_\{P\}\\\!\\left\(x\_\{t,\\eta\},\\dot\{x\}\_\{t,\\eta\}\\right\)^\{2\}
5:

η←η−γ​∇ηℒgeo\\eta\\leftarrow\\eta\-\\gamma\\nabla\_\{\\eta\}\\mathcal\{L\}\_\{\\mathrm\{geo\}\}
6:endwhile

#### Finsler Optimal Transport:

Having learned an approximation of the geodesic connecting arbitrary endpoint pairs, we can approximate the inner minimization in the transport problem[7](https://arxiv.org/html/2610.11318#S4.E7)\. For each\(x0,x1\)\(x\_\{0\},x\_\{1\}\), define the learned Finsler transport cost

cη∗​\(x0,x1\):=∫01FP​\(xt,η∗​\(x0,x1\),x˙t,η∗​\(x0,x1\)\)2​𝑑t\.c\_\{\\eta^\{\*\}\}\(x\_\{0\},x\_\{1\}\):=\\int\_\{0\}^\{1\}F\_\{P\}\\\!\\left\(x\_\{t,\\eta^\{\*\}\}\(x\_\{0\},x\_\{1\}\),\\dot\{x\}\_\{t,\\eta^\{\*\}\}\(x\_\{0\},x\_\{1\}\)\\right\)^\{2\}\\,\\mathrm\{d\}t\.Thus,

cη∗​\(x0,x1\)≈infγ⁡\(0\)=x0γ⁡\(1\)=x1∫01FP​\(γ⁡\(t\),γ˙​\(t\)\)2​𝑑t=cF2​\(x0,x1\)\.c\_\{\\eta^\{\*\}\}\(x\_\{0\},x\_\{1\}\)\\approx\\inf\_\{\\begin\{subarray\}\{c\}\\gamma\(0\)=x\_\{0\}\\\\ \\gamma\(1\)=x\_\{1\}\\end\{subarray\}\}\\int\_\{0\}^\{1\}F\_\{P\}\\\!\\left\(\\gamma\(t\),\\dot\{\\gamma\}\(t\)\\right\)^\{2\}\\,\\mathrm\{d\}t=c\_\{F\}^\{2\}\(x\_\{0\},x\_\{1\}\)\.We can solve the corresponding static optimal transport problem using costs\. In particular, we consider its entropy\-regularized form

πε∗=arg⁡minπ∈Π⁡\(p^0,p^1\)​\{∑i=1m∑j=1mπi​j​ci​j\+ε​KL​\(π∥p^0⊗p^1\)\},\\pi\_\{\\varepsilon\}^\{\*\}=\\arg\\min\_\{\\pi\\in\\Pi\(\\hat\{p\}\_\{0\},\\hat\{p\}\_\{1\}\)\}\\left\\\{\\sum\_\{i=1\}^\{m\}\\sum\_\{j=1\}^\{m\}\\pi\_\{ij\}c\_\{ij\}\+\\varepsilon\\,\\mathrm\{KL\}\\\!\\left\(\\pi\\,\\middle\\\|\\,\\hat\{p\}\_\{0\}\\otimes\\hat\{p\}\_\{1\}\\right\)\\right\\\},\(13\)whereε\>0\\varepsilon\>0controls the strength of the entropic regularization\. This problem can be solved efficiently using Sinkhorn iterations\[[13](https://arxiv.org/html/2610.11318#bib.bib37)\]\. The interpolantxt,η∗x\_\{t,\\eta^\{\*\}\}and the couplingπε∗\\pi\_\{\\varepsilon\}^\{\*\}provide an approximation of the original Finsler OT problem[7](https://arxiv.org/html/2610.11318#S4.E7)up to the approximation errors\.

#### Flow and Stochastic Model:

Assume we have computed\(πε∗,x⋅,η∗\)\\left\(\\pi\_\{\\varepsilon\}^\{\*\},x\_\{\\cdot,\\eta^\{\*\}\}\\right\)\. Drawing\(x0,x1\)∼πε∗\(x\_\{0\},x\_\{1\}\)\\sim\\pi\_\{\\varepsilon\}^\{\*\}and transporting each pair along its learned geodesic gives the probability pathpt​\(x\)≔\(xt,η∗\)\#​πε∗​\(x\)p\_\{t\}\(x\)\\coloneqq\\left\(x\_\{t,\\eta^\{\*\}\}\\right\)\_\{\\\#\}\\pi\_\{\\varepsilon\}^\{\*\}\(x\)\. By construction,p0p\_\{0\}andp1p\_\{1\}are the endpoint marginals\. The corresponding conditional probability distribution path ispt​\(x\|x0,x1\)=δxt,η​\(x0,x1\)p\_\{t\}\(x\|x\_\{0\},x\_\{1\}\)=\\delta\_\{x\_\{t,\\eta\}\(x\_\{0\},x\_\{1\}\)\}\. Furthermore, define the conditional pathμt​\(x0,x1\):=xt,η⋆​\(x0,x1\)\\mu\_\{t\}\(x\_\{0\},x\_\{1\}\):=x\_\{t,\\eta^\{\\star\}\}\(x\_\{0\},x\_\{1\}\)We then train a modelvθv\_\{\\theta\}using the conditional flow matching objective

ℒCFM​\(θ\)=𝔼\(x0,x1\)∼πε⋆t∼𝒰⁡\[0,1\]​\[‖vθ​\(t,μt​\(x0,x1\)\)−μ˙t​\(x0,x1\)‖22\]\.\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{CFM\}\}\(\\theta\)=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}\(x\_\{0\},x\_\{1\}\)\\sim\\pi\_\{\\varepsilon\}^\{\\star\}\\\\ t\\sim\\mathcal\{U\}\[0,1\]\\end\{subarray\}\}\\left\[\\left\\\|v\_\{\\theta\}\\\!\\left\(t,\\mu\_\{t\}\(x\_\{0\},x\_\{1\}\)\\right\)\-\\dot\{\\mu\}\_\{t\}\(x\_\{0\},x\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\.Motivated by[Föllmer \[15\]](https://arxiv.org/html/2610.11318#bib.bib40)we approximate each conditional bridge with an anisotropic Gaussian path\. For\(x0,x1\)∼πε⋆\(x\_\{0\},x\_\{1\}\)\\sim\\pi^\{\\star\}\_\{\\varepsilon\}, defineCρ,t:=Cρ​\(μt\)C\_\{\\rho,t\}:=C\_\{\\rho\}\(\\mu\_\{t\}\)andpt​\(x∣x0,x1\)=𝒩⁡\(x,μt,σ2​t​\(1−t\)​Cρ,t\)\.p\_\{t\}\(x\\mid x\_\{0\},x\_\{1\}\)=\\mathcal\{N\}\\\!\\left\(x;\\,\\mu\_\{t\},\\,\\sigma^\{2\}t\(1\-t\)C\_\{\\rho,t\}\\right\)\.We evaluateCCat the deterministic interpolantμt\\mu\_\{t\}rather than at the statexx, so that the conditional path can be sampled in closed form\. The interpretation of this path is that noise is injected along accessible directions according toPP\. The conditional probability\-flow vector field is

ut​\(x∣x0,x1\)=μ˙t\+1−2​t2​t​\(1−t\)​\(x−μt\)\+12​C˙ρ,t​Cρ,t−1​\(x−μt\),\\displaystyle u\_\{t\}\(x\\mid x\_\{0\},x\_\{1\}\)=\\dot\{\\mu\}\_\{t\}\+\\frac\{1\-2t\}\{2t\(1\-t\)\}\(x\-\\mu\_\{t\}\)\+\\frac\{1\}\{2\}\\dot\{C\}\_\{\\rho,t\}C\_\{\\rho,t\}^\{\-1\}\(x\-\\mu\_\{t\}\),See[21](https://arxiv.org/html/2610.11318#Thmtheorem21)for a derivation\. We distill the conditional bridges into a marginal vector fieldvθv\_\{\\theta\}and scoresψs\_\{\\psi\}using simulation\-free conditional flow and score matching\[[43](https://arxiv.org/html/2610.11318#bib.bib38)\],

ℒCFM​\(θ\)\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{CFM\}\}\(\\theta\)=𝔼\(x0,x1\)∼πε⋆,t∼𝒰⁡\(0,1\)x∼pt\(⋅∣x0,x1\)​\[‖v~θ​\(t,x\)−ut​\(x∣x0,x1\)‖22\],\\displaystyle=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}\(x\_\{0\},x\_\{1\}\)\\sim\\pi\_\{\\varepsilon\}^\{\\star\},\\;t\\sim\\mathcal\{U\}\(0,1\)\\\\ x\\sim p\_\{t\}\(\\cdot\\mid x\_\{0\},x\_\{1\}\)\\end\{subarray\}\}\\left\[\\left\\\|\\tilde\{v\}\_\{\\theta\}\(t,x\)\-u\_\{t\}\(x\\mid x\_\{0\},x\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\right\],ℒCSM​\(ψ\)\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{CSM\}\}\(\\psi\)=𝔼\(x0,x1\)∼πε⋆,t∼𝒰⁡\(0,1\)x∼pt\(⋅∣x0,x1\)​\[λ​\(t\)2​‖sψ​\(t,x\)−st​\(x∣x0,x1\)‖22\]\\displaystyle=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}\(x\_\{0\},x\_\{1\}\)\\sim\\pi\_\{\\varepsilon\}^\{\\star\},\\;t\\sim\\mathcal\{U\}\(0,1\)\\\\ x\\sim p\_\{t\}\(\\cdot\\mid x\_\{0\},x\_\{1\}\)\\end\{subarray\}\}\\left\[\\lambda\(t\)^\{2\}\\left\\\|s\_\{\\psi\}\(t,x\)\-s\_\{t\}\(x\\mid x\_\{0\},x\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\right\]where the final loss is given byℒ\[SF\]2​M=ℒCFM\+ℒCSM\\mathcal\{L\}\_\{\\mathrm\{\[SF\]\}^\{2\}\\mathrm\{M\}\}=\\mathcal\{L\}\_\{\\mathrm\{CFM\}\}\+\\mathcal\{L\}\_\{\\mathrm\{CSM\}\}and the conditional score is

st​\(x∣x0,x1\)=∇x​log​pt​\(x∣x0,x1\)=−1σ2​t​\(1−t\)​Cρ,t−1​\(x−μt\)\.s\_\{t\}\(x\\mid x\_\{0\},x\_\{1\}\)=\\nabla\_\{x\}\\log p\_\{t\}\(x\\mid x\_\{0\},x\_\{1\}\)=\-\\frac\{1\}\{\\sigma^\{2\}t\(1\-t\)\}C\_\{\\rho,t\}^\{\-1\}\(x\-\\mu\_\{t\}\)\.

#### Inference:

We samplex0∼p0x\_\{0\}\\sim p\_\{0\}and integrate the learned dynamics fromt=0t=0tot=1t=1\. In the deterministic setting, we integrate the ODE

d​Xt=vθ​\(t,Xt\)​d​t\.dX\_\{t\}=v\_\{\\theta\}\(t,X\_\{t\}\)\\,dt\.In the stochastic setting, we use the score\-and\-flow SDE introduced in[6](https://arxiv.org/html/2610.11318#S2.E6)and make a slight generalization by setting𝐠⁡\(t,x\)=g⁡\(t\)​Cρ1/2​\(x\),\\mathbf\{g\}\(t,x\)=g\(t\)\\,C\_\{\\rho\}^\{1/2\}\(x\),for some scalar\-valued noise functiong⁡\(t\)g\(t\)\. Substituting this choice into the stochastic dynamics gives

d​Xt=\[v~θ​\(t,Xt\)\+12​g​\(t\)2​Cρ​\(x\)​sψ​\(t,Xt\)\+12​g​\(t\)2​div⁡\(Cρ​\(x\)\)\]​d​t\+g⁡\(t\)​Cρ​\(x\)1/2​d​Wt,\\displaystyle dX\_\{t\}=\\left\[\\tilde\{v\}\_\{\\theta\}\(t,X\_\{t\}\)\+\\frac\{1\}\{2\}g\(t\)^\{2\}\\,C\_\{\\rho\}\(x\)s\_\{\\psi\}\(t,X\_\{t\}\)\+\\frac\{1\}\{2\}g\(t\)^\{2\}\\di\(C\_\{\\rho\}\(x\)\)\\right\]dt\+g\(t\)\\,C\_\{\\rho\}\(x\)^\{1/2\}\\,dW\_\{t\},wherediv⁡\(Cρ​\(x\)\)i=∑jd∂jCi​j​\(x\)\\di\(C\_\{\\rho\}\(x\)\)\_\{i\}=\\sum\_\{j\}^\{d\}\\partial\_\{j\}C\_\{ij\}\(x\)and which we integrate fromX0∼p0X\_\{0\}\\sim p\_\{0\}tot=1t=1\. The scalar scheduleg⁡\(t\)g\(t\)controls the overall noise strength, whereasCρ​\(x\)C\_\{\\rho\}\(x\)determines its local anisotropic geometry\.

## 5 Experiments

#### Experimental Setup:

The transition graph and derived geometric quantities are recomputed after removing the evaluation cells\. Our primary metric is EuclideanW2W\_\{2\}att=1/2t=1/2, supplemented by endpointW2W\_\{2\}and the geometry\-dependentW2,FW\_\{2,F\}\. Becauset=1/2t=1/2is a benchmark convention rather than an established correspondence with biological midpoint time, on Pancreas we additionally reportW¯2:=1\|𝒯\|​∑t∈𝒯W2​\(pt,pwithheld\)\\overline\{W\}\_\{2\}:=\\frac\{1\}\{\|\\mathcal\{T\}\|\}\\sum\_\{t\\in\\mathcal\{T\}\}W\_\{2\}\(p\_\{t\},p\_\{\\mathrm\{withheld\}\}\)\. We compare against OT\-CFM, MFM/LAND, Curly\-FM under matched architectures and training budgets\. More details[AppendixC](https://arxiv.org/html/2610.11318#A3)\.

#### FFM recovers directed and anisotropic transition structure:

To test whether FFM can recover intermediate dynamics when both directionality matters and when the data satisfy the manifold hypothesis, we construct the synthetic Sheet benchmark, a saddle\-shaped manifold\. We score against a narrow corridor, which represents a subset of the manifold corresponding to high probability transition regions\. We define a directed transition kernel that drives motion towards that middle corridor before crossing and away from it afterward\. Curly\-FM receives the first transition moment as its local velocity field, whereas FFM additionally uses the second moment to capture anisotropic accessibility\. FFM substantially improves midpoint recovery, achievingW2=0\.1588W\_\{2\}=0\.1588and0\.14490\.1449for deterministic and stochastic FFM, respectively, compared with0\.52210\.5221for Curly\-FM[Fig\.2](https://arxiv.org/html/2610.11318#S5.F2)\. We further performed and ablation of the Finsler metric, which demonstrates that both the directional and anisotropic components contribute to this improvement[Fig\.4\(b\)](https://arxiv.org/html/2610.11318#A3.F4.sf2)\. On a complementary synthetic bifurcation benchmark, FFM also preserves cell\-level fate assignments across both branches[Fig\.9](https://arxiv.org/html/2610.11318#A3.F9)\.

MethodmidW2↓W\_\{2\}\\downarrowendW2↓W\_\{2\}\\downarrowmidW2,F↓W\_\{2,F\}\\downarrowOT\-CFM0\.7112±\\pm0\.02330\.1734±\\pm0\.02161\.0305±\\pm0\.0263MFM \(LAND\)0\.6843±\\pm0\.00740\.2357±\\pm0\.02580\.8645±\\pm0\.0961Curly\-FM0\.5221±\\pm0\.00160\.1405±\\pm0\.00570\.5236±\\pm0\.1657FFM \(ours\)0\.1588±\\pm0\.00960\.2343±\\pm0\.02920\.2892±\\pm0\.0090FFM \(ours\), with noise0\.1449±\\pm0\.01160\.2636±\\pm0\.06820\.2742±\\pm0\.0314*sampling floor**0\.0646**0\.1896**0\.2812*

![Refer to caption](https://arxiv.org/html/2610.11318v1/sheet_paths.png)

Figure 2:W2,FW\_\{2,F\}denotes the Wasserstein distance induced by the Finsler costcFc\_\{F\}\. Sampling floor is computed between disjoint halves of the corresponding reference cloud\.
#### FFM improves recovery of pancreatic differentiation in high\-dimensional representations:

To test whether FFM’s graph\-derived geometry improves trajectory reconstruction on real RNA\-velocity data, we evaluate FFM on 3,696 pancreatic cells\[[5](https://arxiv.org/html/2610.11318#bib.bib48)\]partitioned into latent\-time tertiles and withhold the intermediate population, which is enriched for Ngn3\-high pre\-endocrine cells\. FFM receives an RNA\-velocity\-derived transition kernel computed using[Lange et al\. \[26\]](https://arxiv.org/html/2610.11318#bib.bib49), whereas Curly\-FM receives per\-cell velocities\. We found that relative performance depended strongly on representation dimensionality\. In two dimensions, Curly\-FM achieved lower midpointW2W\_\{2\}than FFM, whereas FFM achieved substantially lower midpointW2W\_\{2\}in the2020and5050dimensional PCA[Fig\.3](https://arxiv.org/html/2610.11318#S5.F3)\. endpoint rankings vary across methods[Fig\.3](https://arxiv.org/html/2610.11318#S5.F3)

![Refer to caption](https://arxiv.org/html/2610.11318v1/pancreas_paths.png)

MethodW2​\(12\)↓W\_\{2\}\(\\frac\{1\}\{2\}\)\\downarrowW¯2\[1/3,2/3\]↓\\overline\{W\}\_\{2\}^\{\[1/3,2/3\]\}\\downarrowW2​\(1\)↓W\_\{2\}\(1\)\\downarrowW2,F​\(12\)↓W\_\{2,F\}\(\\frac\{1\}\{2\}\)\\downarrowd=2d=2OT\-CFM1\.8062±\\pm0\.01971\.7869±\\pm0\.02300\.1740±\\pm0\.03380\.6420±\\pm0\.0059MFM / LAND1\.4682±\\pm0\.01281\.4642±\\pm0\.01310\.3769±\\pm0\.03460\.5399±\\pm0\.0024Curly\-FM0\.9773±\\pm0\.13110\.9686±\\pm0\.17250\.2581±\\pm0\.03710\.7493±\\pm0\.0258FFM \(ours\), deterministic1\.6692±\\pm0\.01541\.6175±\\pm0\.03480\.3283±\\pm0\.12930\.5681±\\pm0\.0100FFM \(ours\), with noise1\.6957±\\pm0\.03051\.6434±\\pm0\.05550\.2272±\\pm0\.02390\.5730±\\pm0\.0082*sampling floor**0\.1113**0\.1113**0\.1022**0\.4162*d=20d=20OT\-CFM12\.3484±\\pm0\.048512\.4957±\\pm0\.06815\.2262±\\pm0\.158310\.9277±\\pm0\.0412MFM / LAND12\.0705±\\pm0\.009512\.1656±\\pm0\.05227\.8205±\\pm0\.786310\.4768±\\pm0\.0637Curly\-FM12\.2962±\\pm0\.041412\.3920±\\pm0\.06874\.8562±\\pm0\.057611\.0235±\\pm0\.0520FFM \(ours\), deterministic10\.4319±\\pm0\.089610\.7413±\\pm0\.19275\.3793±\\pm0\.18488\.9033±\\pm0\.0400FFM \(ours\), with noise10\.4036±\\pm0\.027910\.7355±\\pm0\.05605\.0605±\\pm0\.16548\.9213±\\pm0\.0453*sampling floor**4\.1781**4\.1781**4\.4483**4\.3608*d=50d=50OT\-CFM13\.5145±\\pm0\.046613\.6811±\\pm0\.13148\.1252±\\pm0\.110315\.5900±\\pm0\.2287MFM / LAND12\.9526±\\pm0\.060513\.0996±\\pm0\.10298\.6687±\\pm0\.388014\.6185±\\pm0\.1085Curly\-FM13\.3381±\\pm0\.148213\.4440±\\pm0\.20697\.2931±\\pm0\.255815\.0427±\\pm0\.1562FFM \(ours\), deterministic12\.5784±\\pm0\.023512\.7715±\\pm0\.07197\.7975±\\pm0\.100513\.9368±\\pm0\.1217FFM \(ours\), with noise12\.5732±\\pm0\.057312\.7723±\\pm0\.09527\.7411±\\pm0\.129713\.7800±\\pm0\.0861*sampling floor**6\.7529**6\.7529**7\.0275**7\.2123*

MethodShared couplingMethod\-specific couplingGraph\-drift FM \(control\)11\.9409±\\pm0\.111011\.9223±\\pm0\.0909Anisotropic fixed\-time FM[Eq\.25](https://arxiv.org/html/2610.11318#A4.E25)11\.5712±\\pm0\.088011\.4344±\\pm0\.1067Second\-moment FFM \(m=0m=0\)11\.5424±\\pm0\.122611\.4967±\\pm0\.0878FFM, one\-form deleted \(existing ablation\)11\.1740±\\pm0\.075911\.1183±\\pm0\.0774FFM \(ours\)10\.3967±\\pm0\.137110\.3960±\\pm0\.1396

Figure 3:Pancreas trajectory reconstruction\. Left: Two\-dimensional representation\. Top right:W2W\_\{2\}att=1/2t=1/2, the averageW¯2\\overline\{W\}\_\{2\}over 11 fixed evaluation times in\[1/3,2/3\]\[1/3,2/3\], endpointW2W\_\{2\}, andW2,FW\_\{2,F\}Bottom right: same\-information objective ablation atd=20d=20\. “Shared” fixes the endpoint coupling “method\-specific” constructs the coupling separately for each method\. Ablation values areW2​\(t=1/2\)W\_\{2\}\(t=1/2\), mean±\\pmpopulation standard deviation over five seeds\.
#### FFM integrates diverse sources of dynamical information through a common transition representation:

To demonstrate that FFM can integrate heterogeneous dynamical information through a common transition representation, we evaluate 2,901 lineage\-recorded cells from the iTracer hindbrain organoid dataset\[[19](https://arxiv.org/html/2610.11318#bib.bib20)\], withholding the middle diffusion\-pseudotime tertile\. Lineage information is encoded by barcode families and finer\-grained scar families\. We construct transition kernels from expression alone and augment them with RNA velocity, lineage similarity derived from barcode and scar families, or both\. Each FFM variant achieves lower mean midpointW2W\_\{2\}than OT\-CFM and MFM across the evaluated representations[Table1](https://arxiv.org/html/2610.11318#S5.T1.fig1)\. We further found that the optimal kernel depends on the representation dimensionality\. Velocity performs best in two dimensions, lineage plus velocity in 20 dimensions, and expression alone in 50 dimensions\. These results show that FFM can incorporate heterogeneous sources of transition information, while the utility of each modality depends on how well it informs the transition kernel\.

Methodd=2d=2midW2W\_\{2\}d=20d=20midW2W\_\{2\}d=50d=50midW2W\_\{2\}OT\-CFM1\.3631±0\.03681\.3631\\pm 0\.036815\.9054±0\.080415\.9054\\pm 0\.080418\.8807±0\.117218\.8807\\pm 0\.1172MFM / LAND1\.3563±0\.05371\.3563\\pm 0\.053715\.0519±0\.225015\.0519\\pm 0\.225018\.2036±0\.266318\.2036\\pm 0\.2663FFM, expression only1\.7162±0\.99191\.7162\\pm 0\.991912\.8597±0\.100712\.8597\\pm 0\.100716\.6979±0\.3020\\mathbf\{16\.6979\\pm 0\.3020\}FFM, \+ velocity1\.0204±0\.1040\\mathbf\{1\.0204\\pm 0\.1040\}12\.7288±0\.125312\.7288\\pm 0\.125316\.7371±0\.256616\.7371\\pm 0\.2566FFM, \+ lineage1\.3074±0\.10831\.3074\\pm 0\.108312\.7368±0\.145512\.7368\\pm 0\.145516\.7415±0\.242616\.7415\\pm 0\.2426FFM, \+ lineage \+ velocity1\.1773±0\.14041\.1773\\pm 0\.140412\.7121±0\.1314\\mathbf\{12\.7121\\pm 0\.1314\}16\.9196±0\.270016\.9196\\pm 0\.2700

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

Appendix[AppendixC](https://arxiv.org/html/2610.11318#A3)reports runtime and parameter\-sensitivity analyses, additional results on erythroid differentiation\[[36](https://arxiv.org/html/2610.11318#bib.bib23)\]as used in used in[Petrović et al\. \[35\]](https://arxiv.org/html/2610.11318#bib.bib10), and further details of the benchmarks above\.

## 6 Discussion

We present Finsler Flow Matching \(FFM\), a framework for transforming discrete Markov transition graphs into continuous stochastic dynamics through a transition\-derived Finsler geometry\. Its joint first\- and second\-moment construction captures directional preference and anisotropic accessibility, which guide source–target coupling, geodesic interpolation, and stochastic conditional paths\. Our experiments suggest that this is particularly useful for strongly directional or anisotropic dynamics\. On the synthetic Sheet benchmark, ablating either component degrades intermediate\-state recovery, while on Pancreas the advantage of FFM becomes more pronounced in higher\-dimensional representations\. The iTracer experiment further demonstrates the flexibility of the Markov transition kernel as an interface between heterogeneous measurements and generative dynamics\. Expression, RNA velocity, lineage information, and their combinations can all be incorporated through the same representation\. Nevertheless, additional modalities do not uniformly improve recovery\. FFM therefore inherits both the information and the limitations of the supplied transition kernel, highlighting the importance of how experimental measurements are translated into transition probabilities\. These findings should be interpreted within the scope of the supplied dynamical prior\. FFM does not resolve the non\-identifiability of cellular dynamics from snapshots alone, and interpreting its geodesics as most\-probable transition curves requires the stated diffusion\-consistency and small\-noise assumptions\. Moreover, the Gaussian conditional paths used for stochastic training are tractable approximations rather than exact bridges of the graph\-implied reference diffusion\. Future work could explicitly model uncertainty in the transition graph and develop richer conditional path families\. Another promising direction is conditional FFM, in which experimental covariates or perturbations parameterize the induced geometry, allowing the model to capture condition\-specific changes in trajectory dynamics\.

### AI use statement

We used generative AI tools to assist with code generation, in particular code for generating synthetic datasets, as well as sentence reformulation and language polishing throughout the manuscript\. Furthermore, we used deep research tools to discover papers\. In particular, these tools identified connections to FW theory, which we subsequently investigated, verified, and developed in the context of our method\. AI tools were further used to assist with completing and checking mathematical proofs\. The theorem statements and underlying proof strategies were developed by the authors, and all AI\-assisted derivations were independently checked for correctness\. We reviewed all AI\-generated output and take full responsibility for all content in this paper\.

### Reproducibility Statement

We provide a code artifact containing the implementation of Finsler Flow Matching, preprocessing and transition\-kernel construction, synthetic\-data generation, training and evaluation scripts, benchmark\-specific configurations, hyperparameter settings, and the records used to generate the reported tables and figures\.

Software dependencies are pinned inrequirements\-lock\.txtto facilitate reconstruction of the experimental environment\. Experimental architecture, optimization settings, training budgets, validation procedures, and numbers of random seeds are reported in the appendix\.

Dataset sources and preprocessing procedures are documented for each benchmark, and validation and test populations are kept disjoint during model and hyperparameter selection\. We additionally provide the configurations used for all reported methods and hyperparameter searches, including the data\-dependent scaling of the Finsler regularization parameters\.

Evaluation settings, including any subsampling used for optimal\-transport calculations, are recorded alongside the experiments\. Because some GPU operations are nondeterministic, repeated training runs are not guaranteed to be bitwise identical; we therefore report results across multiple random seeds\. The complete repository will be made publicly available upon publication\.

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## Appendix APrelimiaries

#### Finsler Geometry:

For a complete introduction to Finsler geometry we refer the reader to[Bao et al\. \[4\]](https://arxiv.org/html/2610.11318#bib.bib50)\. LetMMbe anm−m\-dimensional manifold embedded inℝd\\mathbb\{R\}^\{d\}\. The*tangent bundle*T​M≔⨆x∈MTx​MTM\\coloneqq\\bigsqcup\_\{x\\in M\}T\_\{x\}Mis the2​m2m\-dimensional manifold composed of pairs\(x,v\)\(x,v\)withx∈Mx\\in Mandv∈Tx​Mv\\in T\_\{x\}M, whereTx​MT\_\{x\}Mis the*tangent space*atxx\.

###### Definition 3\(Finsler structure\)\.

A*Finsler structure*onMMis a continuous functionF:T​M→ℝ≥0F:TM\\to\\mathbb\{R\}\_\{\\geq 0\}, which is smooth onT​M∖\{0\}TM\\setminus\\\{0\\\}, such that for all\(x,v\)∈T​M\(x,v\)\\in TM:

1. \(F1\)*Positive homogeneity:*F⁡\(x,λ​v\)=λ​F​\(x,v\)F\(x,\\lambda v\)=\\lambda F\(x,v\)for allλ\>0\\lambda\>0\.
2. \(F2\)*Strong convexity:*the Hessian matrixgi​j​\(x,v\)=12​∂2F2/∂vi​∂vjg\_\{ij\}\(x,v\)=\\tfrac\{1\}\{2\}\\partial^\{2\}F^\{2\}/\\partial v^\{i\}\\partial v^\{j\}is positive definite forv≠0v\\neq 0\.
3. \(F3\)*Positivity:*F⁡\(x,v\)\>0F\(x,v\)\>0forv≠0v\\neq 0\.

In general, we have thatF⁡\(x,−v\)≠F⁡\(x,v\)F\(x,\-v\)\\neq F\(x,v\)hence, the cost of moving in directionvvis different from the cost of moving in direction−v\-v\. A Riemannian metric is the special caseF⁡\(x,v\)=v⊤​G​\(x\)​vF\(x,v\)=\\sqrt\{v^\{\\top\}G\(x\)v\}, which additionally satisfiesF⁡\(x,−v\)=F⁡\(x,v\)F\(x,\-v\)=F\(x,v\)\. Analogously to Riemannian Geometry, one can define a notion of length and geodesics\.

###### Definition 4\(Length and geodesics\)\.

For an absolutely continuous curveγ:\[0,1\]→M\\gamma:\[0,1\]\\to M, define

LF​\(γ\)=∫01F⁡\(γ⁡\(t\),γ˙​\(t\)\)​𝑑t,EF​\(γ\)=∫01F​\(γ⁡\(t\),γ˙​\(t\)\)2​𝑑t\.L\_\{F\}\(\\gamma\)=\\int\_\{0\}^\{1\}F\(\\gamma\(t\),\\dot\{\\gamma\}\(t\)\)\\,\\mathrm\{d\}t,\\qquad E\_\{F\}\(\\gamma\)=\\int\_\{0\}^\{1\}F\(\\gamma\(t\),\\dot\{\\gamma\}\(t\)\)^\{2\}\\,\\mathrm\{d\}t\.A*geodesic*betweenx0,x1∈ℝdx\_\{0\},x\_\{1\}\\in\\mathbb\{R\}^\{d\}is a curve minimizingLFL\_\{F\}subject toγ⁡\(0\)=x0\\gamma\(0\)=x\_\{0\},γ⁡\(1\)=x1\\gamma\(1\)=x\_\{1\}\. Note that energy minimizers are constant\-speed parametrized geodesics\. Furthermore, we define the*length metric*onMMas:

cF​\(x0,x1\)=infγ∫01F⁡\(γt,γ˙t\)​𝑑t,c\_\{F\}\(x\_\{0\},x\_\{1\}\)=\\inf\_\{\\gamma\}\\int\_\{0\}^\{1\}F\(\\gamma\_\{t\},\\dot\{\\gamma\}\_\{t\}\)\\,dt,where the infimum is taken over curves satisfyingγ0=x0,γ1=x1\\gamma\_\{0\}=x\_\{0\},\\ \\gamma\_\{1\}=x\_\{1\}\.

We are particularly interested in the special case called aRanders Metric, whereG⁡\(x\)G\(x\)is a smooth Riemannian metric andβ⁡\(x\)\\beta\(x\)a smooth11\-form and we define

F⁡\(x,v\)=v⊤​G​\(x\)​v\+β​\(x\)⊤​v\.F\(x,v\)=\\sqrt\{v^\{\\top\}G\(x\)v\}\\;\+\\;\\beta\(x\)^\{\\top\}v\.\(14\)
###### Lemma 5\.

\[[Bao et al\. \[4\]](https://arxiv.org/html/2610.11318#bib.bib50)\] The Randers structure \([14](https://arxiv.org/html/2610.11318#A1.E14)\) is a Finsler structure if and only if‖β⁡\(x\)‖G−1​\(x\)<1\\left\\lVert\\beta\(x\)\\right\\rVert\_\{G^\{\-1\}\(x\)\}<1pointwise, where‖u‖G−12=u⊤​G−1​u\\left\\lVert u\\right\\rVert\_\{G^\{\-1\}\}^\{2\}=u^\{\\top\}G^\{\-1\}u\.

###### Definition 7\(PP\-derived Randers metric\)\.

APP\-derived Randers metric is a Randers metric

FP​\(xi,v\)=v⊤​GP​\(xi\)​v\+βP​\(xi\)⊤​vF\_\{P\}\(x\_\{i\},v\)=\\sqrt\{v^\{\\top\}G\_\{P\}\(x\_\{i\}\)v\}\+\\beta\_\{P\}\(x\_\{i\}\)^\{\\top\}vwhose metric tensorGP​\(xi\)G\_\{P\}\(x\_\{i\}\)and one\-formβP​\(xi\)\\beta\_\{P\}\(x\_\{i\}\)are constructed from local transition statistics ofPP, with

GP​\(xi\)∈SPD⁡\(d\),‖βP​\(xi\)‖GP​\(xi\)−1<1\.G\_\{P\}\(x\_\{i\}\)\\in\\mathrm\{SPD\}\(d\),\\qquad\\\|\\beta\_\{P\}\(x\_\{i\}\)\\\|\_\{G\_\{P\}\(x\_\{i\}\)^\{\-1\}\}<1\.

For example, ifCP​\(xi\)C\_\{P\}\(x\_\{i\}\)denotes a local transition second\-moment tensor, one may consider the family

GP,q​\(xi\)=\(τ​I\+CP​\(xi\)\)−q,q\>0,G\_\{P,q\}\(x\_\{i\}\)=\\bigl\(\\tau I\+C\_\{P\}\(x\_\{i\}\)\\bigr\)^\{\-q\},\\qquad q\>0,together with any admissible transition\-derived one\-formβP\\beta\_\{P\}\. Forq=1q=1, this is closely related to local\-kernel geometries, where the second moment determines the induced Riemannian metric\[[8](https://arxiv.org/html/2610.11318#bib.bib43)\]\.

#### Optimal Transport:

Optimal transport \(OT\) was originally formulated by Monge as the problem of finding a deterministic transport mapT:ℝd→ℝdT:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}pushingp0p\_\{0\}ontop1p\_\{1\}while minimizing the transport cost:

infT:T\#​p0=p1∫ℝdc\(x,T\(x\)\)dp0\(x\)\.\\inf\_\{T:\\,T\_\{\\\#\}p\_\{0\}=p\_\{1\}\}\\int\_\{\\mathbb\{R\}^\{d\}\}c\(x,T\(x\)\)\\,\\mathrm\{d\}p\_\{0\}\(x\)\.Since a transport map cannot split mass, such a minimizers does not exist in general\. For example, if

p0=δ0,p1=12​δ−1\+12​δ1,p\_\{0\}=\\delta\_\{0\},\\qquad p\_\{1\}=\\frac\{1\}\{2\}\\delta\_\{\-1\}\+\\frac\{1\}\{2\}\\delta\_\{1\},no mapTTcan satisfyT\#​p0=p1T\_\{\\\#\}p\_\{0\}=p\_\{1\}\. The Kantorovich relaxation resolves this issue by optimizing instead over transport plansπ∈Π⁡\(p0,p1\)\\pi\\in\\Pi\(p\_\{0\},p\_\{1\}\):

infπ∈Π⁡\(p0,p1\)∫ℝd×ℝdc⁡\(x0,x1\)​𝑑π​\(x0,x1\)\.\\inf\_\{\\pi\\in\\Pi\(p\_\{0\},p\_\{1\}\)\}\\int\_\{\\mathbb\{R\}^\{d\}\\times\\mathbb\{R\}^\{d\}\}c\(x\_\{0\},x\_\{1\}\)\\,\\mathrm\{d\}\\pi\(x\_\{0\},x\_\{1\}\)\.In the example above, an admissible plan would be

π=12​δ\(0,−1\)\+12​δ\(0,1\)\.\\pi=\\frac\{1\}\{2\}\\delta\_\{\(0,\-1\)\}\+\\frac\{1\}\{2\}\\delta\_\{\(0,1\)\}\.Under the Euclidean squared distancec\(x0,x1\)=12∥x0−x1∥2c\(x\_\{0\},x\_\{1\}\)=\\frac\{1\}\{2\}\\\|\{\}x\_\{0\}\-x\_\{1\}\\\|\{\}^\{2\}\(absolute continuity ofp0p\_\{0\}\)*Brenier’s theorem*guarantees that the optimal planπ⋆\\pi^\{\\star\}exists and is unique\. Furthermore, there is a unique optimal transport mapT=∇φT=\\nabla\\varphi, for some convex functionφ\\varphi\. The classical*Benamou–Brenier theorem*establishes that this static transport problem is equivalent to a dynamic formulation over density pathsρt\\rho\_\{t\}and velocity fieldsvtv\_\{t\}:

W22​\(p0,p1\)=inf\(ρ,v\)∫01∫ℝd\|vt​\(x\)\|2​d​ρt​\(x\),𝑑ts\.t\.∂tρt\+∇⋅\(ρt​vt\)=0,ρ0=p0,ρ1=p1\.W\_\{2\}^\{2\}\(p\_\{0\},p\_\{1\}\)=\\inf\_\{\(\\rho,v\)\}\\int\_\{0\}^\{1\}\\int\_\{\\mathbb\{R\}^\{d\}\}\|v\_\{t\}\(x\)\|^\{2\}\\mathrm\{d\}\\rho\_\{t\}\(x\),\\mathrm\{d\}t\\quad\\text\{s\.t\.\}\\quad\\partial\_\{t\}\\rho\_\{t\}\+\\nabla\\cdot\(\\rho\_\{t\}v\_\{t\}\)=0,\\quad\\rho\_\{0\}=p\_\{0\},\\quad\\rho\_\{1\}=p\_\{1\}\.When substituting the Euclidean cost with the Finsler geodesic action costcF​\(x0,x1\)c\_\{F\}\(x\_\{0\},x\_\{1\}\)defined in \([1](https://arxiv.org/html/2610.11318#S2.E1)\), the static Kantorovich problem remains uniquely attained\[[34](https://arxiv.org/html/2610.11318#bib.bib47), Theorem 4\.10\]\. Moreover, the static problem coincides with the dynamic Finsler Benamou\-Brenier formulation:

W2,F2​\(p0,p1\)≔infπ∈Π⁡\(p0,p1\)∫ℝd×ℝdcF​\(x0,x1\)2​𝑑π​\(x0,x1\)=inf\(ρ,v\)∫01∫ℝdF​\(x,vt​\(x\)\)2​d​ρt​\(x\)​𝑑t\\displaystyle W\_\{2,F\}^\{2\}\(p\_\{0\},p\_\{1\}\)\\coloneqq\\inf\_\{\\pi\\in\\Pi\(p\_\{0\},p\_\{1\}\)\}\\int\_\{\\mathbb\{R\}^\{d\}\\times\\mathbb\{R\}^\{d\}\}c\_\{F\}\(x\_\{0\},x\_\{1\}\)^\{2\}\\mathrm\{d\}\\pi\(x\_\{0\},x\_\{1\}\)=\\inf\_\{\(\\rho,v\)\}\\int\_\{0\}^\{1\}\\int\_\{\\mathbb\{R\}^\{d\}\}F\(x,v\_\{t\}\(x\)\)^\{2\}\\mathrm\{d\}\\rho\_\{t\}\(x\)\\mathrm\{d\}tsubject to the continuity equation∂tρt\+∇⋅\(ρt​vt\)=0\\partial\_\{t\}\\rho\_\{t\}\+\\nabla\\cdot\(\\rho\_\{t\}v\_\{t\}\)=0with endpoint conditionsρ0=p0\\rho\_\{0\}=p\_\{0\}andρ1=p1\\rho\_\{1\}=p\_\{1\}\[[33](https://arxiv.org/html/2610.11318#bib.bib45), Theorem 7\.3 and Corollary 7\.5\]\.

#### Flow Matching:

We give a brief introduction to Flow Matching for a more in depth treatment consider[Holderrieth and Erives \[21\]](https://arxiv.org/html/2610.11318#bib.bib14)\. The idea behind flow matching is to find a vector field that transports one distribution into another and can generate samples by integrating said vector field\. More formally one can think of it as trying to find a vector fieldut:\[0,1\)×ℝn→ℝnu\_\{t\}:\[0,1\)\\times\\mathbb\{R\}^\{n\}\\rightarrow\\mathbb\{R\}^\{n\}such that

\{dd​t​xt=ut​\(xt\),x0∼p0,x1∼p1\.\\displaystyle\\left\\\{\\begin\{aligned\} &\\frac\{d\}\{dt\}x\_\{t\}=u\_\{t\}\(x\_\{t\}\),\\\\ &x\_\{0\}\\sim p\_\{0\},x\_\{1\}\\sim p\_\{1\}\.\\end\{aligned\}\\right\.Once such a vector field has been learned, samples from

p1p\_\{1\}can be generated by sampling

x0∼p0x\_\{0\}\\sim p\_\{0\}and integrating the vector field from

t=0t=0to

t=1t=1\. In general, this vector field is not unique, which gives us some choice\. For example, if

p0=p1=pp\_\{0\}=p\_\{1\}=p, then any vector field satisfying

∇⋅\(p​u\)=0\\nabla\\cdot\(pu\)=0leaves the distribution

ppinvariant\. More generally, the endpoint distributions

p0p\_\{0\}and

p1p\_\{1\}do not uniquely determine the intermediate probability path\. We exploit this freedom by choosing a probability density path

\(pt\)t∈\[0,1\]\(p\_\{t\}\)\_\{t\\in\[0,1\]\}connecting

p0p\_\{0\}and

p1p\_\{1\}, and then finding a vector field whose flow realizes this path\. Even for a fixed path

ptp\_\{t\}, such a vector field need not be unique \(one could make the choice optimal by choosing

utu\_\{t\}as the minimizer of some action\)\.

In general, we do not have

p0p\_\{0\}and

p1p\_\{1\}available in closed form, and hence the corresponding marginal path

ptp\_\{t\}is intractable\. The idea is to condition the path on samples; then the marginal distributions become Dirac distributions\. We therefore introduce a coupling

π∈Π⁡\(p0,p1\)\\pi\\in\\Pi\(p\_\{0\},p\_\{1\}\), i\.e\., a probability distribution on

ℝn×ℝn\\mathbb\{R\}^\{n\}\\times\\mathbb\{R\}^\{n\}with marginals

p0p\_\{0\}and

p1p\_\{1\}, and define a tractable conditional probability path

pt​\(x∣x0,x1\)p\_\{t\}\(x\\mid x\_\{0\},x\_\{1\}\)with marginals

δx0\\delta\_\{x\_\{0\}\}and

δx1\\delta\_\{x\_\{1\}\}\. The corresponding marginal path is

pt​\(x\)=∫ℝn×ℝnpt​\(x∣x0,x1\)​𝑑π​\(x0,x1\)\.\\displaystyle p\_\{t\}\(x\)=\\int\_\{\\mathbb\{R\}^\{n\}\\times\\mathbb\{R\}^\{n\}\}p\_\{t\}\(x\\mid x\_\{0\},x\_\{1\}\)\\,d\\pi\(x\_\{0\},x\_\{1\}\)\.For every endpoint pair\(x0,x1\)\(x\_\{0\},x\_\{1\}\), we define a conditional vector fieldut​\(x∣x0,x1\)u\_\{t\}\(x\\mid x\_\{0\},x\_\{1\}\)that solves

\{dd​t​xt=ut​\(xt∣x0,x1\),xt∼pt\(⋅∣x0,x1\)for allt∈\[0,1\]\.\\displaystyle\\left\\\{\\begin\{aligned\} &\\frac\{d\}\{dt\}x\_\{t\}=u\_\{t\}\(x\_\{t\}\\mid x\_\{0\},x\_\{1\}\),\\\\ &x\_\{t\}\\sim p\_\{t\}\(\\cdot\\mid x\_\{0\},x\_\{1\}\)\\text\{ for all \}t\\in\[0,1\]\.\\end\{aligned\}\\right\.Surprisingly, although the marginal vector field is generally intractable, the Conditional Flow Matching objective

ℒCFM​\(θ\)=𝔼\(x0,x1\)∼π,t∼𝒰\[0,1\],x∼pt\(⋅∣x0,x1\)​\[‖vθ​\(t,x\)−ut​\(x∣x0,x1\)‖22\]\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{CFM\}\}\(\\theta\)=\\mathbb\{E\}\_\{\\begin\{subarray\}\{c\}\(x\_\{0\},x\_\{1\}\)\\sim\\pi,\\,t\\sim\\mathcal\{U\}\[0,1\],\\\\ x\\sim p\_\{t\}\(\\cdot\\mid x\_\{0\},x\_\{1\}\)\\end\{subarray\}\}\\left\[\\left\\\|v\_\{\\theta\}\(t,x\)\-u\_\{t\}\(x\\mid x\_\{0\},x\_\{1\}\)\\right\\\|\_\{2\}^\{2\}\\right\]has the same gradient with respect toθ\\thetaas the corresponding unconditional Flow Matching objective\. Hence, we can learn the marginal vector field using only samples from tractable conditional probability paths, resulting in a simulation\-free training procedure\.

#### Simulation free Score and Flow Matching:

Here we introduce some of the ideas of[Tong et al\. \[43\]](https://arxiv.org/html/2610.11318#bib.bib38)333Which the authors highly suggest reading in full\.\. Similarly to the previous section, we want to find a vector fieldu~t\\tilde\{u\}\_\{t\}that satisfies\.

\{d​xt=u~t​\(xt\)​d​t\+gt​d​Wt,x0∼p0,x1∼p1\.\\displaystyle\\left\\\{\\begin\{aligned\} &\\mathrm\{d\}x\_\{t\}=\\tilde\{u\}\_\{t\}\(x\_\{t\}\)\\mathrm\{d\}t\+g\_\{t\}dW\_\{t\},\\\\ &x\_\{0\}\\sim p\_\{0\},x\_\{1\}\\sim p\_\{1\}\.\\end\{aligned\}\\right\.where

gtg\_\{t\}is some scalar valued function controlling the diffusion strength and

WtW\_\{t\}is a standard Brownian motion, i\.e\. a continuous stochastic process with independent Gaussian increments\. Hence, instead of an ODE we treat a stochastic differential equation \(SDE\)\.

In a first step we go back again the setting of the previous section and choose a conditional probability path

pt​\(x∣x0,x1\)p\_\{t\}\(x\\mid x\_\{0\},x\_\{1\}\)and learn a conditional vector field such that the corresponding ODE

dd​t​xt=ut​\(xt\)\\displaystyle\\frac\{d\}\{dt\}x\_\{t\}=u\_\{t\}\(x\_\{t\}\)satisfiesxt∼ptx\_\{t\}\\sim p\_\{t\}for everytt\. We additionally introduce the conditional score of the probability path,

st​\(x∣x0,x1\):=∇log⁡pt​\(x∣x0,x1\)\.\\displaystyle s\_\{t\}\(x\\mid x\_\{0\},x\_\{1\}\):=\\nabla\\log p\_\{t\}\(x\\mid x\_\{0\},x\_\{1\}\)\.Note that similarly to flow matching, also the conditional score matching loss has the same gradient unconditional one\. Using this score, the deterministic flow can be transformed into a stochastic process,

d​xt=\[vt​\(xt\)\+12​gt​gt⊤​st​\(xt\)\]​d​t\+gt​d​Wt\.\\displaystyle\\mathrm\{d\}x\_\{t\}=\\left\[v\_\{t\}\(x\_\{t\}\)\+\\frac\{1\}\{2\}g\_\{t\}g\_\{t\}^\{\\top\}s\_\{t\}\(x\_\{t\}\)\\right\]\\mathrm\{d\}t\+g\_\{t\}\\,dW\_\{t\}\.The Fokker\-Planck equation of this SDE is given by

∂tpt\\displaystyle\\partial\_\{t\}p\_\{t\}=−∇⋅\(pt\[vt\+12gtgt⊤st\]\)\+12∇⋅\(gtgt⊤∇pt\)\\displaystyle=\-\\nabla\\cdot\\left\(p\_\{t\}\\left\[v\_\{t\}\+\\frac\{1\}\{2\}g\_\{t\}g\_\{t\}^\{\\top\}s\_\{t\}\\right\]\\right\)\+\\frac\{1\}\{2\}\\nabla\\cdot\\left\(g\_\{t\}g\_\{t\}^\{\\top\}\\nabla p\_\{t\}\\right\)=−∇⋅\(ptvt\),\\displaystyle=\-\\nabla\\cdot\(p\_\{t\}v\_\{t\}\),where we usedpt​st=∇ptp\_\{t\}s\_\{t\}=\\nabla p\_\{t\}\. Hence, the ODE and the corresponding SDE satisfy the same Fokker–Planck equation and therefore have the same marginalsXt∼ptX\_\{t\}\\sim p\_\{t\}\. The score thus allows us to add stochasticity to the learned flow without changing the prescribed probability path\.

## Appendix BTheoretical Contributions

#### Transition\-supported geodesics\.

For a thresholdτ\>0\\tau\>0, define the high\-probability transition edges

Eτ:=\{\(i,j\):Pi​j≥τ\}\.E\_\{\\tau\}:=\\\{\(i,j\):P\_\{ij\}\\geq\\tau\\\}\.For endpointsx0,x1∈𝒟x\_\{0\},x\_\{1\}\\in\\mathcal\{D\}, define the transition\-supported subset

𝒮τ\(x0,x1\):=\{xi∈𝒟:xilies on a directed path fromx0tox1using only edges inEτ\}\.\\mathcal\{S\}\_\{\\tau\}\(x\_\{0\},x\_\{1\}\):=\\left\\\{x\_\{i\}\\in\\mathcal\{D\}:x\_\{i\}\\text\{ lies on a directed path from $x\_\{0\}$ to $x\_\{1\}$ using only edges in \}E\_\{\\tau\}\\right\\\}\.We assume that𝒮τ​\(x0,x1\)\\mathcal\{S\}\_\{\\tau\}\(x\_\{0\},x\_\{1\}\)is non\-empty\. Forr\>0r\>0, define itsrr\-neighborhood by

Ur​\(𝒮τ\):=\{x∈ℝd:dist⁡\(x,𝒮τ\)<r\},dist⁡\(x,𝒮τ\):=miny∈𝒮τ⁡‖x−y‖2\.U\_\{r\}\(\\mathcal\{S\}\_\{\\tau\}\):=\\left\\\{x\\in\\mathbb\{R\}^\{d\}:\\operatorname\{dist\}\(x,\\mathcal\{S\}\_\{\\tau\}\)<r\\right\\\},\\qquad\\operatorname\{dist\}\(x,\\mathcal\{S\}\_\{\\tau\}\):=\\min\_\{y\\in\\mathcal\{S\}\_\{\\tau\}\}\\\|x\-y\\\|\_\{2\}\.Furthermore, define

cr:=minr/2≤dist⁡\(x,𝒮τ\)≤r‖v‖2=1⁡FP​\(x,v\)\>0\.c\_\{r\}:=\\min\_\{\\begin\{subarray\}\{c\}r/2\\leq\\operatorname\{dist\}\(x,\\mathcal\{S\}\_\{\\tau\}\)\\leq r\\\\ \\\|v\\\|\_\{2\}=1\\end\{subarray\}\}F\_\{P\}\(x,v\)\>0\.The positivity follows sinceFPF\_\{P\}is continuous and strictly positive forv≠0v\\neq 0, while the domain of minimization is compact\. For our particularPP\-derived Randers metric \([9](https://arxiv.org/html/2610.11318#S4.E9)\),crc\_\{r\}can be related directly to the transition moments\. By Cauchy–Schwarz,

FP​\(x,v\)≥‖m⁡\(x\)‖Cρ​\(x\)−12\+λ2−‖m⁡\(x\)‖Cρ​\(x\)−1λmax​\(Cρ​\(x\)\)​‖v‖2\.F\_\{P\}\(x,v\)\\geq\\frac\{\\sqrt\{\\\|m\(x\)\\\|\_\{C\_\{\\rho\}\(x\)^\{\-1\}\}^\{2\}\+\\lambda^\{2\}\}\-\\\|m\(x\)\\\|\_\{C\_\{\\rho\}\(x\)^\{\-1\}\}\}\{\\sqrt\{\\lambda\_\{\\max\}\(C\_\{\\rho\}\(x\)\)\}\}\\\|v\\\|\_\{2\}\.Hence, regions and directions with largePP\-weighted transition covariance are relatively cheap, whereas regions with little transition support, corresponding to small eigenvalues ofCρC\_\{\\rho\}, incur a larger cost\. Thus annuli farther from𝒮τ\\mathcal\{S\}\_\{\\tau\}have a larger escape costcrc\_\{r\}\.

###### Proposition 8\(Confinement and directional preference of geodesics\)\.

LetFPF\_\{P\}be the Randers metric induced by the first and second transition momentsmmandCCin \([9](https://arxiv.org/html/2610.11318#S4.E9)\)\. Suppose there exists a curveγ¯\\bar\{\\gamma\}connectingx0x\_\{0\}andx1x\_\{1\}insideUr/2​\(𝒮τ\)U\_\{r/2\}\(\\mathcal\{S\}\_\{\\tau\}\)\. If

LFP​\(γ¯\)<cr​r,L\_\{F\_\{P\}\}\(\\bar\{\\gamma\}\)<c\_\{r\}r,then every minimizing geodesic connectingx0x\_\{0\}andx1x\_\{1\}is contained inUr​\(𝒮τ\)U\_\{r\}\(\\mathcal\{S\}\_\{\\tau\}\)\.

Moreover, for fixed local speed‖v‖Cρ​\(x\)−1\\\|v\\\|\_\{C\_\{\\rho\}\(x\)^\{\-1\}\}, the Finsler cost is minimized whenvvis positively aligned with the first transition momentm⁡\(x\)m\(x\)\.

###### Proof\.

Suppose a minimizing geodesicγ⋆\\gamma^\{\\star\}leavesUr​\(𝒮τ\)U\_\{r\}\(\\mathcal\{S\}\_\{\\tau\}\)\. Sincex0,x1∈𝒮τx\_\{0\},x\_\{1\}\\in\\mathcal\{S\}\_\{\\tau\}, it must cross the annulus

r/2≤dist⁡\(x,𝒮τ\)≤rr/2\\leq\\operatorname\{dist\}\(x,\\mathcal\{S\}\_\{\\tau\}\)\\leq ronce when leaving and once when returning\. Sincex↦dist⁡\(x,𝒮τ\)x\\mapsto\\operatorname\{dist\}\(x,\\mathcal\{S\}\_\{\\tau\}\)is11\-Lipschitz, its total Euclidean length in this annulus is at leastrr\. By positive homogeneity and the definition ofcrc\_\{r\},

LFP​\(γ⋆\)≥cr​r\>LFP​\(γ¯\),L\_\{F\_\{P\}\}\(\\gamma^\{\\star\}\)\\geq c\_\{r\}r\>L\_\{F\_\{P\}\}\(\\bar\{\\gamma\}\),contradicting minimality\. Henceγ⋆⊂Ur​\(𝒮τ\)\\gamma^\{\\star\}\\subset U\_\{r\}\(\\mathcal\{S\}\_\{\\tau\}\)\.

For the directional claim, writeA⁡\(x\)=Cρ​\(x\)−1A\(x\)=C\_\{\\rho\}\(x\)^\{\-1\}\. For fixed‖v‖A⁡\(x\)\\\|v\\\|\_\{A\(x\)\}, the first term of

FP​\(x,v\)=‖m⁡\(x\)‖A⁡\(x\)2\+λ2​‖v‖A⁡\(x\)−⟨v,m⁡\(x\)⟩A⁡\(x\)F\_\{P\}\(x,v\)=\\sqrt\{\\\|m\(x\)\\\|\_\{A\(x\)\}^\{2\}\+\\lambda^\{2\}\}\\,\\\|v\\\|\_\{A\(x\)\}\-\\langle v,m\(x\)\\rangle\_\{A\(x\)\}is constant\. Thus minimizingFPF\_\{P\}is equivalent to maximizing⟨v,m⁡\(x\)⟩A⁡\(x\)\\langle v,m\(x\)\\rangle\_\{A\(x\)\}, which by Cauchy–Schwarz is attained whenvvis a positive multiple ofm⁡\(x\)m\(x\)\. ∎

#### Uniform Convergence of Empirical Drift and Diffusion:

LetYTY\_\{T\}be a stochastic process inℝd\\mathbb\{R\}^\{d\}governed by the Stochastic Differential Equation \(SDE\)

d​YT=b0​\(YT\)​d​t\+Σsde​\(YT\)​d​Wt,\\displaystyle dY\_\{T\}=b\_\{0\}\(Y\_\{T\}\)\\,dt\+\\Sigma\_\{\\mathrm\{sde\}\}\(Y\_\{T\}\)\\,dW\_\{t\},\(15\)whereb0∈C∞​\(ℝd,ℝd\)b\_\{0\}\\in C^\{\\infty\}\(\\mathbb\{R\}^\{d\},\\mathbb\{R\}^\{d\}\)is the drift vector,WtW\_\{t\}is a standarddd\-dimensional Brownian motion andD⁡\(x\)≔Σsde​\(x\)​Σsde​\(x\)⊤D\(x\)\\coloneqq\\Sigma\_\{\\mathrm\{sde\}\}\(x\)\\Sigma\_\{\\mathrm\{sde\}\}\(x\)^\{\\top\}, is the smooth diffusion matrix\.

###### Definition 9\(Transition Operator and Infinitesimal Generator\)\.

The transition operator \(semigroup\)TtT\_\{t\}acting on a test functionf∈Cc∞​\(ℝd\)f\\in C\_\{c\}^\{\\infty\}\(\\mathbb\{R\}^\{d\}\)is defined by

Tt​f​\(x\)≔𝔼⁡\[f⁡\(YT\)∣Y0=x\]\.T\_\{t\}f\(x\)\\coloneqq\\mathbb\{E\}\\left\[f\(Y\_\{T\}\)\\mid Y\_\{0\}=x\\right\]\.The infinitesimal generatorLLof the processYTY\_\{T\}is the operator

Lf\(x\)≔limt→0\+Tt​f​\(x\)−f⁡\(x\)t=∇f\(x\)⊤b0\(x\)\+12Tr\(D\(x\)∇2f\(x\)\)\.Lf\(x\)\\coloneqq\\lim\_\{t\\to 0^\{\+\}\}\\frac\{T\_\{t\}f\(x\)\-f\(x\)\}\{t\}=\\nabla f\(x\)^\{\\top\}b\_\{0\}\(x\)\+\\frac\{1\}\{2\}\\mathrm\{Tr\}\\left\(D\(x\)\\nabla^\{2\}f\(x\)\\right\)\.

Suppose we observe samples\{xi\}i=1N⊂ℝd\\\{x\_\{i\}\\\}\_\{i=1\}^\{N\}\\subset\\mathbb\{R\}^\{d\}drawn i\.i\.d\. from a continuous densityp⁡\(x\)p\(x\)alongside a row\-stochastic transition matrixP∈ℝN×NP\\in\\mathbb\{R\}^\{N\\times N\}constructed at spatial scaleεN\>0\\varepsilon\_\{N\}\>0\.Thus the discrete transition operator is

\(P​f\)​\(xi\)=∑j=1NPi​j​f​\(xj\)\.\(Pf\)\(x\_\{i\}\)=\\sum\_\{j=1\}^\{N\}P\_\{ij\}f\(x\_\{j\}\)\.To link discrete dataset observations to the continuous diffusion process, we need the following assumptions\.

###### Assumption 10\.

The discrete time stepΔ​tN\\Delta t\_\{N\}scales quadratically with the bandwidth parameterεN\\varepsilon\_\{N\}

Δ​tN=c​εN2for a fixed constant​c\>0\.\(Time\-Space Scaling\)\\Delta t\_\{N\}=c\\varepsilon\_\{N\}^\{2\}\\quad\\text\{for a fixed constant \}c\>0\.\\text\{ \(Time\-Space Scaling\)\}

Informally, we say that a Markov chain with transition matrixPPis a discrete observation of a continuous stochastic processYTY\_\{T\}if its discrete transition operator matches the continuous process generatorLLover an infinitesimal time step\. Formally, we assume:

###### Assumption 11\(Uniform discrete\-generator consistency\)\.

Define the empirical generator

LN,εN​f​\(xi\)≔1Δ​tN​\(∑j=1NPi​j​f​\(xj\)−f⁡\(xi\)\)\.L\_\{N,\\varepsilon\_\{N\}\}f\(x\_\{i\}\)\\coloneqq\\frac\{1\}\{\\Delta t\_\{N\}\}\\left\(\\sum\_\{j=1\}^\{N\}P\_\{ij\}f\(x\_\{j\}\)\-f\(x\_\{i\}\)\\right\)\.We assume that for everyf∈Cc∞​\(ℝd\)f\\in C\_\{c\}^\{\\infty\}\(\\mathbb\{R\}^\{d\}\),

supxi∈K\|LN,εN​f​\(xi\)−L​f​\(xi\)\|→0\\sup\_\{x\_\{i\}\\in K\}\\left\|L\_\{N,\\varepsilon\_\{N\}\}f\(x\_\{i\}\)\-Lf\(x\_\{i\}\)\\right\|\\to 0asN→∞N\\to\\inftyandεN→0\\varepsilon\_\{N\}\\to 0\.

###### Assumption 12\.

We assume local support of the transition kernel on the region of interest: for every compactK⊂ℝdK\\subset\\mathbb\{R\}^\{d\}, there exists a compact neighborhoodK\+⊃KK^\{\+\}\\supset Ksuch that, for all sufficiently largeNN,

xi∈K,Pi​j\>0⟹xj∈K\+\.x\_\{i\}\\in K,\\quad P\_\{ij\}\>0\\qquad\\Longrightarrow\\qquad x\_\{j\}\\in K^\{\+\}\.

###### Proposition 13\(Recovery of drift and diffusion fromPP\)\.

Assume thatP=PN,εNP=P\_\{N,\\varepsilon\_\{N\}\}is row\-stochastic and that Assumptions[Assumption10](https://arxiv.org/html/2610.11318#Thmtheorem10)[Assumption11](https://arxiv.org/html/2610.11318#Thmtheorem11)and[Assumption12](https://arxiv.org/html/2610.11318#Thmtheorem12)hold\. Define

b^0,N​\(xi\)=1Δ​tN​∑j=1NPi​j​\(xj−xi\)​and​D^N​\(xi\)=1Δ​tN​∑j=1NPi​j​\(xj−xi\)​\(xj−xi\)⊤\.\\hat\{b\}\_\{0,N\}\(x\_\{i\}\)=\\frac\{1\}\{\\Delta t\_\{N\}\}\\sum\_\{j=1\}^\{N\}P\_\{ij\}\(x\_\{j\}\-x\_\{i\}\)\\text\{ and \}\\hat\{D\}\_\{N\}\(x\_\{i\}\)=\\frac\{1\}\{\\Delta t\_\{N\}\}\\sum\_\{j=1\}^\{N\}P\_\{ij\}\(x\_\{j\}\-x\_\{i\}\)\(x\_\{j\}\-x\_\{i\}\)^\{\\top\}\.Then

maxxi∈K⁡‖b^0,N​\(xi\)−b0​\(xi\)‖→0​and​maxxi∈K​‖D^N​\(xi\)−D⁡\(xi\)‖→0\.\\max\_\{x\_\{i\}\\in K\}\\\|\\hat\{b\}\_\{0,N\}\(x\_\{i\}\)\-b\_\{0\}\(x\_\{i\}\)\\\|\\to 0\\text\{ and \}\\max\_\{x\_\{i\}\\in K\}\\\|\\hat\{D\}\_\{N\}\(x\_\{i\}\)\-D\(x\_\{i\}\)\\\|\\to 0\.

###### Proof\.

Fix a compact setKKand letK\+K^\{\+\}be the compact neighborhood from[Assumption12](https://arxiv.org/html/2610.11318#Thmtheorem12)\. Choose a cutoffχ∈Cc∞​\(ℝd\)\\chi\\in C\_\{c\}^\{\\infty\}\(\\mathbb\{R\}^\{d\}\)such thatχ≡1\\chi\\equiv 1on a neighborhood ofK\+K^\{\+\}\. Fora,b∈\{1,…,d\}a,b\\in\\\{1,\\ldots,d\\\}define

q~a​\(x\):=χ⁡\(x\)​xa,q~a​b​\(x\):=χ⁡\(x\)​xa​xb\.\\widetilde\{q\}\_\{a\}\(x\):=\\chi\(x\)x\_\{a\},\\qquad\\widetilde\{q\}\_\{ab\}\(x\):=\\chi\(x\)x\_\{a\}x\_\{b\}\.Thenq~a,q~a​b∈Cc∞​\(ℝd\)\\widetilde\{q\}\_\{a\},\\widetilde\{q\}\_\{ab\}\\in C\_\{c\}^\{\\infty\}\(\\mathbb\{R\}^\{d\}\), so Assumption[Assumption11](https://arxiv.org/html/2610.11318#Thmtheorem11)applies\. Forxi∈Kx\_\{i\}\\in K[Assumption12](https://arxiv.org/html/2610.11318#Thmtheorem12)implies that everyxjx\_\{j\}withPi​j\>0P\_\{ij\}\>0lies inK\+K^\{\+\}, whereχ≡1\\chi\\equiv 1\. Hence

LN,εN​q~a​\(xi\)=LN,εN​qa​\(xi\),LN,εN​q~a​b​\(xi\)=LN,εN​qa​b​\(xi\)\.L\_\{N,\\varepsilon\_\{N\}\}\\widetilde\{q\}\_\{a\}\(x\_\{i\}\)=L\_\{N,\\varepsilon\_\{N\}\}q\_\{a\}\(x\_\{i\}\),\\qquad L\_\{N,\\varepsilon\_\{N\}\}\\widetilde\{q\}\_\{ab\}\(x\_\{i\}\)=L\_\{N,\\varepsilon\_\{N\}\}q\_\{ab\}\(x\_\{i\}\)\.Moreover, sinceχ≡1\\chi\\equiv 1on a neighborhood ofKK,

L​q~a​\(x\)=L​qa​\(x\),L​q~a​b​\(x\)=L​qa​b​\(x\),x∈K\.L\\widetilde\{q\}\_\{a\}\(x\)=Lq\_\{a\}\(x\),\\qquad L\\widetilde\{q\}\_\{ab\}\(x\)=Lq\_\{ab\}\(x\),\\qquad x\\in K\.Therefore Assumption[Assumption11](https://arxiv.org/html/2610.11318#Thmtheorem11)may be applied to the cutoff observables while yielding the desired generator identities forqaq\_\{a\}andqa​bq\_\{ab\}onKK\. SincePPis row\-stochastic,

LN,εN​qa​\(xi\)=1Δ​tN​∑j=1NPi​j​\(xj,a−xi,a\)=b^0,N,a​\(xi\)\.L\_\{N,\\varepsilon\_\{N\}\}q\_\{a\}\(x\_\{i\}\)=\\frac\{1\}\{\\Delta t\_\{N\}\}\\sum\_\{j=1\}^\{N\}P\_\{ij\}\(x\_\{j,a\}\-x\_\{i,a\}\)=\\hat\{b\}\_\{0,N,a\}\(x\_\{i\}\)\.Moreover,

L​qa​\(x\)=b0,a​\(x\)\.Lq\_\{a\}\(x\)=b\_\{0,a\}\(x\)\.The drift convergence therefore follows directly from Assumption[Assumption11](https://arxiv.org/html/2610.11318#Thmtheorem11)\. Expanding the increments for the second moment gives

LN,εN​qa​b​\(xi\)\\displaystyle L\_\{N,\\varepsilon\_\{N\}\}q\_\{ab\}\(x\_\{i\}\)=xi,a​b^0,N,b​\(xi\)\+xi,b​b^0,N,a​\(xi\)\+D^N,a​b​\(xi\)\.\\displaystyle=x\_\{i,a\}\\hat\{b\}\_\{0,N,b\}\(x\_\{i\}\)\+x\_\{i,b\}\\hat\{b\}\_\{0,N,a\}\(x\_\{i\}\)\+\\hat\{D\}\_\{N,ab\}\(x\_\{i\}\)\.On the other hand, applying the diffusion generator yields

L​qa​b​\(x\)=xa​b0,b​\(x\)\+xb​b0,a​\(x\)\+Da​b​\(x\)\.Lq\_\{ab\}\(x\)=x\_\{a\}b\_\{0,b\}\(x\)\+x\_\{b\}b\_\{0,a\}\(x\)\+D\_\{ab\}\(x\)\.Consequently,

D^N,a​b​\(xi\)−Da​b​\(xi\)\\displaystyle\\hat\{D\}\_\{N,ab\}\(x\_\{i\}\)\-D\_\{ab\}\(x\_\{i\}\)=\(LN,εN​qa​b​\(xi\)−L​qa​b​\(xi\)\)\\displaystyle=\\bigl\(L\_\{N,\\varepsilon\_\{N\}\}q\_\{ab\}\(x\_\{i\}\)\-Lq\_\{ab\}\(x\_\{i\}\)\\bigr\)−xi,a​\(LN,εN​qb​\(xi\)−L​qb​\(xi\)\)\\displaystyle\-x\_\{i,a\}\\bigl\(L\_\{N,\\varepsilon\_\{N\}\}q\_\{b\}\(x\_\{i\}\)\-Lq\_\{b\}\(x\_\{i\}\)\\bigr\)−xi,b​\(LN,εN​qa​\(xi\)−L​qa​\(xi\)\)\.\\displaystyle\-x\_\{i,b\}\\bigl\(L\_\{N,\\varepsilon\_\{N\}\}q\_\{a\}\(x\_\{i\}\)\-Lq\_\{a\}\(x\_\{i\}\)\\bigr\)\.SinceKKis compact, its coordinates are uniformly bounded\. Uniform generator consistency therefore implies uniform convergence of every entry ofD^N\\hat\{D\}\_\{N\}to the corresponding entry ofDD\. Since the dimension is finite, this implies convergence in any matrix norm\. ∎

#### Generator consistency for kernels:

The generator\-consistency assumption[Assumption11](https://arxiv.org/html/2610.11318#Thmtheorem11)is closely related to continuum limits of local graph random walks\[[40](https://arxiv.org/html/2610.11318#bib.bib18), Theorems 2–3\], whose limiting drift and diffusion are determined by the first two local moments\[[8](https://arxiv.org/html/2610.11318#bib.bib43), Lemma 3\.9 and Theorem 3\.12\]\. For neighbourhood kernels, sufficient conditions involve shrinking bandwidths, sufficiently dense sampling from a smooth positive density, and controlled local bandwidth estimates\. For the generator consistency assumption the row\-normalized kernel, including any velocity and lineage kernel, must satisfy

mh=c​h2​b0\+o⁡\(h2\),Ch=c​h2​D\+o⁡\(h2\),m\_\{h\}=ch^\{2\}b\_\{0\}\+o\(h^\{2\}\),\\qquad C\_\{h\}=ch^\{2\}D\+o\(h^\{2\}\),uniformly with higher\-order displacement contributions divided byh2h^\{2\}going to zero ashhgoes to zero\. For the cosine and correlation constructions studied by[Li et al\. \[29\]](https://arxiv.org/html/2610.11318#bib.bib24)gives a first\-order limit on the time scalehh, rather than a diffusion limit on the time scaleh2h^\{2\}\. Their inner\-product construction instead admits a second\-order limit\. A possible diffusive construction is to replace a fixed cosine weightexp⁡\(ν​cos⁡θ\)\\exp\(\\nu\\cos\\theta\)byexp⁡\(νh​cos⁡θ\)\\exp\(\\nu\_\{h\}\\cos\\theta\)withνh/h→ν0\\nu\_\{h\}/h\\to\\nu\_\{0\}, which could be explored in future work\. FFM itself uses the transition moments directly, their interpretation as drift and diffusion\.

#### First\-order velocity\-kernel limits:

Suppose that for some time scaleτh→0\\tau\_\{h\}\\to 0, the discrete generatorsLh=\(Ph−I\)/τhL\_\{h\}=\(P\_\{h\}\-I\)/\\tau\_\{h\}converge to

Lf=b⊤∇f\+12Tr\(A∇2f\)\.Lf=b^\{\\top\}\\nabla f\+\\tfrac\{1\}\{2\}\\operatorname\{Tr\}\(A\\nabla^\{2\}f\)\.Applying this convergence to coordinate and quadratic coordinate functions \(as in[Proposition13](https://arxiv.org/html/2610.11318#Thmtheorem13)\) yields

mh/τh→b,Ch/τh→A\.m\_\{h\}/\\tau\_\{h\}\\to b,\\qquad C\_\{h\}/\\tau\_\{h\}\\to A\.In particular,[Li et al\. \[29\]](https://arxiv.org/html/2610.11318#bib.bib24)show that cosine RNA\-velocity kernels possess a first\-order limit withh=εh=\\sqrt\{\\varepsilon\}

\(Ph−I\)/h→a\(x\)v^\(x\)⊤∇,\(P\_\{h\}\-I\)/h\\to a\(x\)\\widehat\{v\}\(x\)^\{\\top\}\\nabla,witha⁡\(x\)\>0a\(x\)\>0\. Consequentlymh/h→a⁡\(x\)​v^​\(x\)m\_\{h\}/h\\to a\(x\)\\widehat\{v\}\(x\), whileCh/h→0C\_\{h\}/h\\to 0\. Thus the first transition moment asymptotically recovers the RNA\-velocity direction\. For such kernels we therefore interpretChC\_\{h\}as a finite\-scale measure of anisotropic transition accessibility rather than as an estimator of diffusion\.

#### Extension of empirical drift and diffusion\.

The estimators in Proposition[Proposition13](https://arxiv.org/html/2610.11318#Thmtheorem13)are defined only at the observed states\. To evaluate the induced metric along continuous curves, we extend them to a compact regionK⊂ℝdK\\subset\\mathbb\{R\}^\{d\}containing all admissible paths\. LetK\+K^\{\+\}be a compact neighborhood ofKKsuch that

K⊂int⁡\(K\+\)\.K\\subset\\operatorname\{int\}\(K^\{\+\}\)\.We assume that Assumption[Assumption11](https://arxiv.org/html/2610.11318#Thmtheorem11)holds onK\+K^\{\+\}, so that Proposition[Proposition13](https://arxiv.org/html/2610.11318#Thmtheorem13)applies on this neighborhood\.

###### Definition 14\(Uniform nodewise consistency\)\.

LetA⊂ℝdA\\subset\\mathbb\{R\}^\{d\}be compact and letffbe a continuous vector\- or matrix\-valued field onAA\. Suppose thatf^N​\(xi\)\\widehat\{f\}\_\{N\}\(x\_\{i\}\)estimatesf⁡\(xi\)f\(x\_\{i\}\)at each observed statexi∈Ax\_\{i\}\\in A\. We say thatf^N\\widehat\{f\}\_\{N\}is uniformly nodewise consistent forffonAAif

maxi:xi∈A‖f^N\(xi\)−f\(xi\)‖⟶0\\max\_\{i:x\_\{i\}\\in A\}\\left\\\|\\widehat\{f\}\_\{N\}\(x\_\{i\}\)\-f\(x\_\{i\}\)\\right\\\|\\longrightarrow 0along the asymptotic regime under consideration\. For matrix\-valued functions, any matrix norm may be used, since all matrix norms are equivalent in finite dimension\.

In particular, Proposition[Proposition13](https://arxiv.org/html/2610.11318#Thmtheorem13)shows thatb^0,N\\hat\{b\}\_\{0,N\}andD^N\\hat\{D\}\_\{N\}are uniformly nodewise consistent forb0b\_\{0\}andDD, respectively, onK\+K^\{\+\}\.

###### Definition 15\(Kernel extension\)\.

Leth∈Cc∞​\(\[0,∞\)\)h\\in C\_\{c\}^\{\\infty\}\(\[0,\\infty\)\)be nonnegative, supported in\[0,1\]\[0,1\], and strictly positive on a neighborhood of zero\. LetηN→0\\eta\_\{N\}\\to 0and define

kηN​\(x,y\)=h⁡\(‖x−y‖2ηN2\)\.k\_\{\\eta\_\{N\}\}\(x,y\)=h\\left\(\\frac\{\\\|x\-y\\\|^\{2\}\}\{\\eta\_\{N\}^\{2\}\}\\right\)\.Define the fill distance of the sample cloud relative toKKby

rN\(K\)≔supx∈Kmini:xi∈K\+∥x−xi∥\.r\_\{N\}\(K\)\\coloneqq\\sup\_\{x\\in K\}\\min\_\{i:x\_\{i\}\\in K^\{\+\}\}\\\|x\-x\_\{i\}\\\|\.We assume that the samples become dense relative to the kernel bandwidth:

rN​\(K\)ηN⟶0\.\\frac\{r\_\{N\}\(K\)\}\{\\eta\_\{N\}\}\\longrightarrow 0\.Sincehhis positive near zero, this condition ensures that, for all sufficiently largeNN, everyx∈Kx\\in Kreceives positive kernel weight from at least one observed state inK\+K^\{\+\}\. For suchNN, define

wi,N\(x\)=kηN​\(x,xi\)∑j:xj∈K\+kηN\(x,xj\),x∈K,xi∈K\+,w\_\{i,N\}\(x\)=\\frac\{k\_\{\\eta\_\{N\}\}\(x,x\_\{i\}\)\}\{\\displaystyle\\sum\_\{j:x\_\{j\}\\in K^\{\+\}\}k\_\{\\eta\_\{N\}\}\(x,x\_\{j\}\)\},\\qquad x\\in K,\\quad x\_\{i\}\\in K^\{\+\},and extend the nodewise estimators by

b~0,N\(x\)=∑i:xi∈K\+wi,N\(x\)b^0,N\(xi\)\\widetilde\{b\}\_\{0,N\}\(x\)=\\sum\_\{i:x\_\{i\}\\in K^\{\+\}\}w\_\{i,N\}\(x\)\\hat\{b\}\_\{0,N\}\(x\_\{i\}\)and

D~N\(x\)=∑i:xi∈K\+wi,N\(x\)D^N\(xi\)\.\\widetilde\{D\}\_\{N\}\(x\)=\\sum\_\{i:x\_\{i\}\\in K^\{\+\}\}w\_\{i,N\}\(x\)\\hat\{D\}\_\{N\}\(x\_\{i\}\)\.Consequently, for sufficiently largeNN, the extended fields are smooth on a neighborhood ofKK\. Moreover, because everyD^N​\(xi\)\\hat\{D\}\_\{N\}\(x\_\{i\}\)is positive semidefinite,D~N​\(x\)\\widetilde\{D\}\_\{N\}\(x\)is also positive semidefinite\.

###### Proposition 16\(Uniform convergence of the kernel\-extended estimators\)\.

Assume thatb0b\_\{0\}andDDare continuous onK\+K^\{\+\}and thatb^0,N\\hat\{b\}\_\{0,N\}andD^N\\hat\{D\}\_\{N\}are uniformly nodewise consistent onK\+K^\{\+\}in the sense of Definition[Definition14](https://arxiv.org/html/2610.11318#Thmtheorem14)\. IfηN⟶0,rN​\(K\)ηN⟶0,\\eta\_\{N\}\\longrightarrow 0,\\frac\{r\_\{N\}\(K\)\}\{\\eta\_\{N\}\}\\longrightarrow 0,then

supx∈K‖b~0,N​\(x\)−b0​\(x\)‖⟶0​and​supx∈K‖D~N​\(x\)−D⁡\(x\)‖⟶0\.\\sup\_\{x\\in K\}\\left\\\|\\widetilde\{b\}\_\{0,N\}\(x\)\-b\_\{0\}\(x\)\\right\\\|\\longrightarrow 0\\text\{ and \}\\sup\_\{x\\in K\}\\left\\\|\\widetilde\{D\}\_\{N\}\(x\)\-D\(x\)\\right\\\|\\longrightarrow 0\.

###### Proof\.

Modulus of continuity for a continuos functionffonK\+K^\{\+\}is defined as:

ωf​\(r\)≔supx,y∈K\+‖x−y‖≤r‖f⁡\(x\)−f⁡\(y\)‖\.\\omega\_\{f\}\(r\)\\coloneqq\\sup\_\{\\begin\{subarray\}\{c\}x,y\\in K^\{\+\}\\\\ \\\|x\-y\\\|\\leq r\\end\{subarray\}\}\\\|f\(x\)\-f\(y\)\\\|\.SinceK\+K^\{\+\}is compact,ffis uniformly continuous and thereforeωf​\(r\)⟶0​as​r⟶0\.\\omega\_\{f\}\(r\)\\longrightarrow 0\\text\{ as \}r\\longrightarrow 0\.Consider first the drift field\. Since the kernel weights are nonnegative and sum to one,

‖b~0,N​\(x\)−b0​\(x\)‖\\displaystyle\\left\\\|\\widetilde\{b\}\_\{0,N\}\(x\)\-b\_\{0\}\(x\)\\right\\\|≤∑i:xi∈K\+wi,N\(x\)‖b^0,N\(xi\)−b0\(xi\)‖\\displaystyle\\leq\\sum\_\{i:x\_\{i\}\\in K^\{\+\}\}w\_\{i,N\}\(x\)\\left\\\|\\hat\{b\}\_\{0,N\}\(x\_\{i\}\)\-b\_\{0\}\(x\_\{i\}\)\\right\\\|\+∑i:xi∈K\+wi,N\(x\)‖b0\(xi\)−b0\(x\)‖\.\\displaystyle\+\\sum\_\{i:x\_\{i\}\\in K^\{\+\}\}w\_\{i,N\}\(x\)\\left\\\|b\_\{0\}\(x\_\{i\}\)\-b\_\{0\}\(x\)\\right\\\|\.Wheneverwi,N​\(x\)\>0w\_\{i,N\}\(x\)\>0, the support condition onhhimplies

‖xi−x‖≤ηN\.\\\|x\_\{i\}\-x\\\|\\leq\\eta\_\{N\}\.Hence

‖b~0,N\(x\)−b0\(x\)‖≤maxi:xi∈K\+‖b^0,N\(xi\)−b0\(xi\)‖\+ωb0\(ηN\)\.\\left\\\|\\widetilde\{b\}\_\{0,N\}\(x\)\-b\_\{0\}\(x\)\\right\\\|\\leq\\max\_\{i:x\_\{i\}\\in K^\{\+\}\}\\left\\\|\\hat\{b\}\_\{0,N\}\(x\_\{i\}\)\-b\_\{0\}\(x\_\{i\}\)\\right\\\|\+\\omega\_\{b\_\{0\}\}\(\\eta\_\{N\}\)\.Taking the supremum overx∈Kx\\in K, the first term converges to zero by uniform nodewise consistency, while the second converges to zero becauseηN→0\\eta\_\{N\}\\to 0\.

Applying the same argument to the matrix\-valued fieldDDgives

supx∈K‖D~N\(x\)−D\(x\)‖≤maxi:xi∈K\+‖D^N\(xi\)−D\(xi\)‖\+ωD\(ηN\)⟶0\.\\sup\_\{x\\in K\}\\left\\\|\\widetilde\{D\}\_\{N\}\(x\)\-D\(x\)\\right\\\|\\leq\\max\_\{i:x\_\{i\}\\in K^\{\+\}\}\\left\\\|\\hat\{D\}\_\{N\}\(x\_\{i\}\)\-D\(x\_\{i\}\)\\right\\\|\+\\omega\_\{D\}\(\\eta\_\{N\}\)\\longrightarrow 0\.∎

#### Freidlin–Wentzell Theory\.

Consider the small\-noise diffusion

d​YTκ=b0​\(YTκ\)​d​t\+2​κ​D​\(YTκ\)1/2​d​Wt,\\mathrm\{d\}Y\_\{T\}^\{\\kappa\}=b\_\{0\}\(Y\_\{T\}^\{\\kappa\}\)\\,\\mathrm\{d\}t\+\\sqrt\{2\\kappa\}\\,D\(Y\_\{T\}^\{\\kappa\}\)^\{1/2\}\\,\\mathrm\{d\}W\_\{t\},whereb0b\_\{0\}is smooth\. For an absolutely continuous pathϕ:\[0,T\]→ℝd\\phi:\[0,T\]\\to\\mathbb\{R\}^\{d\}, the Freidlin–Wentzell \(FW\)\[[16](https://arxiv.org/html/2610.11318#bib.bib39)\]action is defined as:

ST​\(ϕ\)=14​∫0T\(ϕ˙t−b0​\(ϕt\)\)⊤​D​\(ϕt\)−1​\(ϕ˙t−b0​\(ϕt\)\)​𝑑t\.\\displaystyle S\_\{T\}\(\\phi\)=\\frac\{1\}\{4\}\\int\_\{0\}^\{T\}\(\\dot\{\\phi\}\_\{t\}\-b\_\{0\}\(\\phi\_\{t\}\)\)^\{\\top\}D\(\\phi\_\{t\}\)^\{\-1\}\(\\dot\{\\phi\}\_\{t\}\-b\_\{0\}\(\\phi\_\{t\}\)\)\\,\\mathrm\{d\}t\.\(16\)FW theory gives a path\-space large\-deviation principle with rate functional

STS\_\{T\}\. Informally speaking, among paths satisfying the same endpoint constraints, minimizers of

STS\_\{T\}are the asymptotically most probable transition paths \(See Theorem 4\.1[Freidlin and Wentzell \[16\]](https://arxiv.org/html/2610.11318#bib.bib39)\)\.

The fixed\-time action

STS\_\{T\}depends on the prescribed transition duration

TT\. Minimizing over

TTyields the quasipotential

V\(x0,x1\)≔infT\>0infϕ:ϕ⁡\(0\)=x0,ϕ⁡\(T\)=x1ST\(ϕ\)\.V\(x\_\{0\},x\_\{1\}\)\\coloneqq\\inf\_\{T\>0\}\\inf\_\{\\begin\{subarray\}\{c\}\\phi:\\phi\(0\)=x\_\{0\},\\ \\phi\(T\)=x\_\{1\}\\end\{subarray\}\}S\_\{T\}\(\\phi\)\.By Proposition 2\.1 and Equation \(2\.14\) of[Heymann and Vanden‐Eijnden \[20\]](https://arxiv.org/html/2610.11318#bib.bib36),

V\(x0,x1\)=infT\>0infϕ:ϕ⁡\(0\)=x0,ϕ⁡\(T\)=x1ST\(ϕ\)=12infγ:γ⁡\(0\)=x0,γ⁡\(1\)=x1SFW\(γ\)\.V\(x\_\{0\},x\_\{1\}\)=\\inf\_\{T\>0\}\\inf\_\{\\begin\{subarray\}\{c\}\\phi:\\phi\(0\)=x\_\{0\},\\ \\phi\(T\)=x\_\{1\}\\end\{subarray\}\}S\_\{T\}\(\\phi\)=\\frac\{1\}\{2\}\\inf\_\{\\begin\{subarray\}\{c\}\\gamma:\\gamma\(0\)=x\_\{0\},\\ \\gamma\(1\)=x\_\{1\}\\end\{subarray\}\}S\_\{\\mathrm\{FW\}\}\(\\gamma\)\.\(17\)whereSFW​\(γ\)=∫01FFW​\(γ⁡\(t\),γ˙​\(t\)\)​𝑑tS\_\{\\mathrm\{FW\}\}\(\\gamma\)=\\int\_\{0\}^\{1\}F^\{\\mathrm\{FW\}\}\(\\gamma\(t\),\\dot\{\\gamma\}\(t\)\)\\mathrm\{d\}twith

FFW\(x,v\)≔\(v⊤‖b0​\(x\)‖​DD​\(x\)−12​\(x\)−1⏟GFW​\(x\)v\)1/2\+v⊤\(−D​\(x\)−1​b0​\(x\)\)⏟βFW​\(x\)\.F^\{\\mathrm\{FW\}\}\(x,v\)\\coloneqq\\Bigl\(v^\{\\top\}\\underbrace\{\\\|\{\}b\_\{0\}\(x\)\\\|\{\}\_\{D\(x\)^\{\-1\}\}^\{2\}D\(x\)^\{\-1\}\}\_\{G\_\{\\mathrm\{FW\}\}\(x\)\}v\\Bigl\)^\{1/2\}\+v^\{\\top\}\\underbrace\{\\left\(\-D\(x\)^\{\-1\}b\_\{0\}\(x\)\\right\)\}\_\{\\beta\_\{\\mathrm\{FW\}\}\(x\)\}\.This has almost the form of a Finsler length functional[Definition4](https://arxiv.org/html/2610.11318#A1.Ex16), however, for

b0​\(x\)≠0b\_\{0\}\(x\)\\neq 0,

‖βFW​\(x\)‖GFW​\(x\)−1=1\.\\\|\\beta\_\{\\mathrm\{FW\}\}\(x\)\\\|\_\{G\_\{\\mathrm\{FW\}\}\(x\)^\{\-1\}\}=1\.Thus

FFWF^\{\\mathrm\{FW\}\}is a degenerate Randers\-type structure rather than Finsler metric \(See[Lemma5](https://arxiv.org/html/2610.11318#Thmtheorem5)\)\.

SFWS\_\{\\mathrm\{FW\}\}defines the parameterization\-invariant FW action whose minimizers are the minimum\-action transition curves\. Let

γ∗\\gamma^\{\*\}be the unique minimizer \(up reparametrization\) among curves connecting

x0x\_\{0\}to

x1x\_\{1\}\. We define the optimal transition time scale

T∗=∫01d​tΛ⁡\(γ∗​\(t\),γ˙∗​\(t\)\),where​Λ​\(x,v\)=b0​\(x\)⊤​D​\(x\)−1​b0​\(x\)v⊤​D​\(x\)−1​v,T^\{\*\}=\\int\_\{0\}^\{1\}\\frac\{\\mathrm\{d\}t\}\{\\Lambda\(\\gamma^\{\*\}\(t\),\\dot\{\\gamma\}^\{\*\}\(t\)\)\},\\qquad\\text\{where \}\\Lambda\(x,v\)=\\sqrt\{\\frac\{b\_\{0\}\(x\)^\{\\top\}D\(x\)^\{\-1\}b\_\{0\}\(x\)\}\{v^\{\\top\}D\(x\)^\{\-1\}v\}\},and denote byτδ​\(Xκ\)=inf\{t\>0∣YTκ∈B¯δ​\(x1\)\}\\tau\_\{\\delta\}\(X^\{\\kappa\}\)=\\inf\\\{t\>0\\mid Y\_\{T\}^\{\\kappa\}\\in\\bar\{B\}\_\{\\delta\}\(x\_\{1\}\)\\\}the first hitting time of a closedδ\\delta\-ball centered atx1x\_\{1\}\. Under the hypotheses of[Heymann and Vanden‐Eijnden \[20, Proposition 2\.3\]](https://arxiv.org/html/2610.11318#bib.bib36), including uniqueness of the minimizer and of the corresponding constrained fixed\-time minimizers, together with the stated compactness and convergence conditions, the following conditional concentration statement holds, for any error toleranceη\>0\\eta\>0,

limδ→0\+​T→T∗lim infκ→0ℙx0​\(ρF​\(Xκ\|\[0,τδ​\(Xκ\)\],γ∗\)<η\|τδ​\(Xκ\)≤T\)=1,\\lim\_\{\\delta\\to 0^\{\+\}\\\\ T\\to T^\{\*\}\}\\liminf\_\{\\kappa\\to 0\}\\,\\mathbb\{P\}\_\{x\_\{0\}\}\\\!\\left\(\\rho\_\{F\}\\\!\\left\(X^\{\\kappa\}\\big\|\_\{\[0,\\tau\_\{\\delta\}\(X^\{\\kappa\}\)\]\},\\,\\gamma^\{\*\}\\right\)<\\eta\\,\\,\\middle\|\\,\\,\\tau\_\{\\delta\}\(X^\{\\kappa\}\)\\leq T\\right\)=1,whereρF\\rho\_\{F\}is the Fréchet distance between curves\. Informally speaking, conditioned on the system completing the transition fromx0x\_\{0\}into a small neighborhood ofx1x\_\{1\}within a time frame close toT∗T^\{\*\}, the realizationYTκY\_\{T\}^\{\\kappa\}concentrates with probability11in an arbitrarily narrow geometric tube around the minimizerγ∗\\gamma^\{\*\}asκ→0\\kappa\\to 0\.

#### Empirical Finsler minimizers converge to geometric FW transition curves:

We next show that minimizers of the regularized empirical Finsler action converge to the population geometric FW minimizer, which has the small\-noise transition interpretation established above\.

###### Assumption 17\(Compact path class\)\.

LetΓ=\{γθ:θ∈Θ,∥γ˙\(t\)∥≤M\)\}\\Gamma=\\\{\\gamma\_\{\\theta\}:\\theta\\in\\Theta,\\\|\{\}\\dot\{\\gamma\}\(t\)\\\|\{\}\\leq M\)\\\}be a class of smooth curves withγθ​\(0\)=x0\\gamma\_\{\\theta\}\(0\)=x\_\{0\},γθ​\(1\)=x1\\gamma\_\{\\theta\}\(1\)=x\_\{1\},Θ\\Thetacompact, and\(t,θ\)↦\(γθ​\(t\),γ˙θ​\(t\)\)\(t,\\theta\)\\mapsto\(\\gamma\_\{\\theta\}\(t\),\\dot\{\\gamma\}\_\{\\theta\}\(t\)\)continuous\. SetKΓ=⋃θ∈Θγθ​\(\[0,1\]\)¯K\_\{\\Gamma\}=\\overline\{\\bigcup\_\{\\theta\\in\\Theta\}\\gamma\_\{\\theta\}\(\[0,1\]\)\}\. We assume thatKΓ⊂KK\_\{\\Gamma\}\\subset K, whereKKis the compact set on which the uniform convergence in Proposition[Proposition16](https://arxiv.org/html/2610.11318#Thmtheorem16)holds\.

For the unregularized geometric Freidlin–Wentzell interpretation in this subsection, we additionally assume thatDDis uniformly positive definite on the compact regionK\+K^\{\+\}containing the admissible paths\. For numerical stability and to ensure uniform positive definiteness at finiteNN, we use the ridge\-regularized estimator

D~ρN​\(x\)≔D~N​\(x\)\+ρN​I,\\tilde\{D\}\_\{\\rho\_\{N\}\}\(x\)\\coloneqq\\tilde\{D\}\_\{N\}\(x\)\+\\rho\_\{N\}I,whereρN\>0\\rho\_\{N\}\>0andρN→0\\rho\_\{N\}\\to 0asN→∞N\\to\\infty\. Furthermore, we define the regularized FW metric as:

F~N,λFW​\(x,v\)≔v⊤​D~ρN​\(x\)−1​v​‖b~0,N​\(x\)‖D~ρN​\(x\)−12\+λN2−v⊤​D~ρN​\(x\)−1​b~0,N​\(x\)\.\\tilde\{F\}^\{\\mathrm\{FW\}\}\_\{N,\\lambda\}\(x,v\)\\coloneqq\\sqrt\{v^\{\\top\}\\tilde\{D\}\_\{\\rho\_\{N\}\}\(x\)^\{\-1\}v\}\\,\\sqrt\{\\\|\\tilde\{b\}\_\{0,N\}\(x\)\\\|^\{2\}\_\{\\tilde\{D\}\_\{\\rho\_\{N\}\}\(x\)^\{\-1\}\}\+\\lambda\_\{N\}^\{2\}\}\-v^\{\\top\}\\tilde\{D\}\_\{\\rho\_\{N\}\}\(x\)^\{\-1\}\\tilde\{b\}\_\{0,N\}\(x\)\., note that this ensures thatF~N,λFW\\tilde\{F\}^\{\\mathrm\{FW\}\}\_\{N,\\lambda\}is aPP\-derived Randers[Definition7](https://arxiv.org/html/2610.11318#Thmtheorem7)metric thus a Finsler structure and hence a for some sequenceλN\>0\\lambda\_\{N\}\>0which converges to00asN→∞N\\rightarrow\\infty\.

###### Theorem 18\(Consistency of regularized empirical geometric FW minimizers\)\.

Assume Assumption[Assumption17](https://arxiv.org/html/2610.11318#Thmtheorem17)and the assumptions of Proposition[Proposition16](https://arxiv.org/html/2610.11318#Thmtheorem16)\. LetρN\>0\\rho\_\{N\}\>0andλN\>0\\lambda\_\{N\}\>0satisfyρN→0,λN→0\\rho\_\{N\}\\to 0,\\lambda\_\{N\}\\to 0\. Define

S~N​\(γ\)≔∫01F~N,λNFW​\(γ⁡\(t\),γ˙​\(t\)\)​𝑑t,\\tilde\{S\}\_\{N\}\(\\gamma\)\\coloneqq\\int\_\{0\}^\{1\}\\tilde\{F\}^\{\\mathrm\{FW\}\}\_\{N,\\lambda\_\{N\}\}\\left\(\\gamma\(t\),\\dot\{\\gamma\}\(t\)\\right\)\\,\\mathrm\{d\}t,and let

γ~N∗∈arg⁡minγ∈Γ​S~N​\(γ\)\.\\tilde\{\\gamma\}\_\{N\}^\{\*\}\\in\\arg\\min\_\{\\gamma\\in\\Gamma\}\\tilde\{S\}\_\{N\}\(\\gamma\)\.Then:

1. 1\.supγ∈Γ\|S~N​\(γ\)−SFW​\(γ\)\|⟶0\.\\sup\_\{\\gamma\\in\\Gamma\}\\left\|\\tilde\{S\}\_\{N\}\(\\gamma\)\-S\_\{\\mathrm\{FW\}\}\(\\gamma\)\\right\|\\longrightarrow 0\.
2. 2\.If𝒮FW\\mathcal\{S\}\_\{\\mathrm\{FW\}\}has a unique minimizerγ∗\\gamma^\{\*\}overΓ\\Gamma, up to reparameterization, then ρF​\(γ~N∗,γ∗\)⟶0\.\\rho\_\{F\}\\left\(\\tilde\{\\gamma\}\_\{N\}^\{\*\},\\gamma^\{\*\}\\right\)\\longrightarrow 0\.

Consequently, the regularized empirical Finsler minimizer converges to the population FW minimizer withinΓ\\Gamma\. Ifγ∗\\gamma^\{\*\}is also the global geometric FW minimizer connectingx0x\_\{0\}tox1x\_\{1\}, and the hypotheses of[Heymann and Vanden‐Eijnden \[20, Proposition 2\.3\]](https://arxiv.org/html/2610.11318#bib.bib36)hold, thenγ∗\\gamma^\{\*\}is the asymptotically most probable transition curve in the small\-noise sense stated above, andγ~N∗\\tilde\{\\gamma\}\_\{N\}^\{\*\}converges to this curve\.

###### Proof\.

By Proposition[Proposition16](https://arxiv.org/html/2610.11318#Thmtheorem16), we have uniform convergence onKΓK\_\{\\Gamma\}:

supx∈KΓ‖b~0,N​\(x\)−b0​\(x\)‖→0,supx∈KΓ‖D~N−D⁡\(x\)‖→0\.\\sup\_\{x\\in K\_\{\\Gamma\}\}\\\|\\tilde\{b\}\_\{0,N\}\(x\)\-b\_\{0\}\(x\)\\\|\\to 0,\\qquad\\sup\_\{x\\in K\_\{\\Gamma\}\}\\\|\\tilde\{D\}\_\{N\}\-D\(x\)\\\|\\to 0\.SinceρN→0\\rho\_\{N\}\\to 0, we also have

supx∈KΓ‖D~ρN​\(x\)−D⁡\(x\)‖≤supx∈KΓ‖D~N​\(x\)−D⁡\(x\)‖\+ρN→0\.\\sup\_\{x\\in K\_\{\\Gamma\}\}\\\|\\tilde\{D\}\_\{\\rho\_\{N\}\}\(x\)\-D\(x\)\\\|\\leq\\sup\_\{x\\in K\_\{\\Gamma\}\}\\\|\\tilde\{D\}\_\{N\}\(x\)\-D\(x\)\\\|\+\\rho\_\{N\}\\to 0\.SinceDDis uniformly positive definite onKΓK\_\{\\Gamma\}, and since matrix inversion is continuous on the set of uniformly positive definite matrices, it follows that

supx∈KΓ‖D~ρN​\(x\)−1−D​\(x\)−1‖→0\.\\sup\_\{x\\in K\_\{\\Gamma\}\}\\\|\\tilde\{D\}\_\{\\rho\_\{N\}\}\(x\)^\{\-1\}\-D\(x\)^\{\-1\}\\\|\\to 0\.
By Assumption[Assumption17](https://arxiv.org/html/2610.11318#Thmtheorem17), all curves inΓ\\Gammahave uniformly bounded velocity:

‖γ˙​\(t\)‖≤Mfor all​γ∈Γ,t∈\[0,1\]\.\\\|\\dot\{\\gamma\}\(t\)\\\|\\leq M\\qquad\\text\{for all \}\\gamma\\in\\Gamma,\\ t\\in\[0,1\]\.Therefore all pairs\(γ​\(t\),γ˙​\(t\)\)\(\\gamma\(t\),\\dot\{\\gamma\}\(t\)\)lie in the compact setKΓ×BM​\(0\)¯K\_\{\\Gamma\}\\times\\overline\{B\_\{M\}\(0\)\}\.

On this compact set, the map

\(b,D,v,λN\)↦v⊤​D−1​v​‖b‖D−12\+λN2−v⊤​D−1​b\(b,D,v,\\lambda\_\{N\}\)\\mapsto\\sqrt\{v^\{\\top\}D^\{\-1\}v\}\\,\\sqrt\{\\\|b\\\|^\{2\}\_\{D^\{\-1\}\}\+\\lambda\_\{N\}^\{2\}\}\-v^\{\\top\}D^\{\-1\}bis continuous\. The uniform convergence ofb~0,N\\tilde\{b\}\_\{0,N\},D~ρN\\tilde\{D\}\_\{\\rho\_\{N\}\}andλN→0\\lambda\_\{N\}\\rightarrow 0implies

supx∈KΓ,‖v‖≤M\|F~N,λNFW​\(x,v\)−FFW​\(x,v\)\|→0\.\\sup\_\{x\\in K\_\{\\Gamma\},\\ \\\|v\\\|\\leq M\}\\left\|\\tilde\{F\}^\{\\rm FW\}\_\{N,\\lambda\_\{N\}\}\(x,v\)\-F^\{\\rm FW\}\(x,v\)\\right\|\\to 0\.Consequently, for everyγ∈Γ\\gamma\\in\\Gamma,

\|S~N​\(γ\)−SFW​\(γ\)\|≤∫01\|F~N,λNFW​\(γ⁡\(t\),γ˙​\(t\)\)−FFW​\(γ⁡\(t\),γ˙​\(t\)\)\|​𝑑t\.\|\\tilde\{S\}\_\{N\}\(\\gamma\)\-S\_\{\\rm FW\}\(\\gamma\)\|\\leq\\int\_\{0\}^\{1\}\\left\|\\tilde\{F\}^\{\\rm FW\}\_\{N,\\lambda\_\{N\}\}\(\\gamma\(t\),\\dot\{\\gamma\}\(t\)\)\-F^\{\\rm FW\}\(\\gamma\(t\),\\dot\{\\gamma\}\(t\)\)\\right\|\\,dt\.Taking the supremum overγ∈Γ\\gamma\\in\\Gamma, we obtain

supγ∈Γ\|S~N​\(γ\)−SFW​\(γ\)\|≤supx∈KΓ,‖v‖≤M\|F~N,λNFW​\(x,v\)−FFW​\(x,v\)\|→0\.\\sup\_\{\\gamma\\in\\Gamma\}\|\\tilde\{S\}\_\{N\}\(\\gamma\)\-S\_\{\\rm FW\}\(\\gamma\)\|\\leq\\sup\_\{x\\in K\_\{\\Gamma\},\\ \\\|v\\\|\\leq M\}\\left\|\\tilde\{F\}^\{\\rm FW\}\_\{N,\\lambda\_\{N\}\}\(x,v\)\-F^\{\\rm FW\}\(x,v\)\\right\|\\to 0\.This proves the first claim\. Letγ∗\\gamma^\{\*\}be the unique minimizer ofSFWS\_\{\\rm FW\}overΓ\\Gamma, up to reparameterization\. Fixr\>0r\>0, and define

Γr=\{γ∈Γ:ρF​\(γ,γ∗\)≥r\}\.\\Gamma\_\{r\}=\\\{\\gamma\\in\\Gamma:\\rho\_\{F\}\(\\gamma,\\gamma^\{\*\}\)\\geq r\\\}\.BecauseΓ\\Gammais compact andSSis continuous onΓ\\Gamma, the minimum ofSSoverΓr\\Gamma\_\{r\}is attained\. Sinceγ∗\\gamma^\{\*\}is the unique minimizer, there is a positive gap

Δr≔infγ∈ΓrSFW​\(γ\)−SFW​\(γ∗\)\>0\.\\Delta\_\{r\}\\coloneqq\\inf\_\{\\gamma\\in\\Gamma\_\{r\}\}S\_\{\\rm FW\}\(\\gamma\)\-S\_\{\\rm FW\}\(\\gamma^\{\*\}\)\>0\.By the uniform convergence just proved, for sufficiently largeNN,

supγ∈Γ\|S~N​\(γ\)−SFW​\(γ\)\|<Δr3\.\\sup\_\{\\gamma\\in\\Gamma\}\|\\tilde\{S\}\_\{N\}\(\\gamma\)\-S\_\{\\rm FW\}\(\\gamma\)\|<\\frac\{\\Delta\_\{r\}\}\{3\}\.For anyγ∈Γr\\gamma\\in\\Gamma\_\{r\}, this gives

S~N​\(γ\)≥SFW​\(γ\)−Δr3≥SFW​\(γ∗\)\+Δr−Δr3=SFW​\(γ∗\)\+2​Δr3\.\\tilde\{S\}\_\{N\}\(\\gamma\)\\geq S\_\{\\rm FW\}\(\\gamma\)\-\\frac\{\\Delta\_\{r\}\}\{3\}\\geq S\_\{\\rm FW\}\(\\gamma^\{\*\}\)\+\\Delta\_\{r\}\-\\frac\{\\Delta\_\{r\}\}\{3\}=S\_\{\\rm FW\}\(\\gamma^\{\*\}\)\+\\frac\{2\\Delta\_\{r\}\}\{3\}\.On the other hand,

S~N​\(γ∗\)≤SFW​\(γ∗\)\+Δr3\.\\tilde\{S\}\_\{N\}\(\\gamma^\{\*\}\)\\leq S\_\{\\rm FW\}\(\\gamma^\{\*\}\)\+\\frac\{\\Delta\_\{r\}\}\{3\}\.Therefore

S~N​\(γ\)\>S~N​\(γ∗\)for all​γ∈Γr\.\\tilde\{S\}\_\{N\}\(\\gamma\)\>\\tilde\{S\}\_\{N\}\(\\gamma^\{\*\}\)\\qquad\\text\{for all \}\\gamma\\in\\Gamma\_\{r\}\.Hence no minimizerγ~∗∈arg⁡minγ∈Γ​S~N​\(γ\)\\tilde\{\\gamma\}^\{\*\}\\in\\arg\\min\_\{\\gamma\\in\\Gamma\}\\tilde\{S\}\_\{N\}\(\\gamma\)can lie inΓr\\Gamma\_\{r\}\. Thus,

ρF​\(γ~∗,γ∗\)<r\\rho\_\{F\}\(\\tilde\{\\gamma\}^\{\*\},\\gamma^\{\*\}\)<rfor all sufficiently largeNN\. Sincer\>0r\>0was arbitrary,

ρF​\(γ~∗,γ∗\)→0\.\\rho\_\{F\}\(\\tilde\{\\gamma\}^\{\*\},\\gamma^\{\*\}\)\\to 0\.
Finally, suppose thatγ∗\\gamma^\{\*\}is also the global geometric FW minimizer connectingx0x\_\{0\}tox1x\_\{1\}, and that the hypotheses of[Heymann and Vanden‐Eijnden \[20, Proposition 2\.3\]](https://arxiv.org/html/2610.11318#bib.bib36)hold\. The conditional concentration result above then identifiesγ∗\\gamma^\{\*\}as the asymptotically most probable transition curve in the small\-noise limit\. Since

ρF​\(γ~N∗,γ∗\)⟶0,\\rho\_\{F\}\\left\(\\tilde\{\\gamma\}\_\{N\}^\{\*\},\\gamma^\{\*\}\\right\)\\longrightarrow 0,the regularized empirical Finsler minimizer converges to this transition curve\. ∎

#### Additional Calculation:

###### Proposition 21\(Gaussian probability path\)\.

Letμt∈ℝd\\mu\_\{t\}\\in\\mathbb\{R\}^\{d\}be continuously differentiable and letGt∈SPD⁡\(d\)G\_\{t\}\\in\\mathrm\{SPD\}\(d\)be a continuously differentiable\. Define, fort∈\(0,1\)t\\in\(0,1\),pt​\(x∣x0,x1\)=𝒩⁡\(x,μt,t⁡\(1−t\)​Gt\)\.p\_\{t\}\(x\\mid x\_\{0\},x\_\{1\}\)=\\mathcal\{N\}\\\!\\left\(x;\\mu\_\{t\},\\,t\(1\-t\)G\_\{t\}\\right\)\.Then the vector field

u~t​\(x∣x0,x1\)=μ˙t\+1−2​t2​t​\(1−t\)​\(x−μt\)\+12​G˙t​Gt−1​\(x−μt\)\\displaystyle\\tilde\{u\}\_\{t\}\(x\\mid x\_\{0\},x\_\{1\}\)=\\dot\{\\mu\}\_\{t\}\+\\frac\{1\-2t\}\{2t\(1\-t\)\}\(x\-\\mu\_\{t\}\)\+\\frac\{1\}\{2\}\\dot\{G\}\_\{t\}G\_\{t\}^\{\-1\}\(x\-\\mu\_\{t\}\)satisfiesd​xd​t=u~t​\(xt∣x0,x1\)\\frac\{dx\}\{dt\}=\\tilde\{u\}\_\{t\}\(x\_\{t\}\\mid x\_\{0\},x\_\{1\}\)andxt∼pt\(⋅∣x0,x1\)x\_\{t\}\\sim p\_\{t\}\(\\cdot\\mid x\_\{0\},x\_\{1\}\)for allt∈\(0,1\)\.t\\in\(0,1\)\.

###### Proof\.

LetBtB\_\{t\}be a differentiable matrix factor satisfying

Bt​Bt⊤=Gt,B˙t​Bt−1=12​G˙t​Gt−1,\\displaystyle B\_\{t\}B\_\{t\}^\{\\top\}=G\_\{t\},\\qquad\\dot\{B\}\_\{t\}B\_\{t\}^\{\-1\}=\\frac\{1\}\{2\}\\dot\{G\}\_\{t\}G\_\{t\}^\{\-1\},and letε∼𝒩⁡\(0,I\)\\varepsilon\\sim\\mathcal\{N\}\(0,I\)\. Such a metric exists sinceGtG\_\{t\}is symmetric positive definite \(SPD\) and by using existence and uniqueness of ODEs\. A sample fromptp\_\{t\}can then be written as

xt=μt\+t⁡\(1−t\)​Bt​ε\.\\displaystyle x\_\{t\}=\\mu\_\{t\}\+\\sqrt\{t\(1\-t\)\}\\,B\_\{t\}\\varepsilon\.Solving forε\\varepsilongives

ε=1t⁡\(1−t\)​Bt−1​\(xt−μt\)\.\\displaystyle\\varepsilon=\\frac\{1\}\{\\sqrt\{t\(1\-t\)\}\}B\_\{t\}^\{\-1\}\(x\_\{t\}\-\\mu\_\{t\}\)\.Differentiatingxtx\_\{t\}with respect tottyields

x˙t\\displaystyle\\dot\{x\}\_\{t\}=μ˙t\+1−2​t2​t⁡\(1−t\)​Bt​ε\+t⁡\(1−t\)​B˙t​ε\\displaystyle=\\dot\{\\mu\}\_\{t\}\+\\frac\{1\-2t\}\{2\\sqrt\{t\(1\-t\)\}\}B\_\{t\}\\varepsilon\+\\sqrt\{t\(1\-t\)\}\\,\\dot\{B\}\_\{t\}\\varepsilon=μ˙t\+1−2​t2​t​\(1−t\)​\(xt−μt\)\+B˙t​Bt−1​\(xt−μt\)\\displaystyle=\\dot\{\\mu\}\_\{t\}\+\\frac\{1\-2t\}\{2t\(1\-t\)\}\(x\_\{t\}\-\\mu\_\{t\}\)\+\\dot\{B\}\_\{t\}B\_\{t\}^\{\-1\}\(x\_\{t\}\-\\mu\_\{t\}\)=μ˙t\+1−2​t2​t​\(1−t\)​\(xt−μt\)\+12​G˙t​Gt−1​\(xt−μt\)\.\\displaystyle=\\dot\{\\mu\}\_\{t\}\+\\frac\{1\-2t\}\{2t\(1\-t\)\}\(x\_\{t\}\-\\mu\_\{t\}\)\+\\frac\{1\}\{2\}\\dot\{G\}\_\{t\}G\_\{t\}^\{\-1\}\(x\_\{t\}\-\\mu\_\{t\}\)\.Hencex˙t=ut​\(xt\)\\dot\{x\}\_\{t\}=u\_\{t\}\(x\_\{t\}\)\. Sincext=μt\+t⁡\(1−t\)​Bt​εx\_\{t\}=\\mu\_\{t\}\+\\sqrt\{t\(1\-t\)\}B\_\{t\}\\varepsilonhas distribution𝒩⁡\(μt,t⁡\(1−t\)​Gt\)\\mathcal\{N\}\(\\mu\_\{t\},t\(1\-t\)G\_\{t\}\)by construction, the flow generated byutu\_\{t\}has marginalsptp\_\{t\}for allt∈\(0,1\)t\\in\(0,1\)\. ∎

## Appendix CExperiments

#### Experimental protocol:

All models get the same networks that uses four hidden layers of width 64, SiLU activations, and a 32\-dimensional sinusoidal time embedding\. We use Adam with learning rate2×10−32\\times 10^\{\-3\}, batch size 256, and gradient clipping at 5\. Where applicable, geodesic training uses 2,500 iterations, followed by 1,500 flow\-training iterations, stochastic training uses an additional 1,500 iterations\. Default coupling settings use eight quadrature samples and entropic regularization equal to 0\.1 times the median transport cost, with benchmark\-specific overrides stated separately\. Results report mean±\\pmpopulation standard deviation across three seeds, except Erythroid, which uses five\. Hyperparameters were selected using a validation split of1515percent and using the mean of mid and endW2W\_\{2\}\.

#### Kernel extension of transition moments:

The first and second transition momentsm⁡\(xi\)m\(x\_\{i\}\)andC⁡\(xi\)C\(x\_\{i\}\)are defined only at the observed states\{xi\}i=1N\\\{x\_\{i\}\\\}\_\{i=1\}^\{N\}\. To evaluate the Finsler metric along continuous paths, we extend both moments using normalized non\-negative kernel smoothing\. Letkh:ℝd×ℝd→ℝ≥0k\_\{h\}:\\mathbb\{R\}^\{d\}\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}\_\{\\geq 0\}be a smooth kernel with bandwidthh\>0h\>0\. Assuming that∑ℓ=1Nkh​\(x,xℓ\)\>0\\sum\_\{\\ell=1\}^\{N\}k\_\{h\}\(x,x\_\{\\ell\}\)\>0throughout the region of interest, define

wi​\(x\)≔kh​\(x,xi\)∑ℓ=1Nkh​\(x,xℓ\)\.w\_\{i\}\(x\)\\coloneqq\\frac\{k\_\{h\}\(x,x\_\{i\}\)\}\{\\sum\_\{\\ell=1\}^\{N\}k\_\{h\}\(x,x\_\{\\ell\}\)\}\.The extended moment fields are

m⁡\(x\)≔∑i=1Nwi​\(x\)​m​\(xi\),C⁡\(x\)≔∑i=1Nwi​\(x\)​C​\(xi\)\.m\(x\)\\coloneqq\\sum\_\{i=1\}^\{N\}w\_\{i\}\(x\)m\(x\_\{i\}\),\\qquad C\(x\)\\coloneqq\\sum\_\{i=1\}^\{N\}w\_\{i\}\(x\)C\(x\_\{i\}\)\.\(18\)
We choose kernel smoothing for its simplicity: it requires no additional model training and preserves the positive semidefiniteness of the second\-moment field\. Indeed, the weights are non\-negative and sum to one, soC⁡\(x\)C\(x\)is a convex combination of positive semidefinite matrices\. Both extended fields are smooth wherever the denominator is positive\. Adding the ridge termCρ​\(x\)=C⁡\(x\)\+ρ​IC\_\{\\rho\}\(x\)=C\(x\)\+\\rho I, withρ\>0\\rho\>0, then ensures positive definiteness for the subsequent metric construction\.

#### Sheet Dataset:

We sampleN=4000N=4000points from coordinatesu∼𝒰⁡\(−1\.5,1\.5\)u\\sim\\mathcal\{U\}\(\-1\.5,1\.5\)andw∼𝒰⁡\(−0\.45,0\.45\)w\\sim\\mathcal\{U\}\(\-0\.45,0\.45\)and embed them on the saddle

z⁡\(u,w\)=12​\[\(3​u2\)2−12​\(100​w45\)2\],z\(u,w\)=\\frac\{1\}\{2\}\\left\[\\left\(\\frac\{3u\}\{2\}\\right\)^\{2\}\-\\frac\{1\}\{2\}\\left\(\\frac\{100w\}\{45\}\\right\)^\{2\}\\right\],with Gaussian noise of standard deviation

0\.0120\.012\. The prescribed transition probability matrix

PPforms an hourglass: trajectories are driven towards

u≈0u\\approx 0and go outward after crossing\. Source and target states are defined by

w≤−0\.13w\\leq\-0\.13and

w≥0\.13w\\geq 0\.13, respectively\.

PPis constructed bye directed

1515\-NN kernel combining spatial proximity with alignment to a prescribed vector field \(

λ=6\\lambda=6\), followed by row normalization\. Source and target points are split

70/15/1570/15/15into train/validation/test sets, whereas the withheld region is divided equally between validation and test\. Hyperparameters are selected using the average of endpoint

W2W\_\{2\}and intermediate\-region

W2W\_\{2\}on the validation set\. We also perform a metric ablation where we compare only keeping the first and second moments[Fig\.4\(b\)](https://arxiv.org/html/2610.11318#A3.F4.sf2)\.

\(a\)Sheet benchmark\.
GGDriftceffc\_\{\\mathrm\{eff\}\}midW2↓W\_\{2\}\\downarrowendW2↓W\_\{2\}\\downarrowW2,F↓W\_\{2,F\}\\downarrowIIno0\.0010\.7165±\\pm0\.01360\.1751±\\pm0\.02711\.0840±\\pm0\.0371IIyes0\.9580\.2541±\\pm0\.03750\.2558±\\pm0\.05090\.3676±\\pm0\.0773Cρ−1C\_\{\\rho\}^\{\-1\}no0\.0010\.6672±\\pm0\.01600\.2025±\\pm0\.00790\.9437±\\pm0\.1068Cρ−1C\_\{\\rho\}^\{\-1\}yes0\.9580\.1588±\\pm0\.00960\.2343±\\pm0\.02920\.2892±\\pm0\.0090*sampling floor**0\.0646**0\.1896**0\.2812*

\(b\)Metric ablation\.
![Refer to caption](https://arxiv.org/html/2610.11318v1/sheet_metric_ablation.png)\(c\)Trajectories generated by the different methods\.

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\.
#### Pancreas Dataset:

Hyperparameters are selected independently for each representation using a validation subset disjoint from the reported withheld cells\. Runtime and sensitivity toρ\\rhoandλ\\lambdaare reported in[Table2](https://arxiv.org/html/2610.11318#A3.T2)and[Table3](https://arxiv.org/html/2610.11318#A3.T3)\. We additionally evaluate sensitivity to temporal alignment, the amount of intermediate\-state support used to construct the geometry, and choices in the stochastic construction\. To reduce dependence on the benchmark conventiont=1/2t=1/2, we evaluateW2W\_\{2\}over a fixed 11\-point grid spanningt∈\[1/3,2/3\]t\\in\[1/3,2/3\]and reportW¯2\[1/3,2/3\]\\overline\{W\}\_\{2\}^\{\[1/3,2/3\]\}, the mean over this grid\. We further reportW2,FW\_\{2,F\}across representations, vary the fraction of intermediate\-band cells available for metric construction, and ablate the conditional\-path covariance and inference noise used by stochastic FFM\. Unless otherwise stated, all Pancreas results in[Fig\.3](https://arxiv.org/html/2610.11318#S5.F3)use no intermediate\-band cells for metric construction\.[Fig\.5](https://arxiv.org/html/2610.11318#A3.F5)is an additional sensitivity analysis that deliberately varies this fraction\.

\(a\) Sensitivity to model time

MethodW2​\(1/3\)W\_\{2\}\(1/3\)W2​\(0\.433\)W\_\{2\}\(0\.433\)W2​\(0\.567\)W\_\{2\}\(0\.567\)W2​\(2/3\)W\_\{2\}\(2/3\)W¯2\[1/3,2/3\]\\overline\{W\}\_\{2\}^\{\[1/3,2/3\]\}d=2d=2FFM, deterministic1\.9878±\\pm0\.06311\.8184±\\pm0\.03261\.4429±\\pm0\.01951\.1770±\\pm0\.09191\.6175±\\pm0\.0348FFM, with noise2\.0052±\\pm0\.07051\.8399±\\pm0\.04011\.4723±\\pm0\.06011\.2148±\\pm0\.08511\.6434±\\pm0\.0555OT\-CFM1\.8508±\\pm0\.01721\.7952±\\pm0\.01821\.7607±\\pm0\.01951\.7682±\\pm0\.02011\.7869±\\pm0\.0230MFM / LAND1\.5201±\\pm0\.01191\.4878±\\pm0\.01171\.4438±\\pm0\.01081\.4002±\\pm0\.00981\.4642±\\pm0\.0131Curly\-FM1\.0208±\\pm0\.12770\.9952±\\pm0\.13780\.9480±\\pm0\.14570\.9009±\\pm0\.15050\.9686±\\pm0\.1725d=20d=20FFM, deterministic11\.5969±\\pm0\.199910\.7814±\\pm0\.181110\.3610±\\pm0\.144010\.8331±\\pm0\.147110\.7413±\\pm0\.1927FFM, with noise11\.5392±\\pm0\.164010\.7188±\\pm0\.127010\.3671±\\pm0\.047710\.9998±\\pm0\.216310\.7355±\\pm0\.0560OT\-CFM13\.0727±\\pm0\.132512\.5476±\\pm0\.095912\.2577±\\pm0\.023112\.3903±\\pm0\.040612\.4957±\\pm0\.0681MFM / LAND12\.8496±\\pm0\.107412\.3470±\\pm0\.070611\.8851±\\pm0\.018011\.7331±\\pm0\.027712\.1656±\\pm0\.0522Curly\-FM12\.8636±\\pm0\.050312\.3801±\\pm0\.068712\.2108±\\pm0\.076012\.4096±\\pm0\.106412\.3920±\\pm0\.0687d=50d=50FFM, deterministic13\.2623±\\pm0\.144212\.7423±\\pm0\.096612\.5789±\\pm0\.024012\.9036±\\pm0\.049512\.7715±\\pm0\.0719FFM, with noise13\.2718±\\pm0\.152412\.7538±\\pm0\.110412\.5768±\\pm0\.063912\.8742±\\pm0\.097512\.7723±\\pm0\.0952OT\-CFM14\.1359±\\pm0\.144113\.6933±\\pm0\.124213\.4941±\\pm0\.092313\.6688±\\pm0\.081613\.6811±\\pm0\.1314MFM / LAND13\.8332±\\pm0\.152613\.2557±\\pm0\.096612\.7979±\\pm0\.059312\.7353±\\pm0\.036413\.0996±\\pm0\.1029Curly\-FM13\.8395±\\pm0\.111313\.4287±\\pm0\.152113\.2923±\\pm0\.202013\.4719±\\pm0\.230713\.4440±\\pm0\.2069

\(b\) Finsler\-Wasserstein recovery

Methodd=2d=2d=10d=10d=20d=20d=50d=50OT\-CFM0\.6420±\\pm0\.005910\.9311±\\pm0\.077010\.9277±\\pm0\.041215\.5900±\\pm0\.2287MFM / LAND0\.5399±\\pm0\.00249\.4377±\\pm0\.082410\.4768±\\pm0\.063714\.6185±\\pm0\.1085Curly\-FM0\.7493±\\pm0\.025810\.7482±\\pm0\.130511\.0235±\\pm0\.052015\.0427±\\pm0\.1562FFM, deterministic0\.5681±\\pm0\.01009\.5125±\\pm0\.53988\.9033±\\pm0\.040013\.9368±\\pm0\.1217FFM, with noise0\.5730±\\pm0\.00829\.6113±\\pm0\.53798\.9213±\\pm0\.045313\.7800±\\pm0\.0861*floor**0\.4162**5\.3461**4\.3608**7\.2123*

W2,F​\(t=1/2\)↓W\_\{2,F\}\(t=1/2\)\\downarrow\.

\(c\) Intermediate support used for metric construction

Method0%0\\%5%5\\%35%35\\%d=2d=2OT\-CFM1\.7607±\\pm0\.02621\.7622±\\pm0\.03921\.7666±\\pm0\.0297FFM, deterministic1\.3833±\\pm0\.14101\.3313±\\pm0\.06271\.0927±\\pm0\.2477FFM, with noise1\.3723±\\pm0\.15051\.3525±\\pm0\.04201\.1013±\\pm0\.2611MFM / LAND1\.4506±\\pm0\.00251\.2412±\\pm0\.52830\.6878±\\pm0\.0378Curly\-FM0\.9751±\\pm0\.17040\.9360±\\pm0\.10451\.5865±\\pm0\.1369d=20d=20OT\-CFM12\.3291±\\pm0\.068512\.3264±\\pm0\.044512\.3491±\\pm0\.0850FFM, deterministic10\.4208±\\pm0\.10288\.2742±\\pm0\.35408\.1989±\\pm0\.0481FFM, with noise10\.3679±\\pm0\.06418\.4860±\\pm0\.30988\.2513±\\pm0\.2160MFM / LAND12\.0368±\\pm0\.069611\.8784±\\pm0\.033811\.7247±\\pm0\.1739Curly\-FM12\.2067±\\pm0\.084412\.2163±\\pm0\.109812\.2431±\\pm0\.0665d=50d=50OT\-CFM13\.5319±0\.048013\.5319\\pm 0\.048013\.5269±0\.064213\.5269\\pm 0\.064213\.5577±0\.107513\.5577\\pm 0\.1075FFM, deterministic12\.5043±0\.0544\\mathbf\{12\.5043\\pm 0\.0544\}12\.1225±0\.1252\\mathbf\{12\.1225\\pm 0\.1252\}11\.3633±0\.1356\\mathbf\{11\.3633\\pm 0\.1356\}MFM / LAND13\.5584±0\.039613\.5584\\pm 0\.039613\.5208±0\.027513\.5208\\pm 0\.027513\.5689±0\.123913\.5689\\pm 0\.1239Curly\-FM13\.3220±0\.192713\.3220\\pm 0\.192713\.3291±0\.250113\.3291\\pm 0\.250113\.2897±0\.111613\.2897\\pm 0\.1116

W2​\(t=1/2\)↓W\_\{2\}\(t=1/2\)\\downarrow\.

Figure 5:Pancreas robustness analyses\.\(a\) Intermediate\-population recovery over model time\. We show representative points from the fixed 11\-point grid over\[1/3,2/3\]\[1/3,2/3\]together withW¯2\[1/3,2/3\]\\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\.\(d\) Conditional\-path noise width and covariance

Width mult\.Mt=Cρ​\(μt\)M\_\{t\}=C\_\{\\rho\}\(\\mu\_\{t\}\)W2​\(1/2\)W\_\{2\}\(1/2\)Mt=Cρ​\(μt\)M\_\{t\}=C\_\{\\rho\}\(\\mu\_\{t\}\)W2​\(1\)W\_\{2\}\(1\)Mt=IM\_\{t\}=IW2​\(1/2\)W\_\{2\}\(1/2\)Mt=IM\_\{t\}=IW2​\(1\)W\_\{2\}\(1\)0\.00310\.4874±\\pm0\.04195\.2457±\\pm0\.230110\.4414±\\pm0\.14395\.2980±\\pm0\.15950\.01010\.4940±\\pm0\.11235\.5062±\\pm0\.434410\.5187±\\pm0\.15915\.2420±\\pm0\.26840\.03010\.4873±\\pm0\.09195\.3305±\\pm0\.161910\.4245±\\pm0\.13595\.3180±\\pm0\.13510\.10010\.3694±\\pm0\.09875\.0662±\\pm0\.344910\.3446±\\pm0\.12995\.1521±\\pm0\.17020\.30010\.4630±\\pm0\.21425\.3692±\\pm0\.159010\.4149±\\pm0\.19225\.2433±\\pm0\.24511\.00014\.0089±\\pm0\.06367\.6047±\\pm0\.301513\.8565±\\pm0\.14987\.3363±\\pm0\.16513\.00036\.3654±\\pm0\.565224\.9668±\\pm0\.773033\.9906±\\pm1\.118126\.6449±\\pm1\.138510\.000118\.0193±\\pm4\.788985\.1409±\\pm11\.5354143\.4577±\\pm2\.9533130\.1162±\\pm4\.8789

\(e\) Path and inference covariance

PathInferenceW2​\(1/2\)W\_\{2\}\(1/2\)W2​\(1\)W\_\{2\}\(1\)Cρ​\(μt\)C\_\{\\rho\}\(\\mu\_\{t\}\)Cρ​\(Xt\)C\_\{\\rho\}\(X\_\{t\}\)10\.3694±\\pm0\.09875\.0662±\\pm0\.3449Cρ​\(μt\)C\_\{\\rho\}\(\\mu\_\{t\}\)II10\.3617±\\pm0\.10034\.9788±\\pm0\.2102IICρ​\(Xt\)C\_\{\\rho\}\(X\_\{t\}\)10\.3557±\\pm0\.11125\.1712±\\pm0\.2807IIII10\.3446±\\pm0\.12995\.1521±\\pm0\.1702

\(f\) Inference\-noise schedule

g⁡\(t\)g\(t\)M⁡\(Xt\)=Cρ​\(Xt\)M\(X\_\{t\}\)=C\_\{\\rho\}\(X\_\{t\}\)W2​\(1/2\)W\_\{2\}\(1/2\)M⁡\(Xt\)=Cρ​\(Xt\)M\(X\_\{t\}\)=C\_\{\\rho\}\(X\_\{t\}\)W2​\(1\)W\_\{2\}\(1\)M⁡\(Xt\)=IM\(X\_\{t\}\)=IW2​\(1/2\)W\_\{2\}\(1/2\)M⁡\(Xt\)=IM\(X\_\{t\}\)=IW2​\(1\)W\_\{2\}\(1\)σ\\sigma10\.3694±\\pm0\.09875\.0662±\\pm0\.344910\.3617±\\pm0\.10034\.9788±\\pm0\.2102∝\(1−t\)\\propto\(1\-t\)10\.35805\.007210\.34965\.0492g∼tg\\sim t10\.39225\.197710\.38685\.1169g∼t⁡\(1−t\)g\\sim\\sqrt\{t\(1\-t\)\}10\.36295\.081410\.35155\.003500aftert=0\.8t=0\.810\.36265\.044010\.35434\.99130010\.41735\.195510\.41735\.1955

Figure 6:Pancreas robustness analyses \(continued\)\.\(d\) Sensitivity of stochastic FFM atd=20d=20to 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±\\pmpopulation standard deviation over three seeds unless stated otherwise\. Lower is better\.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\.Methodd=2d=2d=10d=10d=20d=20d=50d=50OT\-CFM3333MFM / LAND2261109254Curly\-FM41106190445FFM \(ours\), deterministic2663109256FFM \(ours\), with noise152948108Table 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\.ddVaried parameterFixed parameterValuesValidation objective2ρ\\rhoλ=0\.1\\lambda=0\.10\.01,0\.03,0\.1,0\.3,1\.00\.01,\\ 0\.03,\\ \\mathbf\{0\.1\},\\ 0\.3,\\ 1\.00\.828,0\.749,0\.535,0\.870,1\.1310\.828,\\ 0\.749,\\ \\mathbf\{0\.535\},\\ 0\.870,\\ 1\.1312λ\\lambdaρ=0\.1\\rho=0\.10\.01,0\.03,0\.1,0\.3,1\.00\.01,\\ 0\.03,\\ \\mathbf\{0\.1\},\\ 0\.3,\\ 1\.01\.226,0\.951,0\.535,0\.826,0\.8731\.226,\\ 0\.951,\\ \\mathbf\{0\.535\},\\ 0\.826,\\ 0\.87320ρ\\rhoλ=1\.0\\lambda=1\.00\.003,0\.01,0\.03,0\.1,0\.30\.003,\\ 0\.01,\\ \\mathbf\{0\.03\},\\ 0\.1,\\ 0\.38\.180,8\.210,8\.135,8\.369,8\.2888\.180,\\ 8\.210,\\ \\mathbf\{8\.135\},\\ 8\.369,\\ 8\.28820λ\\lambdaρ=0\.03\\rho=0\.030\.03,0\.1,0\.3,1\.0,3\.00\.03,\\ 0\.1,\\ 0\.3,\\ \\mathbf\{1\.0\},\\ 3\.08\.990,8\.535,8\.370,8\.135,8\.6678\.990,\\ 8\.535,\\ 8\.370,\\ \\mathbf\{8\.135\},\\ 8\.66750ρ\\rhoλ=0\.03\\lambda=0\.033×10−4,10−3,3×𝟏𝟎−𝟑,10−2,3×10−23\{\\times\}10^\{\-4\},\\ 10^\{\-3\},\\ \\mathbf\{3\{\\times\}10^\{\-3\}\},\\ 10^\{\-2\},\\ 3\{\\times\}10^\{\-2\}10\.455,10\.421,10\.314,10\.390,10\.58910\.455,\\ 10\.421,\\ \\mathbf\{10\.314\},\\ 10\.390,\\ 10\.58950λ\\lambdaρ=0\.003\\rho=0\.0030\.003,0\.01,0\.03,0\.1,0\.30\.003,\\ 0\.01,\\ \\mathbf\{0\.03\},\\ 0\.1,\\ 0\.310\.379,10\.314,10\.314,10\.554,10\.53710\.379,\\ 10\.314,\\ \\mathbf\{10\.314\},\\ 10\.554,\\ 10\.537

#### Erythroid:

We further evaluate FFM on the mouse erythroid differentiation dataset used by[Petrović et al\. \[35\]](https://arxiv.org/html/2610.11318#bib.bib10)\. The dataset consists of 9,815 cells along the erythroid developmental trajectory, partitioned into three temporal snapshots with the intermediate population withheld during training\. We evaluate in the two\-dimensional embedding and ind∈\{20,50\}d\\in\\\{20,50\\\}dimensional representations\. In addition to our methods, we report both the values originally published by[Petrović et al\. \[35\]](https://arxiv.org/html/2610.11318#bib.bib10)and our reproduction of the baselines using a common training harness at their published training budget\.

![Refer to caption](https://arxiv.org/html/2610.11318v1/erythroid_paths.png)

MethodSourceCos\. Dist\.↓\\downarrowL2↓L\_\{2\}\\downarrowW1↓W\_\{1\}\\downarrowW2↓W\_\{2\}\\downarrowd=2d=2OT\-CFMPublished0\.146±\\pm0\.0012\.704±\\pm0\.0190\.646±\\pm0\.006–OT\-CFMReproduced0\.141±\\pm0\.0022\.447±\\pm0\.0310\.645±\\pm0\.0090\.679±\\pm0\.009MFM \(LAND\)Published0\.014±\\pm0\.0011\.999±\\pm0\.0140\.269±\\pm0\.004–MFM \(LAND\)Reproduced0\.010±\\pm0\.0011\.858±\\pm0\.0250\.271±\\pm0\.0070\.286±\\pm0\.007Curly\-FMPublished0\.009±\\pm0\.0001\.663±\\pm0\.2930\.369±\\pm0\.090–Curly\-FMReproduced0\.011±\\pm0\.0012\.151±\\pm0\.0780\.122±\\pm0\.0020\.158±\\pm0\.003FFM \(ours\)Ours0\.021±\\pm0\.0041\.642±\\pm0\.0710\.248±\\pm0\.0340\.291±\\pm0\.023FFM \(ours\), with noiseOurs0\.019±\\pm0\.0031\.710±\\pm0\.0810\.226±\\pm0\.0270\.269±\\pm0\.019d=20d=20OT\-CFMPublished0\.489±\\pm0\.0011\.885±\\pm0\.0206\.103±\\pm0\.074–OT\-CFMReproduced0\.490±\\pm0\.0001\.731±\\pm0\.0146\.106±\\pm0\.0636\.260±\\pm0\.066MFM \(LAND\)Published0\.495±\\pm0\.0011\.627±\\pm0\.0404\.855±\\pm0\.052–MFM \(LAND\)Reproduced0\.494±\\pm0\.0001\.700±\\pm0\.0214\.759±\\pm0\.0564\.855±\\pm0\.054Curly\-FMPublished0\.488±\\pm0\.0011\.721±\\pm0\.0356\.124±\\pm0\.027–Curly\-FMReproduced0\.490±\\pm0\.0001\.691±\\pm0\.0116\.112±\\pm0\.0536\.271±\\pm0\.058FFM \(ours\)Ours0\.491±\\pm0\.0001\.514±\\pm0\.0164\.685±\\pm0\.0124\.782±\\pm0\.014FFM \(ours\), with noiseOurs0\.496±\\pm0\.0000\.757±\\pm0\.0064\.733±\\pm0\.0354\.823±\\pm0\.038d=50d=50OT\-CFMPublished0\.490±\\pm0\.0002\.215±\\pm0\.0227\.969±\\pm0\.029–OT\-CFMReproduced0\.491±\\pm0\.0002\.053±\\pm0\.0177\.839±\\pm0\.0167\.974±\\pm0\.016MFM \(LAND\)Published0\.494±\\pm0\.0001\.971±\\pm0\.0236\.727±\\pm0\.022–MFM \(LAND\)Reproduced0\.495±\\pm0\.0002\.004±\\pm0\.0136\.697±\\pm0\.0526\.819±\\pm0\.058Curly\-FMPublished0\.489±\\pm0\.0002\.045±\\pm0\.0737\.729±\\pm0\.046–Curly\-FMReproduced0\.491±\\pm0\.0001\.990±\\pm0\.0217\.736±\\pm0\.0257\.864±\\pm0\.025FFM \(ours\)Ours0\.491±\\pm0\.0001\.772±\\pm0\.0296\.810±\\pm0\.0226\.907±\\pm0\.025FFM \(ours\), with noiseOurs0\.499±\\pm0\.0000\.915±\\pm0\.0066\.595±\\pm0\.0166\.663±\\pm0\.015

Figure 7:Visualization of the learned erythroid trajectories in 2D and quantitative comparison across dimensionsd=2,20,50d=2,20,50\. “Published” denotes the values reported in Table 4 of[Petrović et al\. \[35\]](https://arxiv.org/html/2610.11318#bib.bib10), 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 asW2W\_\{2\}in Table 4 of CurlyFM corresponds in fact toW1W\_\{1\}\. We therefore report these published values underW1W\_\{1\}and additionally compute the trueW2W\_\{2\}distance for all runs evaluated with our harness\. Ford=20d=20andd=50d=50,L2L\_\{2\}is reported in units of10310^\{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\.
#### iTracer:

Let𝒩15​\(i\)\\mathcal\{N\}\_\{15\}\(i\)denote the 15 nearest neighbours of cellii, excluding itself, and letj7​\(i\)j\_\{7\}\(i\)denote its seventh nearest neighbour\. All four kernel variants share the same directed, unsymmetrized support and self\-tuning bandwidths\. The expression factor is

Ki​jexpr=𝟏\{j∈𝒩15\(i\)\}exp\(−‖xi−xj‖2σi​σj\),σi=∥xi−xj7​\(i\)∥\.K^\{\\mathrm\{expr\}\}\_\{ij\}=\\mathbf\{1\}\_\{\\\{j\\in\\mathcal\{N\}\_\{15\}\(i\)\\\}\}\\exp\\\!\\left\(\-\\frac\{\\\|x\_\{i\}\-x\_\{j\}\\\|^\{2\}\}\{\\sigma\_\{i\}\\sigma\_\{j\}\}\\right\),\\qquad\\sigma\_\{i\}=\\\|x\_\{i\}\-x\_\{j\_\{7\}\(i\)\}\\\|\.\(19\)
For barcode\-family labelsbfi\\mathrm\{bf\}\_\{i\}and scar\-family labelssfi\\mathrm\{sf\}\_\{i\}, define

Ri​j=1−12\(𝟏\{bfi≠bfj\}\+𝟏\{sfi≠sfj\}\),Ki​jlin=exp\(βlinRi​j\)\.R\_\{ij\}=1\-\\frac\{1\}\{2\}\\left\(\\mathbf\{1\}\_\{\\\{\\mathrm\{bf\}\_\{i\}\\neq\\mathrm\{bf\}\_\{j\}\\\}\}\+\\mathbf\{1\}\_\{\\\{\\mathrm\{sf\}\_\{i\}\\neq\\mathrm\{sf\}\_\{j\}\\\}\}\\right\),\\qquad K^\{\\mathrm\{lin\}\}\_\{ij\}=\\exp\(\\beta\_\{\\mathrm\{lin\}\}R\_\{ij\}\)\.\(20\)Pairs involving a cell without a barcode call receiveRi​j=0R\_\{ij\}=0\. Barcode and scar families containing fewer than 100 and 50 cells, respectively, are excluded\. For the RNA\-velocity vectorviv\_\{i\}, the velocity factor is

ai​j=⟨vi,xj−xi⟩‖vi‖​‖xj−xi‖,Ki​jvel=exp⁡\(ν​ai​j\)\.a\_\{ij\}=\\frac\{\\langle v\_\{i\},x\_\{j\}\-x\_\{i\}\\rangle\}\{\\\|v\_\{i\}\\\|\\,\\\|x\_\{j\}\-x\_\{i\}\\\|\},\\qquad K^\{\\mathrm\{vel\}\}\_\{ij\}=\\exp\(\\nu a\_\{ij\}\)\.\(21\)The final transition matrix is

Wi​j=Ki​jexpr​Ki​jlin​Ki​jvel,Pi​j=Wi​j∑ℓWi​ℓ,Pi​i=0\.W\_\{ij\}=K^\{\\mathrm\{expr\}\}\_\{ij\}K^\{\\mathrm\{lin\}\}\_\{ij\}K^\{\\mathrm\{vel\}\}\_\{ij\},\\qquad P\_\{ij\}=\\frac\{W\_\{ij\}\}\{\\sum\_\{\\ell\}W\_\{i\\ell\}\},\\qquad P\_\{ii\}=0\.\(22\)
The different choices

\(βlin,ν\)∈\{\(0,0\),\(0,3\),\(3,0\),\(3,3\)\}\(\\beta\_\{\\mathrm\{lin\}\},\\nu\)\\in\\\{\(0,0\),\(0,3\),\(3,0\),\(3,3\)\\\}correspond to expression only, expression plus velocity, expression plus lineage, and all three factors, respectively\.

![Refer to caption](https://arxiv.org/html/2610.11318v1/itracer_paths_tuned.png)Figure 8:Learned trajectories on iTracer in the two\-dimensional representation\.
#### Bifurcation:

We evaluate a synthetic bifurcation comprising30003000points inℝ3\\mathbb\{R\}^\{3\}, where one source population splits into two target arms and the middle split is withheld[Fig\.9](https://arxiv.org/html/2610.11318#A3.F9)\. Hyperparameters are transferred from Sheet without retuning\. To evaluate cell\-level fate preservation, we integrate the 324 of 969 source cells with resolved initial arm labels and classify their endpoints as lower arm, upper arm, or outside\. The target regions are constructed from balls of radius1/21/2of the distance between arm centres Table[Table4](https://arxiv.org/html/2610.11318#A3.T4)\. FFM maintains high recall on both arms, whereas OT\-CFM and Curly\-FM misassign lower\-arm cells, and MFM loses upper\-arm recall primarily through outside outcomes\. A separate allocation sweep[Table5](https://arxiv.org/html/2610.11318#A3.T5)shows that performance under imbalance remains configuration\-dependent\.

\(a\)Dataset\.![Refer to caption](https://arxiv.org/html/2610.11318v1/forksheet_paths.png)\(b\)Learned trajectories\.

Figure 9:ForkSheet benchmark\.Table 4:ForkSheet fate preservation\. Values are percentages, reported as mean±\\pmpopulation standard deviation over three seeds\.MethodTrue armRecallWrong armOutsideOT\-CFMLower69\.4±\\pm10\.730\.6±\\pm10\.70\.0±\\pm0\.0Upper100\.0±\\pm0\.00\.0±\\pm0\.00\.0±\\pm0\.0MFM / LANDLower91\.4±\\pm5\.64\.4±\\pm3\.94\.2±\\pm3\.4Upper68\.1±\\pm37\.312\.3±\\pm17\.419\.6±\\pm20\.1Curly\-FMLower48\.8±\\pm16\.747\.4±\\pm15\.23\.8±\\pm1\.5Upper100\.0±\\pm0\.00\.0±\\pm0\.00\.0±\\pm0\.0FFM \(ours\), deterministicLower75\.5±\\pm19\.721\.2±\\pm19\.43\.4±\\pm0\.3Upper100\.0±\\pm0\.00\.0±\\pm0\.00\.0±\\pm0\.0FFM \(ours\), with noiseLower68\.3±\\pm19\.820\.8±\\pm22\.710\.9±\\pm12\.7Upper98\.8±\\pm0\.90\.0±\\pm0\.01\.2±\\pm0\.9

Table 5:ForkSheet recall under different nominal lower/upper fate allocations\. Values are percentages, reported as mean±\\pmpopulation standard deviation over three seeds\. Within each cell, the first line reports lower\-arm recall and the second line reports upper\-arm recall\. These runs are separate from the evaluation in Table[Table4](https://arxiv.org/html/2610.11318#A3.T4)\.Method50/5070/3040/6020/8010/90OT\-CFM69\.4±\\pm10\.7100\.0±\\pm0\.071\.1±\\pm2\.5100\.0±\\pm0\.057\.6±\\pm11\.799\.5±\\pm0\.759\.0±\\pm17\.499\.1±\\pm0\.636\.1±\\pm16\.498\.4±\\pm1\.3MFM / LAND91\.2±\\pm9\.576\.0±\\pm16\.699\.9±\\pm0\.284\.8±\\pm1\.892\.0±\\pm8\.398\.5±\\pm2\.273\.2±\\pm23\.674\.0±\\pm34\.950\.9±\\pm15\.196\.9±\\pm4\.4Curly\-FM48\.8±\\pm16\.7100\.0±\\pm0\.068\.0±\\pm7\.1100\.0±\\pm0\.037\.2±\\pm3\.9100\.0±\\pm0\.033\.9±\\pm18\.8100\.0±\\pm0\.00\.0±\\pm0\.0100\.0±\\pm0\.0FFM \(ours\), deterministic95\.6±\\pm3\.494\.1±\\pm5\.292\.2±\\pm10\.489\.1±\\pm13\.989\.4±\\pm5\.794\.2±\\pm8\.298\.9±\\pm0\.897\.7±\\pm0\.877\.8±\\pm10\.498\.3±\\pm0\.3FFM \(ours\), with noise97\.3±\\pm3\.494\.5±\\pm7\.794\.0±\\pm8\.581\.2±\\pm19\.593\.8±\\pm4\.298\.5±\\pm1\.899\.5±\\pm0\.898\.7±\\pm0\.479\.6±\\pm6\.593\.1±\\pm4\.8

## Appendix DRelated Work

#### Relation to Metric Flow Matching\.

Metric Flow Matching \(MFM\)\[[24](https://arxiv.org/html/2610.11318#bib.bib12)\]learns endpoint\-conditioned approximate geodesics of a data\-dependent Riemannian metric

g⁡\(x\)=G⁡\(x,𝒟\)g\(x\)=G\(x;\\mathcal\{D\}\)\(Definition 1\) as interpolants\. This construction is contained in FFM as the symmetric special case in which no directional information is used\. More precisely

G⁡\(x,𝒟\)G\(x;\\mathcal\{D\}\)is a special case of a

P−P\-derived Randers with

FP​\(x,v\)=vT​G​\(x,𝒟\)​vF\_\{P\}\(x,v\)=\\sqrt\{v^\{T\}G\(x;\\mathcal\{D\}\)v\}\.

There is also an explicit connection between the LAND metric\[[2](https://arxiv.org/html/2610.11318#bib.bib16)\]considered in MFM and our

PP\-derived metric \([9](https://arxiv.org/html/2610.11318#S4.E9)\)\. Consider a soft

kk\-NN kernel

wi​j=𝟏\{j∈𝒩k\(i\)\}exp\(−‖xi−xj‖22​σ2\),Pi​j=wi​jZi,Zi=∑jwi​j\.\\displaystyle w\_\{ij\}=\\mathbf\{1\}\_\{\\\{j\\in\\mathcal\{N\}\_\{k\}\(i\)\\\}\}\\exp\\\!\\left\(\-\\frac\{\\\|x\_\{i\}\-x\_\{j\}\\\|^\{2\}\}\{2\\sigma^\{2\}\}\\right\),\\qquad P\_\{ij\}=\\frac\{w\_\{ij\}\}\{Z\_\{i\}\},\\qquad Z\_\{i\}=\\sum\_\{j\}w\_\{ij\}\.The corresponding second transition moment is

CN​\(xi\)=1Zi​∑jwi​j​\(xj−xi\)​\(xj−xi\)⊤\.\\displaystyle C\_\{N\}\(x\_\{i\}\)=\\frac\{1\}\{Z\_\{i\}\}\\sum\_\{j\}w\_\{ij\}\(x\_\{j\}\-x\_\{i\}\)\(x\_\{j\}\-x\_\{i\}\)^\{\\top\}\.On the other hand, defininghα​\(xi\)=∑jwi​j​\(xjα−xiα\)2,h\_\{\\alpha\}\(x\_\{i\}\)=\\sum\_\{j\}w\_\{ij\}\(x\_\{j\}^\{\\alpha\}\-x\_\{i\}^\{\\alpha\}\)^\{2\},we obtain

diag⁡CN​\(xi\)=1Zi​diag⁡\(h⁡\(xi\)\)\.\\operatorname\{diag\}C\_\{N\}\(x\_\{i\}\)=\\frac\{1\}\{Z\_\{i\}\}\\operatorname\{diag\}\(h\(x\_\{i\}\)\)\.\(23\)Consequently,

\(diag⁡CN​\(xi\)\+ρ​I\)−1=Zi​\(diag⁡\(h⁡\(xi\)\)\+ρ​Zi​I\)−1\.\\left\(\\operatorname\{diag\}C\_\{N\}\(x\_\{i\}\)\+\\rho I\\right\)^\{\-1\}=Z\_\{i\}\\left\(\\operatorname\{diag\}\(h\(x\_\{i\}\)\)\+\\rho Z\_\{i\}I\\right\)^\{\-1\}\.\(24\)Thus, apart from the normalization

ZiZ\_\{i\}induced by requiring

PPto be row\-stochastic, the diagonal of our second\-moment geometry has exactly the form of the LAND metric on datapoints\. In particular, when

ZiZ\_\{i\}is approximately constant over the region of interest, choosing

ε=ρ​Z\\varepsilon=\\rho Zrecovers the LAND metric up to the global multiplicative factor

ZZ, which does not change its geodesic curves\.

FFM can therefore be viewed as a generalization of MFM incoporating a sense of directionality when given an transition probability matrix

PP\.

#### Relation to Curly FM:

Curly Flow Matching \(Curly\-FM\)\[[35](https://arxiv.org/html/2610.11318#bib.bib10)\]assumes access to a reference vector fieldft​\(x\)f\_\{t\}\(x\)and learns endpoint conditioned paths whose velocities remain close to this field, using an objective of the form

ℒCurly​\(η\)=𝔼x0,x1,t​\[‖x˙t,η−ft​\(xt,η\)‖22\]\.\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{Curly\}\}\(\\eta\)=\\mathbb\{E\}\_\{x\_\{0\},x\_\{1\},t\}\\left\[\\left\\\|\\dot\{x\}\_\{t,\\eta\}\-f\_\{t\}\(x\_\{t,\\eta\}\)\\right\\\|\_\{2\}^\{2\}\\right\]\.When the dynamics are instead specified by a transition matrixPP, the natural first\-order vector\-field summary is its first spatial momentm⁡\(xi\)=∑j=1NPi​j​\(xj−xi\),m\(x\_\{i\}\)=\\sum\_\{j=1\}^\{N\}P\_\{ij\}\(x\_\{j\}\-x\_\{i\}\),\. If we replace the Euclidean norm in the Curly objective by the second momentCρ−1C\_\{\\rho\}^\{\-1\}

‖x˙t,η−m⁡\(x\)‖C​\(x\)−12=\(x˙t,η−m⁡\(x\)\)T​Cρ​\(x\)−1​\(x˙t,η−m⁡\(x\)\)\\displaystyle\\left\\\|\\dot\{x\}\_\{t,\\eta\}\-m\(x\)\\right\\\|\_\{C\(x\)^\{\-1\}\}^\{2\}=\(\\dot\{x\}\_\{t,\\eta\}\-m\(x\)\)^\{T\}C\_\{\\rho\}\(x\)^\{\-1\}\(\\dot\{x\}\_\{t,\\eta\}\-m\(x\)\)\(25\)which gives us the integrant of the fixed time FW functional[Eq\.16](https://arxiv.org/html/2610.11318#A2.E16)\. For data that follows the manifold hypothesis we have in particular thatC≠IC\\neq I\. This observation also suggests a direct second\-moment generalization of the Curly\-FM path objective\. Specifically, we replace its Euclidean velocity residual by the graph\-derived anisotropic residual

1a​\(x˙−a​m​\(x\)\)⊤​Cρ​\(x\)−1​\(x˙−a​m​\(x\)\),\\frac\{1\}\{a\}\\bigl\(\\dot\{x\}\-a\\,m\(x\)\\bigr\)^\{\\top\}C\_\{\\rho\}\(x\)^\{\-1\}\\bigl\(\\dot\{x\}\-a\\,m\(x\)\\bigr\),so that the control has access to the same first and second transition moments as FFM\. We evaluate this matched control on Pancreas in Fig\.[Fig\.3](https://arxiv.org/html/2610.11318#S5.F3)\. Full FFM obtains lower

W2​\(1/2\)W\_\{2\}\(1/2\)under both a shared endpoint coupling and method\-specific couplings, indicating that the observed improvement is not explained solely by supplying the path objective with second\-moment information\.

Furthermore, FFM can also be used in settings with access to a reference vector field\. Following velocity\-kernel constructions such as those used in CellRank\[[26](https://arxiv.org/html/2610.11318#bib.bib49)\], a vector field

ffcan be converted into a local transition matrix

PfP^\{f\}\. Let

Wi​j\(h\)=\{∥xj−xi∥≤hi\},W^\{\(h\)\}\_\{ij\}=\\mathbf\{1\}\\\!\\left\\\{\\\|x\_\{j\}\-x\_\{i\}\\\|\\leq h\_\{i\}\\right\\\},where

hih\_\{i\}denotes the distance to the

kk\-th nearest neighbor, and define

Pi​jf=Wi​j\(h\)​exp⁡\(κ​f​\(xi\)⊤​\(xj−xi\)‖f⁡\(xi\)‖​‖xj−xi‖\)∑ℓWi​ℓ\(h\)​exp⁡\(κ​f​\(xi\)⊤​\(xℓ−xi\)‖f⁡\(xi\)‖​‖xℓ−xi‖\)\.\\displaystyle P^\{f\}\_\{ij\}=\\frac\{W^\{\(h\)\}\_\{ij\}\\exp\\\!\\left\(\\kappa\\frac\{f\(x\_\{i\}\)^\{\\top\}\(x\_\{j\}\-x\_\{i\}\)\}\{\\\|f\(x\_\{i\}\)\\\|\\,\\\|x\_\{j\}\-x\_\{i\}\\\|\}\\right\)\}\{\\sum\_\{\\ell\}W^\{\(h\)\}\_\{i\\ell\}\\exp\\\!\\left\(\\kappa\\frac\{f\(x\_\{i\}\)^\{\\top\}\(x\_\{\\ell\}\-x\_\{i\}\)\}\{\\\|f\(x\_\{i\}\)\\\|\\,\\\|x\_\{\\ell\}\-x\_\{i\}\\\|\}\\right\)\}\.For dense and locally isotropic sampling, thekk\-NN neighborhood ofxix\_\{i\}is approximately rotationally symmetric\. The cosine kernel assigns a displacement directionuua weight proportional toexp⁡\(κ​f^​\(xi\)⊤​u\)\\exp\(\\kappa\\,\\hat\{f\}\(x\_\{i\}\)^\{\\top\}u\), wheref^=f/‖f‖\\hat\{f\}=f/\\\|f\\\|\. By local isotropy, contributions orthogonal tof^\\hat\{f\}appear in pairs in opposite directions with and hence cancel eachother\. Consequently,

mNf​\(xi\)=∑jPi​jf​\(xj−xi\)=ci​f^​\(xi\)\+o⁡\(1\),ci\>0,\\displaystyle m\_\{N\}^\{f\}\(x\_\{i\}\)=\\sum\_\{j\}P^\{f\}\_\{ij\}\(x\_\{j\}\-x\_\{i\}\)=c\_\{i\}\\,\\hat\{f\}\(x\_\{i\}\)\+o\(1\),\\qquad c\_\{i\}\>0,as the neighborhood becomes dense\. Thus, the first moment of the cosine velocity kernel recovers the orientation of the vector field\. FFM can therefore be applied also when the dynamical information is given as a vector field: its first moment retains the directional information used by Curly\-FM, while its second moment additionally describes the local distribution of accessible transitions\. This is particularly helpful when the vector field measurement is noisy, as is often the case for RNA\-velocity estimates\. Then the construction of the transition kernel can be viewed as a denoising of the vector field\.

#### Multimarginal Flow Matching