Multimarginal flow matching with optimal transport potentials

# Multimarginal flow matching with optimal transport potentials
Source: [https://arxiv.org/html/2606.05327](https://arxiv.org/html/2606.05327)
###### Abstract

Flow matching \(FM\) has emerged as a powerful framework for learning dynamic transport maps between two empirical distributions\. However, less explored is the setting with intermediate observed marginals that can help constrain the flows between the endpoints\. This “multimarginal” regime is central to modeling temporal evolution in dynamical systems in many scientific domains that can sample sequential distributions\. We tackle this problem with a novel approach that leverages the connection between FM and dynamic optimal transport \(OT\), softly steering the flow towards the intermediate marginals through potential terms in the dynamic OT action\. By extending the conditional FM learning target to incorporate these potentials, we derive an efficient, simulation\-free algorithm for multimarginal FM that offers considerable flexibility in the spatiotemporal dynamics of the learned flows\. We demonstrate state\-of\-the\-art performance and training efficiency of OT\-potential FM \(OTP\-FM\) on diverse single\-cell RNA sequencing, oceanographic, and meteorological datasets\. Our code is available at[https://github\.com/Bexorg\-Inc/OTP\-FM](https://github.com/Bexorg-Inc/OTP-FM)\.

Machine Learning, ICML

## 1Introduction

Understanding the complex nonlinear dynamics of physical systems is of central importance in many scientific domains, including transcriptomic state transitions in developmental biology, disease progression in neurodegenerative diseases, and climate modeling\. There has been significant recent progress in these disciplines towards developing a corpus of static snapshots of these systems during their evolution, such as with longitudinal single\-cell RNA sequencing \(scRNA\-seq\) measurements; however, inferring accurate per\-sample trajectories from these, often independent, snapshots remains a critical challenge for mechanistic understanding, therapeutic target identification, and predictive inference\.

Conditional flow matching\(CFM\)\(Lipmanet al\.,[2023](https://arxiv.org/html/2606.05327#bib.bib1); Tonget al\.,[2024](https://arxiv.org/html/2606.05327#bib.bib3); Liuet al\.,[2023](https://arxiv.org/html/2606.05327#bib.bib4); Albergoet al\.,[2025](https://arxiv.org/html/2606.05327#bib.bib2)\)has emerged as a leading framework for this problem, learning continuous\-time transport maps through efficient, simulation\-free regression of simple conditional trajectories, which are commonly conditional optimal transport \(OT\) solutions between paired source and target samples\. When intermediate marginals are available, the natural extension has been to apply CFMpiecewisebetween consecutive marginals, stitching conditional paths end\-to\-end \(Fig\.[1](https://arxiv.org/html/2606.05327#S1.F1)\)\. However, this produces trajectories with unphysical discontinuities at each marginal boundary\. Recent multimarginal methods such as MMFM\(Rohbecket al\.,[2025](https://arxiv.org/html/2606.05327#bib.bib54)\)and 3MSBM\(Theodoropouloset al\.,[2026](https://arxiv.org/html/2606.05327#bib.bib55)\)have tried to smooth these, but with prescriptive, ad\-hoc strategies that we argue need not describe the physical system\.

We propose a principled relaxation: we show that piecewise CFM corresponds to conditional dynamic OT with*hard constraints*on each intermediate marginal, and relax these to smooth, finite\-strength potential energy terms in the dynamic OT action with OT\-potential FM \(OTP\-FM\)—recovering piecewise CFM as a limiting case\. Our contributions are:

- •A multimarginal generalization of dynamic OTwith exact conditional solutions and theoretical bound on alignment to the ground\-truth intermediate marginals, controlled by the potential strength and training loss\.
- •A flexible, simulation\-free training algorithmwith a broad design space in the potential parameters, effectively allowing the data to determine the interpolated dynamics through optimization over this space rather than the prescriptive approaches of prior methods\.
- •State\-of\-the\-art \(SOTA\) performance and training efficiencyon diverse scientific datasets, with systematic ablations and concrete recommendations for applying OTP\-FM to new datasets\.

![Refer to caption](https://arxiv.org/html/2606.05327v1/x1.png)Figure 1:\(Left\)Comparing standard CFM — straight\-line trajectories ignoring intermediate marginals; multimarginal CFM — stitching CFM trajectories piecewise between consecutive marginals; prescriptive approaches such as MMFM and 3MSBM — using fixed interpolation strategies to smooth kinks; and OTP\-FM, whose soft potential\-driven dynamics with tunable strengthww, temporal widthτ\\tau, andλ\\lambdashape yields smooth*and*flexible trajectories\.\(Right\)Method overview: trajectoriesXtX\_\{t\}are decomposed as a base CFM pathXtbaseX\_\{t\}^\{\\mathrm\{base\}\}plus marginal\-driven correctionsXk,tcorrX^\{\\mathrm\{corr\}\}\_\{k,t\}that are gradually scaled by the curriculum parameterα\\alphato converge to the OTP\-FM solution\.
## 2Background and preliminaries

### 2\.1Optimal transport

The original static Monge OT problem solves for a transport mapψ:ℝd→ℝd\\psi\\\!:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}between source and target measuresμ0\\mu\_\{0\}andμ1\\mu\_\{1\}, such that the push\-forward operationψ\#​μ0=μ1\\psi\_\{\\\#\}\\mu\_\{0\}=\\mu\_\{1\}, that is*optimal*with respect to a costc:ℝd×ℝd→ℝc\\colon\\mathbb\{R\}^\{d\}\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}\(Villani,[2009](https://arxiv.org/html/2606.05327#bib.bib7)\):ℒOT:=minψ:ψ\#​μ0=μ1​∫c​\(x,ψ​\(x\)\)​dμ0​\(x\)\\mathcal\{L\}\_\{\\mathrm\{OT\}\}:=\\min\_\{\\psi:\\psi\_\{\\\#\}\\mu\_\{0\}=\\mu\_\{1\}\}\\int\\\!c\(x,\\psi\(x\)\)\\mathrm\{d\}\\mu\_\{0\}\(x\)\. For costcp​\(x,y\)=‖x−y‖pc\_\{p\}\(x,y\)=\\\|x\-y\\\|^\{p\},ℒOT1/p\\mathcal\{L\}\_\{\\mathrm\{OT\}\}^\{1/p\}is the Wassersteinpp\-distance𝒲p\\mathcal\{W\}\_\{p\}\.

Particularly relevant is thedynamic OT\(DOT\) formulation\(Benamou and Brenier,[2000](https://arxiv.org/html/2606.05327#bib.bib6)\), where we define a dynamic probability pathρt:\[0,1\]×ℝd→ℝ\+\\rho\_\{t\}\\colon\[0,1\]\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{\+\}interpolatingμ0\\mu\_\{0\}toμ1\\mu\_\{1\}\. This path is generated by a velocity fieldutu\_\{t\}via the continuity equation∂tρ=−∇⋅\(ρ​u\)\\partial\_\{t\}\\rho=\-\\nabla\\cdot\(\\rho u\), with sample trajectories satisfyingX˙t=ut​\(Xt\)\\dot\{X\}\_\{t\}=u\_\{t\}\(X\_\{t\}\)forX0∼ρ0X\_\{0\}\\sim\\rho\_\{0\}, and the corresponding map, orflow,ψt:\[0,1\]×ℝd→ℝd\\psi\_\{t\}\\colon\[0,1\]\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}transports the samples along the trajectoryXt=ψt​\(X0\)∼ρtX\_\{t\}=\\psi\_\{t\}\(X\_\{0\}\)\\sim\\rho\_\{t\}\. The objective for squared Euclidean costc2/2\\nicefrac\{\{c\_\{2\}\}\}\{\{2\}\}is:111We adopt the conventional factor of1/2\\nicefrac\{\{1\}\}\{\{2\}\}to maintain the analogy with physical kinetic energy\.

ℒDOT:=minρt,ut​∫01∫ℝd12​‖ut​\(x\)‖2​dρt​\(x\)​dt,\\mathcal\{L\}\_\{\\mathrm\{DOT\}\}:=\\min\_\{\\rho\_\{t\},u\_\{t\}\}\\int\_\{0\}^\{1\}\\int\_\{\\mathbb\{R\}^\{d\}\}\\frac\{1\}\{2\}\\\|u\_\{t\}\(x\)\\\|^\{2\}\\mathrm\{d\}\\rho\_\{t\}\(x\)\\mathrm\{d\}t,\(1\)subject to the continuity equation and boundary conditionsρ0=μ0\\rho\_\{0\}=\\mu\_\{0\},ρ1=μ1\\rho\_\{1\}=\\mu\_\{1\}\.

Interestingly, this can be interpreted as minimizing the actionS​\[L\]S\[L\]of a fluid with LagrangianL=T−VL=T\-V, kinetic energyT=12​ρ​‖u‖2T=\\frac\{1\}\{2\}\\rho\\\|u\\\|^\{2\}, and potential energyV=0V=0:

S\[L\]=∫01∫ℝd\[12ρt\(x\)∥ut\(x\)∥2−V\(x,t\)\+φt\(x\)\[∂tρt\(x\)\+∇⋅\(ρt\(x\)ut\(x\)\)\]\]dxdt,S\[L\]=\\int\_\{0\}^\{1\}\\int\_\{\\mathbb\{R\}^\{d\}\}\\Big\[\\frac\{1\}\{2\}\\rho\_\{t\}\(x\)\\\|u\_\{t\}\(x\)\\\|^\{2\}\-V\(x,t\)\+\\\\ \\varphi\_\{t\}\(x\)\[\\partial\_\{t\}\\rho\_\{t\}\(x\)\+\\nabla\\cdot\(\\rho\_\{t\}\(x\)u\_\{t\}\(x\)\)\]\\Big\]\\mathrm\{d\}x\\mathrm\{d\}t,\(2\)whereφ\\varphiis a Lagrange multiplier enforcing the continuity equation\. Minimizers forV=0V=0satisfy the Euler\-Lagrange \(E\-L\) equations:

ut=∇φt,∂tφt​\(x\)\+‖∇φt​\(x\)‖22=0,u\_\{t\}=\\nabla\\varphi\_\{t\},\\qquad\\partial\_\{t\}\\varphi\_\{t\}\(x\)\+\\frac\{\\\|\\nabla\\varphi\_\{t\}\(x\)\\\|^\{2\}\}\{2\}=0,\(3\)and follow straight\-line trajectoriesXt=\(1−t\)​X0\+t​ψ​\(X0\)X\_\{t\}=\(1\-t\)X\_\{0\}\+t\\psi\(X\_\{0\}\),X˙t=ψ​\(X0\)−X0\\dot\{X\}\_\{t\}=\\psi\(X\_\{0\}\)\-X\_\{0\}, whereψ\\psiis the static OT map for costc2c\_\{2\}\. A detailed derivation is provided in App\.[A\.1](https://arxiv.org/html/2606.05327#A1.SS1)\. These straight\-line minimizers correspond to the conditional paths commonly used in CFM, as we describe next, while in OTP\-FM we explore the case of non\-zeroVV\.

### 2\.2Conditional flow matching

Conditional flow matching \(CFM\) similarly aims to find a flow between measures by learning a parametric velocityutθu^\{\\theta\}\_\{t\}\. Obtaining a valid*marginal*training target to which to regress this is often intractable; however,Lipmanet al\.\([2023](https://arxiv.org/html/2606.05327#bib.bib1)\); Tonget al\.\([2024](https://arxiv.org/html/2606.05327#bib.bib3)\)show that we can derive an equivalent, simpler*conditional*target by constructing the marginalρt​\(x\)=∫ρt​\(x\|z\)​q​\(z\)​dz\\rho\_\{t\}\(x\)=\\int\\rho\_\{t\}\(x\|z\)q\(z\)\\mathrm\{d\}zas a mixture of conditional pathsρt​\(x\|z\)\\rho\_\{t\}\(x\|z\), with associated conditional velocityut​\(x\|z\)u\_\{t\}\(x\|z\), conditioned on a latent variablez∼q​\(z\)z\\sim q\(z\), and satisfyingρ0​\(x\)=μ0​\(x\)\\rho\_\{0\}\(x\)=\\mu\_\{0\}\(x\),ρ1​\(x\)=μ1​\(x\)\\rho\_\{1\}\(x\)=\\mu\_\{1\}\(x\)\.

The simplest and most common way to do so is choosingz=\(x0,x1\)∼π​\(x0,x1\)z=\(x\_\{0\},x\_\{1\}\)\\sim\\pi\(x\_\{0\},x\_\{1\}\), whereπ\\piis some joint distribution of the endpoints, andut​\(x\|z\)u\_\{t\}\(x\|z\)theconditional dynamic OT solution\(forV=0V=0\) betweenρ0​\(x\|z\)=δ​\(x−x0\)\\rho\_\{0\}\(x\|z\)=\\delta\(x\-x\_\{0\}\)andρ1​\(x\|z\)=δ​\(x−x1\)\\rho\_\{1\}\(x\|z\)=\\delta\(x\-x\_\{1\}\), defining the CFM objective:

Xt\(x\|z\)=\(1−\\displaystyle X\_\{t\}\(x\|z\)=\(1\-t\)x0\+tx1,\\displaystyle t\)x\_\{0\}\+tx\_\{1\},\(4\)ut\(x\|z\)=X˙t\(x\|\\displaystyle u\_\{t\}\(x\|z\)=\\dot\{X\}\_\{t\}\(x\|z\)=x1−x0,\\displaystyle z\)=x\_\{1\}\-x\_\{0\},\(5\)ℒCFM​\(θ\):=𝔼t,z,x∼ρt​\(x\|z\)\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{CFM\}\}\(\\theta\):=\\mathbb\{E\}\_\{t,z,x\\sim\\rho\_\{t\}\(x\|z\)\}∥uθ\(t,x\)−ut\(x\|z\)∥2\.\\displaystyle\\left\\\|u^\{\\theta\}\(t,x\)\-u\_\{t\}\(x\|z\)\\right\\\|^\{2\}\.\(6\)Perhaps surprisingly, this far simpler objective is gradient\-equivalent to the marginal FM objective, thereby providing an efficient and scalable training algorithm for learning flows\. In OTP\-FM, we generalize this regression target to non\-zero potentialsVVthat flexibly incorporate intermediate marginal constraints\.

### 2\.3Few\-step and consistency models

As we will describe in Sec\.[4\.1](https://arxiv.org/html/2606.05327#S4.SS1), OTP\-FM can require evaluating sample positionsXtkX\_\{t\_\{k\}\}at intermediate times during training\. To avoid costly ODE simulation for this, we train aconsistency modelfor few\-step inference\. Broadly, consistency models aim to learn \(variations of\) theflow mapΨt1,t2:\[0,1\]2×ℝd→ℝd\\Psi\_\{t\_\{1\},t\_\{2\}\}\\colon\[0,1\]^\{2\}\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}— a generalization ofψt\\psi\_\{t\}that can transport samples between two arbitrary time pointst1t\_\{1\}andt2t\_\{2\}:Ψt1,t2​\(Xt1\)=Xt2\\Psi\_\{t\_\{1\},t\_\{2\}\}\(X\_\{t\_\{1\}\}\)=X\_\{t\_\{2\}\}, with boundary conditionΨt,t​\(x\)=x\\Psi\_\{t,t\}\(x\)=x\(Boffiet al\.,[2025](https://arxiv.org/html/2606.05327#bib.bib16),[2026](https://arxiv.org/html/2606.05327#bib.bib17)\)\.

While OTP\-FM is agnostic to the particular training procedure used to learnΨt1,t2\\Psi\_\{t\_\{1\},t\_\{2\}\}, for our experiments we primarily employ that of*improved MeanFlow*\(iMF\)\(Genget al\.,[2026b](https://arxiv.org/html/2606.05327#bib.bib76)\), which, at the time of writing, is SOTA in one\- and two\-step inference\. Namely, we parameterizeΨt1,t2\\Psi\_\{t\_\{1\},t\_\{2\}\}in terms of themean velocitybetweent1t\_\{1\}andt2t\_\{2\},vt1,t2​\(Xt1\)=1t2−t1​∫t1t2ut​\(Xt\)​dtv\_\{t\_\{1\},t\_\{2\}\}\(X\_\{t\_\{1\}\}\)=\\frac\{1\}\{t\_\{2\}\-t\_\{1\}\}\\int\_\{t\_\{1\}\}^\{t\_\{2\}\}u\_\{t\}\(X\_\{t\}\)\\mathrm\{d\}t, from which the regression target is derived:

ℒiMF\(θ\):=𝔼t1,t2,z,x∼ρt1\(⋅\|z\)∥Vt1,t2θ\(x\)−ut1\(x\|z\)∥2\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{iMF\}\}\(\\theta\):=\\mathbb\{E\}\_\{t\_\{1\},t\_\{2\},z,x\\sim\\rho\_\{t\_\{1\}\}\(\\cdot\|z\)\}\\\!\\left\\\|V^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\(x\)\-u\_\{t\_\{1\}\}\\\!\(x\|z\)\\right\\\|^\{2\}\\\!\\\!\\\!\\\!\\\!\\\!\\\!\\\!\(7\)Vt1,t2θ​\(x\)≡vθ​\(x\)−\(t2−t1\)​sg​\[vt1,t1θ​\(x\)​∂xvθ\+∂t1vθ\],\\displaystyle V^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\(x\)\\equiv v^\{\\theta\}\\\!\(x\)\-\(t\_\{2\}\-t\_\{1\}\)\\,\\mathrm\{sg\}\\\!\\left\[v^\{\\theta\}\_\{t\_\{1\},t\_\{1\}\}\\\!\(x\)\\\!\\,\\partial\_\{x\}v^\{\\theta\}\\\!\+\\partial\_\{t\_\{1\}\}\\\!v^\{\\theta\}\\right\],wheresg\\mathrm\{sg\}is the stop\-gradient operator,ut​\(x\|z\)u\_\{t\}\(x\|z\)is theinstantaneousconditional velocity target used in CFM, andVt1,t2θV^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}is parametrization of the modelvt1,t2θv^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}that directly regresses this target\.222Our formulation is largely equivalent to the original ofGenget al\.\([2026b](https://arxiv.org/html/2606.05327#bib.bib76)\)except for a modification of the training target to flowforwardin time\. Details are provided in App\.[C\.2](https://arxiv.org/html/2606.05327#A3.SS2)\.Further discussion of consistency methods and a derivation of our iMF objective appear in App\.[C](https://arxiv.org/html/2606.05327#A3), and results with alternatives in App\.[G](https://arxiv.org/html/2606.05327#A7), demonstrating OTP\-FM’s flexibility to the choice of consistency model\.

## 3The dynamic OTP problem

![Refer to caption](https://arxiv.org/html/2606.05327v1/x2.png)Figure 2:Top:Exact solutions to the marginal dynamic OTP problem for 1D Gaussian marginals for varying potentials, strengths, andλk​\(t\)\\lambda\_\{k\}\(t\)\.Bottom:OTP\-FM solutions for the same marginals and potentials, except the rightmost plot, which demonstrates𝒟=𝒲2∞\\mathcal\{D\}=\\mathcal\{W\}\_\{2\}^\{\\infty\}\.We now generalize CFM to the multimarginal trajectory inference problem: given empirical marginals\{μtk\}k=0K\+1\\\{\\mu\_\{t\_\{k\}\}\\\}\_\{k=0\}^\{K\+1\}\(two endpoint andKKintermediate\) at timestk∈\[0,1\]t\_\{k\}\\in\[0,1\], we aim to learn a velocity fieldutθu^\{\\theta\}\_\{t\}whose flow induces densitiesρt\\rho\_\{t\}aligned withμtk\\mu\_\{t\_\{k\}\}while describing physically plausible interpolated trajectories, as measured by alignment to held\-out marginals \(Sec\.[6](https://arxiv.org/html/2606.05327#S6)\)\. We first observe that standard piecewise CFM targets can be recast as conditional solutions of dynamic OT withhardpenalty terms, or singular potentials \(Sec\.[3\.1](https://arxiv.org/html/2606.05327#S3.SS1)\)\. This motivates a smooth relaxation to what we call the dynamic OT \+ potentials \(OTP\) problem \(Sec\.[3\.2](https://arxiv.org/html/2606.05327#S3.SS2)\), in which the intermediate marginal constraints appear instead assoftpotential terms in the OT action\. Finally, we derive sample trajectories and discuss the design space of potentials \(Secs\.[3\.3](https://arxiv.org/html/2606.05327#S3.SS3)and[3\.4](https://arxiv.org/html/2606.05327#S3.SS4)\)\.

### 3\.1Piecewise CFM as OT with singular potentials

The standard multimarginal extension of CFM stitches conditional OT solutions piecewise between consecutive marginals \(e\.g\.Tonget al\.\([2024](https://arxiv.org/html/2606.05327#bib.bib3),[2023](https://arxiv.org/html/2606.05327#bib.bib69)\), see App\.[A\.2](https://arxiv.org/html/2606.05327#A1.SS2)\), whose target densities satisfyρtk=μtk\\rho\_\{t\_\{k\}\}=\\mu\_\{t\_\{k\}\}exactly at each intermediate time\. This is equivalent to imposinghard constraints\(HC\) on the conditional dynamic OT problem \(Eq\.[1](https://arxiv.org/html/2606.05327#S2.E1)\), to whose solutions we regress:

ℒHC:=minρt,ut​∫01∫ℝd\[12​ρt​‖ut‖2\+φt​\[∂tρt\+∇⋅\(ρt​ut\)\]\]​dx​dt,s\.t\.​ρtk=μtk​∀k\.\\hskip\-10\.0pt\\mathcal\{L\}\_\{\\mathrm\{HC\}\}:=\\min\_\{\\rho\_\{t\},u\_\{t\}\}\\int\_\{0\}^\{1\}\\\!\\\!\\int\_\{\\mathbb\{R\}^\{d\}\}\\\!\\big\[\\tfrac\{1\}\{2\}\\rho\_\{t\}\\\|u\_\{t\}\\\|^\{2\}\+\\varphi\_\{t\}\[\\partial\_\{t\}\\rho\_\{t\}\+\\nabla\\cdot\(\\rho\_\{t\}u\_\{t\}\)\]\\big\]\\mathrm\{d\}x\\mathrm\{d\}t,\\\\ \\text\{s\.t\.\}\\;\\;\\rho\_\{t\_\{k\}\}=\\mu\_\{t\_\{k\}\}\\;\\forall k\.\(8\)Because the kinetic\-energy integral is additive in time and the constraints fix the densities at eachtkt\_\{k\}, this problem decouples across consecutive intervals into independent dynamic OT problems betweenμtk\\mu\_\{t\_\{k\}\}andμtk\+1\\mu\_\{t\_\{k\+1\}\}\(Lemma[A\.3](https://arxiv.org/html/2606.05327#A1.Thmtheorem3)in App\.[A\.2](https://arxiv.org/html/2606.05327#A1.SS2)\), recovering piecewise CFM as its conditional solution \(Lemma[A\.4](https://arxiv.org/html/2606.05327#A1.Thmtheorem4)\)\.

Equivalently, these constraints can be enforced by adding singular penalties based on a metric statistical distance𝒟\\mathcal\{D\}such as𝒲22\\mathcal\{W\}^\{2\}\_\{2\}, time\-localized with Dirac deltas, and taking the penalty strengthwk→∞w\_\{k\}\\to\\infty:

ℒHC:=min∫01∫ℝd\[12ρt∥ut∥2\+∑k=1Kwk​δ​\(t−tk\)​𝒟tk​ρt⏞≡−VHC​\(x,t\)\+φt\[∂tρt\+∇⋅\(ρtut\)\]\]dxdt,\\hskip\-10\.0pt\\mathcal\{L\}\_\{\\mathrm\{HC\}\}:=\\min\\\!\\int\_\{0\}^\{1\}\\\!\\\!\\int\_\{\\mathbb\{R\}^\{d\}\}\\\!\\Big\[\\tfrac\{1\}\{2\}\\rho\_\{t\}\\\|u\_\{t\}\\\|^\{2\}\+\\overbrace\{\\sum\_\{k=1\}^\{K\}w\_\{k\}\\,\\delta\(t\-t\_\{k\}\)\\,\\mathcal\{D\}^\{k\}\_\{t\}\\rho\_\{t\}\}^\{\\equiv\-V\_\{\\mathrm\{HC\}\}\(x,t\)\}\\\\ \+\\varphi\_\{t\}\[\\partial\_\{t\}\\rho\_\{t\}\+\\nabla\\cdot\(\\rho\_\{t\}u\_\{t\}\)\]\\big\]\\mathrm\{d\}x\\mathrm\{d\}t,\(9\)where𝒟tk≡𝒟​\[ρt,μtk\]:𝒫​\(ℝd\)×𝒫​\(ℝd\)→ℝ\+\\mathcal\{D\}^\{k\}\_\{t\}\\equiv\\mathcal\{D\}\[\\rho\_\{t\},\\mu\_\{t\_\{k\}\}\]:\\mathcal\{P\}\(\\mathbb\{R\}^\{d\}\)\\times\\mathcal\{P\}\(\\mathbb\{R\}^\{d\}\)\\to\\mathbb\{R\}^\{\+\}is the statistical distance betweenρt\\rho\_\{t\}andμtk\\mu\_\{t\_\{k\}\}\. The limitwk→∞w\_\{k\}\\to\\inftyenforces𝒟tk=0⇔ρtk=μtk\\mathcal\{D\}^\{k\}\_\{t\}=0\\Leftrightarrow\\rho\_\{t\_\{k\}\}=\\mu\_\{t\_\{k\}\}\(since𝒟tk\\mathcal\{D\}^\{k\}\_\{t\}is a metric\), recovering the hard constraint \(Prop\.[3\.1](https://arxiv.org/html/2606.05327#S3.Thmtheorem1)\)\. Thus, we can interpret the penalties as singular potential terms in the action\.

### 3\.2The OTP problem: a smooth relaxation

This viewpoint suggests a natural relaxation to smooth the resultant unphysical kinks in piecewise CFM: takewk∈ℝw\_\{k\}\\in\\mathbb\{R\}finiteand soften the Dirac deltas to normalized temporal kernelsλk:\[0,1\]→ℝ\+\\lambda\_\{k\}:\[0,1\]\\to\\mathbb\{R\}^\{\+\}centered aroundtkt\_\{k\}with characteristic widthτ∈ℝ\+\\tau\\in\\mathbb\{R\}^\{\+\}\. The choice of statistical distance𝒟\\mathcal\{D\}remains free; in fact, in the soft setting we can further admitnon\-metric divergencessuch as the Kullback\-Leibler divergence \(KLD\), which sacrifice exact constraint enforcement in the hard limit but expand the design space \(Sec\.[3\.4](https://arxiv.org/html/2606.05327#S3.SS4)\)\. These generalizations yield the OTP variational problem:

ℒOTP:=min∫01∫ℝd\[12ρt∥ut∥2\+∑k=1Kwk​λk​\(t\)​𝒟tk​ρt⏞≡−VOTP​\(x,t\)\+φt\[∂tρt\+∇⋅\(ρtut\)\]\]dxdt,\\hskip\-10\.0pt\\mathcal\{L\}\_\{\\mathrm\{OTP\}\}:=\\min\\\!\\int\_\{0\}^\{1\}\\\!\\\!\\int\_\{\\mathbb\{R\}^\{d\}\}\\\!\\Big\[\\tfrac\{1\}\{2\}\\rho\_\{t\}\\\|u\_\{t\}\\\|^\{2\}\+\\overbrace\{\\sum\_\{k=1\}^\{K\}w\_\{k\}\\lambda\_\{k\}\(t\)\\mathcal\{D\}^\{k\}\_\{t\}\\rho\_\{t\}\}^\{\\equiv\-V\_\{\\mathrm\{OTP\}\}\(x,t\)\}\\\\ \+\\varphi\_\{t\}\[\\partial\_\{t\}\\rho\_\{t\}\+\\nabla\\cdot\(\\rho\_\{t\}u\_\{t\}\)\]\\Big\]\\mathrm\{d\}x\\mathrm\{d\}t,\(10\)subject to the boundary conditionsρ0=μ0\\rho\_\{0\}=\\mu\_\{0\},ρ1=μ1\\rho\_\{1\}=\\mu\_\{1\}\. Each termwk​λk​\(t\)​𝒟tkw\_\{k\}\\lambda\_\{k\}\(t\)\\,\\mathcal\{D\}^\{k\}\_\{t\}thus acts as a per\-marginal potential energy in the action: smoothed in time aroundtkt\_\{k\}byλk​\(t\)\\lambda\_\{k\}\(t\); of finite strengthwkw\_\{k\}; and shaped in density space by the choice of𝒟\\mathcal\{D\}\. Proposition[3\.1](https://arxiv.org/html/2606.05327#S3.Thmtheorem1)states that in the singular\-potential limit, the OTP problem recovers piecewise CFM and is thus a strict generalization; the precise statement and proof based onΓ\\Gamma\-convergence are provided in App\.[A\.2](https://arxiv.org/html/2606.05327#A1.SS2)\.

###### Proposition 3\.1\(OTP converges to the hard\-constrained problem in the hard\-potential limit, informal\)\.

In the singular limitwk→∞w\_\{k\}\\to\\infty,λk→δ​\(t−tk\)\\lambda\_\{k\}\\to\\delta\(t\-t\_\{k\}\)for allkk, the OTP problem \(Eq\.[10](https://arxiv.org/html/2606.05327#S3.E10)\) with𝒟=𝒲22\\mathcal\{D\}=\\mathcal\{W\}^\{2\}\_\{2\}converges to the hard\-constrained problem \(Eq\.[8](https://arxiv.org/html/2606.05327#S3.E8)\) and any sequence of OTP minimizers converges to its unique piecewise OT solution between consecutive marginals\.

### 3\.3Minimizers and sample trajectories

To further understand what we have gained with this relaxation, we derive the E\-L equations and corresponding sample trajectories of the OTP problem\.\{restatable\}theoremthmpotentialsolnel Minimizers of the OTP problem \(Eq\.[10](https://arxiv.org/html/2606.05327#S3.E10)\) satisfy the following E\-L equations:

ut​\(x\)\\displaystyle u\_\{t\}\(x\)=∇φt​\(x\),\\displaystyle=\\nabla\\varphi\_\{t\}\(x\),\(11\)∂tφt​\(x\)\+‖∇φt​\(x\)‖22\\displaystyle\\partial\_\{t\}\\varphi\_\{t\}\(x\)\+\\frac\{\\\|\\nabla\\varphi\_\{t\}\(x\)\\\|^\{2\}\}\{2\}=∑k=1Kwk​λk​\(t\)​gk​\(x,t\),\\displaystyle=\\sum\_\{k=1\}^\{K\}w\_\{k\}\\lambda\_\{k\}\(t\)g\_\{k\}\(x,t\),\(12\)along with the continuity equation, wheregk​\(x,t\)≡δ​𝒟tk/δ​ρt​\(x\)g\_\{k\}\(x,t\)\\equiv\\delta\\mathcal\{D\}^\{k\}\_\{t\}/\\delta\\rho\_\{t\}\(x\)is the functional derivative of𝒟tk\\mathcal\{D\}^\{k\}\_\{t\}w\.r\.t\.ρtk​\(x\)\\rho\_\{t\_\{k\}\}\(x\)\.\{restatable\}corollarycorpotentialsample Minimizers of the OTP problem \(Eq\.[10](https://arxiv.org/html/2606.05327#S3.E10)\) satisfy the following sample trajectories:

X¨t=∑k=1Kwk​λk​\(t\)​∇gk​\(Xt,t\)\.\\ddot\{X\}\_\{t\}=\\sum\_\{k=1\}^\{K\}w\_\{k\}\\lambda\_\{k\}\(t\)\\nabla g\_\{k\}\(X\_\{t\},t\)\.\(13\)Proofs are provided in App\.[A\.3](https://arxiv.org/html/2606.05327#A1.SS3)\. We can interpret∇gk​\(Xt,t\)\\nabla g\_\{k\}\(X\_\{t\},t\)as theforceapplied by the marginalμtk\\mu\_\{t\_\{k\}\}to each sample in analogy to Newton’s second law\. Clearly, it is an important object, and in general not trivial to compute; however, we show that it has convenient forms for common choices of𝒟\\mathcal\{D\}such as𝒲22\\mathcal\{W\}^\{2\}\_\{2\}, MMD, and KLD in Table[1](https://arxiv.org/html/2606.05327#S3.T1)\(derived in App\.[B](https://arxiv.org/html/2606.05327#A2)\)\. Solving these trajectories for themarginalproblem remains computationally challenging; however, theconditionalsolutions can be computed highly efficiently, as we show in Sec\.[4](https://arxiv.org/html/2606.05327#S4)\.

Table 1:Functional derivativesgk​\(x,t\)≡δ​𝒟tk/δ​ρt​\(x\)g\_\{k\}\(x,t\)\\equiv\\nicefrac\{\{\\delta\\mathcal\{D\}^\{k\}\_\{t\}\}\}\{\{\\delta\\rho\_\{t\}\(x\)\}\}and their gradients for different statistical distances𝒟tk≡𝒟​\[ρt,μtk\]\\mathcal\{D\}^\{k\}\_\{t\}\\equiv\\mathcal\{D\}\[\\rho\_\{t\},\\mu\_\{t\_\{k\}\}\]\.φ∗\\varphi^\{\*\}andψ∗\\psi^\{\*\}are the OT H\-J potential and map forc2c\_\{2\}cost, respectively\.
### 3\.4Design space of potentials

One case where wecansolve the marginal problem exactly is for isotropic Gaussian marginals \(App\.[D](https://arxiv.org/html/2606.05327#A4)\), illustrated in Fig\.[2](https://arxiv.org/html/2606.05327#S3.F2)for various configurations to visualize our rich design space along three axes: the strengthwkw\_\{k\}, the temporal kernelλk​\(t\)\\lambda\_\{k\}\(t\), and the statistical distance𝒟\\mathcal\{D\}\. Overall, we observe intuitive qualitative behavior: trajectories are driven towards \(or away from\) intermediates depending onwkw\_\{k\}, with the smoothness and temporal dynamics controlled byλk​\(t\)\\lambda\_\{k\}\(t\)\.333Counter\-intuitively, boundary conditions causerepulsiveforces \(negativeVV\) to drive trajectories towards intermediates; the sign convention in Eq\.[10](https://arxiv.org/html/2606.05327#S3.E10)ensures positivewkw\_\{k\}corresponds to “attractive” dynamics\.We discuss practical and performance considerations regarding this design space in the next two sections\.

## 4The OTP\-FM algorithm

We now leverage the OTP dynamics to formulate a practical training algorithm by deriving a CFM\-style regression objective\. We first construct conditional OTP\-FM trajectories—conditioned on endpoint and intermediate\-marginal samples—and obtain an explicit consistency\-model loss \(Sec\.[4\.1](https://arxiv.org/html/2606.05327#S4.SS1)\)\. We then derive the regression targets for different statistical distances \(Sec\.[4\.2](https://arxiv.org/html/2606.05327#S4.SS2)\), identifying𝒲22/∞\\mathcal\{W\}\_\{2\}^\{2/\\infty\}—with either OT or independent couplings—as a particularly clean, efficient, and stable choice that we recommend in practice\. Algorithmic details on the loss and curriculum follow in Sec\.[4\.3](https://arxiv.org/html/2606.05327#S4.SS3), and we conclude with theoretical bounds on the alignment between learned and ground\-truth marginals—controlled by the potential strengths and training loss—in Sec\.[4\.4](https://arxiv.org/html/2606.05327#S4.SS4)\.

### 4\.1Conditional OTP\-FM and training objective

#### Temporal localization\.

Since∇gk​\(Xt,t\)\\nabla g\_\{k\}\(X\_\{t\},t\)in Eq\.[13](https://arxiv.org/html/2606.05327#S3.E13)depends onρt\\rho\_\{t\}andXtX\_\{t\}throughout the support ofλk​\(t\)\\lambda\_\{k\}\(t\), using these dynamics directly as a regression target would require simulating the trajectory at every training step\. Instead, we can exploit the temporal localization ofλk​\(t\)\\lambda\_\{k\}\(t\): on its effective support of widthτ\\tauaroundtkt\_\{k\},Xt−XtkX\_\{t\}\-X\_\{t\_\{k\}\}andρt−ρtk\\rho\_\{t\}\-\\rho\_\{t\_\{k\}\}are both𝒪​\(τ\)\\mathcal\{O\}\(\\tau\)\(assuming bounded velocity\), so∇gk​\(Xt,t\)=∇gk​\(Xtk,tk\)\+𝒪​\(τ\)\\nabla g\_\{k\}\(X\_\{t\},t\)=\\nabla g\_\{k\}\(X\_\{t\_\{k\}\},t\_\{k\}\)\+\\mathcal\{O\}\(\\tau\)\. Substituting this into Eq\.[13](https://arxiv.org/html/2606.05327#S3.E13)yields theOTP\-FM dynamics:

X¨t=∑k=1Kwk​λk​\(t\)​∇gk​\(Xtk,tk\)\.\\ddot\{X\}\_\{t\}=\\sum\_\{k=1\}^\{K\}w\_\{k\}\\lambda\_\{k\}\(t\)\\nabla g\_\{k\}\(X\_\{t\_\{k\}\},t\_\{k\}\)\.\(14\)As the force∇gk​\(Xt,t\)\\nabla g\_\{k\}\(X\_\{t\},t\)now depends only on the trajectory at the discrete marginal times\{tk\}\\\{t\_\{k\}\\\}, we are able to formulate the fullysimulation\-freetraining objective below\. Furthermore, we observe qualitatively identical behavior to the exact OTP solutions \(Fig\.[2](https://arxiv.org/html/2606.05327#S3.F2)\), and retain the important theoretical bounds \(Sec\.[4\.4](https://arxiv.org/html/2606.05327#S4.SS4)\) with this approximation\.

#### Conditional solutions\.

Integrating Eq\.[14](https://arxiv.org/html/2606.05327#S4.E14)once and twice yields the velocity and trajectory:

ut​\(Xt\)=\\displaystyle\\vskip\-10\.00002ptu\_\{t\}\(X\_\{t\}\)=ui\+∑k=1Kwk​∇gk​\(Xtk,tk\)​ℐ​\[λk\]​\(t\),\\displaystyle u\_\{i\}\+\\sum\_\{k=1\}^\{K\}w\_\{k\}\\nabla g\_\{k\}\(X\_\{t\_\{k\}\},t\_\{k\}\)\\mathcal\{I\}\\\!\\left\[\\lambda\_\{k\}\\right\]\\\!\(t\),\(15\)Xt=X0\+ui​t\\displaystyle X\_\{t\}\\\!=\\\!X\_\{0\}\+u\_\{i\}t\+∑k=1Kwk​∇gk​\(Xtk,tk\)​ℐ\(2\)​\[λk\]​\(t\),\\displaystyle\+\\\!\\sum\_\{k=1\}^\{K\}w\_\{k\}\\nabla g\_\{k\}\(X\_\{t\_\{k\}\},t\_\{k\}\)\\mathcal\{I\}^\{\(2\)\}\\\!\\left\[\\lambda\_\{k\}\\right\]\\\!\(t\),\(16\)whereℐ\(n\)​\[f\]​\(t\)\\mathcal\{I\}^\{\(n\)\}\\\!\\left\[f\\right\]\(t\)denotes thenn\-th time integral from0tottandui∈ℝdu\_\{i\}\\in\\mathbb\{R\}^\{d\}is an integration constant fixed by the boundary conditions\. As in CFM, we obtain tractable regression targets by conditioning on endpoints\(x0,x1\)\(x\_\{0\},x\_\{1\}\), as well as, in general, a minibatchℬ\\mathcal\{B\}of samples drawn from a joint distributionπall\\pi\_\{\\mathrm\{all\}\}over all marginals\. ImposingX0=x0X\_\{0\}=x\_\{0\}andX1=x1X\_\{1\}=x\_\{1\}pins downuiu\_\{i\}, and the resulting conditional trajectory and velocity decompose into a CFM base term plus potential\-driven corrections:

ut​\(Xt\|x0,x1,ℬ\)=x1−x0⏟≡ubase\+∑k=1Kwk​∇gk​\(Xtk,tk,ℬ\)​\[ℐ​\[λk\]​\(t\)−ℐ\(2\)​\[λk\]​\(1\)\]⏟≡uk,tcorr\.u\_\{t\}\(X\_\{t\}\|x\_\{0\},x\_\{1\},\\mathcal\{B\}\)=\\underbrace\{x\_\{1\}\-x\_\{0\}\}\_\{\\equiv u^\{\\mathrm\{base\}\}\}\+\\\\ \\sum\_\{k=1\}^\{K\}\\underbrace\{w\_\{k\}\\nabla g\_\{k\}\(X\_\{t\_\{k\}\},t\_\{k\},\\mathcal\{B\}\)\\left\[\\mathcal\{I\}\\\!\\left\[\\lambda\_\{k\}\\right\]\(t\)\-\\mathcal\{I\}^\{\(2\)\}\\\!\\left\[\\lambda\_\{k\}\\right\]\(1\)\\right\]\}\_\{\\equiv u^\{\\mathrm\{corr\}\}\_\{k,t\}\}\.\(17\)Xt​\(x0,x1,ℬ\)=x0\+\(x1−x0\)​t⏟≡Xtbase\+∑k=1Kwk​∇gk​\(Xtk,tk,ℬ\)​\[ℐ\(2\)​\[λk\]​\(t\)−ℐ\(2\)​\[λk\]​\(1\)​t\]⏟≡Xk,tcorr,X\_\{t\}\(x\_\{0\},x\_\{1\},\\mathcal\{B\}\)=\\underbrace\{x\_\{0\}\+\(x\_\{1\}\-x\_\{0\}\)t\}\_\{\\equiv X\_\{t\}^\{\\mathrm\{base\}\}\}\+\\\\ \\sum\_\{k=1\}^\{K\}\\underbrace\{w\_\{k\}\\nabla g\_\{k\}\(X\_\{t\_\{k\}\},t\_\{k\},\\mathcal\{B\}\)\\left\[\\mathcal\{I\}^\{\(2\)\}\\\!\\left\[\\lambda\_\{k\}\\right\]\(t\)\-\\mathcal\{I\}^\{\(2\)\}\\\!\\left\[\\lambda\_\{k\}\\right\]\(1\)\\,t\\right\]\}\_\{\\equiv X^\{\\mathrm\{corr\}\}\_\{k,t\}\},\(18\)The base terms recover the CFM straight\-line path and velocity \(Corollary[A\.2](https://arxiv.org/html/2606.05327#A1.Thmtheorem2)\), while each correctionXk,tcorr,uk,tcorrX^\{\\mathrm\{corr\}\}\_\{k,t\},u^\{\\mathrm\{corr\}\}\_\{k,t\}captures the impulse from thekk\-th potential\.

#### The self\-consistent fixed\-point problem

Note that Eq\.[18](https://arxiv.org/html/2606.05327#S4.E18)is implicit in\{Xtk\}\\\{X\_\{t\_\{k\}\}\\\}: each∇gk​\(Xtk,tk,ℬ\)\\nabla g\_\{k\}\(X\_\{t\_\{k\}\},t\_\{k\},\\mathcal\{B\}\)is evaluated atXtkX\_\{t\_\{k\}\}, which itself depends on the trajectory through∇gk​\(Xtk,tk,ℬ\)\\nabla g\_\{k\}\(X\_\{t\_\{k\}\},t\_\{k\},\\mathcal\{B\}\)\. Solving for\{Xtk\}\\\{X\_\{t\_\{k\}\}\\\}that are self\-consistent with the trajectory they induce is therefore a fixed\-point problem, whose structure and solver depend on the choice of distance𝒟\\mathcal\{D\}and how its force is estimated, as discussed next in Sec\.[4\.2](https://arxiv.org/html/2606.05327#S4.SS2)\.

#### The OTP\-FM training objective

We train a consistency modelvt1,t2θv^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\(Sec\.[2\.3](https://arxiv.org/html/2606.05327#S2.SS3)\) by regressing it onto the conditional OTP\-FM velocityut​\(x\|z\)u\_\{t\}\(x\|z\)\(Eq\.[17](https://arxiv.org/html/2606.05327#S4.E17)\)\. Using the iMF formulation \(Eq\.[7](https://arxiv.org/html/2606.05327#S2.E7)\), to be explicit, we obtain the loss:

ℒOTP−FM\(θ\):=𝔼t1,t2,z,x∥Vt1,t2θ\(x\)−ut\(x\|z\)∥2,\\mathcal\{L\}\_\{\\mathrm\{OTP\-FM\}\}\(\\theta\):=\\mathbb\{E\}\_\{t\_\{1\},t\_\{2\},z,x\}\\\|V^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\(x\)\-u\_\{t\}\(x\|z\)\\\|^\{2\},\\\!\\\!\\\!\(19\)wherez=\(x0,x1,ℬ\)z=\(x\_\{0\},x\_\{1\},\\mathcal\{B\}\)andVt1,t2θ​\(x\)V^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\(x\)is the parametrization ofvθv^\{\\theta\}defined in Eq\.[7](https://arxiv.org/html/2606.05327#S2.E7)\. We reiterate, however, that OTP\-FM is agnostic to the choice of consistency model; we ablate alternative objectives in App\.[C](https://arxiv.org/html/2606.05327#A3)\.

### 4\.2Computing forces and fixed\-points

Eq\.[19](https://arxiv.org/html/2606.05327#S4.E19)introduced the general conditioning variablez=\(x0,x1,ℬ\)z=\(x\_\{0\},x\_\{1\},\\mathcal\{B\}\)withℬ\\mathcal\{B\}a minibatch of sizeMMfrom the joint couplingπall\\pi\_\{\\mathrm\{all\}\}, from which the force estimator∇gk​\(Xtk,tk,ℬ\)\\nabla g\_\{k\}\(X\_\{t\_\{k\}\},t\_\{k\},\\mathcal\{B\}\)is built\. For the𝒲22\\mathcal\{W\}^\{2\}\_\{2\}potential,∇gk​\(x\)=x−ψ∗​\(x\)\\nabla g\_\{k\}\(x\)=x\-\\psi^\{\*\}\(x\)is well\-defined pointwise, so*ℬ\\mathcal\{B\}reduces to a single sample per marginal*and the estimator is linear—together yielding the closed\-form fixed point and a clean training signal\. MMD and KLD, by contrast, are functionals of the full densityρtk\\rho\_\{t\_\{k\}\}and degenerate on Diracs, so their estimators require allMMsamples and yield nonlinear fixed points \(Sec\.[6\.2](https://arxiv.org/html/2606.05327#S6.SS2)\)\.

#### The𝒲22\\mathcal\{W\}^\{2\}\_\{2\}distance

With a single samplextk∼μtkx\_\{t\_\{k\}\}\\sim\\mu\_\{t\_\{k\}\}per marginal, the conditional OT map collapses toψ∗​\(Xtk\|xtk\)=xtk\\psi^\{\*\}\(X\_\{t\_\{k\}\}\|x\_\{t\_\{k\}\}\)=x\_\{t\_\{k\}\}, yielding the one\-sample estimator∇gk​\(Xtk,tk,ℬ\)=Xtk−xtk\\nabla g\_\{k\}\(X\_\{t\_\{k\}\},t\_\{k\},\\mathcal\{B\}\)=X\_\{t\_\{k\}\}\-x\_\{t\_\{k\}\}at𝒪​\(M\)\\mathcal\{O\}\(M\)cost per training step\. Substituted into Eq\.[18](https://arxiv.org/html/2606.05327#S4.E18)att∈\{t1,…,tK\}t\\in\\\{t\_\{1\},\\dots,t\_\{K\}\\\}, this yields a linear system inXT≡\[Xt1,…,XtK\]⊤X\_\{T\}\\equiv\[X\_\{t\_\{1\}\},\\dots,X\_\{t\_\{K\}\}\]^\{\\top\}with the direct closed\-form solution

XTFP=\(𝟙−A\)−1​\(XTbase−A​xT\),Ai​k≡wk​\[ℐ\(2\)​\[λk\]​\(ti\)−ℐ\(2\)​\[λk\]​\(1\)​ti\],X\_\{T\}^\{\\mathrm\{FP\}\}=\(\\mathbbm\{1\}\-A\)^\{\-1\}\\\!\\left\(X^\{\\mathrm\{base\}\}\_\{T\}\-A\\,x\_\{T\}\\right\),\\\\ A\_\{ik\}\\equiv w\_\{k\}\\\!\\left\[\\mathcal\{I\}^\{\(2\)\}\\\!\\left\[\\lambda\_\{k\}\\right\]\(t\_\{i\}\)\-\\mathcal\{I\}^\{\(2\)\}\\\!\\left\[\\lambda\_\{k\}\\right\]\(1\)\\,t\_\{i\}\\right\],\(20\)wherexT≡\[xt1,…,xtK\]⊤x\_\{T\}\\equiv\[x\_\{t\_\{1\}\},\\dots,x\_\{t\_\{K\}\}\]^\{\\top\}and theK×KK\\times Kinverse\(𝟙−A\)−1\(\\mathbbm\{1\}\-A\)^\{\-1\}depends only on the time grid and kernel and is precomputed once per training \(further details in App\.[E\.3](https://arxiv.org/html/2606.05327#A5.SS3)\)\. The remaining design choice is the joint couplingπall\\pi\_\{\\mathrm\{all\}\}across times; we useπallOT≡∏i=0Kπti,ti\+1OT\\pi^\{\\mathrm\{OT\}\}\_\{\\mathrm\{all\}\}\\equiv\\prod\_\{i=0\}^\{K\}\\pi^\{\\mathrm\{OT\}\}\_\{t\_\{i\},t\_\{i\+1\}\}, the product of OT couplings between consecutive marginals, precomputed once across the dataset\. This is closest in spirit to the𝒲22\\mathcal\{W\}^\{2\}\_\{2\}distance on the marginal problem\. We could alternatively condition on the full minibatchℬ\\mathcal\{B\}and estimateψ∗\\psi^\{\*\}via minibatch OT\(Fatraset al\.,[2020](https://arxiv.org/html/2606.05327#bib.bib37)\)at each step, but find the per\-iteration overhead impractical relative to this precomputed variant\.

#### The𝒲2∞\\mathcal\{W\}\_\{2\}^\{\\infty\}distance

While precomputing the OT map has been possible for all experiments in this work, we propose replacingπallOT\\pi^\{\\mathrm\{OT\}\}\_\{\\mathrm\{all\}\}with the independent couplingπallind\\pi^\{\\mathrm\{ind\}\}\_\{\\mathrm\{all\}\}for applications to larger\-scale, higher\-dimensional datasets where the exact OT coupling may not be practical\. This is akin to couplings based on entropically regularized OT,𝒲2ε\\mathcal\{W\}\_\{2\}^\{\\varepsilon\}\(Cuturi,[2013](https://arxiv.org/html/2606.05327#bib.bib36)\), in the limit of the regularization parameterε→∞\\varepsilon\\to\\infty; hence, we refer to this as the𝒲2∞\\mathcal\{W\}\_\{2\}^\{\\infty\}potential\. Eq\.[20](https://arxiv.org/html/2606.05327#S4.E20)applies unchanged; onlyxTx\_\{T\}on the RHS is now drawn fromπallind\\pi^\{\\mathrm\{ind\}\}\_\{\\mathrm\{all\}\}rather thanπallOT\\pi^\{\\mathrm\{OT\}\}\_\{\\mathrm\{all\}\}\. The resulting estimator∇gk​\(Xtk,tk,ℬ\)=Xtk−xtk\\nabla g\_\{k\}\(X\_\{t\_\{k\}\},t\_\{k\},\\mathcal\{B\}\)=X\_\{t\_\{k\}\}\-x\_\{t\_\{k\}\}paired withπallind\\pi^\{\\mathrm\{ind\}\}\_\{\\mathrm\{all\}\}likewise costs𝒪​\(M\)\\mathcal\{O\}\(M\)per training step and is hence extremely efficient; furthermore, it does not require the precomputation of the OT map across all marginals\.

#### Beyond𝒲22\\mathcal\{W\}^\{2\}\_\{2\}: MMD and KLD potentials

As exploratory generalizations of the design space, we also consider MMD and KLD potentials, with corresponding forces∇gk\\nabla g\_\{k\}derived in Table[1](https://arxiv.org/html/2606.05327#S3.T1)\. The MMD admits a natural𝒪​\(M2\)\\mathcal\{O\}\(M^\{2\}\)empirical estimator using a tunable RBF kernel, while the KLD requires score estimators forρtk\\rho\_\{t\_\{k\}\}andμtk\\mu\_\{t\_\{k\}\}, for which we explore an𝒪​\(M\)\\mathcal\{O\}\(M\)Gaussian fit and an𝒪​\(M2\)\\mathcal\{O\}\(M^\{2\}\)RBF kernel\-density estimate \(App\.[B\.2](https://arxiv.org/html/2606.05327#A2.SS2)\)\. Both yield*nonlinear*fixed\-point systems inXtkX\_\{t\_\{k\}\}and noisier finite\-sample gradients than the linear𝒲22\\mathcal\{W\}^\{2\}\_\{2\}estimator, requiring≤5\\leq 5Picard iterations with damped Anderson acceleration\(Anderson,[1965](https://arxiv.org/html/2606.05327#bib.bib25)\)per training step and proving less performant overall \(Sec\.[6\.2](https://arxiv.org/html/2606.05327#S6.SS2)\)\.

### 4\.3Algorithmic details

Algorithm 1Theiith training iteration of OTP\-FM1:Input:

vθ​\(x,t1,t2\)v^\{\\theta\}\(x,t\_\{1\},t\_\{2\}\): model;

MM: batch size;

α​\(i\)\\alpha\(i\): curriculum parameter;

2:Sample

ℬ∼πall\\mathcal\{B\}\\sim\\pi\_\{\\mathrm\{all\}\}\# shape:M×\(K\+2\)×DM\\times\(K\+2\)\\times D

3:Sample

\(t1,t2\)∼pt1,t2\(t\_\{1\},t\_\{2\}\)\\sim p\_\{t\_\{1\},t\_\{2\}\}\# Eq\.[114](https://arxiv.org/html/2606.05327#A5.E114)

4:

x0,x1←ℬ​\[:,0\],ℬ​\[:,−1\]x\_\{0\},x\_\{1\}\\leftarrow\\mathcal\{B\}\[:,0\],\\mathcal\{B\}\[:,\-1\]
5:

ubase←x1−x0u^\{\\mathrm\{base\}\}\\leftarrow x\_\{1\}\-x\_\{0\}
6:Compute fixed points

XtkFP​\(ℬ\)X\_\{t\_\{k\}\}^\{\\mathrm\{FP\}\}\(\\mathcal\{B\}\)\(Eq\.[20](https://arxiv.org/html/2606.05327#S4.E20)for

𝒲22\\mathcal\{W\}^\{2\}\_\{2\}or FP iterations for MMD / KLD\)

7:Compute

uk,t1corru^\{\\mathrm\{corr\}\}\_\{k,t\_\{1\}\}and

Xk,t1corrX^\{\\mathrm\{corr\}\}\_\{k,t\_\{1\}\}using Eqs\.[17](https://arxiv.org/html/2606.05327#S4.E17),[18](https://arxiv.org/html/2606.05327#S4.E18)and

XtkFPX\_\{t\_\{k\}\}^\{\\mathrm\{FP\}\}
8:\#α\\alpha\-weighted Eqs\.[17](https://arxiv.org/html/2606.05327#S4.E17)and[18](https://arxiv.org/html/2606.05327#S4.E18)

9:

ut1←ubase\+α​\(i\)​∑k=1Kuk,t1corru\_\{t\_\{1\}\}\\leftarrow u^\{\\mathrm\{base\}\}\+\\alpha\(i\)\\sum\_\{k=1\}^\{K\}u^\{\\mathrm\{corr\}\}\_\{k,t\_\{1\}\}\#M×DM\\times D

10:

Xt1←x0\+\(x1−x0\)​t1\+α​\(i\)​∑k=1KXk,t1corrX\_\{t\_\{1\}\}\\leftarrow x\_\{0\}\+\(x\_\{1\}\-x\_\{0\}\)t\_\{1\}\+\\alpha\(i\)\\sum\_\{k=1\}^\{K\}X^\{\\mathrm\{corr\}\}\_\{k,t\_\{1\}\}
11:\# MeanFlow target \(Eq\.[19](https://arxiv.org/html/2606.05327#S4.E19)\), computed usingjvp

12:

vtgt←ut1\+\(t2−t1\)​\[ut1​∂x\+∂t1\]​vθ​\(Xt1,t1,t2\)v\_\{\\mathrm\{tgt\}\}\\leftarrow u\_\{t\_\{1\}\}\+\(t\_\{2\}\-t\_\{1\}\)\\left\[u\_\{t\_\{1\}\}\\partial\_\{x\}\+\\partial\_\{t\_\{1\}\}\\right\]v^\{\\theta\}\(X\_\{t\_\{1\}\},t\_\{1\},t\_\{2\}\)
13:Compute loss

ℒOTP−FM\\mathcal\{L\}\_\{\\mathrm\{OTP\-FM\}\}\(

vθ​\(Xt1,t1,t2\),sg​\(vtgt\)v^\{\\theta\}\(X\_\{t\_\{1\}\},t\_\{1\},t\_\{2\}\),\\mathrm\{sg\}\(v\_\{\\mathrm\{tgt\}\}\)\)

14:Update

vθv^\{\\theta\}using

∇ℒOTP−FM\\nabla\\mathcal\{L\}\_\{\\mathrm\{OTP\-FM\}\}

#### Consistency training

We use a consistency model such as iMF\(Genget al\.,[2026b](https://arxiv.org/html/2606.05327#bib.bib76)\)for few\-step inference\. This is crucial for efficient training of the MMD/KLD potentials, which require evaluation of the learnedXtkX\_\{t\_\{k\}\}during training to compute the forces∇gk​\(Xtk,tk,ℬ\)\\nabla g\_\{k\}\(X\_\{t\_\{k\}\},t\_\{k\},\\mathcal\{B\}\)and perform FP iterations \(Sec\.[4\.2](https://arxiv.org/html/2606.05327#S4.SS2)\)\. While not necessary for the𝒲22\\mathcal\{W\}^\{2\}\_\{2\}potential, it nevertheless enables efficient inference\. As is common for image generation, we additionally maintain an exponential moving average \(EMA\) of the model weightsuθEMAu^\{\\mathrm\{EMA\}\}\_\{\\theta\}for both inference and evaluation ofXtkX\_\{t\_\{k\}\}during training\.

#### Loss function

We experiment with three weightings of the squared\-L2 regression target \(Eq\.[19](https://arxiv.org/html/2606.05327#S4.E19)\): 1\) unweighted MSE; 2\) the adaptive weighting of MeanFlow, replacing‖Δ‖22\\\|\\Delta\\\|^\{2\}\_\{2\}with‖Δ‖22/sg​\(‖Δ‖22\+c\)p\\\|\\Delta\\\|^\{2\}\_\{2\}/\\mathrm\{sg\}\(\\\|\\Delta\\\|^\{2\}\_\{2\}\+c\)^\{p\}, whereΔ\\Deltais the regression residual,cca small constant, andppa tunable exponent; and 3\) a learnt log\-variance weighting\(Karraset al\.,[2024](https://arxiv.org/html/2606.05327#bib.bib40)\)\. We find the adaptive weighting withp=1p=1generally the most performant; comparisons are shown in Apps\.[F](https://arxiv.org/html/2606.05327#A6)and[G](https://arxiv.org/html/2606.05327#A7)\.

#### Homotopy curriculum

As the fixed points of Eq\.[18](https://arxiv.org/html/2606.05327#S4.E18)are not guaranteed to be attractive for arbitrary initializations, we find it beneficial to introduce a training curriculum that scales the correction terms by a smoothly increasing parameterα​\(i\)∈\[0,1\]\\alpha\(i\)\\in\[0,1\]over training iterationsii, rewriting Eq\.[18](https://arxiv.org/html/2606.05327#S4.E18)as a family of fixed\-point maps:

Xt=Fα​\(Xt\)≡Xtbase\+α​∑k=1KXk,tcorr,∀t∈\{tk\}\.X\_\{t\}=F\_\{\\alpha\}\(X\_\{t\}\)\\equiv X\_\{t\}^\{\\mathrm\{base\}\}\+\\alpha\\sum\_\{k=1\}^\{K\}X^\{\\mathrm\{corr\}\}\_\{k,t\},\\quad\\forall t\\in\\\{t\_\{k\}\\\}\.\(21\)This is an application of thehomotopy continuationtechnique\(Allgower and Georg,[2003](https://arxiv.org/html/2606.05327#bib.bib24)\): we start by allowing the model to learn atα≈0\\alpha\\approx 0, where the problem reduces to CFM with a trivially attractive fixed point, and gradually transition to the full OTP\-FM dynamics atα=1\\alpha=1\. Under standard regularity conditions, the implicit function theorem guarantees the existence of a locally unique and smooth solution branchα↦XtkFP​\(α\)\\alpha\\mapsto X\_\{t\_\{k\}\}^\{\\mathrm\{FP\}\}\(\\alpha\), and increasingα\\alphagradually allows the training dynamics to track this branch \(App\.[E\.2](https://arxiv.org/html/2606.05327#A5.SS2)\)\. In practice, we use a sigmoid schedule with tunable mean and slope, with ablations in Sec\.[6\.2](https://arxiv.org/html/2606.05327#S6.SS2)\. Interestingly, we find this more performant even for the𝒲22\\mathcal\{W\}^\{2\}\_\{2\}potentials, for which we have a closed\-form solution\.

### 4\.4Theoretical bounds

As in standard CFM,ℒOTP−FM\\mathcal\{L\}\_\{\\mathrm\{OTP\-FM\}\}is gradient\-equivalent to a marginal FM loss regressingvθv^\{\\theta\}onto the induced marginal velocityut​\(x\)=∫ut​\(x\|z\)​ρt​\(z\|x\)​dzu\_\{t\}\(x\)=\\int u\_\{t\}\(x\|z\)\\,\\rho\_\{t\}\(z\|x\)\\,\\mathrm\{d\}zalongρt​\(x\)=∫ρt​\(x\|z\)​q​\(z\)​dz\\rho\_\{t\}\(x\)=\\int\\rho\_\{t\}\(x\|z\)\\,q\(z\)\\,\\mathrm\{d\}z\.444This follows by direct application ofTonget al\.\([2024](https://arxiv.org/html/2606.05327#bib.bib3), Thms\. 3\.1–3\.2\)with the OTP\-FM conditioning variablesz=\(x0,x1,ℬ\)∼qz=\(x\_\{0\},x\_\{1\},\\mathcal\{B\}\)\\sim q\.The following two propositions show that OTP\-FM with the𝒲22\\mathcal\{W\}^\{2\}\_\{2\}potential*guarantees alignment of this inducedρt\\rho\_\{t\}and the ground\-truth intermediate marginalsμtk\\mu\_\{t\_\{k\}\}*, with the𝒲2\\mathcal\{W\}\_\{2\}bound between them*controlled by the potential strengthwwand training lossℒOTP−FM\\mathcal\{L\}\_\{\\mathrm\{OTP\-FM\}\}*\. Proofs are provided in App\.[A\.4](https://arxiv.org/html/2606.05327#A1.SS4)\.

\{restatable\}

propositionpropwbound\(Marginal alignment bound\)\.For𝒲22\\mathcal\{W\}^\{2\}\_\{2\}\-type potentials of strengthww, away from singular configurations:𝒲22​\(ρtk,μtk\)≤Ck/w2\+𝒪​\(w−3\)\\mathcal\{W\}\_\{2\}^\{2\}\(\\rho\_\{t\_\{k\}\},\\mu\_\{t\_\{k\}\}\)\\leq C\_\{k\}/w^\{2\}\+\\mathcal\{O\}\(w^\{\-3\}\), whereCkC\_\{k\}is a problem\-dependent constant independent ofww\.

\{restatable\}

propositionpropendtoend\(End\-to\-end bound\)\.For Eulerian and Lagrangian consistency losses \(App\.[C](https://arxiv.org/html/2606.05327#A3)\), combining Prop\.[4\.4](https://arxiv.org/html/2606.05327#S4.SS4)with flow\-map learning bounds fromBoffiet al\.\([2025](https://arxiv.org/html/2606.05327#bib.bib16)\):𝒲2​\(ρtkθ,μtk\)≤Dk​ℒOTP−FM\+Ck/w\+𝒪​\(w−3/2\)\\mathcal\{W\}\_\{2\}\(\\rho^\{\\theta\}\_\{t\_\{k\}\},\\mu\_\{t\_\{k\}\}\)\\leq D\_\{k\}\\sqrt\{\\mathcal\{L\}\_\{\\mathrm\{OTP\-FM\}\}\}\+\\sqrt\{C\_\{k\}\}/w\+\\mathcal\{O\}\(w^\{\-3/2\}\), whereCkC\_\{k\}andDkD\_\{k\}are problem\- and consistency\-loss\-dependent constants, respectively, both independent ofww\.

## 5Related work

Like OTP\-FM, previous work in trajectory inference has focused on modeling dynamics through neural ordinary or stochastic differential equations \(ODEs or SDEs\)\(Chenet al\.,[2018](https://arxiv.org/html/2606.05327#bib.bib9)\)\. Originally, these methods were trained by repeatedly simulating the ODE or SDE, with examples such as TrajectoryNet\(Tonget al\.,[2020](https://arxiv.org/html/2606.05327#bib.bib46)\), MIOFlow\(Huguetet al\.,[2022](https://arxiv.org/html/2606.05327#bib.bib45)\), and Schrödinger\-bridge methods like DMSB\(Chenet al\.,[2023](https://arxiv.org/html/2606.05327#bib.bib48)\), NLSB\(Koshizuka and Sato,[2023](https://arxiv.org/html/2606.05327#bib.bib49)\), and DeepRUOT\(Zhanget al\.,[2025](https://arxiv.org/html/2606.05327#bib.bib70)\)\. These, while flexible, generally prove computationally prohibitive for large\-scale, high\-dimensional data\.

More recently, flow\- and score\-matching methods successfully pioneered*simulation\-free*training as described above\. However, most focus only on trajectories between two endpoint marginals, simply stitching trajectories piecewise in the case of multiple marginals, as in I\-/OT\-CFM and \[SF\]2M\(Tonget al\.,[2024](https://arxiv.org/html/2606.05327#bib.bib3),[2023](https://arxiv.org/html/2606.05327#bib.bib69)\)\. MMFM\(Rohbecket al\.,[2025](https://arxiv.org/html/2606.05327#bib.bib54)\)and 3MSBM\(Theodoropouloset al\.,[2026](https://arxiv.org/html/2606.05327#bib.bib55)\)are two recent methods that smooth these trajectories, through cubic spline interpolation and lifting to phase space, respectively\. However, as evidenced by our experiments below, these prescribed smoothing strategies need not describe the dynamics of the physical system\. In contrast, OTP\-FM provides a principled framework for deriving smooth, flexible dynamics that lets the data, rather than an ad\-hoc interpolation rule, determine the trajectories through optimization over the broad design space\. New methods such as MFM\(Kapusniaket al\.,[2024](https://arxiv.org/html/2606.05327#bib.bib65)\)and VGFM\(Wanget al\.,[2026](https://arxiv.org/html/2606.05327#bib.bib68)\)represent orthogonal improvements to CFM—learning the data manifold and modeling unbalanced cell growth, respectively—that can be combined with our approach\.

Finally, methods such as WLF\(Neklyudovet al\.,[2024](https://arxiv.org/html/2606.05327#bib.bib66)\), JKOnet\(\*\)\(Bunneet al\.,[2022](https://arxiv.org/html/2606.05327#bib.bib73); Terpinet al\.,[2024](https://arxiv.org/html/2606.05327#bib.bib77)\), and iJKOnet\(Persiianovet al\.,[2026](https://arxiv.org/html/2606.05327#bib.bib67)\)directly solve a multimarginal OT\-like variational problem over the entire dataset, and are promising but fundamentally different computational approaches to solving theconditionalOT problem within the CFM framework, as in OTP\-FM\. We benchmark against all of the above methods\.

## 6Experiments

Table 2:Ablation of OTP\-FM design choices on EB 100D L2O \(MMD¯\\overline\{\\text\{MMD\}\}on held\-out times, lower is better\)\. The optimal value is inbold\.
### 6\.1Synthetic data

We first validate OTP\-FM on 1D Gaussian marginals where exact dynamic OT solutions are available \(App\.[D](https://arxiv.org/html/2606.05327#A4)\)\. Figure[2](https://arxiv.org/html/2606.05327#S3.F2)illustrates that OTP\-FM faithfully reproduces the qualitative behavior across potential types and configurations\. Empirically,𝒲22\\mathcal\{W\}^\{2\}\_\{2\}with theπallOT\\pi^\{\\mathrm\{OT\}\}\_\{\\mathrm\{all\}\}coupling is the most performant and stable to train, with𝒲2∞\\mathcal\{W\}\_\{2\}^\{\\infty\}andπallind\\pi^\{\\mathrm\{ind\}\}\_\{\\mathrm\{all\}\}comparably effective while avoiding the OT precomputation overhead; the kernel\-based MMD and KLD potentials, by contrast, are less stable due to their nonlinear fixed\-point problem and noisier gradient estimates \(further details in App\.[F](https://arxiv.org/html/2606.05327#A6)\)\.

### 6\.2Ablation studies

We ablate the key design choices of OTP\-FM on the EB 100D L2O benchmark \(described below\) in Table[2](https://arxiv.org/html/2606.05327#S6.T2)to show sensitivity to 1\) potential type, 2\) potential strength, 3\)λk​\(t\)\\lambda\_\{k\}\(t\)width and shape, and 4\) curriculum\. We vary one parameter at a time around the optimum \(bold\) and use an 8\-layer MLP ResNet architecture forvθv^\{\\theta\}\(further details in App\.[G\.3](https://arxiv.org/html/2606.05327#A7.SS3)\)\.

Comparing potential types is the central ablation:𝒲22\\mathcal\{W\}^\{2\}\_\{2\}is most performant and𝒲2∞\\mathcal\{W\}\_\{2\}^\{\\infty\}second; combined with their efficiency and the training\-stability issues of MMD/KLD, this establishes𝒲22/∞\\mathcal\{W\}\_\{2\}^\{2/\\infty\}as the preferred potentials for OTP\-FM\. Surprisingly, a sigmoid curriculum outperforms a constant one even for the𝒲22\\mathcal\{W\}^\{2\}\_\{2\}potential—despite its original motivation as homotopy continuation for MMD/KLD \(Sec\.[4\.3](https://arxiv.org/html/2606.05327#S4.SS3)\)—suggesting training still benefits from starting from the simpler CFM target\. The sensitivity to potential strength and temporal dynamics demonstrates the flexibility OTP\-FM offers over prescribed\-interpolation approaches: these are crucial axes for the SOTA performance shown next\. Further ablations on the consistency model and loss function, as well as trajectory visualizations, are in App\.[G](https://arxiv.org/html/2606.05327#A7)\.

### 6\.3Single\-cell RNA sequencing

We perform a comprehensive comparison in Table[3](https://arxiv.org/html/2606.05327#S6.T3)with all baselines in Sec\.[5](https://arxiv.org/html/2606.05327#S5)on two scRNA\-seq datasets: developing embryoid bodies \(EB, 5 time intervalst0t\_\{0\}—t4t\_\{4\}\)\(Moonet al\.,[2019](https://arxiv.org/html/2606.05327#bib.bib44)\)and CITE\-seq of human CD34\+ HSPCs \(CITE, 4 time intervals\)\(Burkhardtet al\.,[2022](https://arxiv.org/html/2606.05327#bib.bib78)\), operating on up to a 100D PCA representation\. To ensure at least one reported point of comparison with each baseline, we adopt six established protocols: EB 5D leave\-one\-marginal\-out \(L1O\) evaluating𝒲1\\mathcal\{W\}\_\{1\}at the held\-out time\(Tonget al\.,[2020](https://arxiv.org/html/2606.05327#bib.bib46)\); EB 100D L0O \(no holdouts, MMD averaged over training times\) and L1O \(MMD over held\-out times\)\(Persiianovet al\.,[2026](https://arxiv.org/html/2606.05327#bib.bib67)\); EB 100D L2O, holding outt1t\_\{1\}andt3t\_\{3\}with MMD averaged over*all four*times aftert0t\_\{0\}\(Theodoropouloset al\.,[2026](https://arxiv.org/html/2606.05327#bib.bib55)\); and CITE 5D/50D L1O, averaging𝒲1\\mathcal\{W\}\_\{1\}over held\-out times\(Kapusniaket al\.,[2024](https://arxiv.org/html/2606.05327#bib.bib65)\); 5D/50D refer to the first 5/50 PCs\. Evaluating on both training and held\-out marginals tests how well each method recovers the training distribution*and*learns physically plausible trajectories that accurately*interpolate*unseen marginals\.

We additionally measure each method’s training time on consistent hardware and plot training time vs\. performance for CITE 5D and EB 100D L2O in Fig\.[3](https://arxiv.org/html/2606.05327#S6.F3)\. Overall, OTP\-FM achieves SOTA results on nearly all metrics in both performance*and*training efficiency, completing training within 3–5 minutes on an NVIDIA L40S GPU in all settings\. The one exception in terms of fidelity is the EB 100D L0O experiment, where it is outperformed by plain piecewise CFM and \[SF\]2M; this is the only setting that does not measure interpolation ability — precisely the limitation of those methods that OTP\-FM is designed to address\. Additional results, trajectory visualizations, and dataset, model, and benchmarking details are in App\.[G](https://arxiv.org/html/2606.05327#A7)\.

Table 3:Results on the EB and CITE scRNA\-seq datasets for different experimental protocols, as defined in the text\. Mean and std\. dev\. over 5 seeds is shown where available \(lower is better for all metrics\)\. Results for all methods described in Sec\.[5](https://arxiv.org/html/2606.05327#S5)are shown\. A \* indicates our own training of the method using the provided code, while gray indicates we were unable to produce reasonable results on that experiment; otherwise, the result is from the prior work\. Best performing method is inboldand second\-best initalics\.
![Refer to caption](https://arxiv.org/html/2606.05327v1/x3.png)

![Refer to caption](https://arxiv.org/html/2606.05327v1/x4.png)

Figure 3:Training time vs\. performance of different methods for the CITE 5D L1O \(top\) and EB 100D L2O \(bottom\) experiments\. Red stars denote OTP\-FM\. Top right is better\.

### 6\.4Gulf of Mexico and Beijing air quality

We further evaluate on two non\-biological datasets: ocean current particle transport in the Gulf of Mexico \(2D, 9 timepoints\)\(Shenet al\.,[2025](https://arxiv.org/html/2606.05327#bib.bib59)\)and hourly particle concentrations from Beijing air quality monitoring stations \(1D, 13 timepoints\)\(Chen,[2017](https://arxiv.org/html/2606.05327#bib.bib61)\)\. FollowingTheodoropouloset al\.\([2026](https://arxiv.org/html/2606.05327#bib.bib55)\), Table[4](https://arxiv.org/html/2606.05327#S6.T4)reports the average𝒲2\\mathcal\{W\}\_\{2\}across four held\-out timepoints each as well as the remaining training timepoints\. We compare against the two simulation\-free methods closest in motivation to OTP\-FM—MMFM and 3MSBM, both of which learn smooth multimarginal trajectories through*prescribed*interpolation strategies \(Fig\.[1](https://arxiv.org/html/2606.05327#S1.F1), Sec\.[5](https://arxiv.org/html/2606.05327#S5)\)\. OTP\-FM significantly outperforms both at interpolating the held\-out times, particularly on Beijing, which we attribute directly to its flexible, data\-driven dynamics; trajectory visualizations and further analysis are in Apps\.[H](https://arxiv.org/html/2606.05327#A8)and[I](https://arxiv.org/html/2606.05327#A9)\.

Table 4:Results on GoM and Beijing air quality datasets showing average𝒲2\\mathcal\{W\}\_\{2\}across held\-out and training timepoints \(lower is better\)\. MMFM results are with our training \(marked with \*\) and 3MSBM from the original paper, which does not report training𝒲2\\mathcal\{W\}\_\{2\}scores on the Beijing air quality dataset\. Best performing method is inboldand second\-best initalics\.

## 7Conclusion

We introduced OTP\-FM, a principled generalization of multimarginal CFM grounded in dynamic OT with soft potential energy terms to steer learnt trajectories toward intermediate marginals\. It recovers piecewise CFM in the singular\-potential limit and unlocks a broad design space for the spatiotemporal dynamics\. We derived exact conditional solutions and an efficient simulation\-free training algorithm, proved Wasserstein bounds on the alignment with ground\-truth marginals controlled by potential strength and training loss, and demonstrated that OTP\-FM pushes the frontier in both performance and training efficiency across biological, oceanographic, and meteorological datasets\.

#### Practical recommendations

Across the design space,𝒲22/∞\\mathcal\{W\}\_\{2\}^\{2/\\infty\}potentials are clearly preferred: they yield closed\-form, low\-variance gradients, a linear fixed\-point system, and the best empirical performance\. Our out\-of\-the\-box recipe is𝒲2∞\\mathcal\{W\}\_\{2\}^\{\\infty\}with strengthw≈1000w\\approx 1000, equally\-spaced Gaussianλk​\(t\)\\lambda\_\{k\}\(t\), and a sigmoid curriculum \(Sec\.[6\.2](https://arxiv.org/html/2606.05327#S6.SS2), App\.[G](https://arxiv.org/html/2606.05327#A7)\);wwis the most impactful parameter to tune, withλk​\(t\)\\lambda\_\{k\}\(t\)shape and width having secondary effects\. Switching to𝒲22\\mathcal\{W\}^\{2\}\_\{2\}with theπallOT\\pi^\{\\mathrm\{OT\}\}\_\{\\mathrm\{all\}\}coupling can provide further improvement when the𝒪​\(N3\)\\mathcal\{O\}\(N^\{3\}\)OT precomputation is feasible\.

#### Limitations and future work

While matching intermediate marginals encourages more plausible trajectories, it does not alone guarantee physically meaningful interpolated states, particularly for sparsely\-sampled or noisily\-labeled data; methods that explicitly learn the data manifold such as MFM\(Kapusniaket al\.,[2024](https://arxiv.org/html/2606.05327#bib.bib65)\)and pixel MeanFlow\(Luet al\.,[2026](https://arxiv.org/html/2606.05327#bib.bib80)\)are important orthogonal directions for improvement\. Future work will also explore incorporating domain\-specific inductive biases such as cell growth in VGFM\(Wanget al\.,[2026](https://arxiv.org/html/2606.05327#bib.bib68)\), and conditional inference as in MMFM\(Rohbecket al\.,[2025](https://arxiv.org/html/2606.05327#bib.bib54)\)\. Like CFM, OTP\-FM solves the*conditional*OTP problem given a joint couplingπall\\pi\_\{\\mathrm\{all\}\}, rather than jointly optimizing the dynamics and coupling toward the full marginal variational problem\. A rectified\-flow\-style retraining\(Liuet al\.,[2023](https://arxiv.org/html/2606.05327#bib.bib4)\)or combining our conditional formulation with new direct multimarginal\-OT solvers such as WLF\(Neklyudovet al\.,[2024](https://arxiv.org/html/2606.05327#bib.bib66)\)are interesting directions toward self\-consistent dynamics–coupling pairs\.

Beyond the𝒲22\\mathcal\{W\}^\{2\}\_\{2\}potential, our MMD and KLD force estimators rely on kernel\-based estimates of the full marginal density, which are noisier than𝒲22\\mathcal\{W\}^\{2\}\_\{2\}’s linear, pointwise gradient and yield nonlinear fixed\-point systems\. Improving them via adaptive kernels or learned score models could unlock a richer family of physically\-motivated potentials\. Finally, while𝒲2∞\\mathcal\{W\}\_\{2\}^\{\\infty\}is a strong default, OTP\-FM’s broad design space imposes a tuning burden absent from prescriptive methods; automating this search, exploring a broader parameter space, and deploying to novel scientific domains are all exciting directions for future work\.

## Impact Statement

This work introduces a principled and computationally efficient framework for learning the latent dynamics of physical systems from discrete empirical snapshots\. In biological contexts, OTP\-FM enables reconstructing continuous trajectories from sparsely sampled, high\-dimensional observations\. In climate science and oceanography, the same formulation improves dynamical modeling from limited temporal data\. The efficiency and performance of OTP\-FM lowers the barrier to trajectory inference on new, large\-scale scientific datasets\.

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## Appendix AProofs of theorems

### A\.1Dynamic OT without a potential

Proofs for the E\-L equations and straight\-line trajectories for the dynamic OT problem can be found in standard texts such asVillani \([2009](https://arxiv.org/html/2606.05327#bib.bib7)\), and are reproduced here for completeness and later reference in App\.[A\.3](https://arxiv.org/html/2606.05327#A1.SS3)\.

###### Theorem A\.1\(E\-L equations\)\.

ForV=0V=0, minimizers of Eq\.[2](https://arxiv.org/html/2606.05327#S2.E2)satisfy the following Euler–Lagrange \(E\-L\) equations:

ut=∇φt,∂tφt​\(x\)\+‖∇φt​\(x\)‖22=0,u\_\{t\}=\\nabla\\varphi\_\{t\},\\qquad\\partial\_\{t\}\\varphi\_\{t\}\(x\)\+\\frac\{\\\|\\nabla\\varphi\_\{t\}\(x\)\\\|^\{2\}\}\{2\}=0,\(22\)along with the continuity equation\.

###### Proof\.

We start with the dynamic OT action without a potential:

S​\[ℒ\]=∫01∫ℝd\[12​‖ut​\(x\)‖2​ρt​\(x\)\+φt​\(x\)​\[∂tρt​\(x\)\+∇⋅\(ρt​\(x\)​ut​\(x\)\)\]\]​dx​dt\.S\[\\mathcal\{L\}\]=\\int\_\{0\}^\{1\}\\int\_\{\\mathbb\{R\}^\{d\}\}\\Big\[\\frac\{1\}\{2\}\\\|u\_\{t\}\(x\)\\\|^\{2\}\\rho\_\{t\}\(x\)\+\\varphi\_\{t\}\(x\)\[\\partial\_\{t\}\\rho\_\{t\}\(x\)\+\\nabla\\cdot\(\\rho\_\{t\}\(x\)u\_\{t\}\(x\)\)\]\\Big\]\\mathrm\{d\}x\\mathrm\{d\}t\.\(23\)Integrating the second term by parts yields:

∫01∫ℝdφt​\(x\)​\[∂tρt​\(x\)\+∇⋅\(ρt​\(x\)​ut​\(x\)\)\]=∫ℝd\[φt​\(x\)​ρt​\(x\)\]01​dx−∫01∫ℝd\[∂tφt​\(x\)\+ut​\(x\)⋅∇φt​\(x\)\]​ρt​\(x\)​dx​dt\\int\_\{0\}^\{1\}\\int\_\{\\mathbb\{R\}^\{d\}\}\\varphi\_\{t\}\(x\)\[\\partial\_\{t\}\\rho\_\{t\}\(x\)\+\\nabla\\cdot\(\\rho\_\{t\}\(x\)u\_\{t\}\(x\)\)\]=\\int\_\{\\mathbb\{R\}^\{d\}\}\\left\[\\varphi\_\{t\}\(x\)\\rho\_\{t\}\(x\)\\right\]^\{1\}\_\{0\}\\mathrm\{d\}x\-\\int\_\{0\}^\{1\}\\int\_\{\\mathbb\{R\}^\{d\}\}\\left\[\\partial\_\{t\}\\varphi\_\{t\}\(x\)\+u\_\{t\}\(x\)\\cdot\\nabla\\varphi\_\{t\}\(x\)\\right\]\\rho\_\{t\}\(x\)\\mathrm\{d\}x\\mathrm\{d\}t\(24\)For fixed boundary conditions, the first term is simply a constant with respect touuandρ\\rho\.

The minima can thus be found by taking the functional derivatives ofSS:

δ​Sδ​φ\\displaystyle\\frac\{\\delta S\}\{\\delta\\varphi\}=∂tρt​\(x\)\+∇⋅\(ρt​\(x\)​ut​\(x\)\)=0\\displaystyle=\\partial\_\{t\}\\rho\_\{t\}\(x\)\+\\nabla\\cdot\(\\rho\_\{t\}\(x\)u\_\{t\}\(x\)\)=0\(Continuity equation\)\(25\)δ​Sδ​u\\displaystyle\\frac\{\\delta S\}\{\\delta u\}=\(ut​\(x\)−∇φt​\(x\)\)​ρt​\(x\)=0\\displaystyle=\\left\(u\_\{t\}\(x\)\-\\nabla\\varphi\_\{t\}\(x\)\\right\)\\rho\_\{t\}\(x\)=0⇒\\displaystyle\\Rightarrowut​\(x\)=∇φt​\(x\)\\displaystyle\\,u\_\{t\}\(x\)=\\nabla\\varphi\_\{t\}\(x\)\(26\)δ​Sδ​ρ\\displaystyle\\frac\{\\delta S\}\{\\delta\\rho\}=‖ut​\(x\)‖22−\[∂tφt​\(x\)\+ut​\(x\)⋅∇φt​\(x\)\]=0\\displaystyle=\\frac\{\\\|u\_\{t\}\(x\)\\\|^\{2\}\}\{2\}\-\\left\[\\partial\_\{t\}\\varphi\_\{t\}\(x\)\+u\_\{t\}\(x\)\\cdot\\nabla\\varphi\_\{t\}\(x\)\\right\]=0⇒\\displaystyle\\Rightarrow∂tφt​\(x\)\+‖∇φt​\(x\)‖22=0\\displaystyle\\,\\partial\_\{t\}\\varphi\_\{t\}\(x\)\+\\frac\{\\\|\\nabla\\varphi\_\{t\}\(x\)\\\|^\{2\}\}\{2\}=0\(H\-J equation\)\(27\)where in Eq\.[27](https://arxiv.org/html/2606.05327#A1.E27)we plugged in Eq\.[26](https://arxiv.org/html/2606.05327#A1.E26)\. ∎

###### Corollary A\.2\(Straight\-line trajectories\)\.

Minimizers of Eq\.[2](https://arxiv.org/html/2606.05327#S2.E2)\(withV=0V=0\) yield straight\-line sample trajectoriesXt=\(1−t\)​X0\+t​ψ​\(X0\)X\_\{t\}=\(1\-t\)X\_\{0\}\+t\\psi\(X\_\{0\}\)withX˙t=ψ​\(X0\)−X0\\dot\{X\}\_\{t\}=\\psi\(X\_\{0\}\)\-X\_\{0\}, whereψ\\psiis the static OT map for costc2c\_\{2\}\.

###### Proof\.

We first take the gradient of the H\-J equation:

∂t∇φ\+\(∇2φ\)​∇φ=0,\\partial\_\{t\}\\nabla\\varphi\+\(\\nabla^\{2\}\\varphi\)\\,\\nabla\\varphi=0,\(28\)where∇2φ\\nabla^\{2\}\\varphiis the Hessian ofφ\\varphi\. Next, the time derivative of the sample velocityX˙t\\dot\{X\}\_\{t\}\(Eq\.[26](https://arxiv.org/html/2606.05327#A1.E26)\):

d​u​\(Xt\)d​t\\displaystyle\\frac\{\\mathrm\{d\}u\(X\_\{t\}\)\}\{\\mathrm\{d\}t\}=dd​t​∇φt​\(Xt\)\\displaystyle=\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\}\\nabla\\varphi\_\{t\}\(X\_\{t\}\)\(29\)=∂t∇φt​\(Xt\)\+\(∇2φ\)​\(Xt\)​X˙t\\displaystyle=\\partial\_\{t\}\\nabla\\varphi\_\{t\}\(X\_\{t\}\)\+\(\\nabla^\{2\}\\varphi\)\(X\_\{t\}\)\\,\\dot\{X\}\_\{t\}=\(i\)∂t∇φt​\(Xt\)\+\(∇2φ\)​\(Xt\)​∇φt​\(Xt\)\\displaystyle\\stackrel\{\{\\scriptstyle\(i\)\}\}\{\{=\}\}\\partial\_\{t\}\\nabla\\varphi\_\{t\}\(X\_\{t\}\)\+\(\\nabla^\{2\}\\varphi\)\(X\_\{t\}\)\\,\\nabla\\varphi\_\{t\}\(X\_\{t\}\)=\(i​i\)0,\\displaystyle\\stackrel\{\{\\scriptstyle\(ii\)\}\}\{\{=\}\}0,where in\(i\)\(i\)we applied Eq\.[26](https://arxiv.org/html/2606.05327#A1.E26)again and in\(i​i\)\(ii\)used Eq\.[28](https://arxiv.org/html/2606.05327#A1.E28)\. Thus, the sample trajectories obey

X¨t=0\.\\ddot\{X\}\_\{t\}=0\.\(30\)As the dynamic OT solution is equivalent to that of the static OT problem with costc2c\_\{2\}, given a sampleX0∼μ0X\_\{0\}\\sim\\mu\_\{0\}, its trajectory must lead toX1=ψ​\(X0\)∼μ1X\_\{1\}=\\psi\(X\_\{0\}\)\\sim\\mu\_\{1\}\. Therefore, integrating Eq\.[30](https://arxiv.org/html/2606.05327#A1.E30)with boundary conditionsX0X\_\{0\}att=0t=0andψ​\(X0\)\\psi\(X\_\{0\}\)att=1t=1yieldsXt=\(1−t\)​X0\+t​ψ​\(X0\)X\_\{t\}=\(1\-t\)X\_\{0\}\+t\\psi\(X\_\{0\}\)andX˙t=ψ​\(X0\)−X0\\dot\{X\}\_\{t\}=\\psi\(X\_\{0\}\)\-X\_\{0\}\. ∎

### A\.2Piecewise multimarginal CFM as the hard\-potential limit of OTP

We make the connection between standard piecewise multimarginal CFM and the OTP variational problem \(Sec\.[3](https://arxiv.org/html/2606.05327#S3)\) precise via two main results: piecewise multimarginal CFM is equivalent to a single CFM training regressing onto conditional solutions of the hard\-constrained problemℒHC\\mathcal\{L\}\_\{\\mathrm\{HC\}\}\(Lemma[A\.4](https://arxiv.org/html/2606.05327#A1.Thmtheorem4)\), and OTP minimizers converge to the solutions of the hard\-constrained problem in the singular, hard\-penalty limit ofwk→∞w\_\{k\}\\to\\infty,λk→δ​\(t−tk\)\\lambda\_\{k\}\\to\\delta\(t\-t\_\{k\}\)\(Prop\.[A\.5](https://arxiv.org/html/2606.05327#A1.Thmtheorem5), stated for𝒟=𝒲22\\mathcal\{D\}=\\mathcal\{W\}^\{2\}\_\{2\}\)\. In order to prove the former, we first establish thatℒHC\\mathcal\{L\}\_\{\\mathrm\{HC\}\}decomposes additively intoK\+1K\+1independent Benamou–Brenier \(BB\) subproblems \(Lemma[A\.3](https://arxiv.org/html/2606.05327#A1.Thmtheorem3)\), whereKKis the number of intermediate marginals\.

Throughout this subsection, we assume0=t0<t1<⋯<tK<tK\+1=10=t\_\{0\}<t\_\{1\}<\\cdots<t\_\{K\}<t\_\{K\+1\}=1and that all marginals\{μtk\}k=0K\+1\\\{\\mu\_\{t\_\{k\}\}\\\}\_\{k=0\}^\{K\+1\}are absolutely continuous probability measures onℝd\\mathbb\{R\}^\{d\}, uniformly bounded in𝒫2​\(ℝd\)\\mathcal\{P\}\_\{2\}\(\\mathbb\{R\}^\{d\}\)— i\.e\., with finite second moments — so that𝒲22​\(μtk,μtk\+1\)<∞\\mathcal\{W\}^\{2\}\_\{2\}\(\\mu\_\{t\_\{k\}\},\\mu\_\{t\_\{k\+1\}\}\)<\\inftyfor allkk\. An*admissible flow*is a pair\(ρt,ut\)t∈\[0,1\]\(\\rho\_\{t\},u\_\{t\}\)\_\{t\\in\[0,1\]\}withρt∈𝒫2​\(ℝd\)\\rho\_\{t\}\\in\\mathcal\{P\}\_\{2\}\(\\mathbb\{R\}^\{d\}\)a curve of probability measures andutu\_\{t\}a velocity field satisfying the continuity equation and endpoint boundary conditionsρ0=μ0\\rho\_\{0\}=\\mu\_\{0\},ρ1=μ1\\rho\_\{1\}=\\mu\_\{1\}\. We define intervalsIk≡\[tk,tk\+1\]I\_\{k\}\\equiv\[t\_\{k\},t\_\{k\+1\}\]and write the rescaled timesk​\(t\):=\(t−tk\)/\(tk\+1−tk\)∈\[0,1\]s\_\{k\}\(t\):=\(t\-t\_\{k\}\)/\(t\_\{k\+1\}\-t\_\{k\}\)\\in\[0,1\]for convenience\.

We refer to*piecewise multimarginal CFM*as the standard procedure from, e\.g\.,Tonget al\.\([2024](https://arxiv.org/html/2606.05327#bib.bib3),[2023](https://arxiv.org/html/2606.05327#bib.bib69)\): pick any joint couplingπall\\pi\_\{\\mathrm\{all\}\}of\(μt0,…,μtK\+1\)\(\\mu\_\{t\_\{0\}\},\\dots,\\mu\_\{t\_\{K\+1\}\}\)and trainK\+1K\+1independent CFM modelsuθkk:ℝd×Ik→ℝdu^\{k\}\_\{\\theta\_\{k\}\}:\\mathbb\{R\}^\{d\}\\times I\_\{k\}\\to\\mathbb\{R\}^\{d\}\(k=0,…,Kk=0,\\dots,K\), one per consecutive intervalIkI\_\{k\}between\(μtk,μtk\+1\)\(\\mu\_\{t\_\{k\}\},\\mu\_\{t\_\{k\+1\}\}\), against the standard CFM regression lossesℒkpiecewise​\(θk\)\\mathcal\{L\}^\{\\mathrm\{piecewise\}\}\_\{k\}\(\\theta\_\{k\}\)with endpoint pairs drawn from the marginal ofπall\\pi\_\{\\mathrm\{all\}\}on\(xtk,xtk\+1\)\(x\_\{t\_\{k\}\},x\_\{t\_\{k\+1\}\}\)and conditional\-OT solutions between them as regression targets \(Eq\.[5](https://arxiv.org/html/2606.05327#S2.E5)\), then stitch theK\+1K\+1learned flows at each\{tk\}k=1K\\\{t\_\{k\}\\\}\_\{k=1\}^\{K\}:

uθpiecewise​\(x,t\)=∑k=0K𝟏Ik​\(t\)​uθkk​\(x,t\),θ≡\(θ0,…,θK\),u^\{\\mathrm\{piecewise\}\}\_\{\\theta\}\(x,t\)=\\sum\_\{k=0\}^\{K\}\\mathbf\{1\}\_\{I\_\{k\}\}\(t\)\\,u^\{k\}\_\{\\theta\_\{k\}\}\(x,t\),\\qquad\\theta\\equiv\(\\theta\_\{0\},\\dots,\\theta\_\{K\}\),\(31\)where𝟏Ik\\mathbf\{1\}\_\{I\_\{k\}\}is the indicator function ofIkI\_\{k\}\.555The overlap at theKKinterior endpoints\{tk\}k=1K\\\{t\_\{k\}\\\}\_\{k=1\}^\{K\}is measure zero and so irrelevant to any time integral\.

###### Lemma A\.3\(Additive decomposition ofℒHC\\mathcal\{L\}\_\{\\mathrm\{HC\}\}\)\.

Under the setup above, the hard\-constrained actionℒHC\\mathcal\{L\}\_\{\\mathrm\{HC\}\}\(Eq\.[8](https://arxiv.org/html/2606.05327#S3.E8)\) decomposes additively intoK\+1K\+1independent Benamou–Brenier dynamic OT subproblems between consecutive marginals,

ℒHC=∑k=0Kmin\(ρ,u\)\|Ik​∫Ik∫12​‖ut‖2​ρt​dx​dt=∑k=0K𝒲22​\(μtk,μtk\+1\)2​\(tk\+1−tk\),\\mathcal\{L\}\_\{\\mathrm\{HC\}\}\\;=\\;\\sum\_\{k=0\}^\{K\}\\min\_\{\(\\rho,u\)\|\_\{I\_\{k\}\}\}\\int\_\{I\_\{k\}\}\\\!\\\!\\int\\tfrac\{1\}\{2\}\\\|u\_\{t\}\\\|^\{2\}\\rho\_\{t\}\\,\\mathrm\{d\}x\\,\\mathrm\{d\}t\\;=\\;\\sum\_\{k=0\}^\{K\}\\frac\{\\mathcal\{W\}^\{2\}\_\{2\}\(\\mu\_\{t\_\{k\}\},\\mu\_\{t\_\{k\+1\}\}\)\}\{2\(t\_\{k\+1\}\-t\_\{k\}\)\},\(32\)where the inner minimization onIkI\_\{k\}is over admissible flows constrained byρtk=μtk\\rho\_\{t\_\{k\}\}=\\mu\_\{t\_\{k\}\}andρtk\+1=μtk\+1\\rho\_\{t\_\{k\+1\}\}=\\mu\_\{t\_\{k\+1\}\}\.

###### Proof\.

Let\(ρt,ut\)\(\\rho\_\{t\},u\_\{t\}\)be an admissible flow that additionally satisfiesρtk=μtk\\rho\_\{t\_\{k\}\}=\\mu\_\{t\_\{k\}\}for allkk\. The kinetic energy is additive over the intervalsIkI\_\{k\},

∫01∫12​‖ut‖2​ρt​dx​dt=∑k=0K∫Ik∫12​‖ut‖2​ρt​dx​dt,\\int\_\{0\}^\{1\}\\\!\\\!\\int\\tfrac\{1\}\{2\}\\\|u\_\{t\}\\\|^\{2\}\\rho\_\{t\}\\,\\mathrm\{d\}x\\,\\mathrm\{d\}t\\;=\\;\\sum\_\{k=0\}^\{K\}\\int\_\{I\_\{k\}\}\\\!\\\!\\int\\tfrac\{1\}\{2\}\\\|u\_\{t\}\\\|^\{2\}\\rho\_\{t\}\\,\\mathrm\{d\}x\\,\\mathrm\{d\}t,\(33\)and the only active constraints over each interval are the continuity equation\(ρt,ut\)\|Ik\(\\rho\_\{t\},u\_\{t\}\)\|\_\{I\_\{k\}\}and the endpoint conditionsρtk=μtk\\rho\_\{t\_\{k\}\}=\\mu\_\{t\_\{k\}\},ρtk\+1=μtk\+1\\rho\_\{t\_\{k\+1\}\}=\\mu\_\{t\_\{k\+1\}\}; no constraint couples distinct intervals\. Hence, the joint minimization decouples intoK\+1K\+1independent subproblems:

ℒHC\|Ik=min\(ρ,u\)\|Ik​∫Ik∫12​‖ut‖2​ρt​dx​dts\.t\.​∂tρ\+∇⋅\(ρ​u\)=0,ρtk=μtk,ρtk\+1=μtk\+1\.\\mathcal\{L\}\_\{\\mathrm\{HC\}\|I\_\{k\}\}=\\min\_\{\(\\rho,u\)\|\_\{I\_\{k\}\}\}\\int\_\{I\_\{k\}\}\\\!\\\!\\int\\tfrac\{1\}\{2\}\\\|u\_\{t\}\\\|^\{2\}\\rho\_\{t\}\\,\\mathrm\{d\}x\\,\\mathrm\{d\}t\\quad\\text\{s\.t\.\}\\;\\;\\partial\_\{t\}\\rho\+\\nabla\\\!\\cdot\\\!\(\\rho u\)=0,\\;\\;\\rho\_\{t\_\{k\}\}=\\mu\_\{t\_\{k\}\},\\;\\;\\rho\_\{t\_\{k\+1\}\}=\\mu\_\{t\_\{k\+1\}\}\.\(34\)Each of these subproblems thus is the standard BB problem betweenμtk\\mu\_\{t\_\{k\}\}andμtk\+1\\mu\_\{t\_\{k\+1\}\}, most easily seen by time rescalings≡sk​\(t\)s\\equiv s\_\{k\}\(t\)as above and definingu~​\(s,x\)≡\(tk\+1−tk\)​u​\(t,x\)\\tilde\{u\}\(s,x\)\\equiv\(t\_\{k\+1\}\-t\_\{k\}\)\\,u\(t,x\)to satisfy the continuity equation inss:

ℒHC\|Ik\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{HC\}\|I\_\{k\}\}=1tk\+1−tk​min\(ρ,u\)​∫01∫12​‖us‖2​ρs​dx​dss\.t\.​∂sρ\+∇⋅\(ρ​u\)=0,ρ0=μtk,ρ1=μtk\+1\.\\displaystyle=\\frac\{1\}\{t\_\{k\+1\}\-t\_\{k\}\}\\min\_\{\(\\rho,u\)\}\\int\_\{0\}^\{1\}\\\!\\\!\\int\\tfrac\{1\}\{2\}\\\|u\_\{s\}\\\|^\{2\}\\rho\_\{s\}\\,\\mathrm\{d\}x\\,\\mathrm\{d\}s\\quad\\text\{s\.t\.\}\\;\\;\\partial\_\{s\}\\rho\+\\nabla\\\!\\cdot\\\!\(\\rho u\)=0,\\;\\;\\rho\_\{0\}=\\mu\_\{t\_\{k\}\},\\;\\;\\rho\_\{1\}=\\mu\_\{t\_\{k\+1\}\}\.\(35\)=𝒲22​\(μtk,μtk\+1\)2​\(tk\+1−tk\)\\displaystyle=\\frac\{\\mathcal\{W\}^\{2\}\_\{2\}\(\\mu\_\{t\_\{k\}\},\\mu\_\{t\_\{k\+1\}\}\)\}\{2\(t\_\{k\+1\}\-t\_\{k\}\)\}Summing over all intervals yields Eq\.[32](https://arxiv.org/html/2606.05327#A1.E32)\. ∎

###### Lemma A\.4\(Piecewise multimarginal CFM is equivalent to CFM regression onto conditional solutions ofℒHC\\mathcal\{L\}\_\{\\mathrm\{HC\}\}\)\.

Under the setup above and any joint couplingπall\\pi\_\{\\mathrm\{all\}\}of\(μt0,…,μtK\+1\)\(\\mu\_\{t\_\{0\}\},\\dots,\\mu\_\{t\_\{K\+1\}\}\), conditioning on a sampled endpoint tuplez=\(xt0,…,xtK\+1\)∼πallz=\(x\_\{t\_\{0\}\},\\dots,x\_\{t\_\{K\+1\}\}\)\\sim\\pi\_\{\\mathrm\{all\}\}reduces each per\-interval subproblem in Eq\.[32](https://arxiv.org/html/2606.05327#A1.E32)to the dynamic OT problem between the Dirac delta functionsδ​\(x−xtk\)\\delta\(x\-x\_\{t\_\{k\}\}\)andδ​\(x−xtk\+1\)\\delta\(x\-x\_\{t\_\{k\+1\}\}\), whose unique conditional solution is the straight\-line trajectory:

XtHC​\(x\|z\)=\(1−sk​\(t\)\)​xtk\+sk​\(t\)​xtk\+1,utHC​\(x\|z\)=xtk\+1−xtktk\+1−tk,t∈Ik,X^\{\\mathrm\{HC\}\}\_\{t\}\(x\|z\)\\;=\\;\(1\-s\_\{k\}\(t\)\)\\,x\_\{t\_\{k\}\}\+s\_\{k\}\(t\)\\,x\_\{t\_\{k\+1\}\},\\qquad u^\{\\mathrm\{HC\}\}\_\{t\}\(x\|z\)\\;=\\;\\frac\{x\_\{t\_\{k\+1\}\}\-x\_\{t\_\{k\}\}\}\{t\_\{k\+1\}\-t\_\{k\}\},\\qquad t\\in I\_\{k\},\(36\)with conditional pathρtHC​\(x∣z\)=δ​\(x−XtHC​\(x∣z\)\)\\rho^\{\\mathrm\{HC\}\}\_\{t\}\(x\\mid z\)=\\delta\(x\-X^\{\\mathrm\{HC\}\}\_\{t\}\(x\\mid z\)\)\. The corresponding CFM regression loss

ℒHC−CFM\(θ\):=𝔼t∼U​\[0,1\]𝔼z∼πall𝔼x∼ρtHC\(⋅∣z\)∥vθ\(x,t\)−utHC\(x∣z\)∥2\\mathcal\{L\}^\{\\mathrm\{HC\-CFM\}\}\(\\theta\)\\;:=\\;\\mathbb\{E\}\_\{t\\sim U\[0,1\]\}\\,\\mathbb\{E\}\_\{z\\sim\\pi\_\{\\mathrm\{all\}\}\}\\,\\mathbb\{E\}\_\{x\\sim\\rho^\{\\mathrm\{HC\}\}\_\{t\}\(\\cdot\\mid z\)\}\\,\\big\\\|v^\{\\theta\}\(x,t\)\-u^\{\\mathrm\{HC\}\}\_\{t\}\(x\\mid z\)\\big\\\|^\{2\}\(37\)of a single shared modelvθ​\(x,t\)v^\{\\theta\}\(x,t\)on\[0,1\]\[0,1\]then equals the total piecewise multimarginal CFM loss underπall\\pi\_\{\\mathrm\{all\}\},

ℒHC−CFM​\(θ\)=∑k=0Kℒkpiecewise​\(θk\),\\mathcal\{L\}^\{\\mathrm\{HC\-CFM\}\}\(\\theta\)\\;=\\;\\sum\_\{k=0\}^\{K\}\\mathcal\{L\}^\{\\mathrm\{piecewise\}\}\_\{k\}\(\\theta\_\{k\}\),when theK\+1K\+1piecewise models are tied to a shared parameterization viauθkk​\(x,t\)=𝟏Ik​vθ​\(x,t\)u^\{k\}\_\{\\theta\_\{k\}\}\(x,t\)=\\mathbf\{1\}\_\{I\_\{k\}\}v^\{\\theta\}\(x,t\)\.

###### Proof\.

*Conditional HC solution\.*Let the CFM conditioning variablez=\(xt0,…,xtK\+1\)∼πallz=\(x\_\{t\_\{0\}\},\\dots,x\_\{t\_\{K\+1\}\}\)\\sim\\pi\_\{\\mathrm\{all\}\}\. By Lemma[A\.3](https://arxiv.org/html/2606.05327#A1.Thmtheorem3),ℒHC\\mathcal\{L\}\_\{\\mathrm\{HC\}\}decomposes intoK\+1K\+1independent BB subproblems \(Eq\.[34](https://arxiv.org/html/2606.05327#A1.E34)\); the conditional version of each replaces the marginal boundary constraintsρtk=μtk\\rho\_\{t\_\{k\}\}=\\mu\_\{t\_\{k\}\},ρtk\+1=μtk\+1\\rho\_\{t\_\{k\+1\}\}=\\mu\_\{t\_\{k\+1\}\}with their conditional counterpartsρtk​\(x∣z\)=δ​\(x−xtk\)\\rho\_\{t\_\{k\}\}\(x\\mid z\)=\\delta\(x\-x\_\{t\_\{k\}\}\),ρtk\+1​\(x∣z\)=δ​\(x−xtk\+1\)\\rho\_\{t\_\{k\+1\}\}\(x\\mid z\)=\\delta\(x\-x\_\{t\_\{k\+1\}\}\), yielding a dynamic OT problem between two Dirac delta functions onIkI\_\{k\}\.

After rescaling to\[0,1\]\[0,1\]as in Lemma[A\.3](https://arxiv.org/html/2606.05327#A1.Thmtheorem3), Corollary[A\.2](https://arxiv.org/html/2606.05327#A1.Thmtheorem2)provides its unique minimizer: the straight\-line trajectoryXs=\(1−s\)​xtk\+s​xtk\+1X\_\{s\}=\(1\-s\)\\,x\_\{t\_\{k\}\}\+s\\,x\_\{t\_\{k\+1\}\}; undoing the rescaling yields Eq\.[36](https://arxiv.org/html/2606.05327#A1.E36)\.

*Loss equality\.*Splitting the time integral in Eq\.[37](https://arxiv.org/html/2606.05327#A1.E37)across theK\+1K\+1subintervals and usingρtHC​\(x∣z\)=δ​\(x−XtHC\)\\rho^\{\\mathrm\{HC\}\}\_\{t\}\(x\\mid z\)=\\delta\(x\-X^\{\\mathrm\{HC\}\}\_\{t\}\)yields:

ℒHC−CFM​\(θ\)=∑k=0K∫Ik𝔼\(xtk,xtk\+1\)∼πalltk,tk\+1​‖vθ​\(XtHC,t\)−xtk\+1−xtktk\+1−tk‖2​dt,\\mathcal\{L\}^\{\\mathrm\{HC\-CFM\}\}\(\\theta\)\\;=\\;\\sum\_\{k=0\}^\{K\}\\\!\\int\_\{I\_\{k\}\}\\\!\\mathbb\{E\}\_\{\(x\_\{t\_\{k\}\},x\_\{t\_\{k\+1\}\}\)\\sim\\pi\_\{\\mathrm\{all\}\}^\{t\_\{k\},t\_\{k\+1\}\}\}\\,\\Big\\\|v^\{\\theta\}\(X^\{\\mathrm\{HC\}\}\_\{t\},t\)\-\\tfrac\{x\_\{t\_\{k\+1\}\}\-x\_\{t\_\{k\}\}\}\{t\_\{k\+1\}\-t\_\{k\}\}\\Big\\\|^\{2\}\\mathrm\{d\}t,\(38\)where we marginalized the components ofzznot appearing in thekk\-th summand\. Each summand is exactly the standard CFM loss onIkI\_\{k\}under the consecutive\-pair couplingπalltk,tk\+1\\pi\_\{\\mathrm\{all\}\}^\{t\_\{k\},t\_\{k\+1\}\}; i\.e\.,ℒkpiecewise\\mathcal\{L\}^\{\\mathrm\{piecewise\}\}\_\{k\}evaluated atuθkk​\(x,t\)=vθ​\(x,t\)u^\{k\}\_\{\\theta\_\{k\}\}\(x,t\)=v^\{\\theta\}\(x,t\)fort∈Ikt\\in I\_\{k\}\. Hence,ℒHC−CFM​\(θ\)=∑k=0Kℒkpiecewise​\(θk\)\\mathcal\{L\}^\{\\mathrm\{HC\-CFM\}\}\(\\theta\)=\\sum\_\{k=0\}^\{K\}\\mathcal\{L\}^\{\\mathrm\{piecewise\}\}\_\{k\}\(\\theta\_\{k\}\)\. ∎

###### Proposition A\.5\(The OTP problem converges to the hard\-constrained problem in the hard\-potential limit\)\.

Consider the OTP problem \(Eq\.[10](https://arxiv.org/html/2606.05327#S3.E10)\) with𝒟=𝒲22\\mathcal\{D\}=\\mathcal\{W\}^\{2\}\_\{2\}and temporal kernels of the formλk​\(t\)=1τ​ηk​\(t−tkτ\)\\lambda\_\{k\}\(t\)=\\frac\{1\}\{\\tau\}\\eta\_\{k\}\\\!\\left\(\\frac\{t\-t\_\{k\}\}\{\\tau\}\\right\), with any fixed nonnegative shapesηk\\eta\_\{k\}normalized so∫ηk=1\\int\\eta\_\{k\}=1\. Then, in the joint limitw→∞w\\to\\infty,τ→0\\tau\\to 0withw​τ→0w\\tau\\to 0,666The hard\-constrained limit only requires thatτ\\taushrink faster than1/w1/walong the sequence considered, e\.g\.,τ=w−2\\tau=w^\{\-2\}\.the OTP problem converges to the hard\-constrained problem \(Eq\.[8](https://arxiv.org/html/2606.05327#S3.E8)\), and any sequence of OTP minimizers converges to the unique piecewise OT solutions of the Benamou–Brenier subproblems of Lemma[A\.3](https://arxiv.org/html/2606.05327#A1.Thmtheorem3)\.

###### Proof\.

We use the fundamental theorem ofΓ\\Gamma\-convergence\(Braides,[2006](https://arxiv.org/html/2606.05327#bib.bib74)\):equi\-coercivity\+\+Γ\\Gamma\-convergence⇒\\Rightarrowconvergence of minimum problems\.Γ\\Gamma\-convergence implies that the OTP energies converge to that of the hard\-constrained problem, while equi\-coercivity is required for the minimizers to converge as well\. We prove both ingredients\. Note that since we assume absolute continuity of the marginals above, byVillani \([2009](https://arxiv.org/html/2606.05327#bib.bib7), Thm\. 9\.4\), the piecewise OT solution of the hard\-constrained problem \(Lemma[A\.3](https://arxiv.org/html/2606.05327#A1.Thmtheorem3)\) is unique\.

Setup\.We define the family of functionals:

Fw,τ​\[ρ,u\]≡∫01∫\[12​‖ut‖2​ρt\+w​∑kλk​\(t\)​𝒲22​\(ρt,μtk\)​ρt\]​dx​dt,F\_\{w,\\tau\}\[\\rho,u\]\\equiv\\int\_\{0\}^\{1\}\\\!\\\!\\int\\big\[\\tfrac\{1\}\{2\}\\\|u\_\{t\}\\\|^\{2\}\\rho\_\{t\}\+w\\\!\\sum\_\{k\}\\lambda\_\{k\}\(t\)\\,\\mathcal\{W\}^\{2\}\_\{2\}\(\\rho\_\{t\},\\mu\_\{t\_\{k\}\}\)\\,\\rho\_\{t\}\\big\]\\,\\mathrm\{d\}x\\,\\mathrm\{d\}t,\(39\)fix sequenceswn→∞w\_\{n\}\\to\\infty,τn→0\\tau\_\{n\}\\to 0withwn​τn→0w\_\{n\}\\tau\_\{n\}\\to 0, and writeFn≡Fwn,τnF\_\{n\}\\equiv F\_\{w\_\{n\},\\tau\_\{n\}\}andλkn\\lambda\_\{k\}^\{n\}for the kernel at widthτn\\tau\_\{n\}\. We define the limit functional to incorporate the hard constraints as follows:

FHC​\(ρ,u\)≡\{∫01∫12​‖ut‖2​ρt​dx​dtif\(ρ,u\)solves the hard\-constrained \(HC\) problem,\+∞otherwise,F\_\{\\mathrm\{HC\}\}\(\\rho,u\)\\equiv\\begin\{cases\}\\int\_\{0\}^\{1\}\\\!\\\!\\int\\tfrac\{1\}\{2\}\\\|u\_\{t\}\\\|^\{2\}\\rho\_\{t\}\\,\\mathrm\{d\}x\\,\\mathrm\{d\}t&\\text\{if $\(\\rho,u\)$ solves the hard\-constrained \(HC\) problem,\}\\\\ \+\\infty&\\text\{otherwise,\}\\end\{cases\}\(40\)where solving the HC problem means that\(ρ,u\)\(\\rho,u\)is admissible as defined above*and*ρtk=μtk\\rho\_\{t\_\{k\}\}=\\mu\_\{t\_\{k\}\}for allkk, somin⁡FHC=ℒHC\\min F\_\{\\mathrm\{HC\}\}=\\mathcal\{L\}\_\{\\mathrm\{HC\}\}\(Eq\.[8](https://arxiv.org/html/2606.05327#S3.E8)\)\.

Equi\-coercivity\.Equi\-coercivity means that all minimizers ofFnF\_\{n\}must be confined to a compact set independent ofnn; i\.e\., they cannot escape to infinity ifFnF\_\{n\}is bounded\. Our proof is based on the formal definition fromDal Maso \([1993](https://arxiv.org/html/2606.05327#bib.bib72), Prop\. 7\.7\): the sequenceFnF\_\{n\}is equi\-coercive if and only if there exists a lower semicontinuous \(l\.s\.c\.\) coercive functionΨ\\Psisuch thatFn≥ΨF\_\{n\}\\geq\\Psifor allnn\. We take the kinetic energy as our candidateΨ\\Psi:

Ψ​\(ρ,u\)=∫01∫12​‖ut‖2​ρt​dx​dt;\\Psi\(\\rho,u\)=\\int\_\{0\}^\{1\}\\\!\\\!\\int\\tfrac\{1\}\{2\}\\\|u\_\{t\}\\\|^\{2\}\\rho\_\{t\}\\,\\mathrm\{d\}x\\,\\mathrm\{d\}t;\(41\)since the OTP penalty is non\-negative,Fn≥ΨF\_\{n\}\\geq\\Psi\. Also, since12​‖ut‖2\\tfrac\{1\}\{2\}\\\|u\_\{t\}\\\|^\{2\}is convex and l\.s\.c\., byAmbrosioet al\.\([2008](https://arxiv.org/html/2606.05327#bib.bib71), Thm\. 5\.4\.4\(ii\)\),Ψ\\Psiis l\.s\.c\. under this convergence\.

What is left to show is thatΨ\\Psiis coercive, for which we useDal Maso \([1993](https://arxiv.org/html/2606.05327#bib.bib72), Def\. 1\.12\):Ψ\\Psiis coercive if the set of all bounded energies\{Ψ​\(ρn,un\)≤E\}\\\{\\Psi\(\\rho^\{n\},u^\{n\}\)\\leq E\\\}is relatively compact777A set is relatively compact if its closure is compact\.for allE∈ℝE\\in\\mathbb\{R\}\. FromAmbrosioet al\.\([2008](https://arxiv.org/html/2606.05327#bib.bib71), Thm\. 8\.3\.1\), ifΨ​\(ρn,un\)≤E\\Psi\(\\rho^\{n\},u^\{n\}\)\\leq E, then

𝒲22​\(ρt1n,ρt2n\)≤\\displaystyle\\mathcal\{W\}^\{2\}\_\{2\}\(\\rho^\{n\}\_\{t\_\{1\}\},\\rho^\{n\}\_\{t\_\{2\}\}\)\\;\\leq\\;\|t1−t2\|​∫t1t2∫‖utn‖2​ρtn​dx​dt≤2​E​\|t1−t2\|\\displaystyle\|t\_\{1\}\-t\_\{2\}\|\\\!\\int\_\{t\_\{1\}\}^\{t\_\{2\}\}\\\!\\\!\\int\\\|u^\{n\}\_\{t\}\\\|^\{2\}\\rho^\{n\}\_\{t\}\\,\\mathrm\{d\}x\\,\\mathrm\{d\}t\\;\\leq\\;2E\\,\|t\_\{1\}\-t\_\{2\}\|\(42\)⇒\\displaystyle\\Rightarrow\\quad𝒲2​\(ρt1n,ρt2n\)≤2​E​\|t1−t2\|1/2,\\displaystyle\\mathcal\{W\}\_\{2\}\(\\rho^\{n\}\_\{t\_\{1\}\},\\rho^\{n\}\_\{t\_\{2\}\}\)\\;\\leq\\;\\sqrt\{2E\}\\,\|t\_\{1\}\-t\_\{2\}\|^\{1/2\},\(43\)which means all bounded\-energy paths are uniformly12\\tfrac\{1\}\{2\}\-Hölder continuous in𝒲2\\mathcal\{W\}\_\{2\}and thus,equi\-continuous\. Since they all start from the same distributionμ0∈𝒫2\\mu\_\{0\}\\in\\mathcal\{P\}\_\{2\}, they also have uniformly bounded second moments throughout time\. Thus, by Ascoli–Arzelà theorem\(Santambrogio,[2015](https://arxiv.org/html/2606.05327#bib.bib75), Box 1\.7\),\{Ψ​\(ρn,un\)≤E\}\\\{\\Psi\(\\rho^\{n\},u^\{n\}\)\\leq E\\\}is relatively compact, andΨ\\Psicoercive\.Ψ\\Psiis therefore an l\.s\.c\. coercive function, which meansFnF\_\{n\}is equi\-coercive\.

Γ\\Gamma\-convergence\.Γ\\Gamma\-convergence requires bothliminfandlimsup inequalities\(Dal Maso,[1993](https://arxiv.org/html/2606.05327#bib.bib72)\), essentially showing that the limit ofFnF\_\{n\}is bounded both below and above byFHCF\_\{\\mathrm\{HC\}\}and, therefore,FnF\_\{n\}converges toFHCF\_\{\\mathrm\{HC\}\}\.

Limsup inequality\.We seek to show for every\(ρ,u\)\(\\rho,u\)there exists a sequence888Commonly referred to as arecovery sequence\.\(ρn,un\)→\(ρ,u\)\(\\rho^\{n\},u^\{n\}\)\\to\(\\rho,u\)withlim supnFn​\(ρn,un\)≤FHC​\(ρ,u\)\\limsup\_\{n\}F\_\{n\}\(\\rho^\{n\},u^\{n\}\)\\leq F\_\{\\mathrm\{HC\}\}\(\\rho,u\)\. If\(ρ,u\)\(\\rho,u\)is not HC\-admissible thenFHC​\(ρ,u\)=\+∞F\_\{\\mathrm\{HC\}\}\(\\rho,u\)=\+\\inftyand the inequality is trivial\. For HC\-admissible\(ρ,u\)\(\\rho,u\), we simply consider the constant sequence\(ρn,un\)=\(ρ,u\)\(\\rho^\{n\},u^\{n\}\)=\(\\rho,u\)\. By definition,FHC​\(ρ,u\)F\_\{\\mathrm\{HC\}\}\(\\rho,u\)equals the kinetic energyΨ​\(ρ,u\)≡E∈ℝ\\Psi\(\\rho,u\)\\equiv E\\in\\mathbb\{R\}\. Subtracting it fromFnF\_\{n\}leaves only the penalty:

Fn​\(ρ,u\)−E=∑k=1Kwn​∫01λkn​\(t\)​𝒲22​\(ρt,μtk\)​dt\.F\_\{n\}\(\\rho,u\)\-E\\;=\\;\\sum\_\{k=1\}^\{K\}w\_\{n\}\\\!\\int\_\{0\}^\{1\}\\lambda\_\{k\}^\{n\}\(t\)\\,\\mathcal\{W\}^\{2\}\_\{2\}\(\\rho\_\{t\},\\mu\_\{t\_\{k\}\}\)\\,\\mathrm\{d\}t\.\(44\)The same bound fromAmbrosioet al\.\([2008](https://arxiv.org/html/2606.05327#bib.bib71), Thm\. 8\.3\.1\)as above further yields𝒲22​\(ρt,μtk\)≤2​E​\|t−tk\|\\mathcal\{W\}^\{2\}\_\{2\}\(\\rho\_\{t\},\\mu\_\{t\_\{k\}\}\)\\leq 2E\\,\|t\-t\_\{k\}\|\. Substituting this and the explicit form of the temporal kernel with the change of variablest~≡t−tkτn\\tilde\{t\}\\equiv\\frac\{t\-t\_\{k\}\}\{\\tau\_\{n\}\}:

Fn​\(ρ,u\)−E≤2​E​∑k=1Kwn​∫01λkn​\(t\)​\|t−tk\|​dt=2​E​∑k=1Kwn​τn​∫\|t~\|​ηk​\(t~\)​dt~→n→∞0,F\_\{n\}\(\\rho,u\)\-E\\;\\leq\\;2E\\\!\\sum\_\{k=1\}^\{K\}w\_\{n\}\\\!\\int\_\{0\}^\{1\}\\lambda\_\{k\}^\{n\}\(t\)\\,\|t\-t\_\{k\}\|\\,\\mathrm\{d\}t\\;=\\;2E\\\!\\sum\_\{k=1\}^\{K\}w\_\{n\}\\tau\_\{n\}\\\!\\int\|\\tilde\{t\}\|\\,\\eta\_\{k\}\(\\tilde\{t\}\)\\,\\mathrm\{d\}\\tilde\{t\}\\;\\xrightarrow\{n\\to\\infty\}\\;0,\(45\)sincewn​τn→0w\_\{n\}\\tau\_\{n\}\\to 0\. Thus,lim supnFn​\(ρ,u\)≤E=FHC​\(ρ,u\)\\limsup\_\{n\}F\_\{n\}\(\\rho,u\)\\leq E=F\_\{\\mathrm\{HC\}\}\(\\rho,u\)\.

Liminf inequality\.We must show that for every sequence\(ρn,un\)→\(ρ,u\)\(\\rho^\{n\},u^\{n\}\)\\to\(\\rho,u\)\(uniformly in𝒲2\\mathcal\{W\}\_\{2\}\),lim infnFn​\(ρn,un\)≥FHC​\(ρ,u\)\\liminf\_\{n\}F\_\{n\}\(\\rho^\{n\},u^\{n\}\)\\geq F\_\{\\mathrm\{HC\}\}\(\\rho,u\)\.*Case A:ρtk=μtk\\rho\_\{t\_\{k\}\}=\\mu\_\{t\_\{k\}\}for allkk\.*Then\(ρ,u\)\(\\rho,u\)is HC\-admissible, so

FHC​\(ρ,u\)=∫01∫12​‖ut‖2​ρt​≤\(i\)​lim infn∫01∫12​‖utn‖2​ρtn⏞≡Ψ​\(ρn,un\)​≤\(ii\)​lim infnFn​\(ρn,un\),F\_\{\\mathrm\{HC\}\}\(\\rho,u\)\\;=\\;\\int\_\{0\}^\{1\}\\\!\\\!\\int\\tfrac\{1\}\{2\}\\\|u\_\{t\}\\\|^\{2\}\\rho\_\{t\}\\;\\overset\{\\mathrm\{\(i\)\}\}\{\\leq\}\\;\\liminf\_\{n\}\\overbrace\{\\int\_\{0\}^\{1\}\\\!\\\!\\int\\tfrac\{1\}\{2\}\\\|u^\{n\}\_\{t\}\\\|^\{2\}\\rho^\{n\}\_\{t\}\}^\{\\equiv\\Psi\(\\rho^\{n\},u^\{n\}\)\}\\;\\overset\{\\mathrm\{\(ii\)\}\}\{\\leq\}\\;\\liminf\_\{n\}F\_\{n\}\(\\rho^\{n\},u^\{n\}\),\(46\)where \(i\) is lower\-semicontinuity of the kinetic energyΨ\\Psiunder\(ρn,un\)→\(ρ,u\)\(\\rho^\{n\},u^\{n\}\)\\to\(\\rho,u\)\(established above\), and \(ii\) follows fromFn≥Ψ​\(ρn,un\)F\_\{n\}\\geq\\Psi\(\\rho^\{n\},u^\{n\}\)since the OTP penalty is non\-negative\.

*Case B:𝒲22​\(ρtk∗,μtk∗\)\>0\\mathcal\{W\}^\{2\}\_\{2\}\(\\rho\_\{t\_\{k^\{\*\}\}\},\\mu\_\{t\_\{k^\{\*\}\}\}\)\>0for somek∗k^\{\*\}\.*Kernel concentration \(λk∗n→δ​\(t−tk∗\)\\lambda\_\{k^\{\*\}\}^\{n\}\\to\\delta\(t\-t\_\{k^\{\*\}\}\)\) and uniform𝒲2\\mathcal\{W\}\_\{2\}\-convergence imply∫01λk∗n​\(t\)​𝒲22​\(ρtn,μtk∗\)​dt→𝒲22​\(ρtk∗,μtk∗\)\>0\\int\_\{0\}^\{1\}\\lambda\_\{k^\{\*\}\}^\{n\}\(t\)\\,\\mathcal\{W\}^\{2\}\_\{2\}\(\\rho^\{n\}\_\{t\},\\mu\_\{t\_\{k^\{\*\}\}\}\)\\,\\mathrm\{d\}t\\to\\mathcal\{W\}^\{2\}\_\{2\}\(\\rho\_\{t\_\{k^\{\*\}\}\},\\mu\_\{t\_\{k^\{\*\}\}\}\)\>0\. SinceFn≥wnF\_\{n\}\\geq w\_\{n\}times this integral andwn→∞w\_\{n\}\\to\\infty, we getFn​\(ρn,un\)→\+∞=FHC​\(ρ,u\)F\_\{n\}\(\\rho^\{n\},u^\{n\}\)\\to\+\\infty=F\_\{\\mathrm\{HC\}\}\(\\rho,u\)\.

∎

### A\.3Dynamic OT with a potential

###### Proof\.

We recall the OTP action \(Eq\.[10](https://arxiv.org/html/2606.05327#S3.E10)\) for convenience:

S​\[ℒ\]=∫01∫ℝd\[12​ρt​\(x\)​‖ut​\(x\)‖2\+∑k=1Kwk​λk​\(t\)​𝒟tk​ρt​\(x\)\+φt​\(x\)​\[∂tρt​\(x\)\+∇⋅\(ρt​\(x\)​ut​\(x\)\)\]\]​dx​dt,S\[\\mathcal\{L\}\]=\\int\_\{0\}^\{1\}\\\!\\\!\\int\_\{\\mathbb\{R\}^\{d\}\}\\\!\\left\[\\tfrac\{1\}\{2\}\\rho\_\{t\}\(x\)\\\|u\_\{t\}\(x\)\\\|^\{2\}\+\\\!\\\!\\sum\_\{k=1\}^\{K\}w\_\{k\}\\lambda\_\{k\}\(t\)\\mathcal\{D\}^\{k\}\_\{t\}\\rho\_\{t\}\(x\)\+\\varphi\_\{t\}\(x\)\\left\[\\partial\_\{t\}\\rho\_\{t\}\(x\)\+\\nabla\\\!\\cdot\\\!\(\\rho\_\{t\}\(x\)u\_\{t\}\(x\)\)\\right\]\\right\]\\mathrm\{d\}x\\mathrm\{d\}t,\(47\)where𝒟tk≡𝒟​\[ρt,μtk\]\\mathcal\{D\}^\{k\}\_\{t\}\\equiv\\mathcal\{D\}\[\\rho\_\{t\},\\mu\_\{t\_\{k\}\}\]is a real\-valued functional ofρt\\rho\_\{t\}that varies smoothly withttthrough its dependence onρt\\rho\_\{t\}\. Comparing this with Eq\.[23](https://arxiv.org/html/2606.05327#A1.E23), the variationsδ​S/δ​φ\\delta S/\\delta\\varphiandδ​S/δ​u\\delta S/\\delta urecover the continuity equation andut=∇φtu\_\{t\}=\\nabla\\varphi\_\{t\}unchanged\.

Variation w\.r\.t\.ρt\\rho\_\{t\}\.Recalling from Eq\.[10](https://arxiv.org/html/2606.05327#S3.E10)that the potential contribution to the action is∫∫\(−VOTP​\(y,s\)\)​dy​ds=∑kwk​∫∫λk​\(s\)​𝒟​\[ρs,μtk\]​ρs​\(y\)​dy​ds\\int\\\!\\\!\\int\(\-V\_\{\\mathrm\{OTP\}\}\(y,s\)\)\\,\\mathrm\{d\}y\\,\\mathrm\{d\}s=\\sum\_\{k\}w\_\{k\}\\int\\\!\\\!\\int\\lambda\_\{k\}\(s\)\\,\\mathcal\{D\}\[\\rho\_\{s\},\\mu\_\{t\_\{k\}\}\]\\,\\rho\_\{s\}\(y\)\\,\\mathrm\{d\}y\\,\\mathrm\{d\}s, direct functional differentiation yields:

δδ​ρt​\(x\)​∫∫\(−VOTP​\(y,s\)\)​dy​ds\\displaystyle\\frac\{\\delta\}\{\\delta\\rho\_\{t\}\(x\)\}\\int\\\!\\\!\\\!\\int\(\-V\_\{\\mathrm\{OTP\}\}\(y,s\)\)\\,\\mathrm\{d\}y\\,\\mathrm\{d\}s=∑kwk​∫∫λk​\(s\)​\[gk​\(x,t\)​δ​\(s−t\)​ρs​\(y\)⏟integrates to 1\+𝒟​\[ρs,μtk\]​δ​\(s−t\)​δ​\(x−y\)\]​dy​ds\\displaystyle=\\sum\_\{k\}w\_\{k\}\\int\\\!\\\!\\\!\\int\\lambda\_\{k\}\(s\)\\Big\[g\_\{k\}\(x,t\)\\,\\delta\(s\-t\)\\,\\underbrace\{\\rho\_\{s\}\(y\)\}\_\{\\text\{integrates to 1\}\}\+\\mathcal\{D\}\[\\rho\_\{s\},\\mu\_\{t\_\{k\}\}\]\\,\\delta\(s\-t\)\\,\\delta\(x\-y\)\\Big\]\\mathrm\{d\}y\\,\\mathrm\{d\}s\(48\)=∑kwk​λk​\(t\)​gk​\(x,t\)\+∑kwk​λk​\(t\)​𝒟tk⏟≡h​\(t\),\\displaystyle=\\sum\_\{k\}w\_\{k\}\\lambda\_\{k\}\(t\)\\,g\_\{k\}\(x,t\)\+\\underbrace\{\\sum\_\{k\}w\_\{k\}\\lambda\_\{k\}\(t\)\\,\\mathcal\{D\}^\{k\}\_\{t\}\}\_\{\\equiv h\(t\)\},wheregk​\(x,t\)≡δ​𝒟tkδ​ρt​\(x\)g\_\{k\}\(x,t\)\\equiv\\frac\{\\delta\\mathcal\{D\}^\{k\}\_\{t\}\}\{\\delta\\rho\_\{t\}\(x\)\}andh​\(t\)h\(t\)is anxx\-independent function ofttthat, as we show below, does not impact the dynamics\. Thus, following the derivation in App\.[A\.1](https://arxiv.org/html/2606.05327#A1.SS1), the third E\-L equation becomes:

∂tφt​\(x\)\+‖∇φt​\(x\)‖22=∑k=1Kwk​λk​\(t\)​gk​\(x,t\)\+h​\(t\)\.\\partial\_\{t\}\\varphi\_\{t\}\(x\)\+\\frac\{\\\|\\nabla\\varphi\_\{t\}\(x\)\\\|^\{2\}\}\{2\}\\;=\\;\\sum\_\{k=1\}^\{K\}w\_\{k\}\\lambda\_\{k\}\(t\)\\,g\_\{k\}\(x,t\)\\;\+\\;h\(t\)\.\(49\)
Gauge re\-parametrization\.Note thath​\(t\)h\(t\)isxx\-independent and can be absorbed into the fieldφt\\varphi\_\{t\}by a gauge transformation:

φt​\(x\)⟼φ~t​\(x\):=φt​\(x\)\+c​\(t\),\\varphi\_\{t\}\(x\)\\;\\longmapsto\\;\\tilde\{\\varphi\}\_\{t\}\(x\)\\;:=\\;\\varphi\_\{t\}\(x\)\+c\(t\),\(50\)leavesut=∇φt=∇φ~tu\_\{t\}=\\nabla\\varphi\_\{t\}=\\nabla\\tilde\{\\varphi\}\_\{t\}\(and hence the E\-L equations of motion and entire flow\) unchanged, since∇c​\(t\)=0\\nabla c\(t\)=0\. Choosingc​\(t\):=−∫0th​\(s\)​dsc\(t\):=\-\\int\_\{0\}^\{t\}h\(s\)\\,\\mathrm\{d\}sso thatc′​\(t\)=−h​\(t\)c^\{\\prime\}\(t\)=\-h\(t\), underφ↦φ~\\varphi\\mapsto\\tilde\{\\varphi\}Eq\.[49](https://arxiv.org/html/2606.05327#A1.E49)becomes:

∂t\[φ~t​\(x\)−c​\(t\)\]\+‖∇φ~t​\(x\)‖22\\displaystyle\\partial\_\{t\}\[\\tilde\{\\varphi\}\_\{t\}\(x\)\-c\(t\)\]\+\\frac\{\\\|\\nabla\\tilde\{\\varphi\}\_\{t\}\(x\)\\\|^\{2\}\}\{2\}=∑k=1Kwk​λk​\(t\)​gk​\(x,t\)\+h​\(t\)\\displaystyle=\\sum\_\{k=1\}^\{K\}w\_\{k\}\\lambda\_\{k\}\(t\)\\,g\_\{k\}\(x,t\)\+h\(t\)\(51\)⇒∂tφ~t​\(x\)\+‖∇φ~t​\(x\)‖22\\displaystyle\\Rightarrow\\partial\_\{t\}\\tilde\{\\varphi\}\_\{t\}\(x\)\+\\frac\{\\\|\\nabla\\tilde\{\\varphi\}\_\{t\}\(x\)\\\|^\{2\}\}\{2\}=∑k=1Kwk​λk​\(t\)​gk​\(x,t\),\\displaystyle=\\sum\_\{k=1\}^\{K\}w\_\{k\}\\lambda\_\{k\}\(t\)\\,g\_\{k\}\(x,t\),i\.e\., Eq\.[12](https://arxiv.org/html/2606.05327#S3.E12)\. Renamingφ~\\tilde\{\\varphi\}back toφ\\varphi\(the gauge choice is conventional\) yields the stated form\. ∎

###### Proof\.

The proof follows that of Corollary[A\.2](https://arxiv.org/html/2606.05327#A1.Thmtheorem2)\. We first take the gradient of the H\-J equation \(Eq\.[12](https://arxiv.org/html/2606.05327#S3.E12)\):

∂t∇φ\+\(∇2φ\)​∇φ=∑k=1Kwk​λk​\(t\)​∇gk​\(x,t\),\\partial\_\{t\}\\nabla\\varphi\+\(\\nabla^\{2\}\\varphi\)\\,\\nabla\\varphi=\\sum\_\{k=1\}^\{K\}w\_\{k\}\\lambda\_\{k\}\(t\)\\,\\nabla g\_\{k\}\(x,t\),\(52\)where∇2φ\\nabla^\{2\}\\varphiis the Hessian ofφ\\varphiand∇gk​\(x,t\)\\nabla g\_\{k\}\(x,t\)is the spatial gradient ofgkg\_\{k\}at fixedtt\. The time derivative ofX˙t\\dot\{X\}\_\{t\}along the flow then gives Eq\.[13](https://arxiv.org/html/2606.05327#S3.E13):

X¨t=d​u​\(Xt\)d​t=∂t∇φt​\(Xt\)\+\(∇2φ\)​\(Xt\)​∇φt​\(Xt\)=∑k=1Kwk​λk​\(t\)​∇gk​\(Xt,t\)\.\\ddot\{X\}\_\{t\}=\\frac\{\\mathrm\{d\}u\(X\_\{t\}\)\}\{\\mathrm\{d\}t\}=\\partial\_\{t\}\\nabla\\varphi\_\{t\}\(X\_\{t\}\)\+\(\\nabla^\{2\}\\varphi\)\(X\_\{t\}\)\\,\\nabla\\varphi\_\{t\}\(X\_\{t\}\)=\\sum\_\{k=1\}^\{K\}w\_\{k\}\\lambda\_\{k\}\(t\)\\,\\nabla g\_\{k\}\(X\_\{t\},t\)\.\(53\)∎

### A\.4Proofs of Propositions[4\.4](https://arxiv.org/html/2606.05327#S4.SS4)–[4\.4](https://arxiv.org/html/2606.05327#S4.SS4)

###### Proof\.

By the one\-sample\-per\-marginal estimator of Sec\.[4\.2](https://arxiv.org/html/2606.05327#S4.SS2),ΨT​\(XT\)=xT≡\[xt1,…,xtK\]⊤\\Psi\_\{T\}\(X\_\{T\}\)=x\_\{T\}\\equiv\[x\_\{t\_\{1\}\},\\dots,x\_\{t\_\{K\}\}\]^\{\\top\}for anyz=\(x0,x1,ℬ\)∼πallz=\(x\_\{0\},x\_\{1\},\\mathcal\{B\}\)\\sim\\pi\_\{\\mathrm\{all\}\}with one GT samplextk∼μtkx\_\{t\_\{k\}\}\\sim\\mu\_\{t\_\{k\}\}per intermediate marginal\. The fixed\-point Eq\.[20](https://arxiv.org/html/2606.05327#S4.E20)is equivalent toXT=XTbase\+A​\(XT−ΨT​\(XT\)\)X\_\{T\}=X^\{\\mathrm\{base\}\}\_\{T\}\+A\(X\_\{T\}\-\\Psi\_\{T\}\(X\_\{T\}\)\)\(App\.[E\.3](https://arxiv.org/html/2606.05327#A5.SS3)\); substitutingΨT​\(XT\)=xT\\Psi\_\{T\}\(X\_\{T\}\)=x\_\{T\}and subtractingxTx\_\{T\}from both sides yields

XT−xT=\(𝟙−A\)−1​\(XTbase−xT\)\.X\_\{T\}\-x\_\{T\}\\;=\\;\(\\mathbbm\{1\}\-A\)^\{\-1\}\\bigl\(X^\{\\mathrm\{base\}\}\_\{T\}\-x\_\{T\}\\bigr\)\.\(54\)
𝒲2\\mathcal\{W\}\_\{2\}bound\.Using Eq\.[54](https://arxiv.org/html/2606.05327#A1.E54):

𝒲22​\(ρtk,μtk\)​≤\(i\)​𝔼​‖Xtk−xtk‖2​≤\(ii\)​‖Pk​\(𝟙−A\)−1‖2​𝔼​‖XTbase−xT‖2\.\\mathcal\{W\}^\{2\}\_\{2\}\(\\rho\_\{t\_\{k\}\},\\mu\_\{t\_\{k\}\}\)\\;\\overset\{\\mathrm\{\(i\)\}\}\{\\leq\}\\;\\mathbb\{E\}\\\|X\_\{t\_\{k\}\}\-x\_\{t\_\{k\}\}\\\|^\{2\}\\;\\overset\{\\mathrm\{\(ii\)\}\}\{\\leq\}\\;\\big\\\|P\_\{k\}\(\\mathbbm\{1\}\-A\)^\{\-1\}\\big\\\|^\{2\}\\;\\mathbb\{E\}\\\|X^\{\\mathrm\{base\}\}\_\{T\}\-x\_\{T\}\\\|^\{2\}\.\(55\)*\(i\)*follows by definition of𝒲22\\mathcal\{W\}^\{2\}\_\{2\}as the*infimum*of𝔼​‖X−Y‖2\\mathbb\{E\}\\\|X\-Y\\\|^\{2\}over all joint couplings of\(ρtk,μtk\)\(\\rho\_\{t\_\{k\}\},\\mu\_\{t\_\{k\}\}\)\(Villani,[2009](https://arxiv.org/html/2606.05327#bib.bib7)\)\. The joint distribution of\(Xtk,xtk\)\(X\_\{t\_\{k\}\},x\_\{t\_\{k\}\}\)induced byz∼πallz\\sim\\pi\_\{\\mathrm\{all\}\}is one such coupling, sinceXtkX\_\{t\_\{k\}\}has marginalρtk\\rho\_\{t\_\{k\}\}andxtkx\_\{t\_\{k\}\}has marginalμtk\\mu\_\{t\_\{k\}\}by construction; plugging this specific joint into the squared cost thus gives an upper bound on the infimum\.*\(ii\)*follows by reading off thekk\-th row of Eq\.[54](https://arxiv.org/html/2606.05327#A1.E54),Xtk−xtk=∑j\[\(𝟙−A\)−1\]k​j​\(Xtjbase−xtj\)X\_\{t\_\{k\}\}\-x\_\{t\_\{k\}\}=\\sum\_\{j\}\[\(\\mathbbm\{1\}\-A\)^\{\-1\}\]\_\{kj\}\\bigl\(X^\{\\mathrm\{base\}\}\_\{t\_\{j\}\}\-x\_\{t\_\{j\}\}\\bigr\), and applying the Cauchy–Schwarz inequality, wherePkP\_\{k\}is the projection operator onto thekk\-th canonical row vector\.

Scaling inww\.RecallingAi​k=wk​\[ℐ\(2\)​\[λk\]​\(ti\)−ℐ\(2\)​\[λk\]​\(1\)​ti\]A\_\{ik\}=w\_\{k\}\\bigl\[\\mathcal\{I\}^\{\(2\)\}\\\!\\left\[\\lambda\_\{k\}\\right\]\(t\_\{i\}\)\-\\mathcal\{I\}^\{\(2\)\}\\\!\\left\[\\lambda\_\{k\}\\right\]\(1\)\\,t\_\{i\}\\bigr\]from Eq\.[20](https://arxiv.org/html/2606.05327#S4.E20), allAi​kA\_\{ik\}scale linearly in the global potential strengthww, soA=w​A~A=w\\tilde\{A\}for a fixed matrixA~\\tilde\{A\}, which depends only on the time grid and temporal kernels\. Away from singular configurations \(i\.e\.,A~\\tilde\{A\}invertible\), the Taylor expansion at largewwyields:

\(𝟙−A\)−1=\(𝟙−w​A~\)−1=−1w​A~−1\+𝒪​\(w−2\),\(\\mathbbm\{1\}\-A\)^\{\-1\}\\;=\\;\(\\mathbbm\{1\}\-w\\tilde\{A\}\)^\{\-1\}\\;=\\;\-\\frac\{1\}\{w\}\\,\\tilde\{A\}^\{\-1\}\\;\+\\;\\mathcal\{O\}\(w^\{\-2\}\),\(56\)soPk​\(𝟙−A\)−1=−Pk​A~−1/w\+𝒪​\(w−2\)P\_\{k\}\(\\mathbbm\{1\}\-A\)^\{\-1\}=\-P\_\{k\}\\tilde\{A\}^\{\-1\}/w\+\\mathcal\{O\}\(w^\{\-2\}\), and hence‖Pk​\(𝟙−A\)−1‖2=‖Pk​A~−1‖2/w2\+𝒪​\(w−3\)\\\|P\_\{k\}\(\\mathbbm\{1\}\-A\)^\{\-1\}\\\|^\{2\}=\\\|P\_\{k\}\\tilde\{A\}^\{\-1\}\\\|^\{2\}/w^\{2\}\+\\mathcal\{O\}\(w^\{\-3\}\)\. The data factor𝔼​‖XTbase−xT‖2\\mathbb\{E\}\\\|X^\{\\mathrm\{base\}\}\_\{T\}\-x\_\{T\}\\\|^\{2\}is independent ofwwand finite by the second\-moment assumption on the GT marginals\. Combining with Eq\.[55](https://arxiv.org/html/2606.05327#A1.E55),

𝒲22​\(ρtk,μtk\)≤Ckw2\+𝒪​\(w−3\),Ck=‖Pk​A~−1‖2⋅𝔼​‖XTbase−xT‖2,\\mathcal\{W\}^\{2\}\_\{2\}\(\\rho\_\{t\_\{k\}\},\\mu\_\{t\_\{k\}\}\)\\;\\leq\\;\\frac\{C\_\{k\}\}\{w^\{2\}\}\+\\mathcal\{O\}\(w^\{\-3\}\),\\qquad C\_\{k\}\\;=\\;\\big\\\|P\_\{k\}\\tilde\{A\}^\{\-1\}\\big\\\|^\{2\}\\cdot\\mathbb\{E\}\\\|X^\{\\mathrm\{base\}\}\_\{T\}\-x\_\{T\}\\\|^\{2\},\(57\)which is the claimed bound\. We note that this bound*applies for any choice of joint couplingπall\\pi\_\{\\mathrm\{all\}\}*\(e\.g\.,πallOT\\pi^\{\\mathrm\{OT\}\}\_\{\\mathrm\{all\}\}orπallind\\pi^\{\\mathrm\{ind\}\}\_\{\\mathrm\{all\}\}, Sec\.[4\.2](https://arxiv.org/html/2606.05327#S4.SS2)\); only the constantCkC\_\{k\}depends on the coupling\. ∎

###### Proof\.

We first decompose the distance by the triangle inequality for𝒲2\\mathcal\{W\}\_\{2\}\(which holds since it is a metric\):

𝒲2​\(ρtkθ,μtk\)≤𝒲2​\(ρtkθ,ρtk\)\+𝒲2​\(ρtk,μtk\),\\mathcal\{W\}\_\{2\}\(\\rho^\{\\theta\}\_\{t\_\{k\}\},\\mu\_\{t\_\{k\}\}\)\\;\\leq\\;\\mathcal\{W\}\_\{2\}\(\\rho^\{\\theta\}\_\{t\_\{k\}\},\\rho\_\{t\_\{k\}\}\)\+\\mathcal\{W\}\_\{2\}\(\\rho\_\{t\_\{k\}\},\\mu\_\{t\_\{k\}\}\),\(58\)whereρtk\\rho\_\{t\_\{k\}\}is the OTP target marginal at strengthwwandρtkθ=\(Ψ^0,tkθ\)\#​ρ0\\rho^\{\\theta\}\_\{t\_\{k\}\}=\(\\hat\{\\Psi\}^\{\\theta\}\_\{0,t\_\{k\}\}\)\_\{\\\#\}\\,\\rho\_\{0\}is the pushforward of the sourceρ0\\rho\_\{0\}under the learned consistency\-model flow map at timetkt\_\{k\}\. Taking the square root of Prop\.[4\.4](https://arxiv.org/html/2606.05327#S4.SS4),𝒲2​\(ρtk,μtk\)≤Ck/w\+𝒪​\(w−3/2\)\\mathcal\{W\}\_\{2\}\(\\rho\_\{t\_\{k\}\},\\mu\_\{t\_\{k\}\}\)\\leq\\sqrt\{C\_\{k\}\}/w\+\\mathcal\{O\}\(w^\{\-3/2\}\)\.

OTP\-FM training error as a*flow\-map distillation error*\.For the first term, we bound the*flow\-map*matching error via the framework ofBoffiet al\.\([2025](https://arxiv.org/html/2606.05327#bib.bib16)\), viewing the OTP\-FM velocity fieldutu\_\{t\}as the teacher \(with associated true two\-time flow mapΨt1,t2\\Psi\_\{t\_\{1\},t\_\{2\}\}\) and the learned MeanFlow consistency modelΨt1,t2θ​\(x\)=x\+\(t2−t1\)​vθ​\(t1,t2,x\)\\Psi^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\(x\)=x\+\(t\_\{2\}\-t\_\{1\}\)v^\{\\theta\}\(t\_\{1\},t\_\{2\},x\)as the distilled student\. As we establish in App\.[C](https://arxiv.org/html/2606.05327#A3), the conditional OTP\-FM training objective \(MeanFlow/ESD or LSD; Eqs\.[85](https://arxiv.org/html/2606.05327#A3.E85),[86](https://arxiv.org/html/2606.05327#A3.E86)\) is by construction the squared self\-distillation residual of one of two consistency conditions onΨt1,t2θ\\Psi^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\(Eulerian or Lagrangian, respectively\)\.

Wasserstein bound\.We assumeutu\_\{t\}satisfies the one\-sided Lipschitz condition ofBoffiet al\.\([2025](https://arxiv.org/html/2606.05327#bib.bib16), Assumption 3\.3\),\(ut​\(x\)−ut​\(y\)\)⋅\(x−y\)≤Ct​‖x−y‖2\(u\_\{t\}\(x\)\-u\_\{t\}\(y\)\)\\cdot\(x\-y\)\\leq C\_\{t\}\\,\\\|x\-y\\\|^\{2\}withCt∈L1​\[0,1\]C\_\{t\}\\in L^\{1\}\[0,1\]for all\(t,x,y\)∈\[0,1\]×ℝd×ℝd\(t,x,y\)\\in\[0,1\]\\times\\mathbb\{R\}^\{d\}\\times\\mathbb\{R\}^\{d\}, to guarantee a unique solution to the flow ODE\. Then, applyingBoffiet al\.\([2025](https://arxiv.org/html/2606.05327#bib.bib16), Prop\. 3\.9 \(Lagrangian\) or Prop\. 3\.10 \(Eulerian\)\)on the time interval\[0,tk\]\[0,t\_\{k\}\]in place of\[0,1\]\[0,1\]\(the proofs run unchanged after restricting all integrals to\[0,tk\]\[0,t\_\{k\}\]\) yields

𝒲22​\(ρtkθ,ρtk\)​≤\(i\)​𝔼​‖Ψ^0,tkθ​\(x0\)−Ψ0,tk​\(x0\)‖2​≤\(ii\)​κk​ℒOTP​\-​FMM​\(θ\),\\mathcal\{W\}\_\{2\}^\{2\}\(\\rho^\{\\theta\}\_\{t\_\{k\}\},\\rho\_\{t\_\{k\}\}\)\\;\\overset\{\\mathrm\{\(i\)\}\}\{\\leq\}\\;\\mathbb\{E\}\\big\\\|\\hat\{\\Psi\}^\{\\theta\}\_\{0,t\_\{k\}\}\(x\_\{0\}\)\-\\Psi\_\{0,t\_\{k\}\}\(x\_\{0\}\)\\big\\\|^\{2\}\\;\\overset\{\\mathrm\{\(ii\)\}\}\{\\leq\}\\;\\kappa\_\{k\}\\,\\mathcal\{L\}^\{\\mathrm\{M\}\}\_\{\\mathrm\{OTP\\text\{\-\}FM\}\}\(\\theta\),\(59\)where*\(i\)*follows because𝒲22\\mathcal\{W\}^\{2\}\_\{2\}, by definition, is the infimum of all joint couplings, of which\(Ψ^0,tkθ​\(x0\),Ψ0,tk​\(x0\)\)\(\\hat\{\\Psi\}^\{\\theta\}\_\{0,t\_\{k\}\}\(x\_\{0\}\),\\Psi\_\{0,t\_\{k\}\}\(x\_\{0\}\)\)is a valid one; and*\(ii\)*is the corresponding bound fromBoffiet al\.\([2025](https://arxiv.org/html/2606.05327#bib.bib16)\)with constants:

κk=etk\+2​∫0tk\|Ct\|​dt\(LSD\),κk=etk\(ESD/MeanFlow\)\.\\kappa\_\{k\}\\;=\\;e^\{\\,t\_\{k\}\+2\\int\_\{0\}^\{t\_\{k\}\}\|C\_\{t\}\|\\,\\mathrm\{d\}t\}\\quad\\text\{\(LSD\),\}\\qquad\\kappa\_\{k\}\\;=\\;e^\{\\,t\_\{k\}\}\\quad\\text\{\(ESD/MeanFlow\)\.\}\(60\)Substituting this into Eq\.[58](https://arxiv.org/html/2606.05327#A1.E58)yields the claimed bound:

𝒲2​\(ρtkθ,μtk\)≤Dk​ℒOTP−FM\+Ckw\+𝒪​\(w−3/2\),\\mathcal\{W\}\_\{2\}\(\\rho^\{\\theta\}\_\{t\_\{k\}\},\\mu\_\{t\_\{k\}\}\)\\leq D\_\{k\}\\sqrt\{\\mathcal\{L\}\_\{\\mathrm\{OTP\-FM\}\}\}\+\\frac\{\\sqrt\{C\_\{k\}\}\}\{w\}\+\\mathcal\{O\}\(w^\{\-3/2\}\),\(61\)whereDk=κkD\_\{k\}=\\sqrt\{\\kappa\_\{k\}\}andCkC\_\{k\}are the constants from Prop\.[4\.4](https://arxiv.org/html/2606.05327#S4.SS4)\. ∎

## Appendix BDistances between probability measures

### B\.1The𝒲22\\mathcal\{W\}^\{2\}\_\{2\}distance, KLD, and MMD

The potentials we consider in this work are based on statistical distances between two probability measures𝒟:𝒫​\(ℝd\)×𝒫​\(ℝd\)→ℝ\+\\mathcal\{D\}:\\mathcal\{P\}\(\\mathbb\{R\}^\{d\}\)\\times\\mathcal\{P\}\(\\mathbb\{R\}^\{d\}\)\\to\\mathbb\{R\}^\{\+\}\. Specifically, we consider three popular statistical distances\. The first is the Kullback\-Leibler divergence \(KLD\):

𝒟KL​\[α,β\]=∫ℝdln⁡α​\(x\)β​\(x\)​d​α​\(x\)\.\\mathcal\{D\}\_\{\\mathrm\{KL\}\}\[\\alpha,\\beta\]=\\int\_\{\\mathbb\{R\}^\{d\}\}\\ln\\frac\{\\alpha\(x\)\}\{\\beta\(x\)\}\\mathrm\{d\}\\alpha\(x\)\.\(62\)The second is the𝒲22\\mathcal\{W\}^\{2\}\_\{2\}distance, a form of the static OT problem discussed in Sec\.[2\.1](https://arxiv.org/html/2606.05327#S2.SS1), and the third the maximum mean discrepancy \(MMD\), a type of integral probability metric \(IPM\)\. IPMs are defined as the difference in expectation of an optimal witness functionh:ℝd→ℝh:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}out of a class of real\-valued measurable functionsℋ\\mathcal\{H\}:

𝒟IPM​\[α,β\]=suph∈ℋ\|∫ℝdh​\(x\)​dα​\(x\)−∫ℝdh​\(x\)​dβ​\(x\)\|\.\\mathcal\{D\}\_\{\\mathrm\{IPM\}\}\[\\alpha,\\beta\]=\\sup\_\{h\\in\\mathcal\{H\}\}\\left\|\\int\_\{\\mathbb\{R\}^\{d\}\}h\(x\)\\mathrm\{d\}\\alpha\(x\)\-\\int\_\{\\mathbb\{R\}^\{d\}\}h\(x\)\\mathrm\{d\}\\beta\(x\)\\right\|\.\(63\)They include the𝒲1\\mathcal\{W\}\_\{1\}distance and have the attractive property of satisfying the definition of a metric for most common choices ofℋ\\mathcal\{H\}\(Müller,[1997](https://arxiv.org/html/2606.05327#bib.bib8)\)\. In the case of MMD,ℋ\\mathcal\{H\}is a unit ball in a reproducing kernel Hilbert space \(RKHS\) with kernelk:ℝd×ℝd→ℝk:\\mathbb\{R\}^\{d\}\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}\. An RKHS satisfies the reproducing property:∀h∈ℋ\\forall h\\in\\mathcal\{H\},h​\(x\)=⟨h,k​\(x,⋅\)⟩ℋh\(x\)=\\langle h,k\(x,\\cdot\)\\rangle\_\{\\mathcal\{H\}\}\. The squared MMD can be expressed conveniently as\(Grettonet al\.,[2012](https://arxiv.org/html/2606.05327#bib.bib10)\):

𝒟MMD2​\[α,β\]=∫∫k​\(x,x′\)​dα​\(x\)​dα​\(x′\)−2​∫∫k​\(x,y\)​dα​\(x\)​dβ​\(y\)\+∫∫k​\(y,y′\)​dβ​\(y\)​dβ​\(y′\)\.\\mathcal\{D\}\_\{\\mathrm\{MMD\}^\{2\}\}\[\\alpha,\\beta\]=\\int\\\!\\\!\\\!\\\!\\int k\(x,x^\{\\prime\}\)\\,\\mathrm\{d\}\\alpha\(x\)\\mathrm\{d\}\\alpha\(x^\{\\prime\}\)\-2\\int\\\!\\\!\\\!\\\!\\int k\(x,y\)\\,\\mathrm\{d\}\\alpha\(x\)\\mathrm\{d\}\\beta\(y\)\+\\int\\\!\\\!\\\!\\\!\\int k\(y,y^\{\\prime\}\)\\,\\mathrm\{d\}\\beta\(y\)\\mathrm\{d\}\\beta\(y^\{\\prime\}\)\.\(64\)

### B\.2Functional derivatives

As highlighted in Sec\.[3](https://arxiv.org/html/2606.05327#S3), the functional derivativesggof statistical distances𝒟\\mathcal\{D\}, and their gradients∇g\\nabla g, are important objects for OTP\-FM\. The expressions for these for the𝒲22\\mathcal\{W\}^\{2\}\_\{2\}distance, MMD, and KL divergence, provided in Table[1](https://arxiv.org/html/2606.05327#S3.T1), can be derived by direct differentiation as we detail here\.

#### The𝒲22\\mathcal\{W\}^\{2\}\_\{2\}distance

The derivative can be calculated most easily using the dynamicdual formof𝒲22\\mathcal\{W\}^\{2\}\_\{2\}\(Villani,[2009](https://arxiv.org/html/2606.05327#bib.bib7)\):

𝒟​\[ρt,μtk\]:=supφt\{∫ℝdφ1​\(x\)​dμtk​\(x\)−∫ℝdφ0​\(x\)​dρt​\(x\)\},\\mathcal\{D\}\[\\rho\_\{t\},\\mu\_\{t\_\{k\}\}\]:=\\sup\_\{\\varphi\_\{t\}\}\\left\\\{\\int\_\{\\mathbb\{R\}^\{d\}\}\\varphi\_\{1\}\(x\)\\mathrm\{d\}\\mu\_\{t\_\{k\}\}\(x\)\-\\int\_\{\\mathbb\{R\}^\{d\}\}\\varphi\_\{0\}\(x\)\\mathrm\{d\}\\rho\_\{t\}\(x\)\\right\\\},\(65\)such that∀t∈\[0,1\]\\forall t\\in\[0,1\],φt\\varphi\_\{t\}satisfies the H\-J equation \(Eq\.[3](https://arxiv.org/html/2606.05327#S2.E3)\)\. Taking the functional derivative with respect toρt​\(x\)\\rho\_\{t\}\(x\), we get:

δ​𝒟δ​ρt≡gk​\(x,t\)=−φ0∗​\(x\),\\frac\{\\delta\\mathcal\{D\}\}\{\\delta\\rho\_\{t\}\}\\equiv g\_\{k\}\(x,t\)=\-\\varphi^\{\*\}\_\{0\}\(x\),\(66\)whereφ∗\\varphi^\{\*\}is the optimal H\-J potential\. Finally, using Theorem[A\.1](https://arxiv.org/html/2606.05327#A1.Thmtheorem1)and Corollary[A\.2](https://arxiv.org/html/2606.05327#A1.Thmtheorem2), we have that:

∇gk​\(x,t\)=−∇φ0∗​\(x\)=−u0​\(x\)=x−ψ∗​\(x\),\\nabla g\_\{k\}\(x,t\)=\-\\nabla\\varphi^\{\*\}\_\{0\}\(x\)=\-u\_\{0\}\(x\)=x\-\\psi^\{\*\}\(x\),\(67\)whereψ∗\\psi^\{\*\}is the optimal \(static\) OT map betweenρt\\rho\_\{t\}andμtk\\mu\_\{t\_\{k\}\}\.

#### MMD

Differentiating Eq\.[64](https://arxiv.org/html/2606.05327#A2.E64)with respect toρt​\(x\)\\rho\_\{t\}\(x\), we get:

δ​𝒟MMD2​\[ρt,μtk\]δ​ρt=2​∫k​\(x,y\)​\[ρt​\(y\)−μtk​\(y\)\]​dy\.\\frac\{\\delta\\mathcal\{D\}\_\{\\mathrm\{MMD\}^\{2\}\}\[\\rho\_\{t\},\\mu\_\{t\_\{k\}\}\]\}\{\\delta\\rho\_\{t\}\}=2\\int k\(x,y\)\\left\[\\rho\_\{t\}\(y\)\-\\mu\_\{t\_\{k\}\}\(y\)\\right\]\\mathrm\{d\}y\.\(68\)The gradient, for arbitrary kernelkk, is then simply:

∇gk​\(x,t\)=2​∫∇xk​\(x,y\)​\[ρt​\(y\)−μtk​\(y\)\]​dy\.\\nabla g\_\{k\}\(x,t\)=2\\int\\nabla\_\{x\}k\(x,y\)\\left\[\\rho\_\{t\}\(y\)\-\\mu\_\{t\_\{k\}\}\(y\)\\right\]\\mathrm\{d\}y\.\(69\)
Specifically, in this work, we primarily consider the well\-known RBF kernel:

kRBF​\(x,y\)=exp⁡\(−‖x−y‖22​σ2\),k\_\{\\mathrm\{RBF\}\}\(x,y\)=\\exp\\left\(\-\\frac\{\\\|x\-y\\\|^\{2\}\}\{2\\sigma^\{2\}\}\\right\),\(70\)whereσ\\sigmais the kernel bandwidth; and experiment briefly with the polynomial kernel \(see App\.[D](https://arxiv.org/html/2606.05327#A4)\):

kpoly​\(x,y\)=\(xT​y\+c\)d,k\_\{\\mathrm\{poly\}\}\(x,y\)=\(x^\{T\}y\+c\)^\{d\},\(71\)whereddis the polynomial degree andcca constant\. The gradients for these kernels are:

∇xkRBF​\(x,y\)=−x−yσ2⋅exp⁡\(−‖x−y‖22​σ2\),\\nabla\_\{x\}k\_\{\\mathrm\{RBF\}\}\(x,y\)=\-\\frac\{x\-y\}\{\\sigma^\{2\}\}\\cdot\\exp\\left\(\-\\frac\{\\\|x\-y\\\|^\{2\}\}\{2\\sigma^\{2\}\}\\right\),\(72\)∇xkpoly​\(x,y\)=d⋅y⋅\(xT​y\+c\)d−1\.\\nabla\_\{x\}k\_\{\\mathrm\{poly\}\}\(x,y\)=d\\cdot y\\cdot\(x^\{T\}y\+c\)^\{d\-1\}\.\(73\)

#### KL divergence

We consider only the “forward” KL divergence as it has a more useful derivative and gradient:

𝒟KL​\[ρt,μtk\]=∫ℝdρt​\(x\)​\[ln⁡ρt​\(x\)−ln⁡μtk​\(x\)\]​dx\.\\mathcal\{D\}\_\{\\mathrm\{KL\}\}\[\\rho\_\{t\},\\mu\_\{t\_\{k\}\}\]=\\int\_\{\\mathbb\{R\}^\{d\}\}\\rho\_\{t\}\(x\)\\left\[\\ln\\rho\_\{t\}\(x\)\-\\ln\\mu\_\{t\_\{k\}\}\(x\)\\right\]\\mathrm\{d\}x\.\(74\)Taking the functional derivative:

δ​𝒟KLδ​ρt=ln⁡ρt​\(x\)\+1−ln⁡μtk​\(x\)\.\\frac\{\\delta\\mathcal\{D\}\_\{\\mathrm\{KL\}\}\}\{\\delta\\rho\_\{t\}\}=\\ln\\rho\_\{t\}\(x\)\+1\-\\ln\\mu\_\{t\_\{k\}\}\(x\)\.\(75\)The gradient is then simply:

∇gk​\(x,t\)=∇ln⁡ρt​\(x\)−∇ln⁡μtk​\(x\)\.\\nabla g\_\{k\}\(x,t\)=\\nabla\\ln\\rho\_\{t\}\(x\)\-\\nabla\\ln\\mu\_\{t\_\{k\}\}\(x\)\.\(76\)

## Appendix CConsistency models and MeanFlow

### C\.1Flow map matching

Consistency models have been presented in the literature in a wide variety of forms, with variations on, among other things: discrete\- vs\. continuous\-time training; the use of one\- vs\. two\-time flow maps; the requirement \(or lack thereof\) for a pre\-trained “teacher” flow matching model; and training objectives\.

Flow map matching\(Boffiet al\.,[2025](https://arxiv.org/html/2606.05327#bib.bib16),[2026](https://arxiv.org/html/2606.05327#bib.bib17)\)provides a useful framework unifying many SOTA continuous\-time approaches, such as that ofSonget al\.\([2023](https://arxiv.org/html/2606.05327#bib.bib14)\), consistency trajectory models\(Kimet al\.,[2024](https://arxiv.org/html/2606.05327#bib.bib19)\), shortcut models\(Franset al\.,[2025](https://arxiv.org/html/2606.05327#bib.bib21)\), and Align Your Flow\(Sabouret al\.,[2025](https://arxiv.org/html/2606.05327#bib.bib22)\)\. These models are generally trained via objectives based on one of three self\-consistent conditions required for a valid flow mapΨt1,t2\\Psi\_\{t\_\{1\},t\_\{2\}\}\.

Semigroup condition:ψt2,t3​\(ψt1,t2​\(x\)\)=ψt1,t3​\(x\),\\displaystyle\\quad\\psi\_\{t\_\{2\},t\_\{3\}\}\(\\psi\_\{t\_\{1\},t\_\{2\}\}\(x\)\)=\\psi\_\{t\_\{1\},t\_\{3\}\}\(x\),\(77\)Eulerian condition:∂t1Ψt1,t2​\(x\)\+∇Ψt1,t2​\(x\)​ut1​\(x\)=0,\\displaystyle\\quad\\partial\_\{t\_\{1\}\}\\Psi\_\{t\_\{1\},t\_\{2\}\}\(x\)\+\\nabla\\Psi\_\{t\_\{1\},t\_\{2\}\}\(x\)u\_\{t\_\{1\}\}\(x\)=0,\(78\)Lagrangian condition:∂t2Ψt1,t2​\(x\)=ut2​\(Ψt1,t2​\(x\)\),\\displaystyle\\quad\\partial\_\{t\_\{2\}\}\\Psi\_\{t\_\{1\},t\_\{2\}\}\(x\)=u\_\{t\_\{2\}\}\(\\Psi\_\{t\_\{1\},t\_\{2\}\}\(x\)\),\(79\)for all\(t1,t2,t3\)∈\[0,1\]3\(t\_\{1\},t\_\{2\},t\_\{3\}\)\\in\[0,1\]^\{3\}andx∈ℝdx\\in\\mathbb\{R\}^\{d\}, whereut​\(x\):\[0,1\]×ℝd→ℝdu\_\{t\}\(x\)\\colon\[0,1\]\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}is the instantaneous velocity field satisfyingut​\(Xt\)=d​Xtd​tu\_\{t\}\(X\_\{t\}\)=\\frac\{\\mathrm\{d\}X\_\{t\}\}\{\\mathrm\{d\}t\}along a trajectoryXt2=Ψt1,t2​\(Xt1\)X\_\{t\_\{2\}\}=\\Psi\_\{t\_\{1\},t\_\{2\}\}\(X\_\{t\_\{1\}\}\)\. Shortcut models, for example, are trained to satisfy the semigroup condition, while most other consistency models the Eulerian; the Lagrangian condition is a relatively newer idea introduced byBoffiet al\.\([2025](https://arxiv.org/html/2606.05327#bib.bib16),[2026](https://arxiv.org/html/2606.05327#bib.bib17)\)\. In our experiments, we consider only the Eulerian and Lagrangian conditions for their desirable theoretical properties\(Boffiet al\.,[2026](https://arxiv.org/html/2606.05327#bib.bib17)\)and superior performance in computer vision\.

By substituting the flow map parametrization in terms of the mean velocityvt1,t2:\[0,1\]2×ℝd→ℝdv\_\{t\_\{1\},t\_\{2\}\}\\colon\[0,1\]^\{2\}\\times\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}:

Ψt1,t2​\(Xt1\)=Xt1\+\(t2−t1\)​vt1,t2​\(Xt1\),\\Psi\_\{t\_\{1\},t\_\{2\}\}\(X\_\{t\_\{1\}\}\)=X\_\{t\_\{1\}\}\+\(t\_\{2\}\-t\_\{1\}\)v\_\{t\_\{1\},t\_\{2\}\}\(X\_\{t\_\{1\}\}\),\(80\)we obtain the Eulerian and Lagrangian conditions in terms ofvt1,t2v\_\{t\_\{1\},t\_\{2\}\}:

Eulerian condition:vt1,t2​\(x\)=ut1​\(x\)\+\(t2−t1\)​\(ut1​\(x\)​∂xvt1,t2\+∂t1vt1,t2\)\.\\displaystyle\\quad v\_\{t\_\{1\},t\_\{2\}\}\(x\)=u\_\{t\_\{1\}\}\(x\)\+\(t\_\{2\}\-t\_\{1\}\)\(u\_\{t\_\{1\}\}\\\!\(x\)\\,\\partial\_\{x\}v\_\{t\_\{1\},t\_\{2\}\}\+\\partial\_\{t\_\{1\}\}\\\!v\_\{t\_\{1\},t\_\{2\}\}\)\.\(81\)Lagrangian condition:vt1,t2​\(x\)=ut2​\(Ψt1,t2​\(x\)\)−\(t2−t1\)​∂t2vt1,t2​\(x\)\.\\displaystyle\\quad v\_\{t\_\{1\},t\_\{2\}\}\(x\)=u\_\{t\_\{2\}\}\(\\Psi\_\{t\_\{1\},t\_\{2\}\}\(x\)\)\-\(t\_\{2\}\-t\_\{1\}\)\\partial\_\{t\_\{2\}\}v\_\{t\_\{1\},t\_\{2\}\}\(x\)\.\(82\)
#### Flow map self\-distillation\.

We can next inject a training target, referred to as a “teacher” flow map, into these conditions, to obtain a training objective to*distill*the teacher into a consistency modelΨt1,t2θ\\Psi^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\(or, equivalently,vt1,t2θv^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\)\. Generally, distillation requires a pre\-trained flow matching model as the teacher; however, in this work we focus exclusively on the*self\-distillation*paradigm, wherein the “teacher” target is formed directly from the CFM\-style conditional velocity targetsut​\(x\|z\)u\_\{t\}\(x\|z\)\(modified in Sec\.[4\.1](https://arxiv.org/html/2606.05327#S4.SS1)for OTP\-FM\)\. Explicitly, we consider the Eulerian and Lagrangian self\-distillation \(ESD and LSD\) training objectives:

ℒESD​\(θ\)\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{ESD\}\}\(\\theta\):=𝔼t1,t2,xt1∼ρt1\(⋅\|z\),z∼q∥vt1,t2θ\(xt1\)−\[ut1\(xt1\|z\)\+\(t2−t1\)\(ut1\(xt1\|z\)∂xvt1,t2θ\+∂t1vt1,t2θ\)\]∥2,\\displaystyle:=\\mathbb\{E\}\_\{t\_\{1\},t\_\{2\},\\,x\_\{t\_\{1\}\}\\sim\\rho\_\{t\_\{1\}\}\(\\cdot\|z\),\\,z\\sim q\}\\left\\\|v^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\(x\_\{t\_\{1\}\}\)\-\\left\[u\_\{t\_\{1\}\}\(x\_\{t\_\{1\}\}\|z\)\+\(t\_\{2\}\-t\_\{1\}\)\(u\_\{t\_\{1\}\}\\\!\(x\_\{t\_\{1\}\}\|z\)\\\!\\,\\partial\_\{x\}v^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\+\\partial\_\{t\_\{1\}\}\\\!v^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\)\\right\]\\right\\\|^\{2\},\(83\)ℒLSD​\(θ\)\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{LSD\}\}\(\\theta\):=𝔼t1,t2,xt1∼ρt1\(⋅\|z\),z∼q∥vt1,t2θ\(xt1\)−\[ut2\(Ψt1,t2\(xt1\)\|z\)−\(t2−t1\)∂t2vt1,t2θ\(xt1\)\]∥2,\\displaystyle:=\\mathbb\{E\}\_\{t\_\{1\},t\_\{2\},\\,x\_\{t\_\{1\}\}\\sim\\rho\_\{t\_\{1\}\}\(\\cdot\|z\),\\,z\\sim q\}\\left\\\|v^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\(x\_\{t\_\{1\}\}\)\-\\left\[u\_\{t\_\{2\}\}\(\\Psi\_\{t\_\{1\},t\_\{2\}\}\(x\_\{t\_\{1\}\}\)\|z\)\-\(t\_\{2\}\-t\_\{1\}\)\\partial\_\{t\_\{2\}\}v^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\(x\_\{t\_\{1\}\}\)\\right\]\\right\\\|^\{2\},\(84\)where we see that we aim to minimize the residual between the LHS and RHS of the Eulerian and Lagrangian conditions through a mean\-squared error regression loss, with the ground\-truth training signal injected viaut​\(x\|z\)u\_\{t\}\(x\|z\)\.

As in many \(continuous\-time\) consistency models, this objective involves derivatives ofvθv^\{\\theta\}with respect to its inputs\(x,t1,t2\)\(x,t\_\{1\},t\_\{2\}\); these can be computed efficiently using the Jacobian\-vector product \(JVP\) operation implemented in modern automatic differentiation libraries such as PyTorch and JAX\. Furthermore, to avoid “double\-backpropagation” through the JVP and improve training stability, it is standard practice to not backpropagate through the RHS term, represented by the “stop\-gradient” operatorsg\\mathrm\{sg\}:

ℒESD​\(θ\)\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{ESD\}\}\(\\theta\):=𝔼t1,t2,xt1∼ρt1\(⋅\|z\),z∼q∥vt1,t2θ\(xt1\)−sg\[ut1\(xt1\|z\)\+\(t2−t1\)\(ut1\(xt1\|z\)∂xvt1,t2θ\+∂t1vt1,t2θ\)\]∥2,\\displaystyle:=\\mathbb\{E\}\_\{t\_\{1\},t\_\{2\},\\,x\_\{t\_\{1\}\}\\sim\\rho\_\{t\_\{1\}\}\(\\cdot\|z\),\\,z\\sim q\}\\left\\\|v^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\(x\_\{t\_\{1\}\}\)\-\\mathrm\{sg\}\\left\[u\_\{t\_\{1\}\}\(x\_\{t\_\{1\}\}\|z\)\+\(t\_\{2\}\-t\_\{1\}\)\(u\_\{t\_\{1\}\}\\\!\(x\_\{t\_\{1\}\}\|z\)\\\!\\,\\partial\_\{x\}v^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\+\\partial\_\{t\_\{1\}\}\\\!v^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\)\\right\]\\right\\\|^\{2\},\(85\)ℒLSD​\(θ\)\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{LSD\}\}\(\\theta\):=𝔼t1,t2,xt1∼ρt1\(⋅\|z\),z∼q∥vt1,t2θ\(xt1\)−sg\[ut2\(Ψt1,t2\(xt1\)\|z\)−\(t2−t1\)∂t2vt1,t2θ\(xt1\)\]∥2,\\displaystyle:=\\mathbb\{E\}\_\{t\_\{1\},t\_\{2\},\\,x\_\{t\_\{1\}\}\\sim\\rho\_\{t\_\{1\}\}\(\\cdot\|z\),\\,z\\sim q\}\\left\\\|v^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\(x\_\{t\_\{1\}\}\)\-\\mathrm\{sg\}\\left\[u\_\{t\_\{2\}\}\(\\Psi\_\{t\_\{1\},t\_\{2\}\}\(x\_\{t\_\{1\}\}\)\|z\)\-\(t\_\{2\}\-t\_\{1\}\)\\partial\_\{t\_\{2\}\}v^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\(x\_\{t\_\{1\}\}\)\\right\]\\right\\\|^\{2\},\(86\)We note our stop\-gradient version of LSD differs slightly from the original inBoffiet al\.\([2026](https://arxiv.org/html/2606.05327#bib.bib17)\)with the time\-derivative moved inside thesg\\mathrm\{sg\}operator for further stability\. Unlike ESD, LSD does not involve a spatial derivative throughΨt1,t2\\Psi\_\{t\_\{1\},t\_\{2\}\}or the learnt model, which can in principle improve training stability\. We experiment with both \(using the MeanFlow and improved MeanFlow \(iMF\) instantiations of ESD below\) and find iMF to be most performant, as shown in Apps\.[F](https://arxiv.org/html/2606.05327#A6)and[G](https://arxiv.org/html/2606.05327#A7)\.

### C\.2MeanFlow and improved MeanFlow

The MeanFlow objective is an instantiation of the Eulerian condition that has proven particularly performant; indeed, as of this writing, its improved variant is SOTA in one\- and two\-step generative modeling\. The original formulation focuses on the average velocity object between two time pointst1t\_\{1\}andt2t\_\{2\}:

vt1,t2​\(Xt1\)=1t2−t1​∫t1t2ut​\(Xt\)​dt,v\_\{t\_\{1\},t\_\{2\}\}\(X\_\{t\_\{1\}\}\)=\\frac\{1\}\{t\_\{2\}\-t\_\{1\}\}\\int\_\{t\_\{1\}\}^\{t\_\{2\}\}u\_\{t\}\(X\_\{t\}\)\\mathrm\{d\}t,\(87\)but is equivalent to the flow\-map based description above and in Sec\.[2\.3](https://arxiv.org/html/2606.05327#S2.SS3)\.

One modification we make to the original MeanFlow objective is to privileget1t\_\{1\}instead oft2t\_\{2\}, in order to flowforwardinstead ofbackwardin time\. The corresponding regression target is derived as follows, following the same steps asGenget al\.\([2026a](https://arxiv.org/html/2606.05327#bib.bib18)\)\.

First, multiplying both sides of Eq\.[87](https://arxiv.org/html/2606.05327#A3.E87)byt2−t1t\_\{2\}\-t\_\{1\}:

\(t2−t1\)​vt1,t2​\(Xt1\)=∫t1t2ut​\(Xt\)​dt\.\(t\_\{2\}\-t\_\{1\}\)v\_\{t\_\{1\},t\_\{2\}\}\(X\_\{t\_\{1\}\}\)=\\int\_\{t\_\{1\}\}^\{t\_\{2\}\}u\_\{t\}\(X\_\{t\}\)\\mathrm\{d\}t\.\(88\)Next, taking the derivative of both sides with respect tot1t\_\{1\}\(instead oft2t\_\{2\}\) and rearranging:

vt1,t2​\(Xt1\)=ut1​\(Xt1\)\+\(t2−t1\)​dd​t1​vt1,t2​\(Xt1\)\.v\_\{t\_\{1\},t\_\{2\}\}\(X\_\{t\_\{1\}\}\)=u\_\{t\_\{1\}\}\(X\_\{t\_\{1\}\}\)\+\(t\_\{2\}\-t\_\{1\}\)\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}t\_\{1\}\}v\_\{t\_\{1\},t\_\{2\}\}\(X\_\{t\_\{1\}\}\)\.\(89\)Finally, expanding out the time derivative in terms of the partial derivatives, and usingd​Xt1d​t1=ut1​\(Xt1\)\\frac\{\\mathrm\{d\}X\_\{t\_\{1\}\}\}\{\\mathrm\{d\}t\_\{1\}\}=u\_\{t\_\{1\}\}\(X\_\{t\_\{1\}\}\),d​t1d​t1=1\\frac\{\\mathrm\{d\}t\_\{1\}\}\{\\mathrm\{d\}t\_\{1\}\}=1, andd​t2d​t1=0\\frac\{\\mathrm\{d\}t\_\{2\}\}\{\\mathrm\{d\}t\_\{1\}\}=0, we obtain the final regression target:

vt1,t2​\(Xt1\)=ut1​\(Xt1\)\+\(t2−t1\)​\(ut1​\(Xt1\)​∂xvt1,t2\+∂t1vt1,t2\)\.v\_\{t\_\{1\},t\_\{2\}\}\(X\_\{t\_\{1\}\}\)=u\_\{t\_\{1\}\}\(X\_\{t\_\{1\}\}\)\+\(t\_\{2\}\-t\_\{1\}\)\(u\_\{t\_\{1\}\}\\\!\(X\_\{t\_\{1\}\}\\\!\)\\,\\partial\_\{x\}v\_\{t\_\{1\},t\_\{2\}\}\+\\partial\_\{t\_\{1\}\}\\\!v\_\{t\_\{1\},t\_\{2\}\}\)\.\(90\)As stated above, this yields the exact same regression target as ESD \(Eq\.[85](https://arxiv.org/html/2606.05327#A3.E85)\)\.

More recently,Genget al\.\([2026b](https://arxiv.org/html/2606.05327#bib.bib76)\)propose the “improved MeanFlow” \(iMF\) objective, with the primary difference being the replacement of the second instance ofut1​\(xt1\|z\)u\_\{t\_\{1\}\}\(x\_\{t\_\{1\}\}\|z\)in the MeanFlow target with the instantaneous velocity prediction from the model itselfvt1,t1θ​\(xt1\)v^\{\\theta\}\_\{t\_\{1\},t\_\{1\}\}\(x\_\{t\_\{1\}\}\)\.999Alternatively, they propose the possibility of using an auxiliary head to predict this instantaneous velocity, but we elect to use a single head inputtingt1t\_\{1\}for both time inputs\.Incorporating this and rearranging terms to match their convention, we obtain the iMF objective:

ℒiMF\(θ\):=𝔼t1,t2,xt1∼ρt1\(⋅\|z\),z∼q∥vt1,t2θ​\(xt1\)−\(t2−t1\)​sg​\[vt1,t1θ​\(xt1\)​∂xvt1,t2θ\+∂t1vt1,t2θ\]⏟≡Vt1,t2θ−ut1\(xt1\|z\)∥2,\\mathcal\{L\}\_\{\\mathrm\{iMF\}\}\(\\theta\):=\\mathbb\{E\}\_\{t\_\{1\},t\_\{2\},\\,x\_\{t\_\{1\}\}\\sim\\rho\_\{t\_\{1\}\}\(\\cdot\|z\),\\,z\\sim q\}\\bigg\\\|\\underbrace\{v^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\(x\_\{t\_\{1\}\}\)\-\(t\_\{2\}\-t\_\{1\}\)\\,\\mathrm\{sg\}\\\!\\left\[v^\{\\theta\}\_\{t\_\{1\},t\_\{1\}\}\(x\_\{t\_\{1\}\}\)\\\!\\,\\partial\_\{x\}v^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\+\\partial\_\{t\_\{1\}\}\\\!v^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\\right\]\}\_\{\\equiv V^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}\}\-\\,u\_\{t\_\{1\}\}\(x\_\{t\_\{1\}\}\|z\)\\bigg\\\|^\{2\},\(91\)whereVt1,t2θV^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}is a compound function ofvt1,t2θv^\{\\theta\}\_\{t\_\{1\},t\_\{2\}\}that directly regresses the conditional instantaneous velocity regression target\.Genget al\.\([2026b](https://arxiv.org/html/2606.05327#bib.bib76)\)argue this substitution ofvt1,t1θ​\(xt1\)v^\{\\theta\}\_\{t\_\{1\},t\_\{1\}\}\(x\_\{t\_\{1\}\}\)forut1​\(xt1\|z\)u\_\{t\_\{1\}\}\(x\_\{t\_\{1\}\}\|z\)reduces the regression loss variance and improves training stability\. Empirically, we find this modestly outperforms MeanFlow and LSD, as shown in App\.[G](https://arxiv.org/html/2606.05327#A7)\.

## Appendix DSolutions for Gaussian marginals

For the special case ofdd\-dimensional isotropic Gaussian marginals, we can derive closed\-form solutions for the OTP problem under the ansatz thatρt\\rho\_\{t\}is an isotropic Gaussian with time\-dependent meanmρtm\_\{\\rho\_\{t\}\}and varianceσρt2​𝟙d\\sigma\_\{\\rho\_\{t\}\}^\{2\}\\mathbbm\{1\}\_\{d\}\. In this case, followingLipmanet al\.\([2023](https://arxiv.org/html/2606.05327#bib.bib1)\), the marginal velocity takes the form:

ut​\(x\)=σ˙ρtσρt​\(x−mρt\)\+m˙ρt\.u\_\{t\}\(x\)=\\frac\{\\dot\{\\sigma\}\_\{\\rho\_\{t\}\}\}\{\\sigma\_\{\\rho\_\{t\}\}\}\(x\-m\_\{\\rho\_\{t\}\}\)\+\\dot\{m\}\_\{\\rho\_\{t\}\}\.\(92\)This means the dynamic OT kinetic energy reduces to:

T​\(x\)=12​∫‖ut​\(x\)‖2​dρt​\(x\)\\displaystyle T\(x\)=\\frac\{1\}\{2\}\\int\\\|u\_\{t\}\(x\)\\\|^\{2\}\\mathrm\{d\}\\rho\_\{t\}\(x\)=12​𝔼​‖ut​\(x\)‖2\\displaystyle=\\frac\{1\}\{2\}\\mathbb\{E\}\\\|u\_\{t\}\(x\)\\\|^\{2\}\(93\)=12​\[σ˙ρt2σρt2​𝔼​‖x−mρt‖2⏟=tr​\(σρt2​𝟙d\)⁣=d​σρt2\+2⋅σ˙ρtσρt⋅m˙ρt⋅𝔼​\(x−mρt\)⏟=0\+‖m˙ρt‖2\]\\displaystyle=\\frac\{1\}\{2\}\\Bigg\[\\frac\{\\dot\{\\sigma\}\_\{\\rho\_\{t\}\}^\{2\}\}\{\\sigma\_\{\\rho\_\{t\}\}^\{2\}\}\\underbrace\{\\mathbb\{E\}\\\|x\-m\_\{\\rho\_\{t\}\}\\\|^\{2\}\}\_\{=\\mathrm\{tr\}\(\\sigma\_\{\\rho\_\{t\}\}^\{2\}\\mathbbm\{1\}\_\{d\}\)=d\\sigma\_\{\\rho\_\{t\}\}^\{2\}\}\+2\\cdot\\frac\{\\dot\{\\sigma\}\_\{\\rho\_\{t\}\}\}\{\\sigma\_\{\\rho\_\{t\}\}\}\\cdot\\dot\{m\}\_\{\\rho\_\{t\}\}\\cdot\\underbrace\{\\mathbb\{E\}\(x\-m\_\{\\rho\_\{t\}\}\)\}\_\{=0\}\+\\\|\\dot\{m\}\_\{\\rho\_\{t\}\}\\\|^\{2\}\\Bigg\]=12​\(‖m˙ρt‖2\+d​σ˙ρt2\)\.\\displaystyle=\\frac\{1\}\{2\}\(\\\|\\dot\{m\}\_\{\\rho\_\{t\}\}\\\|^\{2\}\+d\\dot\{\\sigma\}\_\{\\rho\_\{t\}\}^\{2\}\)\.Each distance𝒟tk≡𝒟​\[ρt,μtk\]\\mathcal\{D\}^\{k\}\_\{t\}\\equiv\\mathcal\{D\}\[\\rho\_\{t\},\\mu\_\{t\_\{k\}\}\]similarly reduces to a function of\(mρt,σρt\)\(m\_\{\\rho\_\{t\}\},\\sigma\_\{\\rho\_\{t\}\}\)alone, as derived explicitly for different distance metrics below\. In general, this means a Lagrangian of the form:

ℒ​\(mρt,σρt,m˙ρt,σ˙ρt,t\)=12​\(‖m˙ρt‖2\+d​σ˙ρt2\)\+∑k=1Kwk​λk​\(t\)​𝒟tk,\\mathcal\{L\}\(m\_\{\\rho\_\{t\}\},\\sigma\_\{\\rho\_\{t\}\},\\dot\{m\}\_\{\\rho\_\{t\}\},\\dot\{\\sigma\}\_\{\\rho\_\{t\}\},t\)=\\frac\{1\}\{2\}\\left\(\\\|\\dot\{m\}\_\{\\rho\_\{t\}\}\\\|^\{2\}\+d\\dot\{\\sigma\}\_\{\\rho\_\{t\}\}^\{2\}\\right\)\+\\sum\_\{k=1\}^\{K\}w\_\{k\}\\lambda\_\{k\}\(t\)\\,\\mathcal\{D\}^\{k\}\_\{t\},\(94\)where the distance𝒟tk\\mathcal\{D\}^\{k\}\_\{t\}depends only on\(mρt,σρt\)\(m\_\{\\rho\_\{t\}\},\\sigma\_\{\\rho\_\{t\}\}\)and the fixed parameters\(mμtk,σμtk\)\(m\_\{\\mu\_\{t\_\{k\}\}\},\\sigma\_\{\\mu\_\{t\_\{k\}\}\}\)of the intermediate marginal\. The E\-L equations for the generalized coordinates\(mρt,σρt\)\(m\_\{\\rho\_\{t\}\},\\sigma\_\{\\rho\_\{t\}\}\)are therefore:

m¨ρt\\displaystyle\\ddot\{m\}\_\{\\rho\_\{t\}\}=∑k=1Kwk​λk​\(t\)​∇mρt𝒟tk,\\displaystyle=\\sum\_\{k=1\}^\{K\}w\_\{k\}\\lambda\_\{k\}\(t\)\\,\\nabla\_\{m\_\{\\rho\_\{t\}\}\}\\mathcal\{D\}^\{k\}\_\{t\},\(95\)σ¨ρt\\displaystyle\\ddot\{\\sigma\}\_\{\\rho\_\{t\}\}=1d​∑k=1Kwk​λk​\(t\)​∂𝒟tk∂σρt\.\\displaystyle=\\frac\{1\}\{d\}\\sum\_\{k=1\}^\{K\}w\_\{k\}\\lambda\_\{k\}\(t\)\\,\\frac\{\\partial\\mathcal\{D\}^\{k\}\_\{t\}\}\{\\partial\\sigma\_\{\\rho\_\{t\}\}\}\.\(96\)
By plugging in the explicit forms for the distances and gradients derived below, this yields second\-order ODEs for the time evolution ofmρtm\_\{\\rho\_\{t\}\}andσρt\\sigma\_\{\\rho\_\{t\}\}that can be efficiently simulated numerically\. To solve the boundary value problem, matchingmρ0=mμ0m\_\{\\rho\_\{0\}\}=m\_\{\\mu\_\{0\}\},σρ0=σμ0\\sigma\_\{\\rho\_\{0\}\}=\\sigma\_\{\\mu\_\{0\}\},mρ1=mμ1m\_\{\\rho\_\{1\}\}=m\_\{\\mu\_\{1\}\}, andσρ1=σμ1\\sigma\_\{\\rho\_\{1\}\}=\\sigma\_\{\\mu\_\{1\}\}, we use the shooting method\(Stoer and Bulirsch,[1980](https://arxiv.org/html/2606.05327#bib.bib32)\)to reduce it to finding the initial conditions that solve an initial value problem yielding the desired final conditions\. Specifically, we use an implicit Runge\-Kutta method\(Hairer and Wanner,[1996](https://arxiv.org/html/2606.05327#bib.bib34)\)\(theRadaumethod implemented in thescipy\.integratelibrary\(Virtanenet al\.,[2020](https://arxiv.org/html/2606.05327#bib.bib35)\)\) to simulate the E\-L equations as first\-order ODEs in the velocities of the mean and standard deviation, and “Broyden’s good method”\(van de Rotten,[2003](https://arxiv.org/html/2606.05327#bib.bib33)\)\(scipy\.optimize\.broyden1\) to solve for the initial velocities\. The exact code is provided in the linkedOTP\-FMrepository\.

### D\.1The𝒲22\\mathcal\{W\}^\{2\}\_\{2\}distance

The𝒲22\\mathcal\{W\}^\{2\}\_\{2\}distance between two Gaussiansρt\\rho\_\{t\}andμtk\\mu\_\{t\_\{k\}\}has the well\-known form — also known as the Fréchet distance — that reduces in the case of isotropic Gaussians to:

𝒟𝒲22​\[ρt,μtk\]=‖mρt−mμtk‖2\+d​\(σρt−σμtk\)2\.\\mathcal\{D\}\_\{\\mathcal\{W\}^\{2\}\_\{2\}\}\[\\rho\_\{t\},\\mu\_\{t\_\{k\}\}\]=\\\|m\_\{\\rho\_\{t\}\}\-m\_\{\\mu\_\{t\_\{k\}\}\}\\\|^\{2\}\+d\(\\sigma\_\{\\rho\_\{t\}\}\-\\sigma\_\{\\mu\_\{t\_\{k\}\}\}\)^\{2\}\.\(97\)The gradients are simply:

∇mρt𝒟tk=2​\(mρt−mμtk\),\\displaystyle\\nabla\_\{m\_\{\\rho\_\{t\}\}\}\\mathcal\{D\}^\{k\}\_\{t\}=2\(m\_\{\\rho\_\{t\}\}\-m\_\{\\mu\_\{t\_\{k\}\}\}\),\(98\)∂𝒟tk∂σρt=2​d​\(σρt−σμtk\)\.\\displaystyle\\frac\{\\partial\\mathcal\{D\}^\{k\}\_\{t\}\}\{\\partial\\sigma\_\{\\rho\_\{t\}\}\}=2d\(\\sigma\_\{\\rho\_\{t\}\}\-\\sigma\_\{\\mu\_\{t\_\{k\}\}\}\)\.\(99\)

### D\.2KL divergence

The KL divergence between two Gaussiansρt\\rho\_\{t\}andμtk\\mu\_\{t\_\{k\}\}similarly has a known form\(Pardo,[2006](https://arxiv.org/html/2606.05327#bib.bib31)\), reducing to:

𝒟KL​\[ρt∥μtk\]=12​\[d​ln⁡σμtk2σρt2\+d​σρt2σμtk2\+‖mρt−mμtk‖2σμtk2−d\],\\mathcal\{D\}\_\{\\mathrm\{KL\}\}\[\\rho\_\{t\}\\\|\\mu\_\{t\_\{k\}\}\]=\\frac\{1\}\{2\}\\left\[d\\ln\\frac\{\\sigma\_\{\\mu\_\{t\_\{k\}\}\}^\{2\}\}\{\\sigma\_\{\\rho\_\{t\}\}^\{2\}\}\+d\\frac\{\\sigma\_\{\\rho\_\{t\}\}^\{2\}\}\{\\sigma\_\{\\mu\_\{t\_\{k\}\}\}^\{2\}\}\+\\frac\{\\\|m\_\{\\rho\_\{t\}\}\-m\_\{\\mu\_\{t\_\{k\}\}\}\\\|^\{2\}\}\{\\sigma\_\{\\mu\_\{t\_\{k\}\}\}^\{2\}\}\-d\\right\],\(100\)with gradients:

∇mρt𝒟tk\\displaystyle\\nabla\_\{m\_\{\\rho\_\{t\}\}\}\\mathcal\{D\}^\{k\}\_\{t\}=mρt−mμtkσμtk2,\\displaystyle=\\frac\{m\_\{\\rho\_\{t\}\}\-m\_\{\\mu\_\{t\_\{k\}\}\}\}\{\\sigma\_\{\\mu\_\{t\_\{k\}\}\}^\{2\}\},\(101\)∂𝒟tk∂σρt\\displaystyle\\frac\{\\partial\\mathcal\{D\}^\{k\}\_\{t\}\}\{\\partial\\sigma\_\{\\rho\_\{t\}\}\}=d​\(σρtσμtk2−1σρt\)\.\\displaystyle=d\\left\(\\frac\{\\sigma\_\{\\rho\_\{t\}\}\}\{\\sigma\_\{\\mu\_\{t\_\{k\}\}\}^\{2\}\}\-\\frac\{1\}\{\\sigma\_\{\\rho\_\{t\}\}\}\\right\)\.\(102\)

### D\.3MMD \(RBF kernel\)

We first consider MMD with the RBF kernel \(Eq\.[70](https://arxiv.org/html/2606.05327#A2.E70)\) with bandwidthγ\\gamma\. The squared MMD between two distributions is \(Eq\.[64](https://arxiv.org/html/2606.05327#A2.E64)\):

𝒟MMD2​\[ρt,μtk\]=𝔼x,x′∼ρt​\[k​\(x,x′\)\]−2​𝔼x∼ρt,y∼μtk​\[k​\(x,y\)\]\+𝔼y,y′∼μtk​\[k​\(y,y′\)\]\.\\mathcal\{D\}\_\{\\mathrm\{MMD\}^\{2\}\}\[\\rho\_\{t\},\\mu\_\{t\_\{k\}\}\]=\\mathbb\{E\}\_\{x,x^\{\\prime\}\\sim\\rho\_\{t\}\}\[k\(x,x^\{\\prime\}\)\]\-2\\mathbb\{E\}\_\{x\\sim\\rho\_\{t\},y\\sim\\mu\_\{t\_\{k\}\}\}\[k\(x,y\)\]\+\\mathbb\{E\}\_\{y,y^\{\\prime\}\\sim\\mu\_\{t\_\{k\}\}\}\[k\(y,y^\{\\prime\}\)\]\.\(103\)
For the self\-terms, ifx,x′∼𝒩​\(m,σ2​Id\)x,x^\{\\prime\}\\sim\\mathcal\{N\}\(m,\\sigma^\{2\}I\_\{d\}\)are independent, thenx−x′∼𝒩​\(0,2​σ2​Id\)x\-x^\{\\prime\}\\sim\\mathcal\{N\}\(0,2\\sigma^\{2\}I\_\{d\}\)\. Using the moment generating function of a chi\-squared distribution:

𝔼x,x′∼𝒩​\(m,σ2​I\)​\[k​\(x,x′\)\]=𝔼​\[exp⁡\(−‖x−x′‖22​γ2\)\]=\(γ2γ2\+2​σ2\)d/2\.\\mathbb\{E\}\_\{x,x^\{\\prime\}\\sim\\mathcal\{N\}\(m,\\sigma^\{2\}I\)\}\[k\(x,x^\{\\prime\}\)\]=\\mathbb\{E\}\\left\[\\exp\\left\(\-\\frac\{\\\|x\-x^\{\\prime\}\\\|^\{2\}\}\{2\\gamma^\{2\}\}\\right\)\\right\]=\\left\(\\frac\{\\gamma^\{2\}\}\{\\gamma^\{2\}\+2\\sigma^\{2\}\}\\right\)^\{d/2\}\.\(104\)
For the cross\-term, ifx∼𝒩​\(m1,σ12​Id\)x\\sim\\mathcal\{N\}\(m\_\{1\},\\sigma\_\{1\}^\{2\}I\_\{d\}\)andy∼𝒩​\(m2,σ22​Id\)y\\sim\\mathcal\{N\}\(m\_\{2\},\\sigma\_\{2\}^\{2\}I\_\{d\}\)are independent, thenx−y∼𝒩​\(m1−m2,\(σ12\+σ22\)​Id\)x\-y\\sim\\mathcal\{N\}\(m\_\{1\}\-m\_\{2\},\(\\sigma\_\{1\}^\{2\}\+\\sigma\_\{2\}^\{2\}\)I\_\{d\}\)\. Hence:

𝔼x∼ρt,y∼μtk​\[k​\(x,y\)\]=\(γ2γ2\+σρt2\+σμtk2\)d/2​exp⁡\(−‖mρt−mμtk‖22​\(γ2\+σρt2\+σμtk2\)\)\.\\mathbb\{E\}\_\{x\\sim\\rho\_\{t\},y\\sim\\mu\_\{t\_\{k\}\}\}\[k\(x,y\)\]=\\left\(\\frac\{\\gamma^\{2\}\}\{\\gamma^\{2\}\+\\sigma\_\{\\rho\_\{t\}\}^\{2\}\+\\sigma\_\{\\mu\_\{t\_\{k\}\}\}^\{2\}\}\\right\)^\{d/2\}\\exp\\left\(\-\\frac\{\\\|m\_\{\\rho\_\{t\}\}\-m\_\{\\mu\_\{t\_\{k\}\}\}\\\|^\{2\}\}\{2\(\\gamma^\{2\}\+\\sigma\_\{\\rho\_\{t\}\}^\{2\}\+\\sigma\_\{\\mu\_\{t\_\{k\}\}\}^\{2\}\)\}\\right\)\.\(105\)
Defining the effective variancesτρ2=γ2\+2​σρt2\\tau\_\{\\rho\}^\{2\}=\\gamma^\{2\}\+2\\sigma\_\{\\rho\_\{t\}\}^\{2\},τμ2=γ2\+2​σμtk2\\tau\_\{\\mu\}^\{2\}=\\gamma^\{2\}\+2\\sigma\_\{\\mu\_\{t\_\{k\}\}\}^\{2\}, andτ\+2=γ2\+σρt2\+σμtk2\\tau\_\{\+\}^\{2\}=\\gamma^\{2\}\+\\sigma\_\{\\rho\_\{t\}\}^\{2\}\+\\sigma\_\{\\mu\_\{t\_\{k\}\}\}^\{2\}, and writingΔ​m=mρt−mμtk\\Delta m=m\_\{\\rho\_\{t\}\}\-m\_\{\\mu\_\{t\_\{k\}\}\}:

𝒟MMD2​\[ρt,μtk\]=\(γ2τρ2\)d/2\+\(γ2τμ2\)d/2−2​\(γ2τ\+2\)d/2​exp⁡\(−‖Δ​m‖22​τ\+2\)\.\\mathcal\{D\}\_\{\\mathrm\{MMD\}^\{2\}\}\[\\rho\_\{t\},\\mu\_\{t\_\{k\}\}\]=\\left\(\\frac\{\\gamma^\{2\}\}\{\\tau\_\{\\rho\}^\{2\}\}\\right\)^\{d/2\}\+\\left\(\\frac\{\\gamma^\{2\}\}\{\\tau\_\{\\mu\}^\{2\}\}\\right\)^\{d/2\}\-2\\left\(\\frac\{\\gamma^\{2\}\}\{\\tau\_\{\+\}^\{2\}\}\\right\)^\{d/2\}\\exp\\left\(\-\\frac\{\\\|\\Delta m\\\|^\{2\}\}\{2\\tau\_\{\+\}^\{2\}\}\\right\)\.\(106\)
Finally, the gradients are:

∇mρt𝒟MMD2\\displaystyle\\nabla\_\{m\_\{\\rho\_\{t\}\}\}\\mathcal\{D\}\_\{\\mathrm\{MMD\}^\{2\}\}=2​Δ​mτ\+2​\(γ2τ\+2\)d/2​exp⁡\(−‖Δ​m‖22​τ\+2\),\\displaystyle=\\frac\{2\\Delta m\}\{\\tau\_\{\+\}^\{2\}\}\\left\(\\frac\{\\gamma^\{2\}\}\{\\tau\_\{\+\}^\{2\}\}\\right\)^\{d/2\}\\exp\\left\(\-\\frac\{\\\|\\Delta m\\\|^\{2\}\}\{2\\tau\_\{\+\}^\{2\}\}\\right\),\(107\)∂𝒟MMD2∂σρt\\displaystyle\\frac\{\\partial\\mathcal\{D\}\_\{\\mathrm\{MMD\}^\{2\}\}\}\{\\partial\\sigma\_\{\\rho\_\{t\}\}\}=−2​d​σρt​\(γ2τρ2\)d/2​1τρ2\+2​σρt​\(γ2τ\+2\)d/2​exp⁡\(−‖Δ​m‖22​τ\+2\)​\[dτ\+2−‖Δ​m‖2τ\+4\]\.\\displaystyle=\-2d\\sigma\_\{\\rho\_\{t\}\}\\left\(\\frac\{\\gamma^\{2\}\}\{\\tau\_\{\\rho\}^\{2\}\}\\right\)^\{d/2\}\\frac\{1\}\{\\tau\_\{\\rho\}^\{2\}\}\+2\\sigma\_\{\\rho\_\{t\}\}\\left\(\\frac\{\\gamma^\{2\}\}\{\\tau\_\{\+\}^\{2\}\}\\right\)^\{d/2\}\\exp\\left\(\-\\frac\{\\\|\\Delta m\\\|^\{2\}\}\{2\\tau\_\{\+\}^\{2\}\}\\right\)\\left\[\\frac\{d\}\{\\tau\_\{\+\}^\{2\}\}\-\\frac\{\\\|\\Delta m\\\|^\{2\}\}\{\\tau\_\{\+\}^\{4\}\}\\right\]\.\(108\)

### D\.4MMD \(polynomial kernel\)

![Refer to caption](https://arxiv.org/html/2606.05327v1/x5.png)Figure 4:A characteristic example of the exact dynamic OT solution \(under the Gaussian ansatz\) with MMD and a polynomial kernel failing to enforce the intermediate marginal at high strength\.We next consider the polynomial kernel \(Eq\.[71](https://arxiv.org/html/2606.05327#A2.E71)\) with degree 2\. Higher\-degree kernels can be computed similarly but are more cumbersome\.

As the quadratic kernel isk​\(x,y\)=\(x⊤​y\+c\)2k\(x,y\)=\(x^\{\\top\}y\+c\)^\{2\}, we first compute𝔼​\[\(x⊤​y\)2\]\\mathbb\{E\}\[\(x^\{\\top\}y\)^\{2\}\]\. For independentx,x′∼𝒩​\(m,σ2​Id\)x,x^\{\\prime\}\\sim\\mathcal\{N\}\(m,\\sigma^\{2\}I\_\{d\}\):

𝔼​\[\(x⊤​x′\)2\]=‖m‖4\+d​σ4\+2​σ2​‖m‖2\.\\mathbb\{E\}\[\(x^\{\\top\}x^\{\\prime\}\)^\{2\}\]=\\\|m\\\|^\{4\}\+d\\sigma^\{4\}\+2\\sigma^\{2\}\\\|m\\\|^\{2\}\.\(109\)For independentx∼𝒩​\(m1,σ12​Id\)x\\sim\\mathcal\{N\}\(m\_\{1\},\\sigma\_\{1\}^\{2\}I\_\{d\}\)andy∼𝒩​\(m2,σ22​Id\)y\\sim\\mathcal\{N\}\(m\_\{2\},\\sigma\_\{2\}^\{2\}I\_\{d\}\):

𝔼​\[\(x⊤​y\)2\]=\(m1⊤​m2\)2\+d​σ12​σ22\+σ12​‖m2‖2\+σ22​‖m1‖2\.\\mathbb\{E\}\[\(x^\{\\top\}y\)^\{2\}\]=\(m\_\{1\}^\{\\top\}m\_\{2\}\)^\{2\}\+d\\sigma\_\{1\}^\{2\}\\sigma\_\{2\}^\{2\}\+\\sigma\_\{1\}^\{2\}\\\|m\_\{2\}\\\|^\{2\}\+\\sigma\_\{2\}^\{2\}\\\|m\_\{1\}\\\|^\{2\}\.\(110\)Hence, combining all terms with𝔼​\[\(x⊤​y\+c\)2\]=𝔼​\[\(x⊤​y\)2\]\+2​c​𝔼​\[x⊤​y\]\+c2\\mathbb\{E\}\[\(x^\{\\top\}y\+c\)^\{2\}\]=\\mathbb\{E\}\[\(x^\{\\top\}y\)^\{2\}\]\+2c\\mathbb\{E\}\[x^\{\\top\}y\]\+c^\{2\}:

𝒟MMD2​\[ρt,μtk\]\\displaystyle\\mathcal\{D\}\_\{\\mathrm\{MMD\}^\{2\}\}\[\\rho\_\{t\},\\mu\_\{t\_\{k\}\}\]=‖mρt‖4\+d​σρt4\+2​σρt2​‖mρt‖2\+2​c​‖mρt‖2\+c2\\displaystyle=\\\|m\_\{\\rho\_\{t\}\}\\\|^\{4\}\+d\\sigma\_\{\\rho\_\{t\}\}^\{4\}\+2\\sigma\_\{\\rho\_\{t\}\}^\{2\}\\\|m\_\{\\rho\_\{t\}\}\\\|^\{2\}\+2c\\\|m\_\{\\rho\_\{t\}\}\\\|^\{2\}\+c^\{2\}\(111\)−2​\[\(mρt⊤​mμtk\)2\+d​σρt2​σμtk2\+σρt2​‖mμtk‖2\+σμtk2​‖mρt‖2\+2​c​mρt⊤​mμtk\+c2\]\\displaystyle\\quad\-2\\Big\[\(m\_\{\\rho\_\{t\}\}^\{\\top\}m\_\{\\mu\_\{t\_\{k\}\}\}\)^\{2\}\+d\\sigma\_\{\\rho\_\{t\}\}^\{2\}\\sigma\_\{\\mu\_\{t\_\{k\}\}\}^\{2\}\+\\sigma\_\{\\rho\_\{t\}\}^\{2\}\\\|m\_\{\\mu\_\{t\_\{k\}\}\}\\\|^\{2\}\+\\sigma\_\{\\mu\_\{t\_\{k\}\}\}^\{2\}\\\|m\_\{\\rho\_\{t\}\}\\\|^\{2\}\+2cm\_\{\\rho\_\{t\}\}^\{\\top\}m\_\{\\mu\_\{t\_\{k\}\}\}\+c^\{2\}\\Big\]\+‖mμtk‖4\+d​σμtk4\+2​σμtk2​‖mμtk‖2\+2​c​‖mμtk‖2\+c2\.\\displaystyle\\quad\+\\\|m\_\{\\mu\_\{t\_\{k\}\}\}\\\|^\{4\}\+d\\sigma\_\{\\mu\_\{t\_\{k\}\}\}^\{4\}\+2\\sigma\_\{\\mu\_\{t\_\{k\}\}\}^\{2\}\\\|m\_\{\\mu\_\{t\_\{k\}\}\}\\\|^\{2\}\+2c\\\|m\_\{\\mu\_\{t\_\{k\}\}\}\\\|^\{2\}\+c^\{2\}\.Finally, the gradients are:

∇mρt𝒟MMD2\\displaystyle\\nabla\_\{m\_\{\\rho\_\{t\}\}\}\\mathcal\{D\}\_\{\\mathrm\{MMD\}^\{2\}\}=4​\[\(‖mρt‖2\+σρt2−σμtk2\+c\)​mρt−\(mρt⊤​mμtk\+c\)​mμtk\],\\displaystyle=4\\Big\[\(\\\|m\_\{\\rho\_\{t\}\}\\\|^\{2\}\+\\sigma\_\{\\rho\_\{t\}\}^\{2\}\-\\sigma\_\{\\mu\_\{t\_\{k\}\}\}^\{2\}\+c\)m\_\{\\rho\_\{t\}\}\-\(m\_\{\\rho\_\{t\}\}^\{\\top\}m\_\{\\mu\_\{t\_\{k\}\}\}\+c\)m\_\{\\mu\_\{t\_\{k\}\}\}\\Big\],\(112\)∂𝒟MMD2∂σρt\\displaystyle\\frac\{\\partial\\mathcal\{D\}\_\{\\mathrm\{MMD\}^\{2\}\}\}\{\\partial\\sigma\_\{\\rho\_\{t\}\}\}=4​σρt​\[d​\(σρt2−σμtk2\)\+\(‖mρt‖2−‖mμtk‖2\)\]\.\\displaystyle=4\\sigma\_\{\\rho\_\{t\}\}\\Big\[d\(\\sigma\_\{\\rho\_\{t\}\}^\{2\}\-\\sigma\_\{\\mu\_\{t\_\{k\}\}\}^\{2\}\)\+\(\\\|m\_\{\\rho\_\{t\}\}\\\|^\{2\}\-\\\|m\_\{\\mu\_\{t\_\{k\}\}\}\\\|^\{2\}\)\\Big\]\.\(113\)
We find this polynomial kernel to not be performant, even for this simple case of Gaussian marginals\. A characteristic example is shown in Fig\.[4](https://arxiv.org/html/2606.05327#A4.F4), for the same marginal configuration as in the second MMD RBF example in Fig\.[2](https://arxiv.org/html/2606.05327#S3.F2)\. As we see, while the force of the polynomial kernel drives the mean as expected, it is not as effective on the variance; hence, we do not pursue it further in this work\.

## Appendix EDetails of the OTP\-FM algorithm

![Refer to caption](https://arxiv.org/html/2606.05327v1/x6.png)

![Refer to caption](https://arxiv.org/html/2606.05327v1/x7.png)

![Refer to caption](https://arxiv.org/html/2606.05327v1/x8.png)

Figure 5:Left:Example loss curves andα​\(i\)\\alpha\(i\)schedule\.Center:Progression of target conditional trajectories during training for the𝒲22\\mathcal\{W\}^\{2\}\_\{2\}distance based on example samples from four 1D marginals\.Right:Example time sampling distributionpt1,t2p\_\{t\_\{1\},t\_\{2\}\}\.### E\.1Time sampling

We sample the training timest1t\_\{1\}andt2t\_\{2\}according to the joint distribution:

pt1,t2=1K\+1​∑k=0K𝟙\[tk,tk\+1\]​\(t1\)tk\+1−tk⋅𝟙\[t1,1\]​\(t2\)1−t1;p\_\{t\_\{1\},t\_\{2\}\}=\\frac\{1\}\{K\+1\}\\sum\_\{k=0\}^\{K\}\\frac\{\\mathbbm\{1\}\_\{\[t\_\{k\},t\_\{k\+1\}\]\}\(t\_\{1\}\)\}\{t\_\{k\+1\}\-t\_\{k\}\}\\cdot\\frac\{\\mathbbm\{1\}\_\{\[t\_\{1\},1\]\}\(t\_\{2\}\)\}\{1\-t\_\{1\}\};\(114\)i\.e\., uniformly samplingt1t\_\{1\}but with each of theK\+1K\+1intervals between consecutive marginals weighted equally, andt2t\_\{2\}uniformly on the interval\[t1,1\]\[t\_\{1\},1\], so thatt2\>t1t\_\{2\}\>t\_\{1\}by construction \(Fig\.[5](https://arxiv.org/html/2606.05327#A5.F5)\)\. This follows the convention of most consistency models of training only the upper or lower triangle ofvθ​\(t1,t2\)v^\{\\theta\}\(t\_\{1\},t\_\{2\}\); however, we note there is no requirement in general to restrictt2\>t1t\_\{2\}\>t\_\{1\}and, indeed, allowingt2<t1t\_\{2\}<t\_\{1\}may be desirable in some applications for flowing “backwards”\. We use positional embeddings\(Vaswaniet al\.,[2017](https://arxiv.org/html/2606.05327#bib.bib23)\)to encodet1t\_\{1\}andΔ​t=t2−t1\\Delta t=t\_\{2\}\-t\_\{1\}as inputs to the model\.

### E\.2Fixed\-point iteration convergence and homotopy continuation

As stated in Sec\.[4\.3](https://arxiv.org/html/2606.05327#S4.SS3), the curriculum on the correction strength in OTP\-FM is an application of the homotopy continuation method\(Allgower and Georg,[2003](https://arxiv.org/html/2606.05327#bib.bib24)\)\. Here, we analyze the convergence of the FP iterations and why the homotopy curriculumα​\(i\)\\alpha\(i\)helps stabilize them\.

#### Contraction analysis

Stacking theKKintermediate positions asXT≡\[Xt1,…,XtK\]⊤X\_\{T\}\\equiv\[X\_\{t\_\{1\}\},\\dots,X\_\{t\_\{K\}\}\]^\{\\top\}, the FP iterations of Eq\.[21](https://arxiv.org/html/2606.05327#S4.E21)can be written asXT\(m\+1\)=Fα​\(XT\(m\)\)X\_\{T\}^\{\(m\+1\)\}=F\_\{\\alpha\}\(X\_\{T\}^\{\(m\)\}\)with

Fα​\(XT\)i=Xtibase\+α​∑k=1KAi​k​∇gk​\(Xtk,tk,ℬ\),F\_\{\\alpha\}\(X\_\{T\}\)\_\{i\}=X\_\{t\_\{i\}\}^\{\\mathrm\{base\}\}\+\\alpha\\sum\_\{k=1\}^\{K\}A\_\{ik\}\\,\\nabla g\_\{k\}\(X\_\{t\_\{k\}\},t\_\{k\},\\mathcal\{B\}\),\(115\)whereAi​kA\_\{ik\}is as in Eq\.[20](https://arxiv.org/html/2606.05327#S4.E20)\. If each force∇gk​\(⋅,tk,ℬ\)\\nabla g\_\{k\}\(\\cdot,t\_\{k\},\\mathcal\{B\}\)is Lipschitz\-continuous with constantLkL\_\{k\}, then the Jacobian ofFαF\_\{\\alpha\}is uniformly bounded:

‖J​Fα‖≤α​maxi​∑k=1K\|Ai​k\|​Lk≡LF\.\\\|JF\_\{\\alpha\}\\\|\\leq\\alpha\\max\_\{i\}\\sum\_\{k=1\}^\{K\}\|A\_\{ik\}\|\\,L\_\{k\}\\equiv L\_\{F\}\.\(116\)By the Banach fixed\-point theorem, wheneverLF<1L\_\{F\}<1,FαF\_\{\\alpha\}is a contraction and the iterationsXT\(m\)X\_\{T\}^\{\(m\)\}converge linearly to a unique fixed point at rate at mostLFL\_\{F\}\.

#### Lipschitz constants

We can computeLkL\_\{k\}directly from the Jacobians of the gradient estimators in Table[1](https://arxiv.org/html/2606.05327#S3.T1):

- •𝒲22\\mathcal\{W\}^\{2\}\_\{2\}:∇gk​\(Xtk,tk,ℬ\)=Xtk−xtk\\nabla g\_\{k\}\(X\_\{t\_\{k\}\},t\_\{k\},\\mathcal\{B\}\)=X\_\{t\_\{k\}\}\-x\_\{t\_\{k\}\}withxtkx\_\{t\_\{k\}\}fixed \(Sec\.[4\.2](https://arxiv.org/html/2606.05327#S4.SS2)\), soLk𝒲22=1L\_\{k\}^\{\\mathcal\{W\}^\{2\}\_\{2\}\}=1andFαF\_\{\\alpha\}is in fact linear inXTX\_\{T\}; we solve it directly by matrix inversion \(App\.[E\.3](https://arxiv.org/html/2606.05327#A5.SS3)\) and FP iterations are unnecessary\.
- •𝒟MMD2\\mathcal\{D\}\_\{\\mathrm\{MMD\}^\{2\}\}with an RBF kernelkσk​\(x,y\)=exp⁡\(−‖x−y‖2/2​σk2\)k\_\{\\sigma\_\{k\}\}\(x,y\)=\\exp\(\-\\\|x\-y\\\|^\{2\}/2\\sigma\_\{k\}^\{2\}\):LkMMD≤4/σk2L\_\{k\}^\{\\mathrm\{MMD\}\}\\leq 4/\\sigma\_\{k\}^\{2\}\.
- •𝒟KL\\mathcal\{D\}\_\{\\mathrm\{KL\}\}with a KDE score estimator:LkKL≲supx‖J​∇ln⁡ρ^tk​\(x\)‖\+supx‖J​∇ln⁡μ^tk​\(x\)‖L\_\{k\}^\{\\mathrm\{KL\}\}\\lesssim\\sup\_\{x\}\\\|J\\nabla\\ln\\hat\{\\rho\}\_\{t\_\{k\}\}\(x\)\\\|\+\\sup\_\{x\}\\\|J\\nabla\\ln\\hat\{\\mu\}\_\{t\_\{k\}\}\(x\)\\\|, controlled by the KDE bandwidth and diverging in low\-density regions\. The Gaussian\-score variant \(Sec\.[4\.2](https://arxiv.org/html/2606.05327#S4.SS2)\) instead yieldsLkKL,Gauss≤‖Σρtk−1‖2\+‖Σμtk−1‖2L\_\{k\}^\{\\mathrm\{KL,Gauss\}\}\\leq\\\|\\Sigma\_\{\\rho\_\{t\_\{k\}\}\}^\{\-1\}\\\|\_\{2\}\+\\\|\\Sigma\_\{\\mu\_\{t\_\{k\}\}\}^\{\-1\}\\\|\_\{2\}, finite away from degenerate covariances\.

The𝒟MMD2\\mathcal\{D\}\_\{\\mathrm\{MMD\}^\{2\}\}bound follows from direct differentiation of the corresponding expression in Table[1](https://arxiv.org/html/2606.05327#S3.T1)\. For𝒟KL\\mathcal\{D\}\_\{\\mathrm\{KL\}\},J​∇gk​\(x,t\)=J​∇ln⁡ρ^tk​\(x\)−J​∇ln⁡μ^tk​\(x\)J\\nabla g\_\{k\}\(x,t\)=J\\nabla\\ln\\hat\{\\rho\}\_\{t\_\{k\}\}\(x\)\-J\\nabla\\ln\\hat\{\\mu\}\_\{t\_\{k\}\}\(x\)and the triangle inequality yields the KDE bound; for the Gaussian estimator,∇ln⁡𝒩​\(x;m,Σ\)=−\(x−m\)⊤​Σ−1\\nabla\\ln\\mathcal\{N\}\(x;m,\\Sigma\)=\-\(x\-m\)^\{\\top\}\\Sigma^\{\-1\}has constant Jacobian−Σ−1\-\\Sigma^\{\-1\}, so‖J​∇gk​\(x,t\)‖2≤‖Σρtk−1‖2\+‖Σμtk−1‖2\\\|J\\nabla g\_\{k\}\(x,t\)\\\|\_\{2\}\\leq\\\|\\Sigma\_\{\\rho\_\{t\_\{k\}\}\}^\{\-1\}\\\|\_\{2\}\+\\\|\\Sigma\_\{\\mu\_\{t\_\{k\}\}\}^\{\-1\}\\\|\_\{2\}\.

Intuitively, narrower kernels and stronger potentials \(largerwkw\_\{k\}, hence larger\|Ai​k\|\|A\_\{ik\}\|\) yield largerLFL\_\{F\}and slower or non\-contractive iterations; KL with KDE scores is particularly sensitive due to the divergence ofLkKLL\_\{k\}^\{\\mathrm\{KL\}\}in sparse regions\. This matches our empirical findings and motivates both damped Anderson acceleration\(Anderson,[1965](https://arxiv.org/html/2606.05327#bib.bib25)\)of the iterations and the homotopy curriculum described next, though, as we discuss in Sec\.[7](https://arxiv.org/html/2606.05327#S7), we recommend the𝒲22\\mathcal\{W\}^\{2\}\_\{2\}potential for efficient, straightforward training, avoiding these complexities altogether\.

#### Homotopy continuation

To help stabilize the FP iterations, we introduce a homotopy curriculumα​\(i\)\\alpha\(i\)that gradually increases the correction strengthα\\alphafrom 0 to 1 over training iterations\. Atα=0\\alpha=0,F0​\(XT\)i=XtibaseF\_\{0\}\(X\_\{T\}\)\_\{i\}=X\_\{t\_\{i\}\}^\{\\mathrm\{base\}\}hasLF=0L\_\{F\}=0and the trivially attractive fixed pointXT=XTbaseX\_\{T\}=X^\{\\mathrm\{base\}\}\_\{T\}\. By continuity ofJ​FαJF\_\{\\alpha\}inα\\alphaand the implicit function theorem, there exists a locally unique, smooth solution branchα↦XTFP​\(α\)\\alpha\\mapsto X\_\{T\}^\{\\mathrm\{FP\}\}\(\\alpha\)that remains attractive on any sub\-interval whereLF<1L\_\{F\}<1\. A sufficiently slow continuation inα\\alphafrom0to11thus allows the training dynamics to track this branch, even whenF1F\_\{1\}alone may not be globally contractive\. The scheduleα​\(i\)\\alpha\(i\)can be fixed as a hyperparameter, or dynamically adjusted during training — for example, based on the residual‖Fα​\(XT\)−XT‖\\\|F\_\{\\alpha\}\(X\_\{T\}\)\-X\_\{T\}\\\|— to ensure a sufficiently slow continuation\. We find a fixed sigmoid schedule \(Fig\.[5](https://arxiv.org/html/2606.05327#A5.F5)\) sufficient and practical, and ablate this choice in Sec\.[6\.2](https://arxiv.org/html/2606.05327#S6.SS2)\. Surprisingly, even for the𝒲22\\mathcal\{W\}^\{2\}\_\{2\}distance, for which we have a closed\-form solution, we find the homotopy curriculum improves the model\.

### E\.3Direct solution of the fixed\-point problem for the𝒲22\\mathcal\{W\}^\{2\}\_\{2\}distance

As described briefly in Sec\.[4\.2](https://arxiv.org/html/2606.05327#S4.SS2), for the𝒲22\\mathcal\{W\}^\{2\}\_\{2\}distance, the fixed\-point problem reduces to a linear system of equations that can be solved directly by matrix inversion\. We see this by writing out the fixed\-point equations for the𝒲22\\mathcal\{W\}^\{2\}\_\{2\}distance, with∇gk​\(Xtk,tk,ℬ\)=Xtk−ψ​\(Xtk\)\\nabla g\_\{k\}\(X\_\{t\_\{k\}\},t\_\{k\},\\mathcal\{B\}\)=X\_\{t\_\{k\}\}\-\\psi\(X\_\{t\_\{k\}\}\), whereψ\\psiis the static OT transport map betweenρtk\\rho\_\{t\_\{k\}\}andμtk\\mu\_\{t\_\{k\}\}:

Xti​\(x0,x1,ℬ\)=Xtibase​\(x0,x1\)\+∑k=1K\(Xtk−ψ​\(Xtk\)\)​\[wk​\(ℐ\(2\)​\[λk\]​\(t\)−ℐ\(2\)​\[λk\]​\(1\)​t\)\]⏟≡Ai,k,∀i∈\{1,…,K\}\.X\_\{t\_\{i\}\}\(x\_\{0\},x\_\{1\},\\mathcal\{B\}\)=X\_\{t\_\{i\}\}^\{\\mathrm\{base\}\}\(x\_\{0\},x\_\{1\}\)\+\\sum\_\{k=1\}^\{K\}\\left\(X\_\{t\_\{k\}\}\-\\psi\(X\_\{t\_\{k\}\}\)\\right\)\\underbrace\{\\left\[w\_\{k\}\\left\(\\mathcal\{I\}^\{\(2\)\}\\\!\\left\[\\lambda\_\{k\}\\right\]\(t\)\-\\mathcal\{I\}^\{\(2\)\}\\\!\\left\[\\lambda\_\{k\}\\right\]\(1\)t\\right\)\\right\]\}\_\{\\equiv A\_\{i,k\}\},\\quad\\forall i\\in\\\{1,\.\.\.,K\\\}\.\(117\)Rewriting this in matrix form, withA∈ℝK×KA\\in\\mathbb\{R\}^\{K\\times K\},XT≡\[Xt1​…​XtK\]T∈ℝK×DX\_\{T\}\\equiv\[X\_\{t\_\{1\}\}\\,\.\.\.\\,X\_\{t\_\{K\}\}\]^\{T\}\\in\\mathbb\{R\}^\{K\\times D\}andΨT​\(XT\)≡\[ψ​\(Xt1\)​…​ψ​\(XtK\)\]T∈ℝK×D\\Psi\_\{T\}\(X\_\{T\}\)\\equiv\[\\psi\(X\_\{t\_\{1\}\}\)\\,\.\.\.\\,\\psi\(X\_\{t\_\{K\}\}\)\]^\{T\}\\in\\mathbb\{R\}^\{K\\times D\}:

XT=XTbase\+A​\(XT−ΨT​\(XT\)\)\\displaystyle X\_\{T\}=X^\{\\mathrm\{base\}\}\_\{T\}\+A\(X\_\{T\}\-\\Psi\_\{T\}\(X\_\{T\}\)\)\(118\)⇒\\displaystyle\\Rightarrow\(𝟙−A\)​XT=XTbase−A​ΨT​\(XT\)\\displaystyle\(\\mathbbm\{1\}\-A\)X\_\{T\}=X^\{\\mathrm\{base\}\}\_\{T\}\-A\\Psi\_\{T\}\(X\_\{T\}\)⇒\\displaystyle\\RightarrowXT=\(𝟙−A\)−1​\(XTbase−A​ΨT​\(XT\)\)\\displaystyle X\_\{T\}=\(\\mathbbm\{1\}\-A\)^\{\-1\}\\left\(X^\{\\mathrm\{base\}\}\_\{T\}\-A\\Psi\_\{T\}\(X\_\{T\}\)\\right\)Thus, the fixed\-point problem reduces to inverting and multiplying by aK×KK\\times Kmatrix, where the inversion can be precomputed\. Though in general each application of this equation may change the mapψ\\psi, thereby still leaving us with a complex fixed\-point problem that needs to be solved iteratively, when conditioned on single samples from each marginal \(as we do in OTP\-FM\), the map is fixed and a single application of Eq\.[118](https://arxiv.org/html/2606.05327#A5.E118)yields the fixed\-point\.

## Appendix FExperimental details and studies on synthetic data

### F\.1Architecture

We use a three\-layer MLP for the MeanFlow velocity model with 256 nodes per hidden layer and the SiLU activation function after each hidden layer\. As stated in Sec\.[4\.3](https://arxiv.org/html/2606.05327#S4.SS3), we embed the timest1t\_\{1\}andΔ​t=t2−t1\\Delta t=t\_\{2\}\-t\_\{1\}using positional embeddings with 64D each, and the position input with a single embedding layer of 64D as well\.

### F\.2Training

We use the Adam optimizer with a learning rate of3⋅10−33\\cdot 10^\{\-3\}and a batch size of 512 samples per marginal\. We use a sigmoid curriculum for the correction strengthα​\(i\)\\alpha\(i\)with a mean at half the total number of training iterationsNNand a slope of12/N12/N:

α​\(i\)=11\+exp⁡\[−12N​\(i−N2\)\],\\alpha\(i\)=\\frac\{1\}\{1\+\\exp\\left\[\-\\frac\{12\}\{N\}\\left\(i\-\\frac\{N\}\{2\}\\right\)\\right\]\},\(119\)as shown in Fig\.[5](https://arxiv.org/html/2606.05327#A5.F5)\. The training epochs vary between 10 and 50 for convergence depending on the potential strengths and distance metrics: generally, stronger potentials and distances with harder FP problems \(MMD and KLD\) require more epochs and/or a slower curriculum\.

### F\.3MeanFlow, LSD and few\-step inference

We train with a 0\.75 probability of samplingt2=t1t\_\{2\}=t\_\{1\}\(i\.e\., on theut1,t2u\_\{t\_\{1\},t\_\{2\}\}diagonal\) and 0\.25 probability of samplingt2\>t1t\_\{2\}\>t\_\{1\}according to the joint distributionpt1,t2p\_\{t\_\{1\},t\_\{2\}\}\(Eq\.[114](https://arxiv.org/html/2606.05327#A5.E114)\)\. We find one or two steps per marginal sufficient for reasonably accurate inference of the intermediateXtkX\_\{t\_\{k\}\}for the MMD and KLD distances, as illustrated in Fig\.[6](https://arxiv.org/html/2606.05327#A6.F6)\. We maintain an EMA of the model weightsuθEMAu^\{\\mathrm\{EMA\}\}\_\{\\theta\}for evaluating theXtkX\_\{t\_\{k\}\}during training as well as for inference, with a decay factor of 0\.99\. We also compare MeanFlow with the LSD training objective \(App\.[C\.1](https://arxiv.org/html/2606.05327#A3.SS1)\) with the same training settings, and find largely similar performance, leaving a more rigorous comparison to future work\. We use 50 steps per marginal for inference in all experiments\.

![Refer to caption](https://arxiv.org/html/2606.05327v1/x9.png)

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Figure 6:Example OTP\-FM trajectories inferred from 1, 2, 5, 10, and 50 steps per marginal \(left to right\), demonstrating the effectiveness of the MeanFlow consistency model\.![Refer to caption](https://arxiv.org/html/2606.05327v1/x19.png)

![Refer to caption](https://arxiv.org/html/2606.05327v1/x20.png)

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Figure 7:Example OTP\-FM trajectories trained with the LSD consistency objective and inferred from 1, 2, 5, 10, and 50 steps per marginal \(left to right\), demonstrating comparable performance to the MeanFlow model\.
### F\.4Weaknesses of kernel\-based distances

As previously discussed in App\.[D](https://arxiv.org/html/2606.05327#A4), MMD with the polynomial kernel is not effective even for the exact solution for 1D Gaussian marginals \(Fig\.[4](https://arxiv.org/html/2606.05327#A4.F4)\)\. The RBF kernel, on the other hand, is performant for the exact solutions, but is limited with respect to the strength of the potential it is able to enforce for OTP\-FM\. This is illustrated in Fig\.[8](https://arxiv.org/html/2606.05327#A6.F8), where we see that for high strength potentials, the dynamic OT MMD solution effectively enforcesρtk\\rho\_\{t\_\{k\}\}to match the intermediate marginal, whereas the OTP\-FM MMD solution does not\. The KLD solution using a KDE for the score estimate faces similar issues\. On the other hand, the𝒲22\\mathcal\{W\}^\{2\}\_\{2\},𝒲2∞\\mathcal\{W\}\_\{2\}^\{\\infty\}, and KLD with a Gaussian score estimator are able to learn this\. This suggests that kernel\-based estimates of the MMD and KLD∇g\\nabla gprovide too noisy and unstable a training target for the OTP\-FM velocity model\.

![Refer to caption](https://arxiv.org/html/2606.05327v1/x29.png)

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![Refer to caption](https://arxiv.org/html/2606.05327v1/x34.png)

Figure 8:The exact dynamic OT solution with MMD and the RBF kernel \(top left\) demonstrates that the intermediate marginal is effectively enforced by the potential\. However, kernel\-based estimates of the MMD \(top center\) and KLD \(top right\)∇g\\nabla gdo not allow OTP\-FM to learn the intermediate marginal well \(even for very high strength potentials\)\. Finally, the𝒲22\\mathcal\{W\}^\{2\}\_\{2\}\(bottom left\),𝒲2∞\\mathcal\{W\}\_\{2\}^\{\\infty\}\(bottom center\), and KLD with a Gaussian score estimator \(bottom right\) do learn the intermediate marginal\.
### F\.5Comparingλk​\(t\)\\lambda\_\{k\}\(t\)shapes and widths

We experiment with three forms of the time\-dependence of the potentialsλk​\(t\)\\lambda\_\{k\}\(t\), with half\-widthτ\\tau:

Triangle:λk​\(t\)​\(τ,t\)=1τ​max⁡\(0,1−\|t−tk\|τ\),\\displaystyle\\text\{Triangle:\}\\quad\\lambda\_\{k\}\(t\)\(\\tau,t\)=\\frac\{1\}\{\\tau\}\\max\\left\(0,1\-\\frac\{\|t\-t\_\{k\}\|\}\{\\tau\}\\right\),\(120\)Rectangle:λk​\(t\)​\(τ,t\)=12​τ​θ​\(τ−\|t−tk\|\),\\displaystyle\\text\{Rectangle:\}\\quad\\lambda\_\{k\}\(t\)\(\\tau,t\)=\\frac\{1\}\{2\\tau\}\\theta\\left\(\\tau\-\|t\-t\_\{k\}\|\\right\),\(121\)Gaussian:λk​\(t\)​\(τ,t\)=1τ​2​π​exp⁡\(−\(t−tk\)22​τ2\)\\displaystyle\\text\{Gaussian:\}\\quad\\lambda\_\{k\}\(t\)\(\\tau,t\)=\\frac\{1\}\{\\tau\\sqrt\{2\\pi\}\}\\exp\\left\(\-\\frac\{\(t\-t\_\{k\}\)^\{2\}\}\{2\\tau^\{2\}\}\\right\)\(122\)whereθ​\(x\)\\theta\(x\)is the Heaviside step function\. Different configurations of shapes and widths are shown in the main text \(Fig\.[2](https://arxiv.org/html/2606.05327#S3.F2)\) and compared directly for the same marginals and potential strengths in Fig\.[9](https://arxiv.org/html/2606.05327#A6.F9)\. The width of the potential has an intuitive effect on the smoothness of the trajectories, with narrower potentials having sharper kinks \(approximating a delta function in the limiting case ofτ→0\\tau\\to 0\)\. We do not observe a significant empirical effect from the shape of theλk​\(t\)\\lambda\_\{k\}\(t\)\.

![Refer to caption](https://arxiv.org/html/2606.05327v1/x35.png)

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Figure 9:Top:Varying the half\-widthτ\\tauof theλk​\(t\)\\lambda\_\{k\}\(t\)s, with Gaussian shape, from 0\.01, 0\.05, 0\.1, 0\.2, to 0\.4 \(left to right\)\.Bottom:Varying the shape of theλk​\(t\)\\lambda\_\{k\}\(t\)s, withτ=0\.1\\tau=0\.1, between Gaussian, triangular, and rectangular \(left to right\)\.
### F\.6Comparing loss functions

Finally, as described in Sec\.[4\.3](https://arxiv.org/html/2606.05327#S4.SS3), we compare three different variations of the regression loss function: 1\) simple squared L2 loss; 2\) adaptive weights as described in the main text withp=1p=1\(Sec\.[4\.3](https://arxiv.org/html/2606.05327#S4.SS3)\); and 3\) a learnt weighting scheme as proposed inKarraset al\.\([2024](https://arxiv.org/html/2606.05327#bib.bib40)\):

ℒ=𝔼​\[e−w​\(t\)​‖vθ​\(t1,t2,x\)−sg​\(vtgt​\(t1,t2,x\)\)‖2\+w​\(t\)\],\\mathcal\{L\}=\\mathbb\{E\}\\left\[e^\{\-w\(t\)\}\\left\\\|v^\{\\theta\}\(t\_\{1\},t\_\{2\},x\)\-\\mathrm\{sg\}\(v\_\{\\mathrm\{tgt\}\}\(t\_\{1\},t\_\{2\},x\)\)\\right\\\|^\{2\}\+w\(t\)\\right\],\(123\)wherew​\(t\)w\(t\)is a learnt log\-variance predicted by the model\. These are motivated by the observation that the flow matching loss is in fact a sum over multiple losses at each time step, each of which may be easier or harder to optimize\. While the naive squared L2 loss simply weights each time step equally \(for uniform time sampling\), the two latter schemes attempt to weight the time steps based on the difficulty, either by simply dividing the squared L2 loss by the magnitude of the loss, or by encouraging the network to learn \(and minimize\) the log\-variance of its own predictions for each time step\. We show an example comparison in Fig\.[10](https://arxiv.org/html/2606.05327#A6.F10)and find the adaptive weights withp=1p=1to be both most performant and most stable to train in this experiment\.

![Refer to caption](https://arxiv.org/html/2606.05327v1/x43.png)

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Figure 10:Top:Example loss curves for the three different loss functions: squared L2 \(top left, log scale\), adaptive weights withp=1p=1\(top center\), and learnt log\-variance \(top right\)\.Bottom:The corresponding OTP\-FM trajectories for each loss function \(for the same marginals and potential strengths\)\.

## Appendix GExperimental details and ablation studies on single\-cell RNA sequencing datasets

### G\.1Embryoid body scRNA\-seq dataset

We use the preprocessed embryoid body \(EB\) data provided byMoonet al\.\([2019](https://arxiv.org/html/2606.05327#bib.bib44)\), which includes the 100\-dimensional PCA representation per cell\. The dataset comprises a total of 16,819 cells collected over 27 days across the five discrete time intervalst0∈\[0,3\]t\_\{0\}\\in\[0,3\],t1∈\[6,9\]t\_\{1\}\\in\[6,9\],t2∈\[12,15\]t\_\{2\}\\in\[12,15\],t3∈\[18,21\]t\_\{3\}\\in\[18,21\], andt4∈\[24,27\]t\_\{4\}\\in\[24,27\]days, with 2,381, 4,163, 3,278, 3,665, and 3,332 cells per interval, respectively\. We consider four experimental protocols on this dataset:

- •EB 5D L1O: training in the 5\-dimensional PCA space with one of the intermediate marginalst1t\_\{1\},t2t\_\{2\}, ort3t\_\{3\}held out per fold, averaging𝒲1\\mathcal\{W\}\_\{1\}in the normalized space over the three held\-out folds, followingNeklyudovet al\.\([2024](https://arxiv.org/html/2606.05327#bib.bib66)\);
- •EB 100D L0O: training in the 100\-dimensional PCA space on all five marginals \(no held\-out marginal\), averaging MMD over the four training times aftert0t\_\{0\}, followingPersiianovet al\.\([2026](https://arxiv.org/html/2606.05327#bib.bib67)\);
- •EB 100D L1O: same as L1O but in the 100\-dimensional PCA space, averaging MMD only at the held\-out times in the unnormalized PCA space, followingPersiianovet al\.\([2026](https://arxiv.org/html/2606.05327#bib.bib67)\);
- •EB 100D L2O: training in the 100\-dimensional PCA space with botht1t\_\{1\}andt3t\_\{3\}held out \(training on\{t0,t2,t4\}\\\{t\_\{0\},t\_\{2\},t\_\{4\}\\\}\), averaging MMD over the four non\-source timest1t\_\{1\}–t4t\_\{4\}, followingTheodoropouloset al\.\([2026](https://arxiv.org/html/2606.05327#bib.bib55)\)\.

Each PC feature is normalized to zero mean and unit variance across the full dataset for training, and evaluation metrics are computed in the original \(un\-normalized\) PCA space, except for EB 5D L1O which evaluates in normalized space following the convention ofNeklyudovet al\.\([2024](https://arxiv.org/html/2606.05327#bib.bib66)\)\. For the EB protocols we evaluate on a random subsample of 2,000 cells per time interval\.

### G\.2CITE\-seq scRNA\-seq dataset

The CITE\-seq dataset\(Burkhardtet al\.,[2022](https://arxiv.org/html/2606.05327#bib.bib78)\)consists of single\-cell RNA sequencing measurements of CD34\+ hematopoietic stem and progenitor cells \(HSPCs\) from four human donors, collected on four days,t0t\_\{0\}—t3t\_\{3\}, over a 10\-day period\. We use the pre\-computed 50\-PC representation provided byWanget al\.\([2026](https://arxiv.org/html/2606.05327#bib.bib68)\), consisting of 31,240 cells distributed across the four time intervals as 7,476, 6,999, 9,511, and 7,254 cells per day, respectively\. For training, the PC features are centered and divided by the maximum per\-feature standard deviation, followingNeklyudovet al\.\([2024](https://arxiv.org/html/2606.05327#bib.bib66)\); this normalization preserves the relative scale across PCs\. We consider two experimental protocols on this dataset, each with two leave\-one\-out folds \(holding out day 3 and day 4 separately, with the source day 2 and target day 7 always present\):

- •CITE 5D L1O: using the first 5 PCs and averaging𝒲1\\mathcal\{W\}\_\{1\}in the normalized PCA space over the two held\-out folds, followingKapusniaket al\.\([2024](https://arxiv.org/html/2606.05327#bib.bib65)\);
- •CITE 50D L1O: using all 50 PCs and averaging𝒲1\\mathcal\{W\}\_\{1\}in the original \(un\-normalized\) PCA space over the two held\-out folds, followingNeklyudovet al\.\([2024](https://arxiv.org/html/2606.05327#bib.bib66)\)\.

For both protocols we evaluate on the full set of cells at each time point for an apples\-to\-apples comparison with the baselines\(Neklyudovet al\.,[2024](https://arxiv.org/html/2606.05327#bib.bib66); Kapusniaket al\.,[2024](https://arxiv.org/html/2606.05327#bib.bib65), and others\)\.

### G\.3Architecture and training

For all single\-cell experiments we use an MLP velocity network with residual connections every two layers, SiLU activations, LayerNorm, and dropout 0\.2, trained with the Adam optimizer at a base learning rate of3⋅10−33\\cdot 10^\{\-3\}and a batch size of 256 samples per marginal\. All training runs are performed on a single NVIDIA L40S GPU and final model checkpoints are chosen based on the minimum MMD or𝒲1\\mathcal\{W\}\_\{1\}scores\. We use 64\-dimensional positional embeddings for botht1t\_\{1\}andΔ​t=t2−t1\\Delta t=t\_\{2\}\-t\_\{1\}, and a 64\-dimensional linear embedding for the position input, following the architecture described in App\.[F](https://arxiv.org/html/2606.05327#A6)\. We use eight to ten hidden layers of 768 nodes each \(4\.9M—6\.5M parameters\), training each of the𝒲22\\mathcal\{W\}^\{2\}\_\{2\},𝒲2∞\\mathcal\{W\}\_\{2\}^\{\\infty\}, MMD, and KLD potentials with the iMF consistency loss \(App\.[C\.1](https://arxiv.org/html/2606.05327#A3.SS1)\), adaptive loss weighting withp=1p=1, tuning the potential strengths andλk​\(t\)\\lambda\_\{k\}\(t\)’s for each dataset \(exact configurations are provided in our linked codebase\)\. We use a sigmoid curriculum forα​\(i\)\\alpha\(i\)with slope 24 and midpoint at iteration0\(i\.e\., the schedule starts atα=0\.5\\alpha=0\.5and quickly saturates to11\)\. Intermediate potential times \{tkt\_\{k\}\} are auto\-computed as evenly spaced fractions of the training interval\. We train for between 200 and 300 epochs depending on the distance metric, with training times shown in Figs\.[3](https://arxiv.org/html/2606.05327#S6.F3)and[12](https://arxiv.org/html/2606.05327#A7.F12)\. We use MMD with the RBF kernel of bandwidth 10 and KLD with KDE score estimates of bandwidth 3, both with a maximum of 5 Picard iterations per training iteration\. We tested the KLD with a Gaussian score estimator as well but this did not converge\.

### G\.4Ablation studies

Table[5](https://arxiv.org/html/2606.05327#A7.T5)continues the ablation study from Sec\.[6\.2](https://arxiv.org/html/2606.05327#S6.SS2), varying the consistency model and loss function on the EB 100D L2O experiment\. The baseline model uses the same parameters specified in Sec\.[6\.2](https://arxiv.org/html/2606.05327#S6.SS2), trained with the iMF formulation and adaptive loss function\. Figure[11](https://arxiv.org/html/2606.05327#A7.F11)shows the ground truth and simulated trajectories on this experiment for the baseline model and different potential parameters\.

Table 5:Ablation of the consistency model choice and loss function on EB 100D L2O \(MMD¯\\overline\{\\text\{MMD\}\}on held\-out times, lower is better\)\. Baseline inbold; others single\-seed\.ConsistencyMMD¯↓\\overline\{\\text\{MMD\}\}\\\!\\downarrowIMF0\.0675MeanFlow0\.0690LSD0\.0696
Loss functionMMD¯↓\\overline\{\\text\{MMD\}\}\\\!\\downarrowAdaptive0\.0675MSE0\.0698Weighted0\.0746

![Refer to caption](https://arxiv.org/html/2606.05327v1/x49.png)Figure 11:Ground truth and simulated trajectories for the EB 100D dataset in the leave\-two\-out setting \(t1t\_\{1\}andt3t\_\{3\}marginals held out\) for different OTP\-FM potential parameters\. The “baseline” model uses𝒟=𝒲22\\mathcal\{D\}=\\mathcal\{W\}^\{2\}\_\{2\}with a Gaussianλ​\(t\)\\lambda\(t\), with half\-widthτ=0\.33\\tau=0\.33and strengthw=1000w=1000\. All other models use the same parameters except for the difference specified\.
### G\.5Additional results

Figure[12](https://arxiv.org/html/2606.05327#A7.F12)shows additional timing vs\. performance plots for the EB 5D, EB 100D L1O, and CITE 50D experiments, and Fig\.[13](https://arxiv.org/html/2606.05327#A7.F13)shows trajectory visualizations for the EB 100D L2O experiment for OTP\-FM with different potentials and all baseline models for which we have access to trajectories, either by our own training \(as specified in App\.[G\.6](https://arxiv.org/html/2606.05327#A7.SS6)\) or provided by the authors\.101010We thank Panagiotis Theodoropoulos for providing the trajectories for 3MSBM\.

![Refer to caption](https://arxiv.org/html/2606.05327v1/x50.png)

![Refer to caption](https://arxiv.org/html/2606.05327v1/x51.png)

![Refer to caption](https://arxiv.org/html/2606.05327v1/x52.png)

Figure 12:Training time vs\. performance of different methods for the EB 5D L1O \(left\), EB 100D L1O \(center\), and CITE 50D L1O \(right\) experiments\. Top right is better\.![Refer to caption](https://arxiv.org/html/2606.05327v1/x53.png)Figure 13:Ground truth and simulated trajectories for the EB 100D dataset in the leave\-two\-out setting \(t1t\_\{1\}andt3t\_\{3\}marginals held out\) in all four PCs across all five time intervals, comparing OTP\-FM with different potentials with baseline models\.
### G\.6Benchmarking methods

Below we describe each baseline method, the codebase and configuration used in our evaluation and timing studies, and the provenance of the results in Table[3](https://arxiv.org/html/2606.05327#S6.T3), in the order that they appear\. All timing measurements \(Figs\.[3](https://arxiv.org/html/2606.05327#S6.F3)and[12](https://arxiv.org/html/2606.05327#A7.F12)\) were performed on a single cloud VM with one NVIDIA L40S GPU and one AMD EPYC 7R13 processor \(8 cores / 16 threads, 32 MiB L3 cache\) and 124 GiB RAM\. Timings report the wall\-clock training time per fold of one held\-out marginal configuration \(Sec\.[6\.3](https://arxiv.org/html/2606.05327#S6.SS3)\), measured on an otherwise idle machine with no concurrent jobs\. We use the author\-provided configurations whenever available for the respective dataset, otherwise adapting the most similar configuration available in their repository\. For methods whose per\-fold training time exceeds ten minutes, we measure the time per training iteration \(or per epoch\) from a ten\-minute run and extrapolate by the total iteration count specified in the original configuration; for all other methods the reported value is of the full training run\. We exclude evaluation time and precomputation of OT couplings from the timing measurements; for the datasets considered in this work, we measured the OT precomputation time to be between 5–15s and, hence, negligible in comparison to the training time\.

#### TrajectoryNet

TrajectoryNet\(Tonget al\.,[2020](https://arxiv.org/html/2606.05327#bib.bib46)\)learns a continuous normalizing flow with dynamic OT regularization for trajectory inference\. We use the provided code111111[https://github\.com/KrishnaswamyLab/TrajectoryNet](https://github.com/KrishnaswamyLab/TrajectoryNet)with the authors’ EB 5D configuration; the EB 5D L1O value is taken directly fromTonget al\.\([2020](https://arxiv.org/html/2606.05327#bib.bib46)\)\. We were not able to run the EB 100D and CITE experiments ourselves because of the prohibitively slow training time\. For the same reason, timing values for EB 100D, CITE 5D, and CITE 50D are extrapolated from limited\-iteration runs using configurations adapted from the EB 5D config\.

#### NLSB

Neural Lagrangian Schrödinger Bridge \(NLSB\)\(Koshizuka and Sato,[2023](https://arxiv.org/html/2606.05327#bib.bib49)\)models population dynamics via a Lagrangian\-regularized neural SDE\. We use the provided code121212[https://github\.com/take\-koshizuka/NLSB](https://github.com/take-koshizuka/NLSB)with the authors’ EB 5D configuration; the EB 5D L1O value is taken fromKoshizuka and Sato \([2023](https://arxiv.org/html/2606.05327#bib.bib49)\), the EB 100D L0O and L1O values fromChenet al\.\([2023](https://arxiv.org/html/2606.05327#bib.bib48)\), and we ran the CITE 5D L1O setting ourselves with a configuration adapted from EB 5D\. We were not able to complete the EB 100D L2O and CITE 50D runs because of the prohibitively slow training time\. Their timings are extrapolated from ten\-minute runs using configurations adapted from the EB 5D config\.

#### MIOFlow

MIOFlow\(Huguetet al\.,[2022](https://arxiv.org/html/2606.05327#bib.bib45)\)combines a Geodesic Autoencoder \(GAGA\) with a neural ODE for trajectory inference; crucially it trains the neural ODE by matching predicted and observed marginals under the𝒲22\\mathcal\{W\}^\{2\}\_\{2\}distance instead of the maximum\-likelihood loss used by continuous normalizing flows \(CNFs\)\. This sidesteps the instantaneous change\-of\-variables computation that CNFs require at every integration step, leading to much faster training than conventional simulation\-based approaches\. We use the provided code131313[https://github\.com/KrishnaswamyLab/MIOFlow](https://github.com/KrishnaswamyLab/MIOFlow)\. The repository does not provide scRNA\-specific configurations matching our benchmark, so we adapt the 5D demo configuration for all four datasets\. The CITE 50D L1O values are taken fromHuguetet al\.\([2022](https://arxiv.org/html/2606.05327#bib.bib45)\), the EB 100D L1O and L0O values fromChenet al\.\([2023](https://arxiv.org/html/2606.05327#bib.bib48)\), and we ran the EB 5D L1O, EB 100D L2O, and CITE 5D L1O settings ourselves, bypassing the GAGA encoding and decoding\.

#### DMSB

DMSB\(Chenet al\.,[2023](https://arxiv.org/html/2606.05327#bib.bib48)\)is a neural SDE method that lifts multi\-marginal Schrödinger bridges to phase space to smooth the dynamics\. We use the provided code141414[https://github\.com/TianrongChen/DMSB](https://github.com/TianrongChen/DMSB)with the authors’ EB 100D and CITE 50D configurations; the EB 5D L1O value is reported fromChenet al\.\([2023](https://arxiv.org/html/2606.05327#bib.bib48)\), the CITE 5D L1O value fromKapusniaket al\.\([2024](https://arxiv.org/html/2606.05327#bib.bib65)\), and we ran the EB 100D and CITE 50D L1O settings ourselves\. We note thatChenet al\.\([2023](https://arxiv.org/html/2606.05327#bib.bib48)\)provides EB 100D L1O \+ L0O in the z\-score*normalized*space while we report unnormalized MMD values; hence, we retrain DMSB to obtain unnormalized values for this setting as well\. The EB 5D and CITE 5D timings use configurations adapted from the EB 100D and CITE 50D configs, respectively\.

#### DeepRUOT

DeepRUOT\(Zhanget al\.,[2025](https://arxiv.org/html/2606.05327#bib.bib70)\)is a neural SDE method that jointly learns velocity, growth, and score networks under a regularized unbalanced OT objective\. We use the provided code151515[https://github\.com/zhenyiizhang/DeepRUOT](https://github.com/zhenyiizhang/DeepRUOT)and adapt the original four\-phase EB 5D training schedule for all datasets\. The EB 5D L1O, CITE 5D L1O, and CITE 50D L1O values are taken fromZhanget al\.\([2025](https://arxiv.org/html/2606.05327#bib.bib70)\), and we ran the EB 100D settings ourselves\. Timing values are obtained by extrapolating the per\-phase iteration timing measured within 10 minutes per dataset\.

#### WLF\-UOT

WLF\-UOT\(Neklyudovet al\.,[2024](https://arxiv.org/html/2606.05327#bib.bib66)\)represents an alternative paradigm for trajectory inference, solving the variational problem directly\. We use the provided code161616[https://github\.com/necludov/wl\-mechanics](https://github.com/necludov/wl-mechanics)with the authors’ EB 5D and CITE 50D unbalanced\-OT configurations; the EB 5D L1O value is taken fromNeklyudovet al\.\([2024](https://arxiv.org/html/2606.05327#bib.bib66)\), the CITE 5D and 50D L1O values fromKapusniaket al\.\([2024](https://arxiv.org/html/2606.05327#bib.bib65)\), and we ran the EB 100D settings ourselves with configurations adapted from those provided\. For consistency with the other baselines, for evaluation we generate all samples starting fromt=0t=0\. We note thatNeklyudovet al\.\([2024](https://arxiv.org/html/2606.05327#bib.bib66)\)also report values incorporating potentials based on the*held\-out*marginals, which we do not include for fairness\.

#### JKOnet\*

JKOnet\(Bunneet al\.,[2022](https://arxiv.org/html/2606.05327#bib.bib73)\)interprets diffusion as energy\-minimizing trajectories in Wasserstein space, following the work of Jordan, Kinderlehrer, and Otto \(JKO\) and solves this minimization problem\. We experiment with an improved version, JKOnet\*\(Terpinet al\.,[2024](https://arxiv.org/html/2606.05327#bib.bib77)\)\. We use the provided code171717[https://github\.com/antonioterpin/jkonet\-star](https://github.com/antonioterpin/jkonet-star)with the authors’ EB 5D configuration; the other datasets are run with configurations adapted from the EB 5D config\. For EB 5D L1O, EB 100D L2O, CITE 5D L1O, and CITE 50D L1O we ran the method ourselves, while the EB 100D L0O and L1O values are fromPersiianovet al\.\([2026](https://arxiv.org/html/2606.05327#bib.bib67)\)\.

#### iJKOnet

iJKOnet\(Persiianovet al\.,[2026](https://arxiv.org/html/2606.05327#bib.bib67)\)proposes an alternative, adversarial scheme for the JKO optimization problem above\. We use the provided code181818[https://github\.com/MuXauJl11110/iJKOnet](https://github.com/MuXauJl11110/iJKOnet)with the authors’ EB 5D and CITE 50D configurations\. For EB 5D L1O, EB 100D L2O, CITE 5D L1O, and CITE 50D L1O we re\-ran the method, while the EB 100D L0O and L1O values are fromPersiianovet al\.\([2026](https://arxiv.org/html/2606.05327#bib.bib67)\)\. EB 100D and CITE 5D timings use configurations adapted from EB 5D and CITE 50D, respectively\.

#### 3MSBM

3MSBM\(Theodoropouloset al\.,[2026](https://arxiv.org/html/2606.05327#bib.bib55)\)extends multi\-marginal Schrödinger bridges to phase space via a score\-matching\-style objective to avoid simulation\-based training, with the score targets calculated by solving a dynamic programming problem during training\. We completed one training run of 3MSBM on the EB 100D L2O setting following the code provided191919[https://github\.com/panostheo98/3MSBM](https://github.com/panostheo98/3MSBM)and the authors’ EB\-specific configuration but it did not converge\. We therefore only report results directly fromTheodoropouloset al\.\([2026](https://arxiv.org/html/2606.05327#bib.bib55)\)\(EB 100D L2O\) and include trajectories that are provided by the authors in Fig\.[13](https://arxiv.org/html/2606.05327#A7.F13)\.

#### I\-CFM, OT\-CFM, and \[SF\]2M

I\-CFM and OT\-CFM\(Tonget al\.,[2024](https://arxiv.org/html/2606.05327#bib.bib3)\)train flow\-matching velocity models on independently sampled or OT\-aligned pairs of consecutive marginals, respectively, while \[SF\]2M\(Tonget al\.,[2023](https://arxiv.org/html/2606.05327#bib.bib69)\)uses score\-matching to learn a Schrödinger bridge between marginals\. In all cases, trajectories are stitched together piecewise between consecutive training marginals, as described in App\.[A\.2](https://arxiv.org/html/2606.05327#A1.SS2)\. We use the providedtorchcfmcode,202020[https://github\.com/atong01/conditional\-flow\-matching](https://github.com/atong01/conditional-flow-matching)with hyperparameters adapted from the single\-cell tutorials\. For OT\-CFM we precompute a single full\-dataset EMD coupling per consecutive time pair rather than re\-solving per minibatch, for a fair comparison with OTP\-FM\. The EB 5D L1O value is taken fromTonget al\.\([2024](https://arxiv.org/html/2606.05327#bib.bib3),[2023](https://arxiv.org/html/2606.05327#bib.bib69)\), the CITE 5D and 50D L1O values fromKapusniaket al\.\([2024](https://arxiv.org/html/2606.05327#bib.bib65)\), and the EB 100D experiments we ran ourselves\.

#### OT\-MFM

OT\-MFM\(Kapusniaket al\.,[2024](https://arxiv.org/html/2606.05327#bib.bib65)\)encourages CFM trajectories to follow the data manifold by replacing straight interpolants with approximate geodesics of a data\-dependent Riemannian metric\. We use the provided code212121[https://github\.com/kksniak/metric\-flow\-matching](https://github.com/kksniak/metric-flow-matching)with the authors’ EB 5D, CITE 5D, and CITE 50D LAND\-metric configurations; the EB 5D L1O and CITE 5D and 50D L1O values are taken fromKapusniaket al\.\([2024](https://arxiv.org/html/2606.05327#bib.bib65)\), and we ran the EB 100D experiments ourselves using a configuration adapted from the EB 5D config\.

#### VGFM

VGFM\(Wanget al\.,[2026](https://arxiv.org/html/2606.05327#bib.bib68)\)jointly learns velocity and*growth*fields using a loss derived from semi\-relaxed OT on top of the CFM loss\. We use the provided code222222[https://github\.com/DongyiWang\-66/VGFM](https://github.com/DongyiWang-66/VGFM)with the authors’ EB 5D, CITE 5D, and CITE 50D notebook configurations \(both the “warm\-up” and training phases\); the EB 5D L1O and CITE 5D and 50D L1O values are taken fromWanget al\.\([2026](https://arxiv.org/html/2606.05327#bib.bib68)\), and we ran the EB 100D experiments ourselves with a configuration adapted from the EB 5D notebook\.

#### MMFM

MMFM\(Rohbecket al\.,[2025](https://arxiv.org/html/2606.05327#bib.bib54)\)trains CFM between multiple marginals using a cubic\-spline interpolation as the conditional velocity regression target\. We train MMFM following the code provided,232323[https://github\.com/Genentech/MMFM](https://github.com/Genentech/MMFM)using an identical model architecture to OTP\-FM’s for each respective configuration, over 5 training seeds on all five evaluation settings\.

## Appendix HExperimental details on the Gulf of Mexico dataset

### H\.1Dataset

This dataset comprises high\-resolution bathymetric measurements of the Gulf of Mexico \(GoM\) in the region between 98∘E and 77∘E in longitude and from 18∘N to 32∘N in latitude\. The original dataset was released to the public by the United States Department of Defense and contains hourly observations between January 1st, 2001 and August 31st, 2024\(HYCOM Consortium and Center for Ocean\-Atmospheric Prediction Studies \(COAPS\),[2024](https://arxiv.org/html/2606.05327#bib.bib60)\)\.Shenet al\.\([2025](https://arxiv.org/html/2606.05327#bib.bib59)\)then extracted the velocity field around a vortex from the June 1st, 2024, 5pm timepoint, which they then used to simulate 1,000 particle trajectories\. Finally, these trajectories were each randomly sampled at one of nine time pointst0t\_\{0\}—t8t\_\{8\}, such that each particle is only observed at a single time point, yielding 111 samples per marginal\. These are the ground truth marginals shown in Fig\.[14](https://arxiv.org/html/2606.05327#A8.F14)\.

### H\.2Architecture and training

We use a ten\-layer MLP with residual connections every two layers and 128 nodes per hidden layer for the velocity model\. We use the Adam optimizer with a learning rate of10−310^\{\-3\}, a batch size of 64 samples per marginal, and the same sigmoid curriculum forα​\(i\)\\alpha\(i\)as in the synthetic data experiments\. In this case, we find the simple MSE loss to be most performant\. We train for a total of 2000 epochs for each distance metric, corresponding to a total training time of 4 minutes for the𝒲22\\mathcal\{W\}^\{2\}\_\{2\}and𝒲2∞\\mathcal\{W\}\_\{2\}^\{\\infty\}distances and 12 minutes for the MMD and KLD distances on a single NVIDIA L40S GPU\. We use MMD with the RBF kernel and KLD with KDEs for the score estimates, both with a bandwidth of 3 and a maximum of 20 Picard iterations per training iteration\. The final model checkpoints are chosen based on the average𝒲22\\mathcal\{W\}^\{2\}\_\{2\}distance over the held\-out times\.

### H\.3Additional results

Table[6](https://arxiv.org/html/2606.05327#A8.T6)shows the per\-timepoint results on the GoM dataset, including OTP\-FM with the KL and MMD2potentials, and Fig\.[14](https://arxiv.org/html/2606.05327#A8.F14)shows the ground truth and simulated trajectories for the GoM dataset, comparing MMFM and OTP\-FM with different potentials\.

Table 6:Per\-timepoint results on the GoM dataset, showing the mean and standard deviation across five seeds of the𝒲2\\mathcal\{W\}\_\{2\}distance to the four held\-out marginals and the average distance to the rest \(lower is better\)\. 3MSBM results are reported fromTheodoropouloset al\.\([2026](https://arxiv.org/html/2606.05327#bib.bib55)\), while MMFM results are from our training runs\. The best performing method per timepoint is inboldand the second\-best initalics\.![Refer to caption](https://arxiv.org/html/2606.05327v1/x54.png)Figure 14:Ground truth and simulated trajectories for the GoM dataset, comparing MMFM and OTP\-FM with different potentials\.

## Appendix IExperimental details on the Beijing air quality dataset

### I\.1Dataset

This final dataset includes hourly measurements of the concentrations of six main air pollutants and six meteorological variables from 12 monitoring sites in Beijing, between 2013 and 2017\(Chen,[2017](https://arxiv.org/html/2606.05327#bib.bib61)\)\. We follow the experimental setup ofTheodoropouloset al\.\([2026](https://arxiv.org/html/2606.05327#bib.bib55)\), whereby we focus on one station — Dingling’s — measurements of one important pollutant: particulate matter smaller than 2\.5μ\\mum, or PM2\.5\. We aggregate the hourly measurements by month, and sample every other month in the period of March 2013 — March 2015; i\.e\., 13 total marginals:t0t\_\{0\}—t12t\_\{12\}\. We hold outt2t\_\{2\},t5t\_\{5\},t8t\_\{8\}, andt11t\_\{11\}from the training to test the interpolation ability of the model, leaving nine marginals \(seven intermediate potentials\) for training\. Each marginal contains around 740 samples\.

### I\.2Architecture and training

We use a four\-layer MLP with residual connections every two layers and 128 nodes per hidden layer for the velocity model\. We use the Adam optimizer with a learning rate of3⋅10−33\\cdot 10^\{\-3\}, a batch size of 128 samples per marginal, and the same sigmoid curriculum forα​\(i\)\\alpha\(i\)as in the synthetic data experiments\. We find simple MSE to again be most performant\. We train for a total of 600 epochs for the𝒲22\\mathcal\{W\}^\{2\}\_\{2\}and𝒲2∞\\mathcal\{W\}\_\{2\}^\{\\infty\}distances and 100 epochs for the MMD and KLD distances, corresponding to a total training time of 5 minutes for the former and 2 minutes for the latter distances on a single NVIDIA L40S GPU\. We use MMD with the RBF kernel of bandwidth 3 and KLD with the Gaussian score estimates, both with a maximum of 5 Picard iterations per training iteration\. The final model checkpoints are chosen based on the average𝒲22\\mathcal\{W\}^\{2\}\_\{2\}distance over the held\-out times\.

### I\.3Additional results

Table[7](https://arxiv.org/html/2606.05327#A9.T7)shows the per\-timepoint results on the Beijing air quality dataset, and Fig\.[15](https://arxiv.org/html/2606.05327#A9.F15)shows the ground truth and simulated trajectories for the Beijing air quality dataset, comparing MMFM and OTP\-FM\. This dataset interestingly highlights a key failure mode of prescribed approaches such as MMFM: as evident from Fig\.[15](https://arxiv.org/html/2606.05327#A9.F15), the cubic spline interpolation assumed in MMFM manifestly does not capture the data\. In contrast, the flexibility of the OTP\-FM design space allows the data to dictate the optimal interpolation, yielding a better fit to the held\-out marginals\.

Table 7:Per\-timepoint results on the Beijing air quality dataset, showing the mean and standard deviation across five seeds of the𝒲2\\mathcal\{W\}\_\{2\}distance to the four held\-out marginals and the average distance to the rest \(lower is better\)\. 3MSBM results are reported fromTheodoropouloset al\.\([2026](https://arxiv.org/html/2606.05327#bib.bib55)\), who do not report the training scores, while MMFM results are from our training runs\. The best performing method per timepoint is inboldand the second\-best initalics\.![Refer to caption](https://arxiv.org/html/2606.05327v1/x55.png)Figure 15:Visualizing the Beijing PM2\.5 air quality dataset\. \(a\) Ground truth marginal distributions as violin plots showing the density at each timepoint, the median, and the interquartile range\. The marginals are heavily concentrated at low PM2\.5 but with long tails\. \(b\) Cubic spline interpolation through OT\-coupled ground\-truth samples, excluding held\-out marginals; this represents MMFM’s training targets\. \(c\) Trajectories inferred by MMFM\. \(d\) Trajectories inferred by OTP\-FM \(𝒲22\\mathcal\{W\}\_\{2\}^\{2\}potential\)\.