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Finsler Flow Matching: Dynamics-Aware Geodesic Interpolation for Single-Snapshot Trajectory Inference
Niklas Canova, Jonas Simon Fleck
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 91%
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Summary
The paper introduces Finsler Flow Matching (FFM), a framework for single-cell trajectory inference that learns continuous stochastic dynamics from discrete Markov transition graphs. FFM constructs a Finsler geometry (specifically a Randers metric) using the first and second local moments of the transition kernel to capture directional preference and anisotropic accessibility. It then uses neural networks to approximate directed geodesics, constructs source-target couplings based on Finsler costs, and employs simulation-free score and flow matching to generate continuous trajectories. The method improves the recovery of intermediate cellular states compared to existing methods, particularly in strongly directional or anisotropic dynamics.
Entities (8)
Relation Signals (6)
Finsler Flow Matching → constructs → Finsler geometry
confidence 95% · We use the first and second local moments to construct a Finsler structure motivated by the Freidlin--Wentzell action
Finsler Flow Matching → uses → Markov transition graph
confidence 95% · We introduce Finsler Flow Matching (FFM), a framework for learning continuous stochastic dynamics from discrete Markov transition graphs.
Finsler Flow Matching → employs → Score and flow matching
confidence 90% · define geometry-aware stochastic conditional paths that can be distilled into a continuous generative process through simulation-free score and flow matching.
Finsler Flow Matching → improves → Single-cell Trajectory Inference
confidence 90% · Across synthetic and single-cell trajectory inference benchmarks, FFM improves recovery of withheld intermediate populations
Finsler geometry → includes → Randers metric
confidence 90% · Freidlin–Wentzell theory then motivates a Randers metric, in which local diffusion determines anisotropic accessibility and drift introduces directional preference.
Finsler geometry → motivatedby → Freidlin-Wentzell action
confidence 85% · We use the first and second local moments to construct a Finsler structure motivated by the Freidlin--Wentzell action
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Abstract
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.
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- Source: https://arxiv.org/abs/2610.11318v1
- Canonical: https://arxiv.org/abs/2610.11318v1
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Finsler Flow Matching: Dynamics-Aware Geodesic Interpolation for Single-Snapshot Trajectory Inference Niklas Canova Jonas Simon Fleck Affiliation: Institute of Human Biology (IHB), Roche, Basel, Switzerland Email: niklas.canova@roche.com 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 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]. 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]. Several experimental and computational strategies can provide additional information about developmental directionality and dynamics. RNA velocity infers the direction of local transcriptional change [25, 7], lineage tracing tracks ancestry [48, 46, 31] and metabolic or fluorescent labelling can introduce an absolute timescale [6, 11, 17]. 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]. 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, 47]. Such kernels have proven highly effective for inferring terminal states, lineage relationships, and fate probabilities [14, 27]. 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]. Given samples at multiple time points, optimal transport formulations recover a coupling between the observed marginals [38, 13], while flow and Schrödinger Bridge matching provide simulation-free frameworks for learning continuous dynamics [43, 30, 37, 41]. Some variants additionally incorporate the geometry of the state or distribution space [12, 18]. Metric Flow Matching replaces Euclidean interpolation with geodesics under a learned Riemannian metric [24], 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], 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, 20] theory then motivates a Randers metric [4], 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], 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], 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 Appendix A Finsler Geometry: Riemannian geometry measures local distance via a symmetric metric tensor G(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)= v G(x)v+β(x) v, where G(x)∈SPD(d)G(x)∈ (d) is a Riemannian metric tensor and β(x)β(x) is a 11-form giving directionality. For an absolutely continuous path γ:[0,1]→ℝdγ:[0,1] ^d, the Finsler length LF(γ)L_F(γ) and energy EF(γ)E_F(γ) functionals are defined as: LF(γ)≔∫01F(γ(t),γ˙(t))t,EF(γ)≔∫01F(γ(t),γ˙(t))2t.L_F(γ) _0^1F(γ(t), γ(t))dt, E_F(γ) _0^1F(γ(t), γ(t))^2dt. Curves minimizing LFL_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)=x0γ(1)=x1∫01F(γ(t),γ˙(t))t.c_F(x_0,x_1) _ subarraycγ(0)=x_0\ γ(1)=x_1 subarray _0^1F(γ(t), γ(t))dt. (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]. Given source and target densities p0,p1∈(ℝd)p_0,p_1 (R^d) and a cost function c:ℝd×ℝd→ℝ≥0c:R^d×R^d _≥ 0, the Kantorovich problem seeks an optimal transport plan π∈Π(p0,p1)π∈ (p_0,p_1) minimizing total cost: π⋆=arginfπ∈Π(p0,p1)∫ℝd×ℝdc(x0,x1)dπ(x0,x1),π = _π∈ (p_0,p_1) _R^d×R^dc(x_0,x_1)dπ(x_0,x_1), where Π(p0,p1) (p_0,p_1) is the set of joint distributions with marginals p0p_0 and p1p_1. Using the squared Finsler geodesic distance as cost function and the elementary fact cF(x0,x1)2=infγ:γ(0)=x0,γ(1)=x1EF(γ)c_F(x_0,x_1)^2= _γ:γ(0)=x_0,γ(1)=x1E_F(γ) 11 1 It 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)) dt dπ(x0,x1). W_2,F^2(p_0,p_1)= _π∈ (p_0,p_1) _γ:γ(0)=x_0,γ(1)=x_1 _R^d×R^d _0^1F^2(γ(t), γ(t)) dt dπ(x_0,x_1). (2) A stochastic counterpart to the OT problem is given by the Schrödinger Bridge Problem (SBP) [28]. Given a reference measure on Ω=C([0,1],ℝd) =C([0,1],R^d) of a stochastic process R, the SBP searches for the closest path measure to R that has endpoint marginals given by p0p_0 and p1p_1. More precisely Q∗=argminQ:Q0=p0,;Q1=p1KL(Q∥R). Q^*= _Q:Q_0=p_0,;Q_1=p_1KL(Q\|R). (3) A classical result due to Föllmer [15] tells us that the solution can be decomposed as Q∗=∫Rx0,x1dπ∗(x0,x1), where π∗=argminπ∈Π(p0,p1)KL(π∥R01), Q^*= R^x_0,x_1dπ^*(x_0,x_1), where π^*= _π∈ (p_0,p_1)KL (π\|R_01 ), (4) where Rx0,x1=R(⋅∣Y0=x0,Y1=x1)R^x_0,x_1=R(· Y_0=x_0,Y_1=x_1) denotes the reference process conditioned on its endpoints and R01R_01 its endpoint law. For a Brownian reference process dYt=σBdWtdY_t= _BdW_t, one can see that Rtx0,x1=((1−t)x0+tx1,σ2t(1−t)I).R_t^x_0,x_1=N ((1-t)x_0+tx_1,σ^2t(1-t)I ). For non-Brownian reference processes, such as SDEs with anisotropic diffusion and non-trivial drift dYt=b0(Yt),dt+Σ(Yt)dWt, dY_t=b_0(Y_t),dt+ (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 Appendix B. Score and Flow Matching: Flow Matching (FM) is a simulation-free method for training continuous-time generative models. Given a target probability path pt(x)p_t(x) pushed forward by an unknown vector field ut(x)u_t(x), FM fits a neural vector field vθ(t,x)v_θ(t,x) via a regression objective: ℒFM(θ)=∫01x∼pt(x)[|vθ(t,x)−ut(x)|2]t.L_FM(θ)= _0^1E_x p_t(x) [|v_θ(t,x)-u_t(x)|^2 ]dt. (5) Conditional Flow Matching (CFM) circumvents the intractability of the marginal vector field ut(x)u_t(x) by conditioning on endpoint pairs (x0,x1)∼q(x0,x1)(x_0,x_1) q(x_0,x_1) [30, 1]. The network is trained on tractable conditional vector fields vt(x∣x0,x1)v_t(x x_0,x_1) pushing probability paths along conditional trajectories μt(x0,x1) _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]. Following the approach of Tong et al. [43] one may jointly training the flow and score through the combined objective ℒ[SF]2M(θ)=ℒCFM(θ)+λℒCSM(θ). _[SF]^2M(θ)=L_CFM(θ)+ _CSM(θ). The learned flow vθv_θ and score sψs_ψ can be used with an arbitrary matrix-valued diffusion schedule :[0,1]→ℝd×dg:[0,1] ^d× d to define the stochastic generative process dXt=[vθ(t,Xt)+12(t)(t)⊤sψ(t,Xt)]dt+(t)dWt, dX_t= [v_θ(t,X_t)+ 12g(t)g(t) s_ψ(t,X_t) ]dt+g(t)\,dW_t, (6) which satisfies Xt∼ptX_t p_t for all t∈[0,1].t∈[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, 42, 10, 50]. Relatedly, Huguet et al. [23] 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, 47]. 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]. Related work studies closed-form Gaussian Schrödinger bridges and smooth Schrödinger bridges for trajectory inference [9, 22]. 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] 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] 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] 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] 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=xii=1N⊆ℝdD=\x_i\_i=1^N ^d be a collection of observations sampled from an unknown distribution p. Let D0=xi0i=1N0D_0=\x_i^0\_i=1^N_0 and D1=xi1i=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 ^N× 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∈[0,1] that are consistent with the dynamics encoded by P. More formally we can define this as an OT problem of the form 2 (π∗,γ∗)=argmin(π,γ)∫ℝd×ℝd∫01FP2(γ(t),γ˙(t)) dt dπ(x0,x1) where π∈Π(p0,p1) and γ(0)=x0,γ(1)=x1 aligned &(π^*,γ^*)= _(π,γ) _R^d×R^d _0^1F_P^2(γ(t), γ(t)) dt dπ(x_0,x_1)\\ & where π∈ (p_0,p_1) and γ(0)=x_0,γ(1)=x_1 aligned (7) and FP(x,v)=v⊤GP(x)v+βP(x)⊤vF_P(x,v)= v G_P(x)v+ _P(x) v is a P-derived Randers metric (Definition 7). 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 P-derived Randers metric is constructed from spatial moments of P. More precisely we define at each observed state xix_i moments as: m(xi)≔∑j=1NPij(xj−xi), and C(xi)≔∑j=1NPij(xj−xi)(xj−xi)⊤. m(x_i) _j=1^NP_ij(x_j-x_i), and C(x_i) _j=1^NP_ij(x_j-x_i)(x_j-x_i) . (8) 22 2 At finite graph scale C=CovP(Δx)+mm⊤C=Cov_P( x)+m , we interpret C as an anisotropic transition-accessibility rather then diffusion covariance. m(xi)m(x_i) captures the net preferred displacement and therefore provides directionality whereas C(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 of m and C, for example using kernel smoothing, neural networks, or support vector regression [39], while preserving positive semidefiniteness of C. We use kernel smoothing for simplicity details are provided in C. To obtain a non-degenerate Finsler structure, we define the regularized Freidlin–Wentzell (FW) action [16]. Firstly, set Cρ(x)≔C(x)+ρIC_ρ(x) C(x)+ρ I with ρ>0ρ>0 and define FP(x,v)≔v⊤Cρ(x)−1v⏟Anisotropic accessibilitym(x)⊤Cρ(x)−1m(x)+λ2⏟first-moment calibration−v⊤Cρ(x)−1m(x)⏟directional alignment,F_P(x,v) v C_ρ(x)^-1v_Anisotropic accessibility\, m(x) C_ρ(x)^-1m(x)+λ^2_first-moment calibration- v C_ρ(x)^-1m(x)_directional alignment, (9) where λ>0λ>0 ensures Randers admissibility 5. 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))t.S(γ)= _0^1F_P\! (γ(t), γ(t) )\,dt. (10) The following proposition formalizes how geodesics under FPF_P align with the transition structure encoded by P: Proposition 1 (Informal 8). 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 minimizing FPF_P-geodesic remains within that neighborhood. Moreover, locally the metric favors alignment with the first transition moment. In the special case where P is the discrete observation of an SDE 14, we can conclude even more about geodesics. By Freidlin and Wentzell [16] and Heymann and Vanden‐Eijnden [20] we can show that geodesics under the resulting metric FPF_P follow the maximum likelihood paths of the stochastic process. Theorem 2 (Informal 18). If P is 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. Connections to the limits of local graph kernels, and the scope of these results for velocity-based constructions, are discussed in B. Learning Geodesics: For endpoints (x0,x1)∼p0⊗p1(x_0,x_1) p_0 p_1 and t∈[0,1]t∈[0,1], parameterize a candidate curve by xt,η(x0,x1)=(1−t)x0+tx1+t(1−t)φη(t,x0,x1),x_t,η(x_0,x_1)\;=\;(1-t)\,x_0+t\,x_1+t(1-t)\, _η(t,x_0,x_1), (11) where φη:[0,1]×ℝd×ℝd→ℝd _η:[0,1]×R^d×R^d ^d is some neural network with parameters η. The boundary correction t(1−t)t(1-t) enforces x0,η=x0x_0,η=x_0 and x1,η=x1x_1,η=x_1. The time derivative is given by, x˙t,η=x1−x0+(1−2t)φη(t,x0,x1)+t(1−t)∂tφη(t,x0,x1). x_t,η=x_1-x_0+(1-2t) _η(t,x_0,x_1)+t(1-t) _t _η(t,x_0,x_1). This interpolant was previously proposed in Kapusniak et al. [24], and similarly to their approach, we minimize the Finsler Energy functional, ℒgeo(η)=(x0,x1)∼p0⊗p1t∼[0,1][FP(xt,η(x0,x1),x˙t,η(x0,x1))2].L_geo(η)=E_ subarrayc(x_0,x_1) p_0 p_1\\ t [0,1] subarray [F_P\! (x_t,η(x_0,x_1), x_t,η(x_0,x_1) )^2 ]. (12) Since energy minimizers are constant-speed reparameterizations of geodesics, minimizing (12) over η yields an approximation of the pairwise P-derived Finsler geodesics. We denote the resulting parameters by η∗η^*. Algorithm 1 Training the Finsler Geodesic Model 1: Source and target distributions p0,p1p_0,p_1, Finsler metric FPF_P, interpolant network φη _η, LR γ. 2: while not converged do 3: Sample x0∼p0x_0 p_0, x1∼p1x_1 p_1, t∼[0,1]t [0,1] and compute xt,η(x0,x1)x_t,η(x_0,x_1) and x˙t,η(x0,x1) x_t,η(x_0,x_1) 4: ℒgeo←FP(xt,η,x˙t,η)2L_geo← F_P\! (x_t,η, x_t,η )^2 5: η←η−γ∇ηℒgeoη←η-γ _ηL_geo 6: end while 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. 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))2t.c_η^*(x_0,x_1):= _0^1F_P\! (x_t,η^*(x_0,x_1), x_t,η^*(x_0,x_1) )^2\,dt. Thus, cη∗(x0,x1)≈infγ(0)=x0γ(1)=x1∫01FP(γ(t),γ˙(t))2t=cF2(x0,x1).c_η^*(x_0,x_1)≈ _ subarraycγ(0)=x_0\\ γ(1)=x_1 subarray _0^1F_P\! (γ(t), γ(t) )^2\,dt=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∗=argminπ∈Π(p^0,p^1)∑i=1m∑j=1mπijcij+εKL(π∥p^0⊗p^1), _ ^*= _π∈ ( p_0, p_1) \ _i=1^m _j=1^m _ijc_ij+ \,KL\! (π\, \|\, p_0 p_1 ) \, (13) where ε>0 >0 controls the strength of the entropic regularization. This problem can be solved efficiently using Sinkhorn iterations [13]. The interpolant xt,η∗x_t,η^* and the coupling πε∗ _ ^* provide an approximation of the original Finsler OT problem 7 up to the approximation errors. Flow and Stochastic Model: Assume we have computed (πε∗,x⋅,η∗) ( _ ^*,x_·,η^* ). Drawing (x0,x1)∼πε∗(x_0,x_1) _ ^* and transporting each pair along its learned geodesic gives the probability path pt(x)≔(xt,η∗)#πε∗(x)p_t(x) (x_t,η^* )_\# _ ^*(x). By construction, p0p_0 and p1p_1 are the endpoint marginals. The corresponding conditional probability distribution path is pt(x|x0,x1)=δxt,η(x0,x1)p_t(x|x_0,x_1)= _x_t,η(x_0,x_1). Furthermore, define the conditional path μt(x0,x1):=xt,η⋆(x0,x1) _t(x_0,x_1):=x_t,η (x_0,x_1) We then train a model vθv_θ using the conditional flow matching objective ℒCFM(θ)=(x0,x1)∼πε⋆t∼[0,1][‖vθ(t,μt(x0,x1))−μ˙t(x0,x1)‖22]. _CFM(θ)=E_ subarrayc(x_0,x_1) _ \\ t [0,1] subarray [ \|v_θ\! (t, _t(x_0,x_1) )- μ_t(x_0,x_1) \|_2^2 ]. Motivated by Föllmer [15] we approximate each conditional bridge with an anisotropic Gaussian path. For (x0,x1)∼πε⋆(x_0,x_1) π _ , define Cρ,t:=Cρ(μt)C_ρ,t:=C_ρ( _t) and pt(x∣x0,x1)=(x,μt,σ2t(1−t)Cρ,t).p_t(x x_0,x_1)=N\! (x;\, _t,\,σ^2t(1-t)C_ρ,t ). We evaluate C at the deterministic interpolant μt _t rather than at the state x, 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 to P. The conditional probability-flow vector field is ut(x∣x0,x1)=μ˙t+1−2t2t(1−t)(x−μt)+12C˙ρ,tCρ,t−1(x−μt), u_t(x x_0,x_1)= μ_t+ 1-2t2t(1-t)(x- _t)+ 12 C_ρ,tC_ρ,t^-1(x- _t), See 21 for a derivation. We distill the conditional bridges into a marginal vector field vθv_θ and score sψs_ψ using simulation-free conditional flow and score matching [43], ℒCFM(θ) _CFM(θ) =(x0,x1)∼πε⋆,t∼(0,1)x∼pt(⋅∣x0,x1)[‖v~θ(t,x)−ut(x∣x0,x1)‖22], =E_ subarrayc(x_0,x_1) _ ,\;t (0,1)\\ x p_t(· x_0,x_1) subarray [ \| v_θ(t,x)-u_t(x x_0,x_1) \|_2^2 ], ℒCSM(ψ) _CSM(ψ) =(x0,x1)∼πε⋆,t∼(0,1)x∼pt(⋅∣x0,x1)[λ(t)2‖sψ(t,x)−st(x∣x0,x1)‖22] =E_ subarrayc(x_0,x_1) _ ,\;t (0,1)\\ x p_t(· x_0,x_1) subarray [λ(t)^2 \|s_ψ(t,x)-s_t(x x_0,x_1) \|_2^2 ] where the final loss is given by ℒ[SF]2M=ℒCFM+ℒCSML_[SF]^2M=L_CFM+L_CSM and the conditional score is st(x∣x0,x1)=∇xlogpt(x∣x0,x1)=−1σ2t(1−t)Cρ,t−1(x−μt).s_t(x x_0,x_1)= _x p_t(x x_0,x_1)=- 1σ^2t(1-t)C_ρ,t^-1(x- _t). Inference: We sample x0∼p0x_0 p_0 and integrate the learned dynamics from t=0t=0 to t=1t=1. In the deterministic setting, we integrate the ODE dXt=vθ(t,Xt)dt.dX_t=v_θ(t,X_t)\,dt. In the stochastic setting, we use the score-and-flow SDE introduced in 6 and make a slight generalization by setting (t,x)=g(t)Cρ1/2(x),g(t,x)=g(t)\,C_ρ^1/2(x), for some scalar-valued noise function g(t)g(t). Substituting this choice into the stochastic dynamics gives dXt=[v~θ(t,Xt)+12g(t)2Cρ(x)sψ(t,Xt)+12g(t)2div(Cρ(x))]dt+g(t)Cρ(x)1/2dWt, dX_t= [ v_θ(t,X_t)+ 12g(t)^2\,C_ρ(x)s_ψ(t,X_t)+ 12g(t)^2 (C_ρ(x)) ]dt+g(t)\,C_ρ(x)^1/2\,dW_t, where div(Cρ(x))i=∑jd∂jCij(x) (C_ρ(x))_i= _j^d _jC_ij(x) and which we integrate from X0∼p0X_0 p_0 to t=1t=1. The scalar schedule g(t)g(t) controls the overall noise strength, whereas Cρ(x)C_ρ(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 Euclidean W2W_2 at t=1/2t=1/2, supplemented by endpoint W2W_2 and the geometry-dependent W2,FW_2,F. Because t=1/2t=1/2 is a benchmark convention rather than an established correspondence with biological midpoint time, on Pancreas we additionally report W¯2:=1||∑t∈W2(pt,pwithheld) W_2:= 1|T| _t W_2(p_t,p_withheld). We compare against OT-CFM, MFM/LAND, Curly-FM under matched architectures and training budgets. More details Appendix C. 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, achieving W2=0.1588W_2=0.1588 and 0.14490.1449 for deterministic and stochastic FFM, respectively, compared with 0.52210.5221 for Curly-FM Fig. 2. 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). On a complementary synthetic bifurcation benchmark, FFM also preserves cell-level fate assignments across both branches Fig. 9. Method mid W2↓W_2 end W2↓W_2 mid W2,F↓W_2,F OT-CFM 0.7112 ± 0.0233 0.1734 ± 0.0216 1.0305 ± 0.0263 MFM (LAND) 0.6843 ± 0.0074 0.2357 ± 0.0258 0.8645 ± 0.0961 Curly-FM 0.5221 ± 0.0016 0.1405 ± 0.0057 0.5236 ± 0.1657 FFM (ours) 0.1588 ± 0.0096 0.2343 ± 0.0292 0.2892 ± 0.0090 FFM (ours), with noise 0.1449 ± 0.0116 0.2636 ± 0.0682 0.2742 ± 0.0314 sampling floor 0.0646 0.1896 0.2812 Figure 2: W2,FW_2,F denotes the Wasserstein distance induced by the Finsler cost cFc_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] 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], whereas Curly-FM receives per-cell velocities. We found that relative performance depended strongly on representation dimensionality. In two dimensions, Curly-FM achieved lower midpoint W2W_2 than FFM, whereas FFM achieved substantially lower midpoint W2W_2 in the 2020 and 5050dimensional PCA Fig. 3. endpoint rankings vary across methods Fig. 3 Method W2(12)↓W_2( 12) W¯2[1/3,2/3]↓ W_2^[1/3,2/3] W2(1)↓W_2(1) W2,F(12)↓W_2,F( 12) d=2d=2 OT-CFM 1.8062 ± 0.0197 1.7869 ± 0.0230 0.1740 ± 0.0338 0.6420 ± 0.0059 MFM / LAND 1.4682 ± 0.0128 1.4642 ± 0.0131 0.3769 ± 0.0346 0.5399 ± 0.0024 Curly-FM 0.9773 ± 0.1311 0.9686 ± 0.1725 0.2581 ± 0.0371 0.7493 ± 0.0258 FFM (ours), deterministic 1.6692 ± 0.0154 1.6175 ± 0.0348 0.3283 ± 0.1293 0.5681 ± 0.0100 FFM (ours), with noise 1.6957 ± 0.0305 1.6434 ± 0.0555 0.2272 ± 0.0239 0.5730 ± 0.0082 sampling floor 0.1113 0.1113 0.1022 0.4162 d=20d=20 OT-CFM 12.3484 ± 0.0485 12.4957 ± 0.0681 5.2262 ± 0.1583 10.9277 ± 0.0412 MFM / LAND 12.0705 ± 0.0095 12.1656 ± 0.0522 7.8205 ± 0.7863 10.4768 ± 0.0637 Curly-FM 12.2962 ± 0.0414 12.3920 ± 0.0687 4.8562 ± 0.0576 11.0235 ± 0.0520 FFM (ours), deterministic 10.4319 ± 0.0896 10.7413 ± 0.1927 5.3793 ± 0.1848 8.9033 ± 0.0400 FFM (ours), with noise 10.4036 ± 0.0279 10.7355 ± 0.0560 5.0605 ± 0.1654 8.9213 ± 0.0453 sampling floor 4.1781 4.1781 4.4483 4.3608 d=50d=50 OT-CFM 13.5145 ± 0.0466 13.6811 ± 0.1314 8.1252 ± 0.1103 15.5900 ± 0.2287 MFM / LAND 12.9526 ± 0.0605 13.0996 ± 0.1029 8.6687 ± 0.3880 14.6185 ± 0.1085 Curly-FM 13.3381 ± 0.1482 13.4440 ± 0.2069 7.2931 ± 0.2558 15.0427 ± 0.1562 FFM (ours), deterministic 12.5784 ± 0.0235 12.7715 ± 0.0719 7.7975 ± 0.1005 13.9368 ± 0.1217 FFM (ours), with noise 12.5732 ± 0.0573 12.7723 ± 0.0952 7.7411 ± 0.1297 13.7800 ± 0.0861 sampling floor 6.7529 6.7529 7.0275 7.2123 Method Shared coupling Method-specific coupling Graph-drift FM (control) 11.9409 ± 0.1110 11.9223 ± 0.0909 Anisotropic fixed-time FM Eq. 25 11.5712 ± 0.0880 11.4344 ± 0.1067 Second-moment FFM (m=0m=0) 11.5424 ± 0.1226 11.4967 ± 0.0878 FFM, one-form deleted (existing ablation) 11.1740 ± 0.0759 11.1183 ± 0.0774 FFM (ours) 10.3967 ± 0.1371 10.3960 ± 0.1396 Figure 3: Pancreas trajectory reconstruction. Left: Two-dimensional representation. Top right: W2W_2 at t=1/2t=1/2, the average W¯2 W_2 over 11 fixed evaluation times in [1/3,2/3][1/3,2/3], endpoint W2W_2, and W2,FW_2,F Bottom right: same-information objective ablation at d=20d=20. “Shared” fixes the endpoint coupling “method-specific” constructs the coupling separately for each method. Ablation values are W2(t=1/2)W_2(t=1/2), mean ± population 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], 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 midpoint W2W_2 than OT-CFM and MFM across the evaluated representations Table 1. 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. Method d=2d=2 mid W2W_2 d=20d=20 mid W2W_2 d=50d=50 mid W2W_2 OT-CFM 1.3631±0.03681.3631± 0.0368 15.9054±0.080415.9054± 0.0804 18.8807±0.117218.8807± 0.1172 MFM / LAND 1.3563±0.05371.3563± 0.0537 15.0519±0.225015.0519± 0.2250 18.2036±0.266318.2036± 0.2663 FFM, expression only 1.7162±0.99191.7162± 0.9919 12.8597±0.100712.8597± 0.1007 16.6979±0.302016.6979± 0.3020 FFM, + velocity 1.0204±0.10401.0204± 0.1040 12.7288±0.125312.7288± 0.1253 16.7371±0.256616.7371± 0.2566 FFM, + lineage 1.3074±0.10831.3074± 0.1083 12.7368±0.145512.7368± 0.1455 16.7415±0.242616.7415± 0.2426 FFM, + lineage + velocity 1.1773±0.14041.1773± 0.1404 12.7121±0.131412.7121± 0.1314 16.9196±0.270016.9196± 0.2700 Table 1: Intermediate-distribution recovery on iTracer R2 for different transition-kernel constructions. Lower is better. Additional analyses: Appendix Appendix C reports runtime and parameter-sensitivity analyses, additional results on erythroid differentiation [36] as used in used in Petrović et al. [35], 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 in requirements-lock.txt to 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. References [1] M. S. Albergo and E. Vanden-Eijnden (2023) Building Normalizing Flows with Stochastic Interpolants. In International Conference on Learning Representations, Note: arXiv:2209.15571 [cs.LG] External Links: Link, Document Cited by: §2. [2] G. Arvanitidis, L. K. Hansen, and S. Hauberg (2016) A Locally Adaptive Normal Distribution. In Conference on Neural Information Processing Systems, (en). External Links: Link Cited by: Appendix D. [3] J. Bamberger, I. Jones, D. Duncan, M. Bronstein, P. Vandergheynst, and A. Gosztolai (2026) Carré du champ flow matching: better quality-generalisation tradeoff in generative models. International Conference on Learning Representations 2026, p. 135754–135779 (en). External Links: Link Cited by: §3. [4] D. Bao, S.-S. Chern, and Z. Shen (2000) An Introduction to Riemann-Finsler Geometry. Graduate Texts in Mathematics, Vol. 200, Springer, New York, NY. External Links: ISBN 978-1-4612-7070-6 978-1-4612-1268-3, Link, Document Cited by: Appendix A, §1, Lemma 5. [5] A. Bastidas-Ponce, S. Tritschler, L. Dony, K. Scheibner, M. Tarquis-Medina, C. Salinno, S. Schirge, I. Burtscher, A. Böttcher, F. J. Theis, H. Lickert, and M. Bakhti (2019) Comprehensive single cell mRNA profiling reveals a detailed roadmap for pancreatic endocrinogenesis. Development 146 (12), p. dev173849 (eng). External Links: ISSN 1477-9129, Document Cited by: §5. [6] N. Battich, J. Beumer, B. de Barbanson, L. Krenning, C. S. Baron, M. E. Tanenbaum, H. Clevers, and A. van Oudenaarden (2020) Sequencing metabolically labeled transcripts in single cells reveals mRNA turnover strategies. Science 367 (6482), p. 1151–1156. External Links: Link, Document Cited by: §1. [7] V. Bergen, M. Lange, S. Peidli, F. A. Wolf, and F. J. Theis (2020) Generalizing RNA velocity to transient cell states through dynamical modeling. Nature Biotechnology 38 (12), p. 1408–1414 (en). External Links: ISSN 1546-1696, Link, Document Cited by: §1. [8] T. Berry and T. Sauer (2015) Local Kernels and the Geometric Structure of Data. arXiv. Note: arXiv:1407.1426 [math.CA] External Links: Link, Document Cited by: Appendix A, Appendix B. [9] C. Bunne, Y. Hsieh, M. Cuturi, and A. Krause (2023) The Schrödinger Bridge between Gaussian Measures has a Closed Form. In Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, p. 5802–5833 (en). External Links: ISSN 2640-3498, Link Cited by: §3. [10] C. Bunne, L. Papaxanthos, A. Krause, and M. Cuturi (2022) Proximal Optimal Transport Modeling of Population Dynamics. In Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, p. 6511–6528 (en). External Links: ISSN 2640-3498, Link Cited by: §3. [11] J. Cao, W. Zhou, F. Steemers, C. Trapnell, and J. Shendure (2020) Sci-fate characterizes the dynamics of gene expression in single cells. Nature Biotechnology 38 (8), p. 980–988 (eng). External Links: ISSN 1546-1696, Document Cited by: §1. [12] R. T. Q. Chen and Y. Lipman (2023) Flow Matching on General Geometries. In International Conference on Learning Representations, (en). External Links: Link Cited by: §1. [13] M. Cuturi (2013) Sinkhorn Distances: Lightspeed Computation of Optimal Transport. In Advances in Neural Information Processing Systems, Vol. 26. External Links: Link Cited by: §1, §4. [14] J. S. Fleck, S. M. J. Jansen, D. Wollny, F. Zenk, M. Seimiya, A. Jain, R. Okamoto, M. Santel, Z. He, J. G. Camp, and B. Treutlein (2023) Inferring and perturbing cell fate regulomes in human brain organoids. Nature 621 (7978), p. 365–372 (en). External Links: ISSN 1476-4687, Link, Document Cited by: §1. [15] H. Föllmer (1988) Random fields and diffusion processes. In École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–87, P. Diaconis, D. Elworthy, H. Föllmer, E. Nelson, G. Papanicolaou, S. R. S. Varadhan, and P. Hennequin (Eds.), Berlin, Heidelberg, p. 101–203 (en). External Links: ISBN 978-3-540-46042-8, Document Cited by: §2, §4. [16] M. I. Freidlin and A. D. Wentzell (2012) Random Perturbations of Dynamical Systems. Grundlehren der mathematischen Wissenschaften, Vol. 260, Springer, Berlin, Heidelberg (en). External Links: ISBN 978-3-642-25846-6 978-3-642-25847-3, Link, Document Cited by: Appendix B, Appendix B, 1st item, §1, §4, §4. [17] H. Gehart, J. H. van Es, K. Hamer, J. Beumer, K. Kretzschmar, J. F. Dekkers, A. Rios, and H. Clevers (2019) Identification of Enteroendocrine Regulators by Real-Time Single-Cell Differentiation Mapping. Cell 176 (5), p. 1158–1173.e16 (eng). External Links: ISSN 1097-4172, Document Cited by: §1. [18] D. Haviv, A. Pooladian, D. Pe’er, and B. Amos (2025) Wasserstein Flow Matching: Generative modeling over families of distributions. arXiv. Note: arXiv:2411.00698 [cs.LG] External Links: Link, Document Cited by: §1. [19] Z. He, A. Maynard, A. Jain, T. Gerber, R. Petri, H. Lin, M. Santel, K. Ly, J. Dupré, L. Sidow, F. Sanchis Calleja, S. M. J. Jansen, S. Riesenberg, J. G. Camp, and B. Treutlein (2022) Lineage recording in human cerebral organoids. Nature Methods 19 (1), p. 90–99 (en). External Links: ISSN 1548-7105, Link, Document Cited by: §5. [20] M. Heymann and E. Vanden‐Eijnden (2008) The geometric minimum action method: A least action principle on the space of curves. Communications on Pure and Applied Mathematics 61 (8), p. 1052–1117 (en). External Links: ISSN 0010-3640, 1097-0312, Link, Document Cited by: Appendix B, Appendix B, Appendix B, §1, §4, Theorem 18. [21] P. Holderrieth and E. Erives (2026) An Introduction to Flow Matching and Diffusion Models. (en). External Links: Link Cited by: Appendix A. [22] W. Hong, Y. Shi, and J. Niles-Weed (2025) Trajectory Inference with Smooth Schrödinger Bridges. In Proceedings of the 42nd International Conference on Machine Learning, p. 23722–23750 (en). External Links: ISSN 2640-3498, Link Cited by: §3. [23] G. Huguet, D. S. Magruder, A. Tong, O. Fasina, M. Kuchroo, G. Wolf, and S. Krishnaswamy (2022) Manifold Interpolating Optimal-Transport Flows for Trajectory Inference. In Neural Information Processing Systems, (en). Cited by: §3. [24] K. Kapusniak, P. Potaptchik, T. Reu, L. Zhang, A. Tong, M. Bronstein, A. J. Bose, and F. D. Giovanni (2024) Metric Flow Matching for Smooth Interpolations on the Data Manifold. In Conference on Neural Information Processing Systems, (en). Cited by: Appendix D, §1, §3, §4. [25] G. La Manno, R. Soldatov, A. Zeisel, E. Braun, H. Hochgerner, V. Petukhov, K. Lidschreiber, M. E. Kastriti, P. Lönnerberg, A. Furlan, J. Fan, L. E. Borm, Z. Liu, D. van Bruggen, J. Guo, X. He, R. Barker, E. Sundström, G. Castelo-Branco, P. Cramer, I. Adameyko, S. Linnarsson, and P. V. Kharchenko (2018) RNA velocity of single cells. Nature 560 (7719), p. 494–498 (en). External Links: ISSN 1476-4687, Link, Document Cited by: §1. [26] M. Lange, V. Bergen, M. Klein, M. Setty, B. Reuter, M. Bakhti, H. Lickert, M. Ansari, J. Schniering, H. B. Schiller, D. Pe’er, and F. J. Theis (2022) CellRank for directed single-cell fate mapping. Nature Methods 19 (2), p. 159–170 (en). External Links: ISSN 1548-7105, Link, Document Cited by: Appendix D, §1, §3, §5. [27] M. Lange, Z. Piran, M. Klein, B. Spanjaard, D. Klein, J. P. Junker, F. J. Theis, and M. Nitzan (2024) Mapping lineage-traced cells across time points with moslin. Genome Biology 25 (1), p. 277 (en). External Links: ISSN 1474-760X, Link, Document Cited by: §1. [28] C. Léonard (2013) A survey of the Schrödinger problem and some of its connections with optimal transport. arXiv. Note: arXiv:1308.0215 [math.PR] External Links: Link, Document Cited by: §2. [29] T. Li, J. Shi, Y. Wu, and P. Zhou (2021) On the Mathematics of RNA Velocity I: Theoretical Analysis. CSIAM Transactions on Applied Mathematics 2 (1), p. 1–55 (en). External Links: ISSN 2708-0560, 2708-0579, Link, Document Cited by: Appendix B, Appendix B. [30] Y. Lipman, R. T. Q. Chen, H. Ben-Hamu, M. Nickel, and M. Le (2023) Flow Matching for Generative Modeling. In International Conference on Learning Representations, (en). External Links: Link Cited by: §1, §2. [31] A. McKenna, G. M. Findlay, J. A. Gagnon, M. S. Horwitz, A. F. Schier, and J. Shendure (2016) Whole-organism lineage tracing by combinatorial and cumulative genome editing. Science 353 (6298), p. aaf7907. External Links: Link, Document Cited by: §1. [32] K. Neklyudov, R. Brekelmans, D. Severo, and A. Makhzani (2023) Action Matching: Learning Stochastic Dynamics from Samples. In Proceedings of the 40th International Conference on Machine Learning, p. 25858–25889 (en). External Links: ISSN 2640-3498, Link Cited by: §1. [33] S. Ohta and K. Sturm (2009) Heat flow on Finsler manifolds. Communications on Pure and Applied Mathematics 62 (10), p. 1386–1433 (en). External Links: ISSN 1097-0312, Link, Document Cited by: Appendix A. [34] S. Ohta (2009) Finsler interpolation inequalities. Calculus of Variations and Partial Differential Equations 36 (2), p. 211–249 (en). External Links: ISSN 0944-2669, 1432-0835, Link, Document Cited by: Appendix A. [35] K. Petrović, L. Atanackovic, V. Moro, K. Kapuśniak, İ. İ. Ceylan, M. Bronstein, A. J. Bose, and A. Tong (2025) Curly Flow Matching for Learning Non-gradient Field Dynamics. In Conference on Neural Information Processing Systems, Note: arXiv:2510.26645 [cs.LG] External Links: Link Cited by: Figure 7, Figure 7, Appendix C, Appendix D, §1, §3, §5. [36] B. Pijuan-Sala, J. A. Griffiths, C. Guibentif, T. W. Hiscock, W. Jawaid, F. J. Calero-Nieto, C. Mulas, X. Ibarra-Soria, R. C. V. Tyser, D. L. L. Ho, W. Reik, S. Srinivas, B. D. Simons, J. Nichols, J. C. Marioni, and B. Göttgens (2019) A single-cell molecular map of mouse gastrulation and early organogenesis. Nature 566 (7745), p. 490–495 (eng). External Links: ISSN 1476-4687, Document Cited by: §5. [37] M. Rohbeck, E. D. Brouwer, C. Bunne, A. Biton, K. Y. Chen, A. Regev, and R. Lopez (2025) Modeling Complex System Dynamics with Flow Matching Across Time and Conditions. In International Conference on Learning Representations, (en). Cited by: §1. [38] G. Schiebinger, J. Shu, M. Tabaka, B. Cleary, V. Subramanian, A. Solomon, J. Gould, S. Liu, S. Lin, P. Berube, L. Lee, J. Chen, J. Brumbaugh, P. Rigollet, K. Hochedlinger, R. Jaenisch, A. Regev, and E. S. Lander (2019) Optimal-Transport Analysis of Single-Cell Gene Expression Identifies Developmental Trajectories in Reprogramming. Cell 176 (4), p. 928–943.e22 (eng). External Links: ISSN 1097-4172, Document Cited by: §1, §1, §3. [39] A. J. Smola and B. Schölkopf (2004) A tutorial on support vector regression. Statistics and Computing 14 (3), p. 199–222 (en). External Links: ISSN 1573-1375, Link, Document Cited by: §4. [40] D. Ting, L. Huang, and M. Jordan (2011) An Analysis of the Convergence of Graph Laplacians. arXiv. Note: arXiv:1101.5435 [stat.ML] External Links: Link, Document Cited by: Appendix B. [41] A. Tong, K. Fatras, N. Malkin, G. Huguet, Y. Zhang, J. Rector-Brooks, G. Wolf, and Y. Bengio (2024) Improving and generalizing flow-based generative models with minibatch optimal transport. Note: arXiv:2302.00482 [cs.LG] External Links: Link, Document Cited by: §1, §2. [42] A. Tong, J. Huang, G. Wolf, D. van Dijk, and S. Krishnaswamy (2020) TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics. Proceedings of machine learning research 119, p. 9526–9536. External Links: ISSN 2640-3498, Link Cited by: §3. [43] A. Y. Tong, N. Malkin, K. Fatras, L. Atanackovic, Y. Zhang, G. Huguet, G. Wolf, and Y. Bengio (2024) Simulation-Free Schrödinger Bridges via Score and Flow Matching. In Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, p. 1279–1287 (en). External Links: ISSN 2640-3498, Link Cited by: Appendix A, §1, §1, §2, §3, §4, Remark 22. [44] C. Trapnell, D. Cacchiarelli, J. Grimsby, P. Pokharel, S. Li, M. Morse, N. J. Lennon, K. J. Livak, T. S. Mikkelsen, and J. L. Rinn (2014) The dynamics and regulators of cell fate decisions are revealed by pseudotemporal ordering of single cells. Nature Biotechnology 32 (4), p. 381–386 (eng). External Links: ISSN 1546-1696, Document Cited by: §1. [45] C. VillaniM. Berger, B. Eckmann, P. De La Harpe, F. Hirzebruch, N. Hitchin, L. Hörmander, A. Kupiainen, G. Lebeau, M. Ratner, D. Serre, Ya. G. Sinai, N. J. A. Sloane, A. M. Vershik, and M. Waldschmidt (Eds.) (2009) Optimal Transport. Grundlehren der mathematischen Wissenschaften, Vol. 338, Springer, Berlin, Heidelberg. External Links: ISBN 978-3-540-71049-3 978-3-540-71050-9, Link, Document Cited by: §2. [46] D. E. Wagner and A. M. Klein (2020) Lineage tracing meets single-cell omics: opportunities and challenges. Nature Reviews. Genetics 21 (7), p. 410–427 (eng). External Links: ISSN 1471-0064, Document Cited by: §1. [47] P. Weiler, M. Lange, M. Klein, D. Pe’er, and F. Theis (2024) CellRank 2: unified fate mapping in multiview single-cell data. Nature Methods 21 (7), p. 1196–1205 (eng). External Links: ISSN 1548-7105, Document Cited by: §1, §3. [48] C. Weinreb, A. Rodriguez-Fraticelli, F. D. Camargo, and A. M. Klein (2020) Lineage tracing on transcriptional landscapes links state to fate during differentiation. Science 367 (6479), p. eaaw3381. External Links: Link, Document Cited by: §1. [49] C. Weinreb, S. Wolock, B. K. Tusi, M. Socolovsky, and A. M. Klein (2018) Fundamental limits on dynamic inference from single-cell snapshots. Proceedings of the National Academy of Sciences 115 (10), p. E2467–E2476. External Links: Link, Document Cited by: §1. [50] Z. Zhang, T. Li, and P. Zhou (2025) Learning stochastic dynamics from snapshots through regularized unbalanced optimal transport. In International Conference on Learning Representations, External Links: Link, Document Cited by: §3. [51] A. Zweig, M. Zhang, D. A. Knowles, and E. Azizi (2026) Learning Lineage-guided Geodesics with Finsler Geometry. arXiv. Note: arXiv:2603.16708 [cs.LG] External Links: Link, Document Cited by: §3. Appendix A Prelimiaries Finsler Geometry: For a complete introduction to Finsler geometry we refer the reader to Bao et al. [4]. Let M be an m−m-dimensional manifold embedded in ℝdR^d. The tangent bundle TM≔⨆x∈MTxMTM _x∈ MT_xM is the 2m2m-dimensional manifold composed of pairs (x,v)(x,v) with x∈Mx∈ M and v∈TxMv∈ T_xM, where TxMT_xM is the tangent space at x. Definition 3 (Finsler structure). A Finsler structure on M is a continuous function F:TM→ℝ≥0F:TM _≥ 0, which is smooth on TM∖0TM \0\, such that for all (x,v)∈TM(x,v)∈ TM: (F1) Positive homogeneity: F(x,λv)=λF(x,v)F(x,λ v)=λ F(x,v) for all λ>0λ>0. (F2) Strong convexity: the Hessian matrix gij(x,v)=12∂2F2/∂vi∂vjg_ij(x,v)= 12∂^2F^2/∂ v^i∂ v^j is positive definite for v≠0v≠ 0. (F3) Positivity: F(x,v)>0F(x,v)>0 for v≠0v≠ 0. In general, we have that F(x,−v)≠F(x,v)F(x,-v)≠ F(x,v) hence, the cost of moving in direction v is different from the cost of moving in direction −v-v. A Riemannian metric is the special case F(x,v)=v⊤G(x)vF(x,v)= v G(x)v, which additionally satisfies F(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 curve γ:[0,1]→Mγ:[0,1]→ M, define LF(γ)=∫01F(γ(t),γ˙(t))t,EF(γ)=∫01F(γ(t),γ˙(t))2t.L_F(γ)= _0^1F(γ(t), γ(t))\,dt, E_F(γ)= _0^1F(γ(t), γ(t))^2\,dt. A geodesic between x0,x1∈ℝdx_0,x_1 ^d is a curve minimizing LFL_F subject to γ(0)=x0γ(0)=x_0, γ(1)=x1γ(1)=x_1. Note that energy minimizers are constant-speed parametrized geodesics. Furthermore, we define the length metric on M as: cF(x0,x1)=infγ∫01F(γt,γ˙t)t,c_F(x_0,x_1)= _γ _0^1F( _t, γ_t)\,dt, where the infimum is taken over curves satisfying γ0=x0,γ1=x1 _0=x_0,\ _1=x_1. We are particularly interested in the special case called a Randers Metric, where G(x)G(x) is a smooth Riemannian metric and β(x)β(x) a smooth 11-form and we define F(x,v)=v⊤G(x)v+β(x)⊤v.F(x,v)= v G(x)v\;+\;β(x) v. (14) Lemma 5. [Bao et al. [4]] The Randers structure (14) is a Finsler structure if and only if ‖β(x)‖G−1(x)<1 β(x) _G^-1(x)<1 pointwise, where ‖u‖G−12=u⊤G−1u u _G^-1^2=u G^-1u. Remark 6 (Approximation of the data manifold). Although the underlying dynamics may be concentrated on an unknown low-dimensional manifold ℳ⊂ℝdM ^d, we do not estimate ℳM. Instead, we construct the Finsler metric on ℝdR^d so that motion along directions supported by the local data and transition geometry has low cost, whereas motion away from the data manifold, in particular along weakly supported normal directions, has high cost. Thus, the Finsler structures in this work will be defined on ℝdR^d. Definition 7 (P-derived Randers metric). A P-derived Randers metric is a Randers metric FP(xi,v)=v⊤GP(xi)v+βP(xi)⊤vF_P(x_i,v)= v G_P(x_i)v+ _P(x_i) v whose metric tensor GP(xi)G_P(x_i) and one-form βP(xi) _P(x_i) are constructed from local transition statistics of P, with GP(xi)∈SPD(d),‖βP(xi)‖GP(xi)−1<1.G_P(x_i) (d), \| _P(x_i)\|_G_P(x_i)^-1<1. For example, if CP(xi)C_P(x_i) denotes a local transition second-moment tensor, one may consider the family GP,q(xi)=(τI+CP(xi))−q,q>0,G_P,q(x_i)= (τ I+C_P(x_i) )^-q, q>0, together with any admissible transition-derived one-form βP _P. For q=1q=1, this is closely related to local-kernel geometries, where the second moment determines the induced Riemannian metric [8]. Optimal Transport: Optimal transport (OT) was originally formulated by Monge as the problem of finding a deterministic transport map T:ℝd→ℝdT:R^d ^d pushing p0p_0 onto p1p_1 while minimizing the transport cost: infT:T#p0=p1∫ℝdc(x,T(x))dp0(x). _T:\,T_\#p_0=p_1 _R^dc(x,T(x))\,dp_0(x). Since a transport map cannot split mass, such a minimizers does not exist in general. For example, if p0=δ0,p1=12δ−1+12δ1,p_0= _0, p_1= 12 _-1+ 12 _1, no map T can satisfy T#p0=p1T_\#p_0=p_1. The Kantorovich relaxation resolves this issue by optimizing instead over transport plans π∈Π(p0,p1)π∈ (p_0,p_1): infπ∈Π(p0,p1)∫ℝd×ℝdc(x0,x1)π(x0,x1). _π∈ (p_0,p_1) _R^d×R^dc(x_0,x_1)\,dπ(x_0,x_1). In the example above, an admissible plan would be π=12δ(0,−1)+12δ(0,1).π= 12 _(0,-1)+ 12 _(0,1). Under the Euclidean squared distance c(x0,x1)=12∥x0−x1∥2c(x_0,x_1)= 12\|x_0-x_1\|^2 (absolute continuity of p0p_0) Brenier’s theorem guarantees that the optimal plan π⋆π exists and is unique. Furthermore, there is a unique optimal transport map T=∇φT=∇ , for some convex function φ . The classical Benamou–Brenier theorem establishes that this static transport problem is equivalent to a dynamic formulation over density paths ρt _t and velocity fields vtv_t: W22(p0,p1)=inf(ρ,v)∫01∫ℝd|vt(x)|2dρt(x),ts.t.∂tρt+∇⋅(ρtvt)=0,ρ0=p0,ρ1=p1.W_2^2(p_0,p_1)= _(ρ,v) _0^1 _R^d|v_t(x)|^2d _t(x),dt .t. _t _t+∇·( _tv_t)=0, _0=p_0, _1=p_1. When substituting the Euclidean cost with the Finsler geodesic action cost cF(x0,x1)c_F(x_0,x_1) defined in (1), the static Kantorovich problem remains uniquely attained [34, 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)2π(x0,x1)=inf(ρ,v)∫01∫ℝdF(x,vt(x))2dρt(x)t W_2,F^2(p_0,p_1) _π∈ (p_0,p_1) _R^d×R^dc_F(x_0,x_1)^2dπ(x_0,x_1)= _(ρ,v) _0^1 _R^dF(x,v_t(x))^2d _t(x)dt subject to the continuity equation ∂tρt+∇⋅(ρtvt)=0 _t _t+∇·( _tv_t)=0 with endpoint conditions ρ0=p0 _0=p_0 and ρ1=p1 _1=p_1 [33, 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]. 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 field ut:[0,1)×ℝn→ℝnu_t:[0,1)×R^n ^n such that dtxt=ut(xt),x0∼p0,x1∼p1. \ aligned & ddtx_t=u_t(x_t),\\ &x_0 p_0,x_1 p_1. aligned . Once such a vector field has been learned, samples from p1p_1 can be generated by sampling x0∼p0x_0 p_0 and integrating the vector field from t=0t=0 to t=1t=1. In general, this vector field is not unique, which gives us some choice. For example, if p0=p1=p_0=p_1=p, then any vector field satisfying ∇⋅(pu)=0∇·(pu)=0 leaves the distribution p invariant. 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∈[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)π∈ (p_0,p_1), i.e., a probability distribution on ℝn×ℝnR^n×R^n with marginals p0p_0 and p1p_1, and define a tractable conditional probability path pt(x∣x0,x1)p_t(x x_0,x_1) with marginals δx0 _x_0 and δx1 _x_1. The corresponding marginal path is pt(x)=∫ℝn×ℝnpt(x∣x0,x1)π(x0,x1). p_t(x)= _R^n×R^np_t(x x_0,x_1)\,dπ(x_0,x_1). For every endpoint pair (x0,x1)(x_0,x_1), we define a conditional vector field ut(x∣x0,x1)u_t(x x_0,x_1) that solves dtxt=ut(xt∣x0,x1),xt∼pt(⋅∣x0,x1) for all t∈[0,1]. \ aligned & ddtx_t=u_t(x_t x_0,x_1),\\ &x_t p_t(· x_0,x_1) for all t∈[0,1]. aligned . Surprisingly, although the marginal vector field is generally intractable, the Conditional Flow Matching objective ℒCFM(θ)=(x0,x1)∼π,t∼[0,1],x∼pt(⋅∣x0,x1)[‖vθ(t,x)−ut(x∣x0,x1)‖22] _CFM(θ)=E_ subarrayc(x_0,x_1) π,\,t [0,1],\\ x p_t(· x_0,x_1) subarray [ \|v_θ(t,x)-u_t(x x_0,x_1) \|_2^2 ] has the same gradient with respect to θ as 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] 33 3 Which the authors highly suggest reading in full.. Similarly to the previous section, we want to find a vector field u~t u_t that satisfies. dxt=u~t(xt)dt+gtdWt,x0∼p0,x1∼p1. \ aligned &dx_t= u_t(x_t)dt+g_tdW_t,\\ &x_0 p_0,x_1 p_1. aligned . 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 x_0,x_1) and learn a conditional vector field such that the corresponding ODE dtxt=ut(xt) ddtx_t=u_t(x_t) satisfies xt∼ptx_t p_t for every t. We additionally introduce the conditional score of the probability path, st(x∣x0,x1):=∇logpt(x∣x0,x1). s_t(x x_0,x_1):=∇ p_t(x 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, dxt=[vt(xt)+12gtgt⊤st(xt)]dt+gtdWt. _t= [v_t(x_t)+ 12g_tg_t s_t(x_t) ]dt+g_t\,dW_t. The Fokker-Planck equation of this SDE is given by ∂tpt _tp_t =−∇⋅(pt[vt+12gtgt⊤st])+12∇⋅(gtgt⊤∇pt) =-∇· (p_t [v_t+ 12g_tg_t s_t ] )+ 12∇· (g_tg_t ∇ p_t ) =−∇⋅(ptvt), =-∇·(p_tv_t), where we used ptst=∇ptp_ts_t=∇ p_t. Hence, the ODE and the corresponding SDE satisfy the same Fokker–Planck equation and therefore have the same marginals Xt∼ptX_t p_t. The score thus allows us to add stochasticity to the learned flow without changing the prescribed probability path. Appendix B Theoretical Contributions Transition-supported geodesics. For a threshold τ>0τ>0, define the high-probability transition edges Eτ:=(i,j):Pij≥τ.E_τ:=\(i,j):P_ij≥τ\. For endpoints x0,x1∈x_0,x_1 , define the transition-supported subset τ(x0,x1):=xi∈:xi lies on a directed path from x0 to x1 using only edges in Eτ.S_τ(x_0,x_1):= \x_i :x_i lies on a directed path from $x_0$ to $x_1$ using only edges in E_τ \. We assume that τ(x0,x1)S_τ(x_0,x_1) is non-empty. For r>0r>0, define its r-neighborhood by Ur(τ):=x∈ℝd:dist(x,τ)<r,dist(x,τ):=miny∈τ‖x−y‖2.U_r(S_τ):= \x ^d:dist(x,S_τ)<r \, (x,S_τ):= _y _τ\|x-y\|_2. Furthermore, define cr:=minr/2≤dist(x,τ)≤r‖v‖2=1FP(x,v)>0.c_r:= _ subarraycr/2 (x,S_τ)≤ r\\ \|v\|_2=1 subarrayF_P(x,v)>0. The positivity follows since FPF_P is continuous and strictly positive for v≠0v≠ 0, while the domain of minimization is compact. For our particular P-derived Randers metric (9), 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)≥ \|m(x)\|_C_ρ(x)^-1^2+λ^2-\|m(x)\|_C_ρ(x)^-1 _ (C_ρ(x))\|v\|_2. Hence, regions and directions with large P-weighted transition covariance are relatively cheap, whereas regions with little transition support, corresponding to small eigenvalues of CρC_ρ, incur a larger cost. Thus annuli farther from τS_τ have a larger escape cost crc_r. Proposition 8 (Confinement and directional preference of geodesics). Let FPF_P be the Randers metric induced by the first and second transition moments m and C in (9). Suppose there exists a curve γ¯ γ connecting x0x_0 and x1x_1 inside Ur/2(τ)U_r/2(S_τ). If LFP(γ¯)<crr,L_F_P( γ)<c_rr, then every minimizing geodesic connecting x0x_0 and x1x_1 is contained in Ur(τ)U_r(S_τ). Moreover, for fixed local speed ‖v‖Cρ(x)−1\|v\|_C_ρ(x)^-1, the Finsler cost is minimized when v is positively aligned with the first transition moment m(x)m(x). Proof. Suppose a minimizing geodesic γ⋆γ leaves Ur(τ)U_r(S_τ). Since x0,x1∈τx_0,x_1 _τ, it must cross the annulus r/2≤dist(x,τ)≤r/2 (x,S_τ)≤ r once when leaving and once when returning. Since x↦dist(x,τ)x (x,S_τ) is 11-Lipschitz, its total Euclidean length in this annulus is at least r. By positive homogeneity and the definition of crc_r, LFP(γ⋆)≥crr>LFP(γ¯),L_F_P(γ )≥ c_rr>L_F_P( γ), contradicting minimality. Hence γ⋆⊂Ur(τ)γ ⊂ U_r(S_τ). For the directional claim, write A(x)=Cρ(x)−1A(x)=C_ρ(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)= \|m(x)\|_A(x)^2+λ^2\,\|v\|_A(x)- v,m(x) _A(x) is constant. Thus minimizing FPF_P is equivalent to maximizing ⟨v,m(x)⟩A(x) v,m(x) _A(x), which by Cauchy–Schwarz is attained when v is a positive multiple of m(x)m(x). ∎ Uniform Convergence of Empirical Drift and Diffusion: Let YTY_T be a stochastic process in ℝdR^d governed by the Stochastic Differential Equation (SDE) dYT=b0(YT)dt+Σsde(YT)dWt, dY_T=b_0(Y_T)\,dt+ _sde(Y_T)\,dW_t, (15) where b0∈C∞(ℝd,ℝd)b_0∈ C^∞(R^d,R^d) is the drift vector, WtW_t is a standard d-dimensional Brownian motion and D(x)≔Σsde(x)Σsde(x)⊤D(x) _sde(x) _sde(x) , is the smooth diffusion matrix. Definition 9 (Transition Operator and Infinitesimal Generator). The transition operator (semigroup) TtT_t acting on a test function f∈Cc∞(ℝd)f∈ C_c^∞(R^d) is defined by Ttf(x)≔[f(YT)∣Y0=x].T_tf(x) [f(Y_T) Y_0=x ]. The infinitesimal generator L of the process YTY_T is the operator Lf(x)≔limt→0+Ttf(x)−f(x)t=∇f(x)⊤b0(x)+12Tr(D(x)∇2f(x)).Lf(x) _t→ 0^+ T_tf(x)-f(x)t=∇ f(x) b_0(x)+ 12Tr (D(x)∇^2f(x) ). Suppose we observe samples xii=1N⊂ℝd\x_i\_i=1^N ^d drawn i.i.d. from a continuous density p(x)p(x) alongside a row-stochastic transition matrix P∈ℝN×NP ^N× N constructed at spatial scale εN>0 _N>0.Thus the discrete transition operator is (Pf)(xi)=∑j=1NPijf(xj).(Pf)(x_i)= _j=1^NP_ijf(x_j). To link discrete dataset observations to the continuous diffusion process, we need the following assumptions. Assumption 10. The discrete time step ΔtN t_N scales quadratically with the bandwidth parameter εN _N ΔtN=cεN2for a fixed constant c>0. (Time-Space Scaling) t_N=c _N^2 a fixed constant c>0. (Time-Space Scaling) Informally, we say that a Markov chain with transition matrix P is a discrete observation of a continuous stochastic process YTY_T if its discrete transition operator matches the continuous process generator L over an infinitesimal time step. Formally, we assume: Assumption 11 (Uniform discrete-generator consistency). Define the empirical generator LN,εNf(xi)≔1ΔtN(∑j=1NPijf(xj)−f(xi)).L_N, _Nf(x_i) 1 t_N ( _j=1^NP_ijf(x_j)-f(x_i) ). We assume that for every f∈Cc∞(ℝd)f∈ C_c^∞(R^d), supxi∈K|LN,εNf(xi)−Lf(xi)|→0 _x_i∈ K |L_N, _Nf(x_i)-Lf(x_i) |→ 0 as N→∞N→∞ and εN→0 _N→ 0. Assumption 12. We assume local support of the transition kernel on the region of interest: for every compact K⊂ℝdK ^d, there exists a compact neighborhood K+⊃K^+⊃ K such that, for all sufficiently large N, xi∈K,Pij>0⟹xj∈K+.x_i∈ K, P_ij>0 x_j∈ K^+. Proposition 13 (Recovery of drift and diffusion from P). Assume that P=PN,εNP=P_N, _N is row-stochastic and that Assumptions Assumption 10 Assumption 11 and Assumption 12 hold. Define b^0,N(xi)=1ΔtN∑j=1NPij(xj−xi) and D^N(xi)=1ΔtN∑j=1NPij(xj−xi)(xj−xi)⊤. b_0,N(x_i)= 1 t_N _j=1^NP_ij(x_j-x_i) and D_N(x_i)= 1 t_N _j=1^NP_ij(x_j-x_i)(x_j-x_i) . Then maxxi∈K‖b^0,N(xi)−b0(xi)‖→0 and maxxi∈K‖D^N(xi)−D(xi)‖→0. _x_i∈ K\| b_0,N(x_i)-b_0(x_i)\|→ 0 and _x_i∈ K\| D_N(x_i)-D(x_i)\|→ 0. Proof. Fix a compact set K and let K+K^+ be the compact neighborhood from Assumption 12. Choose a cutoff χ∈Cc∞(ℝd)χ∈ C_c^∞(R^d) such that χ≡1χ≡ 1 on a neighborhood of K+K^+. For a,b∈1,…,da,b∈\1,…,d\ define q~a(x):=χ(x)xa,q~ab(x):=χ(x)xaxb. q_a(x):=χ(x)x_a, q_ab(x):=χ(x)x_ax_b. Then q~a,q~ab∈Cc∞(ℝd) q_a, q_ab∈ C_c^∞(R^d), so Assumption Assumption 11 applies. For xi∈Kx_i∈ K Assumption 12 implies that every xjx_j with Pij>0P_ij>0 lies in K+K^+, where χ≡1χ≡ 1. Hence LN,εNq~a(xi)=LN,εNqa(xi),LN,εNq~ab(xi)=LN,εNqab(xi).L_N, _N q_a(x_i)=L_N, _Nq_a(x_i), L_N, _N q_ab(x_i)=L_N, _Nq_ab(x_i). Moreover, since χ≡1χ≡ 1 on a neighborhood of K, Lq~a(x)=Lqa(x),Lq~ab(x)=Lqab(x),x∈K.L q_a(x)=Lq_a(x), L q_ab(x)=Lq_ab(x), x∈ K. Therefore Assumption Assumption 11 may be applied to the cutoff observables while yielding the desired generator identities for qaq_a and qabq_ab on K. Since P is row-stochastic, LN,εNqa(xi)=1ΔtN∑j=1NPij(xj,a−xi,a)=b^0,N,a(xi).L_N, _Nq_a(x_i)= 1 t_N _j=1^NP_ij(x_j,a-x_i,a)= b_0,N,a(x_i). Moreover, Lqa(x)=b0,a(x).Lq_a(x)=b_0,a(x). The drift convergence therefore follows directly from Assumption Assumption 11. Expanding the increments for the second moment gives LN,εNqab(xi) L_N, _Nq_ab(x_i) =xi,ab^0,N,b(xi)+xi,bb^0,N,a(xi)+D^N,ab(xi). =x_i,a b_0,N,b(x_i)+x_i,b b_0,N,a(x_i)+ D_N,ab(x_i). On the other hand, applying the diffusion generator yields Lqab(x)=xab0,b(x)+xbb0,a(x)+Dab(x).Lq_ab(x)=x_ab_0,b(x)+x_bb_0,a(x)+D_ab(x). Consequently, D^N,ab(xi)−Dab(xi) D_N,ab(x_i)-D_ab(x_i) =(LN,εNqab(xi)−Lqab(xi)) = (L_N, _Nq_ab(x_i)-Lq_ab(x_i) ) −xi,a(LN,εNqb(xi)−Lqb(xi)) -x_i,a (L_N, _Nq_b(x_i)-Lq_b(x_i) ) −xi,b(LN,εNqa(xi)−Lqa(xi)). -x_i,b (L_N, _Nq_a(x_i)-Lq_a(x_i) ). Since K is compact, its coordinates are uniformly bounded. Uniform generator consistency therefore implies uniform convergence of every entry of D^N D_N to the corresponding entry of D. Since the dimension is finite, this implies convergence in any matrix norm. ∎ Generator consistency for kernels: The generator-consistency assumption Assumption 11 is closely related to continuum limits of local graph random walks [40, Theorems 2–3], whose limiting drift and diffusion are determined by the first two local moments [8, 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=ch2b0+o(h2),Ch=ch2D+o(h2),m_h=ch^2b_0+o(h^2), C_h=ch^2D+o(h^2), uniformly with higher-order displacement contributions divided by h2h^2 going to zero as h goes to zero. For the cosine and correlation constructions studied by Li et al. [29] gives a first-order limit on the time scale h, rather than a diffusion limit on the time scale h2h^2. Their inner-product construction instead admits a second-order limit. A possible diffusive construction is to replace a fixed cosine weight exp(νcosθ) (ν θ) by exp(νhcosθ) ( _h θ) with νh/h→ν0 _h/h→ _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 scale τh→0 _h→ 0, the discrete generators Lh=(Ph−I)/τhL_h=(P_h-I)/ _h converge to Lf=b⊤∇f+12Tr(A∇2f).Lf=b ∇ f+ 12Tr(A∇^2f). Applying this convergence to coordinate and quadratic coordinate functions (as in Proposition 13) yields mh/τh→b,Ch/τh→A.m_h/ _h→ b, C_h/ _h→ A. In particular, Li et al. [29] show that cosine RNA-velocity kernels possess a first-order limit with h=εh= (Ph−I)/h→a(x)v^(x)⊤∇,(P_h-I)/h→ a(x) v(x) ∇, with a(x)>0a(x)>0. Consequently mh/h→a(x)v^(x)m_h/h→ a(x) v(x), while Ch/h→0C_h/h→ 0. Thus the first transition moment asymptotically recovers the RNA-velocity direction. For such kernels we therefore interpret ChC_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 Proposition 13 are defined only at the observed states. To evaluate the induced metric along continuous curves, we extend them to a compact region K⊂ℝdK ^d containing all admissible paths. Let K+K^+ be a compact neighborhood of K such that K⊂int(K+).K (K^+). We assume that Assumption Assumption 11 holds on K+K^+, so that Proposition Proposition 13 applies on this neighborhood. Definition 14 (Uniform nodewise consistency). Let A⊂ℝdA ^d be compact and let f be a continuous vector- or matrix-valued field on A. Suppose that f^N(xi) f_N(x_i) estimates f(xi)f(x_i) at each observed state xi∈Ax_i∈ A. We say that f^N f_N is uniformly nodewise consistent for f on A if maxi:xi∈A‖f^N(xi)−f(xi)‖⟶0 _i:x_i∈ A \| f_N(x_i)-f(x_i) \| 0 along 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 Proposition 13 shows that b^0,N b_0,N and D^N D_N are uniformly nodewise consistent for b0b_0 and D, respectively, on K+K^+. Definition 15 (Kernel extension). Let h∈Cc∞([0,∞))h∈ C_c^∞([0,∞)) be nonnegative, supported in [0,1][0,1], and strictly positive on a neighborhood of zero. Let ηN→0 _N→ 0 and define kηN(x,y)=h(‖x−y‖2ηN2).k_ _N(x,y)=h ( \|x-y\|^2 _N^2 ). Define the fill distance of the sample cloud relative to K by rN(K)≔supx∈Kmini:xi∈K+∥x−xi∥.r_N(K) _x∈ K _i:x_i∈ K^+\|x-x_i\|. We assume that the samples become dense relative to the kernel bandwidth: rN(K)ηN⟶0. r_N(K) _N 0. Since h is positive near zero, this condition ensures that, for all sufficiently large N, every x∈Kx∈ K receives positive kernel weight from at least one observed state in K+K^+. For such N, define wi,N(x)=kηN(x,xi)∑j:xj∈K+kηN(x,xj),x∈K,xi∈K+,w_i,N(x)= k_ _N(x,x_i) _j:x_j∈ K^+k_ _N(x,x_j), x∈ K, x_i∈ K^+, and extend the nodewise estimators by b~0,N(x)=∑i:xi∈K+wi,N(x)b^0,N(xi) b_0,N(x)= _i:x_i∈ K^+w_i,N(x) b_0,N(x_i) and D~N(x)=∑i:xi∈K+wi,N(x)D^N(xi). D_N(x)= _i:x_i∈ K^+w_i,N(x) D_N(x_i). Consequently, for sufficiently large N, the extended fields are smooth on a neighborhood of K. Moreover, because every D^N(xi) D_N(x_i) is positive semidefinite, D~N(x) D_N(x) is also positive semidefinite. Proposition 16 (Uniform convergence of the kernel-extended estimators). Assume that b0b_0 and D are continuous on K+K^+ and that b^0,N b_0,N and D^N D_N are uniformly nodewise consistent on K+K^+ in the sense of Definition Definition 14. If ηN⟶0,rN(K)ηN⟶0, _N 0, r_N(K) _N 0, then supx∈K‖b~0,N(x)−b0(x)‖⟶0 and supx∈K‖D~N(x)−D(x)‖⟶0. _x∈ K \| b_0,N(x)-b_0(x) \| 0 and _x∈ K \| D_N(x)-D(x) \| 0. Proof. Modulus of continuity for a continuos function f on K+K^+ is defined as: ωf(r)≔supx,y∈K+‖x−y‖≤r‖f(x)−f(y)‖. _f(r) _ subarraycx,y∈ K^+\\ \|x-y\|≤ r subarray\|f(x)-f(y)\|. Since K+K^+ is compact, f is uniformly continuous and therefore ωf(r)⟶0 as r⟶0. _f(r) 0 as r 0. Consider first the drift field. Since the kernel weights are nonnegative and sum to one, ‖b~0,N(x)−b0(x)‖ \| b_0,N(x)-b_0(x) \| ≤∑i:xi∈K+wi,N(x)‖b^0,N(xi)−b0(xi)‖ ≤ _i:x_i∈ K^+w_i,N(x) \| b_0,N(x_i)-b_0(x_i) \| +∑i:xi∈K+wi,N(x)‖b0(xi)−b0(x)‖. + _i:x_i∈ K^+w_i,N(x) \|b_0(x_i)-b_0(x) \|. Whenever wi,N(x)>0w_i,N(x)>0, the support condition on h implies ‖xi−x‖≤ηN.\|x_i-x\|≤ _N. Hence ‖b~0,N(x)−b0(x)‖≤maxi:xi∈K+‖b^0,N(xi)−b0(xi)‖+ωb0(ηN). \| b_0,N(x)-b_0(x) \|≤ _i:x_i∈ K^+ \| b_0,N(x_i)-b_0(x_i) \|+ _b_0( _N). Taking the supremum over x∈Kx∈ K, the first term converges to zero by uniform nodewise consistency, while the second converges to zero because ηN→0 _N→ 0. Applying the same argument to the matrix-valued field D gives supx∈K‖D~N(x)−D(x)‖≤maxi:xi∈K+‖D^N(xi)−D(xi)‖+ωD(ηN)⟶0. _x∈ K \| D_N(x)-D(x) \|≤ _i:x_i∈ K^+ \| D_N(x_i)-D(x_i) \|+ _D( _N) 0. ∎ Freidlin–Wentzell Theory. Consider the small-noise diffusion dYTκ=b0(YTκ)dt+2κD(YTκ)1/2dWt,dY_T^κ=b_0(Y_T^κ)\,dt+ 2κ\,D(Y_T^κ)^1/2\,dW_t, where b0b_0 is smooth. For an absolutely continuous path ϕ:[0,T]→ℝdφ:[0,T] ^d, the Freidlin–Wentzell (FW) [16] action is defined as: ST(ϕ)=14∫0T(ϕ˙t−b0(ϕt))⊤D(ϕt)−1(ϕ˙t−b0(ϕt))t. S_T(φ)= 14 _0^T( φ_t-b_0( _t)) D( _t)^-1( φ_t-b_0( _t))\,dt. (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]). The fixed-time action STS_T depends on the prescribed transition duration T. Minimizing over T yields the quasipotential V(x0,x1)≔infT>0infϕ:ϕ(0)=x0,ϕ(T)=x1ST(ϕ).V(x_0,x_1) _T>0 _ subarraycφ:φ(0)=x_0,\ φ(T)=x_1 subarrayS_T(φ). By Proposition 2.1 and Equation (2.14) of Heymann and Vanden‐Eijnden [20], 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)= _T>0 _ subarraycφ:φ(0)=x_0,\ φ(T)=x_1 subarrayS_T(φ)= 12 _ subarraycγ:γ(0)=x_0,\ γ(1)=x_1 subarrayS_FW(γ). (17) where SFW(γ)=∫01FFW(γ(t),γ˙(t))tS_FW(γ)= _0^1F^FW(γ(t), γ(t))dt with FFW(x,v)≔(v⊤‖b0(x)‖D(x)−12(x)−1⏟GFW(x)v)1/2+v⊤(−D(x)−1b0(x))⏟βFW(x).F^FW(x,v) (v \|b_0(x)\|_D(x)^-1^2D(x)^-1_G_FW(x)v )^1/2+v (-D(x)^-1b_0(x) )_ _FW(x). This has almost the form of a Finsler length functional Definition 4, however, for b0(x)≠0b_0(x)≠ 0, ‖βFW(x)‖GFW(x)−1=1.\| _FW(x)\|_G_FW(x)^-1=1. Thus FFWF^FW is a degenerate Randers-type structure rather than Finsler metric (See Lemma 5). SFWS_FW defines the parameterization-invariant FW action whose minimizers are the minimum-action transition curves. Let γ∗γ^* be the unique minimizer (up reparametrization) among curves connecting x0x_0 to x1x_1. We define the optimal transition time scale T∗=∫01dtΛ(γ∗(t),γ˙∗(t)),where Λ(x,v)=b0(x)⊤D(x)−1b0(x)v⊤D(x)−1v,T^*= _0^1 dt (γ^*(t), γ^*(t)), (x,v)= b_0(x) D(x)^-1b_0(x)v D(x)^-1v, and denote by τδ(Xκ)=inft>0∣YTκ∈B¯δ(x1) _δ(X^κ)= \t>0 Y_T^κ∈ B_δ(x_1)\ the first hitting time of a closed δ-ball centered at x1x_1. Under the hypotheses of Heymann and Vanden‐Eijnden [20, Proposition 2.3], 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 tolerance η>0η>0, limδ→0+T→T∗lim infκ→0ℙx0(ρF(Xκ|[0,τδ(Xκ)],γ∗)<η|τδ(Xκ)≤T)=1, _δ→ 0^+\\ T→ T^* _κ→ 0\,P_x_0\! ( _F\! (X^κ |_[0, _δ(X^κ)],\,γ^* )<η\,\, |\,\, _δ(X^κ)≤ T )=1, where ρF _F is the Fréchet distance between curves. Informally speaking, conditioned on the system completing the transition from x0x_0 into a small neighborhood of x1x_1 within a time frame close to T∗T^*, the realization YTκY_T^κ concentrates with probability 11 in an arbitrarily narrow geometric tube around the minimizer γ∗γ^* as κ→0κ→ 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). Let Γ=γθ:θ∈Θ,∥γ˙(t)∥≤M) =\ _θ:θ∈ ,\| γ(t)\|≤ M)\ be a class of smooth curves with γθ(0)=x0 _θ(0)=x_0, γθ(1)=x1 _θ(1)=x_1, Θ compact, and (t,θ)↦(γθ(t),γ˙θ(t))(t,θ) ( _θ(t), γ_θ(t)) continuous. Set KΓ=⋃θ∈Θγθ([0,1])¯K_ = _θ∈ _θ([0,1]). We assume that KΓ⊂K_ ⊂ K, where K is the compact set on which the uniform convergence in Proposition Proposition 16 holds. For the unregularized geometric Freidlin–Wentzell interpretation in this subsection, we additionally assume that D is uniformly positive definite on the compact region K+K^+ containing the admissible paths. For numerical stability and to ensure uniform positive definiteness at finite N, we use the ridge-regularized estimator D~ρN(x)≔D~N(x)+ρNI, D_ _N(x) D_N(x)+ _NI, where ρN>0 _N>0 and ρN→0 _N→ 0 as N→∞N→∞. Furthermore, we define the regularized FW metric as: F~N,λFW(x,v)≔v⊤D~ρN(x)−1v‖b~0,N(x)‖D~ρN(x)−12+λN2−v⊤D~ρN(x)−1b~0,N(x). F^FW_N,λ(x,v) v D_ _N(x)^-1v\, \| b_0,N(x)\|^2_ D_ _N(x)^-1+ _N^2-v D_ _N(x)^-1 b_0,N(x). , note that this ensures that F~N,λFW F^FW_N,λ is a P-derived Randers Definition 7 metric thus a Finsler structure and hence a for some sequence λN>0 _N>0 which converges to 00 as N→∞N→∞. Theorem 18 (Consistency of regularized empirical geometric FW minimizers). Assume Assumption Assumption 17 and the assumptions of Proposition Proposition 16. Let ρN>0 _N>0 and λN>0 _N>0 satisfy ρN→0,λN→0 _N→ 0, _N→ 0. Define S~N(γ)≔∫01F~N,λNFW(γ(t),γ˙(t))t, S_N(γ) _0^1 F^FW_N, _N (γ(t), γ(t) )\,dt, and let γ~N∗∈argminγ∈ΓS~N(γ). γ_N^*∈ _γ∈ S_N(γ). Then: 1. supγ∈Γ|S~N(γ)−SFW(γ)|⟶0. _γ∈ | S_N(γ)-S_FW(γ) | 0. 2. If FWS_FW has a unique minimizer γ∗γ^* over Γ , up to reparameterization, then ρF(γ~N∗,γ∗)⟶0. _F ( γ_N^*,γ^* ) 0. Consequently, the regularized empirical Finsler minimizer converges to the population FW minimizer within Γ . If γ∗γ^* is also the global geometric FW minimizer connecting x0x_0 to x1x_1, and the hypotheses of Heymann and Vanden‐Eijnden [20, Proposition 2.3] hold, then γ∗γ^* is the asymptotically most probable transition curve in the small-noise sense stated above, and γ~N∗ γ_N^* converges to this curve. Proof. By Proposition Proposition 16, we have uniform convergence on KΓK_ : supx∈KΓ‖b~0,N(x)−b0(x)‖→0,supx∈KΓ‖D~N−D(x)‖→0. _x∈ K_ \| b_0,N(x)-b_0(x)\|→ 0, _x∈ K_ \| D_N-D(x)\|→ 0. Since ρN→0 _N→ 0, we also have supx∈KΓ‖D~ρN(x)−D(x)‖≤supx∈KΓ‖D~N(x)−D(x)‖+ρN→0. _x∈ K_ \| D_ _N(x)-D(x)\|≤ _x∈ K_ \| D_N(x)-D(x)\|+ _N→ 0. Since D is uniformly positive definite on KΓK_ , 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. _x∈ K_ \| D_ _N(x)^-1-D(x)^-1\|→ 0. By Assumption Assumption 17, all curves in Γ have uniformly bounded velocity: ‖γ˙(t)‖≤Mfor all γ∈Γ,t∈[0,1].\| γ(t)\|≤ M all γ∈ ,\ t∈[0,1]. Therefore all pairs (γ(t),γ˙(t))(γ(t), γ(t)) lie in the compact set KΓ×BM(0)¯K_ × B_M(0) . On this compact set, the map (b,D,v,λN)↦v⊤D−1v‖b‖D−12+λN2−v⊤D−1b(b,D,v, _N) v D^-1v\, \|b\|^2_D^-1+ _N^2-v D^-1b is continuous. The uniform convergence of b~0,N b_0,N, D~ρN D_ _N and λN→0 _N→ 0 implies supx∈KΓ,‖v‖≤M|F~N,λNFW(x,v)−FFW(x,v)|→0. _x∈ K_ ,\ \|v\|≤ M | F FW_N, _N(x,v)-F FW(x,v) |→ 0. Consequently, for every γ∈Γγ∈ , |S~N(γ)−SFW(γ)|≤∫01|F~N,λNFW(γ(t),γ˙(t))−FFW(γ(t),γ˙(t))|t.| S_N(γ)-S_ FW(γ)|≤ _0^1 | F FW_N, _N(γ(t), γ(t))-F FW(γ(t), γ(t)) |\,dt. Taking the supremum over γ∈Γγ∈ , we obtain supγ∈Γ|S~N(γ)−SFW(γ)|≤supx∈KΓ,‖v‖≤M|F~N,λNFW(x,v)−FFW(x,v)|→0. _γ∈ | S_N(γ)-S_ FW(γ)|≤ _x∈ K_ ,\ \|v\|≤ M | F FW_N, _N(x,v)-F FW(x,v) |→ 0. This proves the first claim. Let γ∗γ^* be the unique minimizer of SFWS_ FW over Γ , up to reparameterization. Fix r>0r>0, and define Γr=γ∈Γ:ρF(γ,γ∗)≥r. _r=\γ∈ : _F(γ,γ^*)≥ r\. Because Γ is compact and S is continuous on Γ , the minimum of S over Γr _r is attained. Since γ∗γ^* is the unique minimizer, there is a positive gap Δr≔infγ∈ΓrSFW(γ)−SFW(γ∗)>0. _r _γ∈ _rS_ FW(γ)-S_ FW(γ^*)>0. By the uniform convergence just proved, for sufficiently large N, supγ∈Γ|S~N(γ)−SFW(γ)|<Δr3. _γ∈ | S_N(γ)-S_ FW(γ)|< _r3. For any γ∈Γrγ∈ _r, this gives S~N(γ)≥SFW(γ)−Δr3≥SFW(γ∗)+Δr−Δr3=SFW(γ∗)+2Δr3. S_N(γ)≥ S_ FW(γ)- _r3≥ S_ FW(γ^*)+ _r- _r3=S_ FW(γ^*)+ 2 _r3. On the other hand, S~N(γ∗)≤SFW(γ∗)+Δr3. S_N(γ^*)≤ S_ FW(γ^*)+ _r3. Therefore S~N(γ)>S~N(γ∗)for all γ∈Γr. S_N(γ)> S_N(γ^*) all γ∈ _r. Hence no minimizer γ~∗∈argminγ∈ΓS~N(γ) γ^*∈ _γ∈ S_N(γ) can lie in Γr _r. Thus, ρF(γ~∗,γ∗)<r _F( γ^*,γ^*)<r for all sufficiently large N. Since r>0r>0 was arbitrary, ρF(γ~∗,γ∗)→0. _F( γ^*,γ^*)→ 0. Finally, suppose that γ∗γ^* is also the global geometric FW minimizer connecting x0x_0 to x1x_1, and that the hypotheses of Heymann and Vanden‐Eijnden [20, Proposition 2.3] hold. The conditional concentration result above then identifies γ∗γ^* as the asymptotically most probable transition curve in the small-noise limit. Since ρF(γ~N∗,γ∗)⟶0, _F ( γ_N^*,γ^* ) 0, the regularized empirical Finsler minimizer converges to this transition curve. ∎ Remark 19 (Fixed-ridge geometry). The positive-definiteness assumption above is needed only for passing to the unregularized limit ρN→0 _N→ 0 and recovering the geometric FW action of the original diffusion. For any fixed ρ¯>0 ρ>0, the regularized population tensor Dρ¯(x):=D(x)+ρ¯ID_ ρ(x):=D(x)+ ρI satisfies is unifromly positive definite. Hence, Proposition Proposition 16 implies supx∈K‖(D~N(x)+ρ¯I)−1−(D(x)+ρ¯I)−1‖⟶0. _x∈ K \|( D_N(x)+ ρI)^-1-(D(x)+ ρI)^-1 \| 0. Consequently, for fixed ρ¯>0 ρ>0 and λ¯>0 λ>0, the same argument as in Theorem Theorem 18 gives uniform convergence of the empirical regularized action to the corresponding population regularized action, without requiring D itself to be uniformly positive definite. Remark 20. For a common graph time step Δt>0 t>0, the diffusion-normalized moments are b^=m/Δt b=m/ t and D^=C/Δt D=C/ t. Writing Fρ,λ[m,C]F_ρ,λ[m,C] for the regularized metric constructed from the moments m,Cm,C, direct substitution gives Fρ,λ[m,C](x,v)=Fρ/Δt,λ/Δt[mΔt,CΔt](x,v).F_ρ,λ[m,C](x,v)=F_ρ/ t,\,λ/ t [ m t, C t ](x,v). Thus, raw and diffusion-normalized moments yield exactly the same metric after rescaling the regularization parameters, and we absorb the time-step normalization into these hyperparameters in the implementation. The vanishing-regularization assumptions in the consistency theorem apply to the normalized parameters; in raw-moment notation, they require ρN/ΔtN→0 _N/ t_N→ 0 and λN/ΔtN→0 _N/ t_N→ 0. Additional Calculation: Proposition 21 (Gaussian probability path). Let μt∈ℝd _t ^d be continuously differentiable and let Gt∈SPD(d)G_t (d) be a continuously differentiable. Define, for t∈(0,1)t∈(0,1), pt(x∣x0,x1)=(x,μt,t(1−t)Gt).p_t(x x_0,x_1)=N\! (x; _t,\,t(1-t)G_t ). Then the vector field u~t(x∣x0,x1)=μ˙t+1−2t2t(1−t)(x−μt)+12G˙tGt−1(x−μt) u_t(x x_0,x_1)= μ_t+ 1-2t2t(1-t)(x- _t)+ 12 G_tG_t^-1(x- _t) satisfies dxdt=u~t(xt∣x0,x1) dxdt= u_t(x_t x_0,x_1) and xt∼pt(⋅∣x0,x1)x_t p_t(· x_0,x_1) for all t∈(0,1).t∈(0,1). Proof. Let BtB_t be a differentiable matrix factor satisfying BtBt⊤=Gt,B˙tBt−1=12G˙tGt−1, B_tB_t =G_t, B_tB_t^-1= 12 G_tG_t^-1, and let ε∼(0,I) (0,I). Such a metric exists since GtG_t is symmetric positive definite (SPD) and by using existence and uniqueness of ODEs. A sample from ptp_t can then be written as xt=μt+t(1−t)Btε. x_t= _t+ t(1-t)\,B_t . Solving for ε gives ε=1t(1−t)Bt−1(xt−μt). = 1 t(1-t)B_t^-1(x_t- _t). Differentiating xtx_t with respect to t yields x˙t x_t =μ˙t+1−2t2t(1−t)Btε+t(1−t)B˙tε = μ_t+ 1-2t2 t(1-t)B_t + t(1-t)\, B_t =μ˙t+1−2t2t(1−t)(xt−μt)+B˙tBt−1(xt−μt) = μ_t+ 1-2t2t(1-t)(x_t- _t)+ B_tB_t^-1(x_t- _t) =μ˙t+1−2t2t(1−t)(xt−μt)+12G˙tGt−1(xt−μt). = μ_t+ 1-2t2t(1-t)(x_t- _t)+ 12 G_tG_t^-1(x_t- _t). Hence x˙t=ut(xt) x_t=u_t(x_t). Since xt=μt+t(1−t)Btεx_t= _t+ t(1-t)B_t has distribution (μt,t(1−t)Gt)N( _t,t(1-t)G_t) by construction, the flow generated by utu_t has marginals ptp_t for all t∈(0,1)t∈(0,1). ∎ Remark 22. Note that in the case where Gt=IdG_t= and μt=tx1+(1−t)x0 _t=tx_1+(1-t)x_0we recover the formula from Tong et al. [43]. Appendix C Experiments 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 rate 2×10−32× 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 ± population standard deviation across three seeds, except Erythroid, which uses five. Hyperparameters were selected using a validation split of 1515 percent and using the mean of mid and end W2W_2. Kernel extension of transition moments: The first and second transition moments m(xi)m(x_i) and C(xi)C(x_i) are defined only at the observed states xii=1N\x_i\_i=1^N. To evaluate the Finsler metric along continuous paths, we extend both moments using normalized non-negative kernel smoothing. Let kh:ℝd×ℝd→ℝ≥0k_h:R^d×R^d _≥ 0 be a smooth kernel with bandwidth h>0h>0. Assuming that ∑ℓ=1Nkh(x,xℓ)>0 _ =1^Nk_h(x,x_ )>0 throughout the region of interest, define wi(x)≔kh(x,xi)∑ℓ=1Nkh(x,xℓ).w_i(x) k_h(x,x_i) _ =1^Nk_h(x,x_ ). The extended moment fields are m(x)≔∑i=1Nwi(x)m(xi),C(x)≔∑i=1Nwi(x)C(xi).m(x) _i=1^Nw_i(x)m(x_i), C(x) _i=1^Nw_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, so C(x)C(x) is a convex combination of positive semidefinite matrices. Both extended fields are smooth wherever the denominator is positive. Adding the ridge term Cρ(x)=C(x)+ρIC_ρ(x)=C(x)+ρ I, with ρ>0ρ>0, then ensures positive definiteness for the subsequent metric construction. Sheet Dataset: We sample N=4000N=4000 points from coordinates u∼(−1.5,1.5)u (-1.5,1.5) and w∼(−0.45,0.45)w (-0.45,0.45) and embed them on the saddle z(u,w)=12[(3u2)2−12(100w45)2],z(u,w)= 12 [ ( 3u2 )^2- 12 ( 100w45 )^2 ], with Gaussian noise of standard deviation 0.0120.012. The prescribed transition probability matrix P forms an hourglass: trajectories are driven towards u≈0u≈ 0 and go outward after crossing. Source and target states are defined by w≤−0.13w≤-0.13 and w≥0.13w≥ 0.13, respectively. P is constructed bye directed 1515-N kernel combining spatial proximity with alignment to a prescribed vector field (λ=6λ=6), followed by row normalization. Source and target points are split 70/15/1570/15/15 into 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). (a) Sheet benchmark. G Drift ceffc_eff mid W2↓W_2 end W2↓W_2 W2,F↓W_2,F I no 0.001 0.7165 ± 0.0136 0.1751 ± 0.0271 1.0840 ± 0.0371 I yes 0.958 0.2541 ± 0.0375 0.2558 ± 0.0509 0.3676 ± 0.0773 Cρ−1C_ρ^-1 no 0.001 0.6672 ± 0.0160 0.2025 ± 0.0079 0.9437 ± 0.1068 Cρ−1C_ρ^-1 yes 0.958 0.1588 ± 0.0096 0.2343 ± 0.0292 0.2892 ± 0.0090 sampling floor 0.0646 0.1896 0.2812 (b) Metric ablation. (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 to ρ and λ are reported in Table 2 and Table 3. 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 convention t=1/2t=1/2, we evaluate W2W_2 over a fixed 11-point grid spanning t∈[1/3,2/3]t∈[1/3,2/3] and report W¯2[1/3,2/3] W_2^[1/3,2/3], the mean over this grid. We further report W2,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 use no intermediate-band cells for metric construction. Fig. 5 is an additional sensitivity analysis that deliberately varies this fraction. (a) Sensitivity to model time Method W2(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] W_2^[1/3,2/3] d=2d=2 FFM, deterministic 1.9878 ± 0.0631 1.8184 ± 0.0326 1.4429 ± 0.0195 1.1770 ± 0.0919 1.6175 ± 0.0348 FFM, with noise 2.0052 ± 0.0705 1.8399 ± 0.0401 1.4723 ± 0.0601 1.2148 ± 0.0851 1.6434 ± 0.0555 OT-CFM 1.8508 ± 0.0172 1.7952 ± 0.0182 1.7607 ± 0.0195 1.7682 ± 0.0201 1.7869 ± 0.0230 MFM / LAND 1.5201 ± 0.0119 1.4878 ± 0.0117 1.4438 ± 0.0108 1.4002 ± 0.0098 1.4642 ± 0.0131 Curly-FM 1.0208 ± 0.1277 0.9952 ± 0.1378 0.9480 ± 0.1457 0.9009 ± 0.1505 0.9686 ± 0.1725 d=20d=20 FFM, deterministic 11.5969 ± 0.1999 10.7814 ± 0.1811 10.3610 ± 0.1440 10.8331 ± 0.1471 10.7413 ± 0.1927 FFM, with noise 11.5392 ± 0.1640 10.7188 ± 0.1270 10.3671 ± 0.0477 10.9998 ± 0.2163 10.7355 ± 0.0560 OT-CFM 13.0727 ± 0.1325 12.5476 ± 0.0959 12.2577 ± 0.0231 12.3903 ± 0.0406 12.4957 ± 0.0681 MFM / LAND 12.8496 ± 0.1074 12.3470 ± 0.0706 11.8851 ± 0.0180 11.7331 ± 0.0277 12.1656 ± 0.0522 Curly-FM 12.8636 ± 0.0503 12.3801 ± 0.0687 12.2108 ± 0.0760 12.4096 ± 0.1064 12.3920 ± 0.0687 d=50d=50 FFM, deterministic 13.2623 ± 0.1442 12.7423 ± 0.0966 12.5789 ± 0.0240 12.9036 ± 0.0495 12.7715 ± 0.0719 FFM, with noise 13.2718 ± 0.1524 12.7538 ± 0.1104 12.5768 ± 0.0639 12.8742 ± 0.0975 12.7723 ± 0.0952 OT-CFM 14.1359 ± 0.1441 13.6933 ± 0.1242 13.4941 ± 0.0923 13.6688 ± 0.0816 13.6811 ± 0.1314 MFM / LAND 13.8332 ± 0.1526 13.2557 ± 0.0966 12.7979 ± 0.0593 12.7353 ± 0.0364 13.0996 ± 0.1029 Curly-FM 13.8395 ± 0.1113 13.4287 ± 0.1521 13.2923 ± 0.2020 13.4719 ± 0.2307 13.4440 ± 0.2069 (b) Finsler-Wasserstein recovery Method d=2d=2 d=10d=10 d=20d=20 d=50d=50 OT-CFM 0.6420 ± 0.0059 10.9311 ± 0.0770 10.9277 ± 0.0412 15.5900 ± 0.2287 MFM / LAND 0.5399 ± 0.0024 9.4377 ± 0.0824 10.4768 ± 0.0637 14.6185 ± 0.1085 Curly-FM 0.7493 ± 0.0258 10.7482 ± 0.1305 11.0235 ± 0.0520 15.0427 ± 0.1562 FFM, deterministic 0.5681 ± 0.0100 9.5125 ± 0.5398 8.9033 ± 0.0400 13.9368 ± 0.1217 FFM, with noise 0.5730 ± 0.0082 9.6113 ± 0.5379 8.9213 ± 0.0453 13.7800 ± 0.0861 floor 0.4162 5.3461 4.3608 7.2123 W2,F(t=1/2)↓W_2,F(t=1/2) . (c) Intermediate support used for metric construction Method 0%0\% 5%5\% 35%35\% d=2d=2 OT-CFM 1.7607 ± 0.0262 1.7622 ± 0.0392 1.7666 ± 0.0297 FFM, deterministic 1.3833 ± 0.1410 1.3313 ± 0.0627 1.0927 ± 0.2477 FFM, with noise 1.3723 ± 0.1505 1.3525 ± 0.0420 1.1013 ± 0.2611 MFM / LAND 1.4506 ± 0.0025 1.2412 ± 0.5283 0.6878 ± 0.0378 Curly-FM 0.9751 ± 0.1704 0.9360 ± 0.1045 1.5865 ± 0.1369 d=20d=20 OT-CFM 12.3291 ± 0.0685 12.3264 ± 0.0445 12.3491 ± 0.0850 FFM, deterministic 10.4208 ± 0.1028 8.2742 ± 0.3540 8.1989 ± 0.0481 FFM, with noise 10.3679 ± 0.0641 8.4860 ± 0.3098 8.2513 ± 0.2160 MFM / LAND 12.0368 ± 0.0696 11.8784 ± 0.0338 11.7247 ± 0.1739 Curly-FM 12.2067 ± 0.0844 12.2163 ± 0.1098 12.2431 ± 0.0665 d=50d=50 OT-CFM 13.5319±0.048013.5319± 0.0480 13.5269±0.064213.5269± 0.0642 13.5577±0.107513.5577± 0.1075 FFM, deterministic 12.5043±0.054412.5043± 0.0544 12.1225±0.125212.1225± 0.1252 11.3633±0.135611.3633± 0.1356 MFM / LAND 13.5584±0.039613.5584± 0.0396 13.5208±0.027513.5208± 0.0275 13.5689±0.123913.5689± 0.1239 Curly-FM 13.3220±0.192713.3220± 0.1927 13.3291±0.250113.3291± 0.2501 13.2897±0.111613.2897± 0.1116 W2(t=1/2)↓W_2(t=1/2) . 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 with W¯2[1/3,2/3] 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_ρ( _t) W2(1/2)W_2(1/2) Mt=Cρ(μt)M_t=C_ρ( _t) W2(1)W_2(1) Mt=IM_t=I W2(1/2)W_2(1/2) Mt=IM_t=I W2(1)W_2(1) 0.003 10.4874 ± 0.0419 5.2457 ± 0.2301 10.4414 ± 0.1439 5.2980 ± 0.1595 0.010 10.4940 ± 0.1123 5.5062 ± 0.4344 10.5187 ± 0.1591 5.2420 ± 0.2684 0.030 10.4873 ± 0.0919 5.3305 ± 0.1619 10.4245 ± 0.1359 5.3180 ± 0.1351 0.100 10.3694 ± 0.0987 5.0662 ± 0.3449 10.3446 ± 0.1299 5.1521 ± 0.1702 0.300 10.4630 ± 0.2142 5.3692 ± 0.1590 10.4149 ± 0.1922 5.2433 ± 0.2451 1.000 14.0089 ± 0.0636 7.6047 ± 0.3015 13.8565 ± 0.1498 7.3363 ± 0.1651 3.000 36.3654 ± 0.5652 24.9668 ± 0.7730 33.9906 ± 1.1181 26.6449 ± 1.1385 10.000 118.0193 ± 4.7889 85.1409 ± 11.5354 143.4577 ± 2.9533 130.1162 ± 4.8789 (e) Path and inference covariance Path Inference W2(1/2)W_2(1/2) W2(1)W_2(1) Cρ(μt)C_ρ( _t) Cρ(Xt)C_ρ(X_t) 10.3694 ± 0.0987 5.0662 ± 0.3449 Cρ(μt)C_ρ( _t) I 10.3617 ± 0.1003 4.9788 ± 0.2102 I Cρ(Xt)C_ρ(X_t) 10.3557 ± 0.1112 5.1712 ± 0.2807 I I 10.3446 ± 0.1299 5.1521 ± 0.1702 (f) Inference-noise schedule g(t)g(t) M(Xt)=Cρ(Xt)M(X_t)=C_ρ(X_t) W2(1/2)W_2(1/2) M(Xt)=Cρ(Xt)M(X_t)=C_ρ(X_t) W2(1)W_2(1) M(Xt)=IM(X_t)=I W2(1/2)W_2(1/2) M(Xt)=IM(X_t)=I W2(1)W_2(1) σ 10.3694 ± 0.0987 5.0662 ± 0.3449 10.3617 ± 0.1003 4.9788 ± 0.2102 ∝(1−t) (1-t) 10.3580 5.0072 10.3496 5.0492 g∼tg t 10.3922 5.1977 10.3868 5.1169 g∼t(1−t)g t(1-t) 10.3629 5.0814 10.3515 5.0035 00 after t=0.8t=0.8 10.3626 5.0440 10.3543 4.9913 00 10.4173 5.1955 10.4173 5.1955 Figure 6: Pancreas robustness analyses (continued). (d) Sensitivity of stochastic FFM at d=20d=20 to conditional-path noise width and covariance. (e) Sensitivity to conditional-path and inference covariance. (f) Sensitivity to the scalar inference-noise schedule. Values are mean ± population 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. Method d=2d=2 d=10d=10 d=20d=20 d=50d=50 OT-CFM 3 3 3 3 MFM / LAND 22 61 109 254 Curly-FM 41 106 190 445 FFM (ours), deterministic 26 63 109 256 FFM (ours), with noise 15 29 48 108 Table 3: One-at-a-time sensitivity of FFM on Pancreas. The reported value is the validation selection objective; lower is better. The selected value is shown in bold. d Varied parameter Fixed parameter Values Validation objective 2 ρ λ=0.1λ=0.1 0.01, 0.03, 0.1, 0.3, 1.00.01,\ 0.03,\ 0.1,\ 0.3,\ 1.0 0.828, 0.749, 0.535, 0.870, 1.1310.828,\ 0.749,\ 0.535,\ 0.870,\ 1.131 2 λ ρ=0.1ρ=0.1 0.01, 0.03, 0.1, 0.3, 1.00.01,\ 0.03,\ 0.1,\ 0.3,\ 1.0 1.226, 0.951, 0.535, 0.826, 0.8731.226,\ 0.951,\ 0.535,\ 0.826,\ 0.873 20 ρ λ=1.0λ=1.0 0.003, 0.01, 0.03, 0.1, 0.30.003,\ 0.01,\ 0.03,\ 0.1,\ 0.3 8.180, 8.210, 8.135, 8.369, 8.2888.180,\ 8.210,\ 8.135,\ 8.369,\ 8.288 20 λ ρ=0.03ρ=0.03 0.03, 0.1, 0.3, 1.0, 3.00.03,\ 0.1,\ 0.3,\ 1.0,\ 3.0 8.990, 8.535, 8.370, 8.135, 8.6678.990,\ 8.535,\ 8.370,\ 8.135,\ 8.667 50 ρ λ=0.03λ=0.03 3×10−4, 10−3, 3×−, 10−2, 3×10−23×10^-4,\ 10^-3,\ 3×10^-3,\ 10^-2,\ 3×10^-2 10.455, 10.421, 10.314, 10.390, 10.58910.455,\ 10.421,\ 10.314,\ 10.390,\ 10.589 50 λ ρ=0.003ρ=0.003 0.003, 0.01, 0.03, 0.1, 0.30.003,\ 0.01,\ 0.03,\ 0.1,\ 0.3 10.379, 10.314, 10.314, 10.554, 10.53710.379,\ 10.314,\ 10.314,\ 10.554,\ 10.537 Erythroid: We further evaluate FFM on the mouse erythroid differentiation dataset used by Petrović et al. [35]. 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 in d∈20,50d∈\20,50\ dimensional representations. In addition to our methods, we report both the values originally published by Petrović et al. [35] and our reproduction of the baselines using a common training harness at their published training budget. Method Source Cos. Dist. ↓ L2↓L_2 W1↓W_1 W2↓W_2 d=2d=2 OT-CFM Published 0.146 ± 0.001 2.704 ± 0.019 0.646 ± 0.006 – OT-CFM Reproduced 0.141 ± 0.002 2.447 ± 0.031 0.645 ± 0.009 0.679 ± 0.009 MFM (LAND) Published 0.014 ± 0.001 1.999 ± 0.014 0.269 ± 0.004 – MFM (LAND) Reproduced 0.010 ± 0.001 1.858 ± 0.025 0.271 ± 0.007 0.286 ± 0.007 Curly-FM Published 0.009 ± 0.000 1.663 ± 0.293 0.369 ± 0.090 – Curly-FM Reproduced 0.011 ± 0.001 2.151 ± 0.078 0.122 ± 0.002 0.158 ± 0.003 FFM (ours) Ours 0.021 ± 0.004 1.642 ± 0.071 0.248 ± 0.034 0.291 ± 0.023 FFM (ours), with noise Ours 0.019 ± 0.003 1.710 ± 0.081 0.226 ± 0.027 0.269 ± 0.019 d=20d=20 OT-CFM Published 0.489 ± 0.001 1.885 ± 0.020 6.103 ± 0.074 – OT-CFM Reproduced 0.490 ± 0.000 1.731 ± 0.014 6.106 ± 0.063 6.260 ± 0.066 MFM (LAND) Published 0.495 ± 0.001 1.627 ± 0.040 4.855 ± 0.052 – MFM (LAND) Reproduced 0.494 ± 0.000 1.700 ± 0.021 4.759 ± 0.056 4.855 ± 0.054 Curly-FM Published 0.488 ± 0.001 1.721 ± 0.035 6.124 ± 0.027 – Curly-FM Reproduced 0.490 ± 0.000 1.691 ± 0.011 6.112 ± 0.053 6.271 ± 0.058 FFM (ours) Ours 0.491 ± 0.000 1.514 ± 0.016 4.685 ± 0.012 4.782 ± 0.014 FFM (ours), with noise Ours 0.496 ± 0.000 0.757 ± 0.006 4.733 ± 0.035 4.823 ± 0.038 d=50d=50 OT-CFM Published 0.490 ± 0.000 2.215 ± 0.022 7.969 ± 0.029 – OT-CFM Reproduced 0.491 ± 0.000 2.053 ± 0.017 7.839 ± 0.016 7.974 ± 0.016 MFM (LAND) Published 0.494 ± 0.000 1.971 ± 0.023 6.727 ± 0.022 – MFM (LAND) Reproduced 0.495 ± 0.000 2.004 ± 0.013 6.697 ± 0.052 6.819 ± 0.058 Curly-FM Published 0.489 ± 0.000 2.045 ± 0.073 7.729 ± 0.046 – Curly-FM Reproduced 0.491 ± 0.000 1.990 ± 0.021 7.736 ± 0.025 7.864 ± 0.025 FFM (ours) Ours 0.491 ± 0.000 1.772 ± 0.029 6.810 ± 0.022 6.907 ± 0.025 FFM (ours), with noise Ours 0.499 ± 0.000 0.915 ± 0.006 6.595 ± 0.016 6.663 ± 0.015 Figure 7: Visualization of the learned erythroid trajectories in 2D and quantitative comparison across dimensions d=2,20,50d=2,20,50. “Published” denotes the values reported in Table 4 of Petrović et al. [35], while “Reproduced” denotes our reproduction of the corresponding baselines using a common training harness at the published training budget. Due to a bug in the original Curly-FM evaluation, the Wasserstein quantity reported as W2W_2 in Table 4 of CurlyFM corresponds in fact to W1W_1. We therefore report these published values under W1W_1 and additionally compute the true W2W_2 distance for all runs evaluated with our harness. For d=20d=20 and d=50d=50, L2L_2 is reported in units of 10310^3. 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: Let 15(i)N_15(i) denote the 15 nearest neighbours of cell i, excluding itself, and let j7(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 Kijexpr=j∈15(i)exp(−‖xi−xj‖2σiσj),σi=∥xi−xj7(i)∥.K^expr_ij=1_\j _15(i)\ \! (- \|x_i-x_j\|^2 _i _j ), _i=\|x_i-x_j_7(i)\|. (19) For barcode-family labels bfibf_i and scar-family labels sfisf_i, define Rij=1−12(bfi≠bfj+sfi≠sfj),Kijlin=exp(βlinRij).R_ij=1- 12 (1_\bf_i _j\+1_\sf_i _j\ ), K^lin_ij= ( _linR_ij). (20) Pairs involving a cell without a barcode call receive Rij=0R_ij=0. Barcode and scar families containing fewer than 100 and 50 cells, respectively, are excluded. For the RNA-velocity vector viv_i, the velocity factor is aij=⟨vi,xj−xi⟩‖vi‖‖xj−xi‖,Kijvel=exp(νaij).a_ij= v_i,x_j-x_i \|v_i\|\,\|x_j-x_i\|, K^vel_ij= (ν a_ij). (21) The final transition matrix is Wij=KijexprKijlinKijvel,Pij=Wij∑ℓWiℓ,Pii=0.W_ij=K^expr_ijK^lin_ijK^vel_ij, P_ij= W_ij _ W_i , P_i=0. (22) The different choices (βlin,ν)∈(0,0),(0,3),(3,0),(3,3)( _lin,ν)∈\(0,0),(0,3),(3,0),(3,3)\ correspond to expression only, expression plus velocity, expression plus lineage, and all three factors, respectively. Figure 8: Learned trajectories on iTracer in the two-dimensional representation. Bifurcation: We evaluate a synthetic bifurcation comprising 30003000 points in ℝ3R^3, where one source population splits into two target arms and the middle split is withheld Fig. 9. 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 radius 1/21/2 of the distance between arm centres Table Table 4. 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 Table 5 shows that performance under imbalance remains configuration-dependent. (a) Dataset. (b) Learned trajectories. Figure 9: ForkSheet benchmark. Table 4: ForkSheet fate preservation. Values are percentages, reported as mean ± population standard deviation over three seeds. Method True arm Recall Wrong arm Outside OT-CFM Lower 69.4 ± 10.7 30.6 ± 10.7 0.0 ± 0.0 Upper 100.0 ± 0.0 0.0 ± 0.0 0.0 ± 0.0 MFM / LAND Lower 91.4 ± 5.6 4.4 ± 3.9 4.2 ± 3.4 Upper 68.1 ± 37.3 12.3 ± 17.4 19.6 ± 20.1 Curly-FM Lower 48.8 ± 16.7 47.4 ± 15.2 3.8 ± 1.5 Upper 100.0 ± 0.0 0.0 ± 0.0 0.0 ± 0.0 FFM (ours), deterministic Lower 75.5 ± 19.7 21.2 ± 19.4 3.4 ± 0.3 Upper 100.0 ± 0.0 0.0 ± 0.0 0.0 ± 0.0 FFM (ours), with noise Lower 68.3 ± 19.8 20.8 ± 22.7 10.9 ± 12.7 Upper 98.8 ± 0.9 0.0 ± 0.0 1.2 ± 0.9 Table 5: ForkSheet recall under different nominal lower/upper fate allocations. Values are percentages, reported as mean ± population 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 Table 4. Method 50/50 70/30 40/60 20/80 10/90 OT-CFM 69.4 ± 10.7 100.0 ± 0.0 71.1 ± 2.5 100.0 ± 0.0 57.6 ± 11.7 99.5 ± 0.7 59.0 ± 17.4 99.1 ± 0.6 36.1 ± 16.4 98.4 ± 1.3 MFM / LAND 91.2 ± 9.5 76.0 ± 16.6 99.9 ± 0.2 84.8 ± 1.8 92.0 ± 8.3 98.5 ± 2.2 73.2 ± 23.6 74.0 ± 34.9 50.9 ± 15.1 96.9 ± 4.4 Curly-FM 48.8 ± 16.7 100.0 ± 0.0 68.0 ± 7.1 100.0 ± 0.0 37.2 ± 3.9 100.0 ± 0.0 33.9 ± 18.8 100.0 ± 0.0 0.0 ± 0.0 100.0 ± 0.0 FFM (ours), deterministic 95.6 ± 3.4 94.1 ± 5.2 92.2 ± 10.4 89.1 ± 13.9 89.4 ± 5.7 94.2 ± 8.2 98.9 ± 0.8 97.7 ± 0.8 77.8 ± 10.4 98.3 ± 0.3 FFM (ours), with noise 97.3 ± 3.4 94.5 ± 7.7 94.0 ± 8.5 81.2 ± 19.5 93.8 ± 4.2 98.5 ± 1.8 99.5 ± 0.8 98.7 ± 0.4 79.6 ± 6.5 93.1 ± 4.8 Appendix D Related Work Relation to Metric Flow Matching. Metric Flow Matching (MFM) [24] learns endpoint-conditioned approximate geodesics of a data-dependent Riemannian metric g(x)=G(x,)g(x)=G(x;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,)G(x;D) is a special case of a P−P-derived Randers with FP(x,v)=vTG(x,)vF_P(x,v)= v^TG(x;D)v. There is also an explicit connection between the LAND metric [2] considered in MFM and our P-derived metric (9). Consider a soft k-N kernel wij=j∈k(i)exp(−‖xi−xj‖22σ2),Pij=wijZi,Zi=∑jwij. w_ij=1_\j _k(i)\ \! (- \|x_i-x_j\|^22σ^2 ), P_ij= w_ijZ_i, Z_i= _jw_ij. The corresponding second transition moment is CN(xi)=1Zi∑jwij(xj−xi)(xj−xi)⊤. C_N(x_i)= 1Z_i _jw_ij(x_j-x_i)(x_j-x_i) . On the other hand, defining hα(xi)=∑jwij(xjα−xiα)2,h_α(x_i)= _jw_ij(x_j^α-x_i^α)^2, we obtain diagCN(xi)=1Zidiag(h(xi)).diagC_N(x_i)= 1Z_idiag(h(x_i)). (23) Consequently, (diagCN(xi)+ρI)−1=Zi(diag(h(xi))+ρZiI)−1. (diagC_N(x_i)+ρ I )^-1=Z_i (diag(h(x_i))+ρ Z_iI )^-1. (24) Thus, apart from the normalization ZiZ_i induced by requiring P to 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=ρZ =ρ Z recovers the LAND metric up to the global multiplicative factor Z, 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 P. Relation to Curly FM: Curly Flow Matching (Curly-FM) [35] assumes access to a reference vector field ft(x)f_t(x) and learns endpoint conditioned paths whose velocities remain close to this field, using an objective of the form ℒCurly(η)=x0,x1,t[‖x˙t,η−ft(xt,η)‖22]. _Curly(η)=E_x_0,x_1,t [ \| x_t,η-f_t(x_t,η) \|_2^2 ]. When the dynamics are instead specified by a transition matrix P, the natural first-order vector-field summary is its first spatial moment m(xi)=∑j=1NPij(xj−xi),m(x_i)= _j=1^NP_ij(x_j-x_i),. If we replace the Euclidean norm in the Curly objective by the second moment Cρ−1C_ρ^-1 ‖x˙t,η−m(x)‖C(x)−12=(x˙t,η−m(x))TCρ(x)−1(x˙t,η−m(x)) \| x_t,η-m(x) \|_C(x)^-1^2=( x_t,η-m(x))^TC_ρ(x)^-1( x_t,η-m(x)) (25) which gives us the integrant of the fixed time FW functional Eq. 16. For data that follows the manifold hypothesis we have in particular that C≠IC≠ 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˙−am(x))⊤Cρ(x)−1(x˙−am(x)), 1a ( x-a\,m(x) ) C_ρ(x)^-1 ( x-a\,m(x) ), 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. 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], a vector field f can be converted into a local transition matrix PfP^f. Let Wij(h)=∥xj−xi∥≤hi,W^(h)_ij=1\! \\|x_j-x_i\|≤ h_i \, where hih_i denotes the distance to the k-th nearest neighbor, and define Pijf=Wij(h)exp(κf(xi)⊤(xj−xi)‖f(xi)‖‖xj−xi‖)∑ℓWiℓ(h)exp(κf(xi)⊤(xℓ−xi)‖f(xi)‖‖xℓ−xi‖). P^f_ij= W^(h)_ij \! (κ f(x_i) (x_j-x_i)\|f(x_i)\|\,\|x_j-x_i\| ) _ W^(h)_i \! (κ f(x_i) (x_ -x_i)\|f(x_i)\|\,\|x_ -x_i\| ). For dense and locally isotropic sampling, the k-N neighborhood of xix_i is approximately rotationally symmetric. The cosine kernel assigns a displacement direction u a weight proportional to exp(κf^(xi)⊤u) (κ\, f(x_i) u), where f^=f/‖f‖ f=f/\|f\|. By local isotropy, contributions orthogonal to f f appear in pairs in opposite directions with and hence cancel eachother. Consequently, mNf(xi)=∑jPijf(xj−xi)=cif^(xi)+o(1),ci>0, m_N^f(x_i)= _jP^f_ij(x_j-x_i)=c_i\, f(x_i)+o(1), 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