Title: Curly Flow Matching for Learning Non-gradient Field Dynamics
Venue: NeurIPS 2025
Accepted: yes
Keywords: Single-cell

Trajectory Inference

Schrödinger Bridge

Optimal Transport

Flow Matching
Abstract: Modeling the transport dynamics of natural processes from population-level observations is a ubiquitous problem in the natural sciences. Such models rely on key assumptions about the underlying process in order to enable faithful learning of governing dynamics that mimic the actual system behavior. The de facto assumption in current approaches relies on the principle of least action that results in gradient field dynamics and leads to trajectories minimizing an energy functional between two probability measures. However, many real-world systems, such as cell cycles in single-cell RNA, are known to exhibit non-gradient, periodic behavior, which fundamentally cannot be captured by current state-of-the-art methods such as flow and bridge matching. In this paper, we introduce Curly Flow Matching (Curly-FM), a novel approach that is capable of learning non-gradient field dynamics by designing and solving a Schrödinger bridge problem with a non-zero drift reference process---in stark contrast to typical zero-drift reference processes---which is constructed using inferred velocities in addition to population snapshot data. We showcase Curly-FM by solving the trajectory inference problems for single cells, computational fluid dynamics, and ocean currents with approximate velocities. We demonstrate that Curly-FM can learn trajectories that better match both the reference process and population marginals. Curly-FM expands flow matching models beyond the modeling of populations and towards the modeling of known periodic behavior in physical systems. Our code repository is accessible
at: https://github.com/kpetrovicc/curly-flow-matching.git

Curly Flow Matching for Learning Non-gradient Field Dynamics

Abstract

Modeling the transport dynamics of natural processes from population-level observations is a ubiquitous problem in the natural sciences. Such models rely on key assumptions about the underlying process in order to enable faithful learning of governing dynamics that mimic the actual system behavior. The de facto assumption in current approaches relies on the principle of least action that results in gradient field dynamics and leads to trajectories minimizing an energy functional between two probability measures. However, many real-world systems, such as cell cycles in single-cell RNA, are known to exhibit non-gradient, periodic behavior, which fundamentally cannot be captured by current state-of-the-art methods such as flow and bridge matching. In this paper, we introduce CURLY FLOW MATCHING (CURLY-FM), a novel approach that is capable of learning non-gradient field dynamics by designing and solving a Schrödinger bridge problem with a non-zero drift reference process-in stark contrast to typical zero-drift reference processeswhich is constructed using inferred velocities in addition to population snapshot data. We showcase CURLY-FM by solving the trajectory inference problems for single cells, computational fluid dynamics, and ocean currents with approximate velocities. We demonstrate that CURLY-FM can learn trajectories that better match both the reference process and population marginals. CURLY-FM expands flow matching models beyond the modeling of populations and towards the modeling of known periodic behavior in physical systems. Our code repository is accessible at: https://github.com/kpetrovicc/curly-flow-matching.git .

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1 Introduction

Understanding the temporal evolution of multi-body systems remains a central challenge across many applications in life sciences [Lange et al., 2024, Pan and Zhang, 2024], as such systems are often characterized by complex dynamics governing their evolutionary behavior. For example, biological tissues are complex systems that evolve through tissue differentiation characterized by cell divisions and deaths as well as morphological changes. Learning evolutionary behavior in such systems can be formulated as a trajectory inference problem [Hashimoto et al., 2016, Lavenant et al., 2024], where the goal is to recover full trajectories of particles given partial and noisy population-level snapshots.

The dominant paradigm in solving the trajectory inference problems in scientific applications involves leveraging tools from computational optimal transport (OT) [Peyré and Cuturi, 2019] to learn neural dynamical systems, e.g. NeuralODE [Chen et al., 2018], such that sampled trajectories under the model optimize a notion of likeliness of being observed [Bunne et al., 2024b]. For instance, in single-cell trajectory inference, such methods broadly follow a pipeline that first infers 'optimal' cell trajectories that follow the gradient of some potential function (often termed a Waddington Landscape [Waddington, 1942]), before searching for important regulators of a biological process in development [Schiebinger et al., 2019, Shahan et al., 2022] or disease [Tong et al., 2023, Klein et al., 2025]. Despite the ability to produce approximately optimal trajectories w.r.t. the energy landscape, current methods are limited in their ability to model only gradient-field dynamics. Consequently, trajectories inferred under the model are not realistic and fail to model crucial system dynamics such as periodic behavior that arises in natural systems, e.g., cell cycling [Riba et al., 2022] where periodic behavior is observed at the scale of months, days, hours, and minutes. These behaviors cannot be addressed by current OT-based methods, as gradient field dynamics cannot capture periodic behavior.

Present work . In this paper, we tackle the modeling of systems governed by non-gradient field dynamics. We introduce CURLY FLOW MATCHING (CURLY-FM), a novel approach for learning non-gradient field dynamics by solving a Schrödinger bridge problem with a designed reference process that induces the learning of periodic behavior. Specifically, we consider reference processes with non-zero drift-in stark contrast to zero drift processes in approaches such as Diffusion Schrödinger Bridges [De Bortoli et al., 2021b, Shi et al., 2024]. Such a modification elevates the established (entropic optimal) mass transport problem to a new class of problems that require matching the reference drift while also transporting mass between time marginals associated with observations. As a result, solutions to this Schrödinger bridge problem are capable of learning non-gradient field dynamics and exhibit behaviors such as periodicity as found in cell cycling.

In addition to conventional population snapshot data, we design CURLY-FM by leveraging approximate velocity information that is used to construct the drift of a reference process. Consequently, to model periodic dynamics CURLY-FM solves the Schrödinger bridge problem by decomposing it into a two-stage algorithm. The first stage learns a neural path interpolant by regressing against the drift of our constructed reference process. Unlike straight paths in optimal transform conditional flow matching trajectories, the neural path interpolant exhibits cyclic behavior due to the optimization objective of matching a constructed reference drift. In stage two, we learn, in a simulation-free manner, to construct the generative process that solves the mass transport problem as a mixture of conditional bridges built using optimal transport-based couplings that minimize the length of the velocity field of the neural path interpolant. The combination of our two-stage approach enables CURLY-FM to learn dynamics that do not affect the population data, but do affect individual particles, which include observed periodic non-gradient field dynamics, unlike other methods (see table 1).

We instantiate CURLY-FM on modeling a suite of problems in natural sciences that exhibit known non-gradient field dynamics, such as cell cycle systems found in scRNA-seq data with RNAvelocity [La Manno et al., 2018], ocean currents, and computational fluid dynamics for PDEs simulated using the Lagrangian particle discretizations [Toshev et al., 2023]. In each case, the system under consideration is under the influence of a known reference process, e.g., RNA-velocity gives an approximation of the instantaneous velocity of a cell, which must be adhered to when solving the trajectory inference problem. More precisely, using the reference process drift and population snapshots CURLY-FM solves the Schrödinger bridge problem by searching for a bridge that matches the reference process as closely as possible, but also matches the marginal distributions at both timepoints of the system dynamics. Consequently, we show that previous flow matching approaches fail to model this cyclic behavior as they are not able to take advantage of the additional reference process information. Furthermore, while there exist simulation-based methods that are in principle able to learn the correct dynamics [Tong et al., 2020], we show that in practice CURLY-FM performs

significantly better both in terms of accuracy to match the reference drift-enabling modeling of cyclic behavior (c.f. fig. 1)-while being computationally cheaper due to its simulation-free training nature.

2 Background and preliminaries

Given two distributions ρ 0 and ρ 1 , the distributional matching problem seeks to find a push-forward map ψ : R d → R d that transports the initial distribution to the desired endpoint, ρ 1 = [ ψ ] # ( ρ 0 ) . Such a problem setup is pervasive in many areas of machine learning and notably encompasses the standard generative modeling and optimal transport settings [De Bortoli et al., 2021a, Peyré et al., 2019]. In this paper, we consider the setting where each distribution ρ 0 and ρ 1 is an empirical distribution that is accessible through a dataset of observations { x i 0 } N i =1 ∼ ρ 0 ( x 0 ) and { x j 1 } N j =1 ∼ ρ 1 ( x 1 ) . Thus the modeling task is to learn the (approximate) transport map ψ .

2.1 Continuous normalizing flows and flow matching

One common choice for modeling ψ is as a deterministic dynamic system with a time-dependent generator ψ t : [0 , 1] × R d → R d . The solution to this dynamical system is an ordinary differential equation (ODE) and the learned transport map is known as a continuous normalizing flow (CNF). A CNF is a time-indexed neural transport map ψ t , for all time t ∈ [0 , 1] , that is trained to push forward samples from prior µ 0 to a desired target µ 1 . Specifically, a CNF models the ODE d dt ψ t ( x ) = f t ( ψ t ( x )) with initial conditions ψ 0 ( x 0 ) = x 0 and f t : [0 , 1] × R d → R d being the time-dependent vector field associated with the ODE and transports samples from µ 0 → µ 1 .

The most scalable way to train CNFs is to utilize a simulation-free training objective which regresses a learned neural vector field v t,θ ( x t ) : [0 , 1] × R d → R d to the desired target vector field f t ( x t ) for all time. This technique is commonly known as flow-matching [Liu, 2022, Albergo and Vanden-Eijnden, 2023, Lipman et al., 2023, Tong et al., 2024a] and has the neural transport map ψ t,θ which is obtained through a neural differential equation [Chen et al., 2018] d dt ψ t,θ ( x ) = v t,θ ( ψ t,θ ( x )) . Specifically, flow-matching regresses v t,θ ( x t ) to the target conditional vector field f t ( x t | z ) associated to the target flow ψ t ( x t | z ) . We say that this conditional vector field f t ( x t | z ) , generates the target density µ 1 ( x 1 ) by interpolating along the probability path µ t ( x t | z ) in time. We often do not have closed-form access to the generating marginal vector field f t ( x t ) . Still, with conditioning, e.g., z = ( x 0 , x 1 ) , we can obtain a simple analytic expression of a conditional vector field that achieves the same goals. The conditional flow-matching (CFM) objective can then be stated as a simple simulation-free regression,

The conditioning distribution q ( z ) can be chosen from any valid coupling, for instance, the independent coupling q ( z ) = µ 0 ( x 0 ) µ 1 ( x 1 ) . To generate samples and their corresponding log density according to the CNF, we may solve the following flow ODE numerically with initial conditions x 0 = ψ 0 ( x 0 ) and c = log µ 0 ( x 0 ) , which is the log density under the prior:

In the next section, we outline a different methodology to build a transport map leveraging stochastic dynamics. This allows us to frame the mass transport problem as a Schrödinger bridge, which is well suited to modeling noisy measurements in applications such as single-cell evolution.

3 Schrödinger Bridge with non-zero reference field

The complex nature of particle dynamics can be captured as a mass transport problem under a prescribed reference process. Specifically, we model particle evolution using a parametrized stochastic differential equation (SDE), with drift v t,θ : [0 , 1] × R d → R d , diffusion coefficient g t > 0 :

where B t is a standard Brownian motion and by convention time t ∈ [0 , 1] flows from t = 0 to t = 1 such that marginal distribution at the endpoints are ρ 0 and ρ 1 . These endpoints are provided as empirical distributions and represent endpoint observations along a transport trajectory. The SDE in eq. (3) induces a path measure in the space of Markov path measures ( P t,θ ) t ∈ [0 , 1] ∈ P ( C [0 , 1] , R d )

such that the marginal density p t evolves according to the following Fokker-Plank equation:

In addition, our modeling of particle dynamics is informed by a reference process which is defined by the following SDE with corresponding drift f t : R d → R d and diffusion coefficient g t > 0 :

We denote the induced path measure of eq. (5) as ( Q t ) t ∈ [0 , 1] .

Note further that we assume the diffusion coefficient, g t , to be the same for both processes P t and Q t to simplify the setting and facilitate easier exposition of the setting considered in this paper.

Schrödinger bridge with zero-drift . We now state the Schrödinger bridge problem, which finds an optimal path measure P ∗ that is the solution to the following KL-divergence minimization problem:

In settings where eq. (5) is zero-drift and with constant diffusion coefficient-i.e. d X t = g t B t -the Schrödinger bridge problem [Schrödinger, 1932] devolves into the Diffusion Schrödinger Bridge problem [De Bortoli et al., 2021a, Bunne et al., 2023]. In this special case, the Schrödinger bridge problem admits a unique solution and is linked to the entropic optimal transport plan through the seminal result of Föllmer [1988]. Specifically, P ∗ is a mixture of conditional Brownian bridges Q t ( ·| x 0 , x 1 ) weighted by the entropic OT-plan π ∗ ∈ Π( ρ 0 ⊗ ρ 1 ) which is a valid coupling in the product measure ρ 0 ⊗ ρ 1 , in other words ∫ π ( x 0 , · ) = µ 0 ( x 0 ) , ∫ π ( · , x 1 ) = µ ( x 1 ) ,

This holds when g t = σ for some constant σ (i.e. g t is 'fixed'), which assume for the remainder of this work. In this case, operationally, the conditional Brownian bridges take the form of a Normal distribution Q t ( ·| x 0 , x 1 ) = N ( x t ; tx 1 +(1 -t ) x 0 , t (1 -t ) σ 2 ) with the mean given as an interpolation between two endpoints. Furthermore, when σ → 0 , the entropic OT problem reduces to the regular OT problem. We note that this Schrödinger bridge problem can be reinterpreted as a stochastic optimal control problem where the control cost is the drift v t,θ . That is the stochastic optimal control perspective minimizes average kinetic energy 3 of the learned process which leads to the following optimization problem:

Weapproximate P ∗ using mini-batch OT [Fatras et al., 2020, 2021] and simulation-free matching algorithms [Tong et al., 2024a,b, Pooladian et al., 2023], iterative proportional and Markov fitting [De Bortoli et al., 2021a, Shi et al., 2024], and generalized Schrödinger bridge matching [Liu et al., 2023a].

3.1 Schrödinger Bridges with non-zero drift

We now consider the more general case where the drift of the reference process Q is non-zero. In this case, existing computational approaches, which rely on gradient field dynamics, no longer apply. As detailed in table 1, we consider the case of a constant g t > 0 and tailor our approach to small g t as previous work has shown this to be preferred empirically [Tong et al.,

2024b]. While it is possible to also learn g t as in GSBM [Liu et al., 2023a], this is computationally expensive; in this work, we focus on the setting of fixed or small g t and vectorfields with Curl, as motivated by applications in our experiments section 4. In this case, we can approximate the

marginal process using a mixture of conditional bridges, albeit not necessarily Brownian. Specifically, we consider the case modeling the marginal process P t = ∫ Q t ( ·| x 0 , x 1 ) dπ ( x 0 , x 1 ) , where we use Q t ( ·| x 0 , x 1 ) to denote the stochastic bridge pinned at x 0 , x 1 at times 0 and 1 respectively and π is a valid coupling. With this decomposition, it remains to specify the parameterization of Q t ( ·| x 0 , x 1 ) and π ( x 0 , x 1 ) . As a modeling choice, we parameterize Q t ( ·| x 0 , x 1 ) using a mixture of Brownian bridges with learnable parameters η :

with the idea that,

We approximate the mean of Q t by designing a neural path interpolant φ t,η with parameters η :

We optimize φ t,η by minimizing the following simulation-free objective of the relative kinetic energy:

Here κ t ( µ t,η , x 0 ) is any smooth function, e.g. a nearest neighbor based distance kernel κ t ( µ t,η , x 0 ) = ∥ µ t,η -x i 0 ∥ 2 / ∑ N i ∥ µ t,η -x i 0 ∥ 2 . Then we use a kernel definition of f t based on data as it is uncommon to have access to f t for the practical applications we consider. We further discuss our assumptions for κ t in the appendix §G.4. We also note ∂µ t,η /∂t can be computed using automatic differentiation:

The pseudocode for learning the neural path is presented in algorithm 1. To approximate P t we next learn to approximate the optimal mixture of conditional bridges P ∗ t = E x 0 ,x 1 ∼ π ∗ ( x 0 ,x 1 ) [ x t,η ] . However, this necessitates the feasibility of computing the OT-plan, which is defined below:

Indeed, this optimal transport cost c ( x 0 , x 1 ) can be computed through simulating the entire trajectory. However, we opt for using a stochastic estimator of the cost with K samples:

This cost ensures that we choose a coupling which minimizes the total cost in eq. (10). We highlight that the optimal plan π ∗ is intractable and we instead use a biased minibatch approximation of the plan (see §G.3). We use the cost in eq. (13) to estimate a transport plan π ( x 0 , x 1 ) to construct the approximated mixture of conditional bridges P t which is needed to learn the drift v t,θ ( x t ) of eq. (3),

The procedure for this marginal (flow) matching objective is presented in algorithm 2. In the case that g t := σ > 0 , for a constant σ , we also need to learn the marginal score s t,θ ≈ ∇ log p t ( x t ) . This can be learned using a conditional score matching objective where λ t = 2 √ t (1 -t ) /σ and

In totality, the combined loss is given by thus L ( θ ) = L flow ( θ ) + L score ( θ ) .

Algorithm 2 Marginal Score and Flow Matching

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4 Experiments

We investigate the application of CURLY-FM on multiple applications which exhibit non-gradient field dynamics including a simple toy example, an ocean currents modeling application, a computational fluid mechanics dataset, and an application to single-cell trajectory inference. We benchmark CURLY-FM against both simulation-free flow matching approaches: Conditional flow matching (CFM) [Liu et al., 2023b, Peluchetti, 2023, Lipman et al., 2023, Albergo et al., 2023], optimal transport conditional flow matching (OT-CFM) [Tong et al., 2024a] and when possible metric flow matching [Kapu´ sniak et al., 2024] which cannot model non-zero drift dynamics, as well as simulation-based methods in TrajectoryNet and SBIRR [Shen et al., 2025] which can model non-zero drift dynamics but are much slower [Tong et al., 2020] and numerically unstable.

We evaluate CURLY-FM using metrics both on held out samples (2-Wasserstein ( W 2 )) as well as metrics which directly measure how well the learned drift f θ field matches the reference drift (Cosine distance and L 2 cost). We note that in many cases, it is not possible to match the reference drift exactly as the model is forced to match the marginals.

4.1 Synthetic Experiments

We start our experimental study of learning cyclical patterns from population-level observed populations by considering a synthetic example. We construct source and target distributions on asymmetrically arranged circles ( fig. 1a), each with higher particle population density on one side. Given a circular reference velocity field f t ( x t , ω ) with constant rotational speed, the goal is to learn the velocity-field v t,θ ( x t ) and trajectories ψ t,θ ( x t ) for t ∈ [0 , 1] . We find that previous flow matching methods with zero-reference field f ∗ t result in straight paths between source and target distributions, thereby failing to capture cycling patterns in the underlying data (see 1b and fig. 1c).

4.2 Modeling Ocean Currents

We model ocean currents in the Gulf of Mexico using a resolution of 1 km of bathymetry data from HYbrid Coordinate Ocean Model (HYCOM), which allows us to obtain a reference field. We follow the data processing pipeline of Shen et al. [2025] and observe 111 particles per time-point (see §D for exact dataset details). We report our quantitative results in table 2 and observe that across the left-out time points, CURLY-FM obtains the best results for the majority of the reported metrics,

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and also outperforms the previous state-of-the-art SBIRR [Shen et al., 2025] on the EMD metric. Moreover, we note that CURLY-FM is computationally fast and achieves these results in minutes compared to 4 hrs for the simulation-based SBIRR. These findings are also substantiated in fig. 2, where we see trajectories that look more natural at modeling periodic behavior than OT-CFM.

4.3 Experiments on Single-Cell Data

To show that CURLY-FM is effective in learning dynamic behavior in single-cell data, we leverage two biologically rich datasets consisting of cell cycles in human cell fibroblasts [Riba et al., 2022] and erythroblast development in mouse [Pijuan-Sala et al., 2019]. We aim to learn cell state trajectories and development paths considering the respective RNA-velocity fields, providing information about

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Figure 3: Ground truth data.

Cell cycle dynamics in human fibroblasts . We study the nature of cell cycling in human fibroblasts and reconstruct cyclical patterns in spliced-unspliced RNA space for single genes. We leverage RNAvelocities in figure 3a to construct cell state transition paths in figure 3 by estimating RNA velocity field between marginals using k -nn algorithm. Further dataset details are included in §E.1. Figure 4 shows learned velocity fields v t,θ ( x t ) and trajectories ψ t,θ ( x t ) between cell cycle distributions at t = 0 and t = 1 . In table 3, we show results on the trajectory inference task comparing CURLY-FM to CFM, OT-CFM, and TrajectoryNet. Given the underlying cell cycle process, the aim is to learn circular trajectories resulting from a divergence-free velocity field. While traditional methods are successful in generating end points near ground truth, they fail at learning cyclic patterns, as shown in figures 4f.

Our results show that considering a non-zero reference field and velocity inference captures non-gradient dynamics in data. Figure 4a and fig. 4d show learned behavior using CURLY-FM. We observe that the trajectory ψ t,θ ( x t ) inferred with CURLY-FM closely matches expected cycling patterns in the fibroblast dataset, in contrast to trajectories inferred using CFM and OT-CFM. This

is quantified in table 3, where we can see the cosine distance to the reference field is significantly lower for CURLY-FM.

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Reconstructing cell differentiation in mouse Erythroid development. . Mouse erythroid cells develop in a curved trajectory over time. We show that CURLY-FM can adhere to this developmental path purely based on velocity data for the first time. Earlier works have used manifold-based penalties to follow curved structures. We show that this is no longer necessary with clever usage of velocity information. We observe 9,815 erythroid cells undergoing differentiation and partition the data into three temporal snapshots, withholding the central marginal to assess trajectory inference (see §E.2).

We visualize trajectories in figure 5 showing that CURLY-FM clearly follows the developmental pathway of mouse erythroid cells, whereas OT-CFM fails to capture dynamics between marginals. To assess CURLY-FM performance, we measure cosine-distance and L 2 norm between learnt and ground truth velocities as well as W 2 distance between points on left-out marginal. Our quantitative results show that CURLY-FM outperforms OT-CFM at reconstructing the underlying RNA-velocity field and cell trajectories in majority of selected dimensions. MFM continues to achieve

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4.4 Experiments on Computational Fluid Mechanics Data

We evaluate CURLY-FM on a particle-based PDE dataset generated by a Lagrangian solver. Unlike grid-based (Eulerian) methods, Lagrangian approaches discretize the fluid as a set of particles that move with the flow. These particles evolve under the dynamics of the PDE which provides the particles' positions over time; other quantities of interest, such as velocity or energy, are then computed from these positions. We use data from LagrangeBench [Toshev et al., 2023], specifically the two-dimensional decaying Taylor-Green vortex (2DTGV) dataset(see §F).

In table 5, we report quantitative results on left-out particles for each marginal from a test set. We evaluate performance using (i) the cosine distance between the learned velocity field and the reference field; (ii) mean squared error (MSE) between the predicted particle positions at marginal t +1 and the ground truth positions, using the known coupling (ordering) between particles across marginals; and (iii) precision@ k , measuring how often the predicted position is among the k nearest neighbors of the corresponding ground truth particle. CURLY-FM outperforms baselines in terms of cosine distance and precision@ k while matching or outperforming the baseline methods on MSE. The smaller cosine distance for CURLY-FM shows that CURLY-FM produces velocity fields that better align with the reference field. Equal or lower MSE paired with higher precision@k shows that CURLY-FM more accurately recovers the true particle coupling and generates more faithful trajectories (c.f. fig. 10).

4.5 Further analysis of CURLY-FM performance

On higher stochasticity . We extend our discussion in section 3.1 on stochasticity levels σ to consider an ablation where σ > 0 . Despite our work being motivated in the little to no stochasticity regime, we demonstrate that considering σ > 0 does not impact CURLY-FM efficiency.

We find, similar to previous work [Tong et al., 2024b], low values of σ perform the best on all metrics. As a result, we recommend setting σ to zero unless some reference σ value is known. Therefore, all of our experiments are performed under σ = g t = 0 assumption.

We consider ocean currents dataset and find that larger stochasticity monotonically decreases performance on our tasks, thus justifying our choice of σ for our empirical work.

On computational efficiency . We further provide the computational cost in wall clock time for TrajectoryNet, SBIRR and CURLY-FM in table 7 (see §H.2 for further baseline comparison). We observe that CURLY-FM completes the Ocean currents problem in minutes with higher accuracy in trajectory and velocity field inference task, while SBIRR and TrajectoryNet are in the order of multiple hours and unlike CURLY-FM are simulation-based.

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5 Related Work

Flow matching . Flow matching [Lipman et al., 2023], also known as rectified flows [Liu, 2022, Liu et al., 2023b] or stochastic interpolants [Albergo and Vanden-Eijnden, 2023, Albergo et al., 2023], has emerged as the default method for training continuous normalizing flow (CNF) models [Chen et al., 2018, Grathwohl et al., 2019]. However, FM can lead to unnatural dynamics less, and therefore many works attempt to derive methods for using minimum energy [Tong et al., 2024a, Pooladian et al., 2023] and more flexible conditional paths [Neklyudov et al., 2024, Kapu´ sniak et al., 2024].

Schrödinger bridges with deep learning . To tackle the Schrödinger bridge problem in high dimensions many methods propose simulation-based [De Bortoli et al., 2021b, Chen et al., 2022, Koshizuka and Sato, 2023, Liu et al., 2022] and simulation-free [Shi et al., 2024, Tong et al., 2024b, Pooladian and Niles-Weed, 2023, Liu et al., 2023a] set-ups with various additional components incorporating variable growth rates [Zhang et al., 2025, Pariset et al., 2023, Sha et al., 2024], stochasticity, and manifold structure [Huguet et al., 2022] proposed based on neural ODE and neural SDE [Li et al., 2020, Kidger et al., 2021] frameworks. However, very few methods are able to incorporate approximate velocity data, and either match marginals using simulation [Tong et al., 2020], or do not attempt to match marginals [Qiu et al., 2022]. Finally, Schrödinger bridges with non-zero reference field have also been considered by Bartosh et al. [2024] and concurrently by Bartosh et al. [2025] and Shen et al. [2025], however, they do not employ a two-stage simulation-free approximation as CURLY-FM. We include further details on related work comparison in appendix §B.

RNA-velocity methods on discrete manifolds . A common strategy to regularize and interpret RNA-velocity [La Manno et al., 2018, Bergen et al., 2020] is to restrict it to a Markov process on a graph of cells representing a discrete manifold or compute higher-level statistics on it [Qiu et al., 2022]. However, these approaches are not equipped to match the marginal cell distribution over time. CURLY-FM can be seen as a method that unites these approaches with marginal-matching approaches.

6 Conclusion

In this work, we introduced CURLY-FM, a method capable of learning non-gradient field dynamics by solving a Schrödinger bridge problem with a non-zero reference process drift. In contrast to prior work, CURLY-FM is simulation-free, greatly improving numerical stability and efficiency. We showed the utility of this method in learning more accurate dynamics in a cell cycle system with known periodic behavior, computational fluid dynamics under Lagrangian solvers, and ocean currents. CURLY-FM opens up the possibility of moving beyond modeling population dynamics with simulation-free training methods and towards reconstructing the underlying governing dynamics [Xing, 2022]. Nevertheless, CURLY-FM is currently limited in its ability to discover the underlying dynamics by accurate inference of the reference field, which is an inherently difficult problem, especially over longer timescales. Exciting directions for future work involve additional verification of trajectories through lineage tracing [McKenna and Gagnon, 2019, Wagner and Klein, 2020], and improved modeling across non-stationary populations with the additional incorporation of unbalanced transport or multiomics datatypes [Baysoy et al., 2023].

Acknowledgments

The authors acknowledge funding from UNIQUE, CIFAR, NSERC, Intel, and Samsung. The research was enabled in part by computational resources provided by the Digital Research Alliance of Canada ( https://alliancecan.ca ), the Province of Ontario, companies sponsoring the Vector Institute ( http://vectorinstitute.ai/partners/ ), Mila ( https://mila.quebec ), and NVIDIA. KK is supported by the EPSRC CDT in Health Data Science (EP/S02428X/1). AJB and LA are partially supported by NSERC Post-doc fellowships. LA is supported by the Eric and Wendy Schmidt Center at the Broad Institute of MIT and Harvard. This research is partially supported by EP- SRC Turing AI World-Leading Research Fellowship No. EP/X040062/1 and EPSRC AI Hub on Mathematical Foundations of Intelligence: An 'Erlangen Programme' for AI No. EP/Y028872/1. We thank Renato Berlinghieri, author of SBIRR, for valuable discussions and for generously sharing his ocean currents code and data, which made our ocean current experiments possible.

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Appendix

A Complementary Orthogonal Work on Single Cell

There are many recent works tackling the single-cell trajectory inference problem on single-cell RNA-sequencing data. In this work we focus on incorporating a reference velocity field and learning non-gradient field dynamics. There are many other related works that may be used in conjunction with ideas in this paper. Here we detail some of these complementary works and how they could be combined with ideas in CURLY-FM. Specifically a number of areas have been identified as improving how cells are modeled by flow-based networks.

Optimal transport / minimal energy : Cells are modeled by optimal transport over short enough time scales and is therefore desirable in almost all applications of single-cell trajectory inference [Hashimoto et al., 2016, Schiebinger et al., 2019, Tong et al., 2020, 2024a].

Density or manifold assumptions : Cells are also known to lie on a low dimensional manifold within gene space [Moon et al., 2018]. This knowledge has been exploited in a number of works that are both require simulation during training [Tong et al., 2020, Huguet et al., 2022] and more recent methods which do not [Neklyudov et al., 2024, Kapu´ sniak et al., 2024]

Unbalanced transport (modeling cell birth and death) : By default, flow models assume conservation of mass over time through the continuity equation. Over long time scales this is not a good model of a population of cells as [Zhang et al., 2025]

Stochasticity : Cells move stochastically based on unobserved factors. This has led to a number of methods that attempt to model cells with stochastic dynamics [Koshizuka and Sato, 2023, Tong et al., 2024b, Schiebinger, 2021], and in particular with Schrödinger bridges, as the stochastic extension of dynamic optimal transport. While there have been many works on learning Schrödinger bridges simulation-free there is little work on efficient learning of general reference drift functions which we tackle here.

Velocity estimates : RNA-velocity [La Manno et al., 2018] exploits particularities about the RNA collection data to measure both older and newer RNA transcripts at a single timepoint. This allows the approximation of RNA-velocity- the approximate instantaneous change of the RNA expression of each cell. First used in Tong et al. [2020], velocity estimates are relatively underutilized in the trajectory inference problem as data is more scarce and more difficult to process. To our knowledge, CURLY-FM is the first method to provide a simulation-free method for incorporating velocity information into a learned flow.

Distribution conditioning Where most flow-based frameworks model cells as a noninteracting point cloud, recent work has also considered flows which include terms for the interaction between cells [Atanackovic et al., 2025, Haviv et al., 2025]. This allows the modeling of more complex interactions between cells which is an extremely prevelent dynamic in real cell systems.

While CURLY-FM focuses on the incorporation of optimal transport, stochasticity, and in particular velocity , it does not address problems of unbalanced transport or manifold structure of the single-cell datatype. Future work will incorporate ideas from CURLY-FM in combination with existing ideas on how to model density and unbalanced transport for more expressive, accurate, and useful models of cell dynamics towards virtual cells [Bunne et al., 2024a].

B Further Details on Related Work

Iterative Algorithms . Shen et al. [2025] propose an iterative bi-level algorithm, first solving a Schrödinger bridge and estimating the forward-backward drift given a current guess of the reference drift. Given the solution to step 1, trajectories are simulated to obtain an updated estimate of the reference drift. This bi-level approach is considered since the reference drift belongs to a family of possible reference drifts rather than a single prescribed one. In contrast, CURLY-FM employs a twostage algorithm that is not iterative and also does not search over a family of reference drifts. More critically, CURLY-FM is also simulation-free, which is achieved by making the modeling assumption that the conditional mixture of bridges is modelled as Brownian bridges. In practice, this means that

CURLY-FM is significantly more efficient to train as shown in table 15. Finally, note, both Shen et al. [2025] and CURLY-FM can model non-gradient field dynamics, but only the latter is simulation-free.

Latent SDEs . Bartosh et al. [2024] and Bartosh et al. [2025] consider Latent SDEs for learning stochastic dynamics with simulation-free training without explicitly solving Schrödinger Bridge problem as considered by CURLY-FM framework. This means that unlike CURLY-FM, work on Latent SDEs does not consider minimizing KL divergence to find an optimal path measure with respect to the reference process. Further, SDEs in CURLY-FM are not Latent SDEs. Whilst building a computational approach to solving the Schrödinger Bridge problem with latent SDEs may be possible, this is an orthogonal research direction to our work.

C Algorithmic Details

In this section we detail python code for training and inference for reproducibility.

D Ocean Currents Dataset

We assess performance of CURLY-FM on real ocean currents dataset acquired from a HYbrid Coordinate Ocean Model (HYCOM) reanalysis released by US Department of Defense.

Dataset . This data consists of real ocean currents measurements in Gulf of Mexico acquired at 1km bathymetry, providing hourly ocean currents velocity fields for the geographic region between 98E and 77E in longitude and 18N to 32N in latitude at each day since January 1st, 2001.

Experimental Set-up . We leverage experimental set-up as presented in Shen et al. [2025], and focus on a specific time point at 17:00 UTC on June 1st, 2024, extracting ocean surface velocity field that contains a vortex. From a point near its center, we uniformly draw 1,000 initial positions whithin radius 0.05 and evolve them across nine time steps such that ∆ t = 0 . 9 , computing velocities by the nearest grid node for ∼ 111 observations per time step.

Ablations . Tables 8 and 9 show ablations on using coupling cost in algorithm 2 for trajectory inference between drifter observations in ocean currents experiments. We observe that there is limited effect in integrating coupling cost over increased numbers of time steps between marginals.

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E Single Cell Datasets

E.1 Human Fibroblasts Dataset

We consider the human fibroblasts dataset [Riba et al., 2022] that contains genomic information about 5,367 cells observed across a fibroblast cell cycle. Cell data further contains information about cycling genes, and more specifically, their RNA velocities, which are used to estimate the reference RNA velocity field. Figure 7 shows a distribution of cell rotations and phases during a cell cycle process.

Pre-processing . Data is pre-processed by selecting top d variable genes from the data. Further, we use scvelo package [Bergen et al., 2020] to compute imputed unspliced (Mu) and spliced (Ms) expressions as well as velocity graph. We construct a cell-cell k -nn graph using scv.pp.neighbours(adata) in the joint spliced and unspliced expression space using default scvelo hyperparameters. For each gene and cell, we compute averaged fits moment and the centered second moment of spliced and unspliced genes as well as relared attributes. RNA velocities are computed using scv.tl.velocity(adata) fitting the stochastic transcriptional dynamics model [La Manno et al., 2018]. Finally, we compute low-dimensional embeddings for velocities using UMAP with scv.tl.velocity_embedding(adata) for our visualizations in figure 4d. For selecting top d highly variable genes, we use sc.pp.highly_variable_genes(adata, n_top_genes=d) .

E.2 Mouse Erythroid

We consider a dataset showing mouse gastrulation subset to erythroid lineage [Pijuan-Sala et al., 2019], representing the developmental pathway during which embryonic cells diversify into lineage-specific precursors, evolving into adult organisms. The data consists of 9,815 cells evolving through five lineage stages as shown in Figure 8, and it is available through scvelo package API.

Pre-processing . Data is pre-processed using scvelo and unitvelo [Gao et al., 2022] package to compute imputed unspliced (Mu) and spliced (Ms) expressions as well as the velocity graph. With unitvelo , we construct the cell latent time used to approximate the differentiated time experienced by cells. Mouse erythroid velocity field is computed using unitvelo.run_model() following instructions from unitvelo documentation.

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Filtering RNA Velocities . To address tail-effects of noisy single-cell data, we filter RNA velocities by down-weighting distant neighbors in the k -NN estimate at x t and injecting small Gaussian noise. We construct new estimate f ∗ t by interpolating between k -nn velocity estimate f t and noise such that f ∗ t = (1 -w γ ( x t )) ∗ f t + w γ ( x t ) ∗ N (0 , 0 . 1) . The weight w γ ( x t ) ∈ [0 , 1] is given by a sigmoid of the distance between the k -NN distance and a threshold hyperparameter γ , so that larger distances yield larger w γ ( x t ) and thus stronger penalization of distant neighbors.

F Computational Fluid Dynamics Dataset

Experimental details for CFD . We conducted our CFD experiments using the two-dimensional decaying Taylor-Green vortex (2DTGV) dataset provided by LagrangeBench [Toshev et al., 2023]. We subsampled 2000 particles and considered five equispaced marginals (snapshot distributions over particle positions). The goal was to perform trajectory inference from unordered population snapshots. Since instantaneous velocity is not directly observed, we constructed the reference drift field using finite differences. This aligns with how derived physical quantities-such as velocity or energy-are computed from particle positions in Lagrangian PDE solvers.

We considered a dataset split of [80% , 20%] across the train and test sets, respectively. For the 2000 particles, this resulted in 1600 being used for training and 400 for testing. All other hyperparameters of CURLY-FM were the same as for the other experiments.

F.1 Ablations on CFD

Figure 9 illustrates the trajectories of 25 particles under CFM, OT-CFM, and CURLY-FM. Both CFM and OT-CFM tend to produce straight, relatively short paths, most notably in the case of OT-CFM, indicating a preference for minimal transport effort. In contrast, CURLY-FM learns longer, more intricate trajectories that better resemble the expected fluid dynamics.

Figure 10 visualizes the velocity fields inferred by each method. While CFM and OT-CFM yield smoother and simpler velocity patterns, CURLY-FM captures a richer and more structured field. This complexity reflects closer alignment with the reference field and suggests improved physical fidelity. The resulting velocity field from CURLY-FM more accurately models the underlying dynamics, adhering to both data-driven transport and the governing reference flow.

Tables 10 and 11 show CURLY-FM without coupling and for different number of times used to evaluate the coupling cost, respectively. In particular, table 10 shows that the coupling based on minimizing the kinetic energy only marginally improves performance for CFD experiments. Furthermore, table 11 shows that there is little to no benefit in using additional times to approximate the coupling cost in algorithm 2.

G Further Experimental Details

G.1 Single-marginal Set-up

φ t,η ( x 0 , x 1 , t ) and v t,θ ( x t , η ) design . Both φ t,η ( x 0 , x 1 , t ) and v t,θ ( x t , η ) are designed as MLP models with 3 layers. We select MLP dimensions based on the number of chosen highly variable genes d . For φ t,η ( x 0 , x 1 , t ) we choose d in = 2 × d and d out = d . For v t,θ ( x t , η ) , we choose dimensions of d in = d . The dataset is split as [80% , 10% , 10%] across training, validation, and test.

Training . All CURLY-FM and baseline experiments are run using lr = 10 -4 learning rate and Adam optimizer with default β 1 , β 2 , and ϵ values across three seeds and with 1,000 epochs split into 500 epochs to train φ t,η followed by 500 epochs to train v t,θ .

Baselines . TrajectoryNet was run with 250 epochs with the Euler integrator with 20 timesteps per timepoint. We use 250 epochs to limit the experimental time and the number of function evaluations to roughly 5 × that of simulation-free methods. We use a batch size of 256 samples. We use a Dormand-Prince 4-5 (dopri5) adaptive step size ODE solver to sample trajectories with absolute and relative tolerances of 10 -4 .

Compute . All experiments were conducted using a mixture of CPUs and A10 GPUs.

G.2 Multi-marginal set-up

In the multi-marginal setting, we train by randomly sampling interpolation times t within intervals corresponding to each adjacent pair of marginals-for example these intervals will be [0,1] and [1,2] if our marginals lie at t ∈ { 0 , 1 , 2 } . For each interval [ t i , t i +1 ] , we sample a random time t , then compute both the neural interpolant x t and its derivative ˙ x t . We also compute global time ˆ t = t i + t , if say t = 0 . 5 , the global time for the second marginal pair will be ˆ t = 1+0 . 5 = 1 . 5 . This effectively ensures that neural interpolant and its derivative match the global axis of time. We then connect all the marginals by concatenating neural interpolant, its derivative and global times into single tensors and use these as inputs to train our neural path interpolant in algorithm 1 and the drift in algorithm 2.

φ t,η ( x 0 , x 1 , t ) and v t,θ ( x t , η ) design . We design φ t,η ( x 0 , x 1 , t ) and v t,θ ( x t , η ) as MLPs as shown in table 12 across all multi-marginal experiments. CFD experiments additionally use four residual connection blocks with no dropout. We select MLP dimensions based on the number of chosen highly variable genes d . For φ t,η ( x 0 , x 1 , t ) we choose d in = 2 × d and d out = d . For v t,θ ( x t , η ) , we choose dimensions of d in = d .

Training . We show training details for multimarginal set-up in table 12 and compute test metrics on left-out marginals. We select k = 20 neighbours for ocean currents and CFD experiments and k = 30 neighbors for mouse erythroid experiments to compute ground-truth velocities.

Compute . All experiments were conducted using a mixture of CPUs and A10 NVIDIA GPUs.

G.3 Cost of OT plan

The cost of the optimal plan π ∗ is intractable as it would in general require computing costs over stochastic paths. Consequently, we make several approximations to these couplings that enable faster throughput as offered in simulation-free training. In particular, we make three approximations to the cost c ( x 0 , x 1 ) between two points, and one on the coupling given this cost:

Algorithm 1 has learned a path or set of paths for the means of Gaussians in proximity to the optimal Q t ( x t | x 0 , x 1 )

We consider low stochasticity σ setting, and therefore the cost is close to the distance of the means travel

The integral of the squared length of the curve is approximated empirically using a Monte Carlo estimate

After approximating c ( x 0 , x 1 ) , we use we use mini-batch OT to approximate the entropic OT problem following set-up in Tong et al. [2024b]. Interestingly, in the low stochasticity setting, Tong et al. [2024b] found that mini-batch OT with no stochasticity was empirically a better approximator of π ∗ in the Gaussian case. This is because mini-batch transport adds some amount of 'entropy' to the plan due to its approximation. We follow this approximation in our setting.

G.4 Further details on ground-truth velocity estimate

On use of kernels . We highlight that in practical settings we investigate, a continuous ground truth reference field does not exist. In our applications, we only have access to the reference field at discrete points in space and time that correspond to ground truth data at the different time marginals. Consequently, we use a kernel to build a continuous reference field f t ( x t,η ) .

Reference drift velocity estimate f t . In our considered setting, we assume access to the ground truth reference drift at samples x 0 ∼ µ 0 and x 1 ∼ µ 1 as part of the problem setup. Note that in the high-impact application domains we consider, such as trajectory inference in single-cell data, we have access to the RNA velocity [Riba et al., 2022, Bergen et al., 2020], which is assumed to be a reasonable estimate (up to a scaling factor) of the SDE velocity [Tong et al., 2020]. As we need to estimate CURLY-FM everywhere in space and time during training, we construct a smoothed version of the reference drift by using a kernel κ t . This allows us to construct the reference drift f t ( x t ) -using knowledge from the ground truth reference drift in the existing dataset-in places where there are no ground truth samples. Consequently, we take f t in these intermediate points as our ground truth reference drift. We further highlight that kernel estimate is not used in cases where ground-truth velocities are given on a continuous domain.

Ablation on kernel estimate accuracy . To show that estimate accuracy does not effect our findings, we conduct an ablation study by constructing a noisy reference drift f noise t for various noise levels β ∈ [0 , 1] . We show results in table 13 for the Ocean Currents dataset. The noisy reference drift is obtained as a linear combination of the ground truth reference drift f t and noise from a standard gaussian distribution ( f noise t = (1 -β ) ∗ f t + β ∗ noise ). We find that the performance between β = 0 (no noise, i.e. regular CURLY-FM) and β = 0 . 25 are similar whilst the performance for β = 0 . 5 and higher gradually becomes worse, as the noise dominates over the ground truth reference drift in f noise t . This shows that CurlyFM is robust to moderate amounts of noise added to the ground truth reference drift.

Reconstructed field . We provide comparison to ground truth velocity field and k -nn estimate for the human fibroblasts data. From figure 12 it is clear that our approach faithfully reconstructs ground-truth RNA velocities.

(a) Ground truth velocity

(b) k -nn velocity estimate

G.5 On the tradeoff between directly computing OT plan vs. iterative refinement

Two main strategies exist for approximating the static (possibly regularized) optimal transport coupling. In this work we primarily use the mini-batch approximation. However, there are also iterative-refinement type approaches that are unbiased in the infinite limit like those presented in De Bortoli et al. [2021b], Shi et al. [2024]. These methods work by creating an 'outer loop' where the bridge is simulated in one direction, then matched in the other to create an iterative refinement to match marginals. While this is possible to incorporate into our framework, it is not as suitable for our application domain, where we assume small stochasticity level. Iterative approaches are not suitable for small stochasticity levels because the number of iterations of iterative proportional fitting or iterative Markov fitting (IPF / IMF) for suitable convergence depends is inversely proportional to the stochasticity [Tong et al., 2024b, Shi et al., 2024]. Indeed, at the zero noise limit, the IMF approach collapses to a rectified flow, which only solve the OT problem in limited domains e.g. 1D [Liu et al., 2023b], and a few select Gaussian settings [Bansal et al., 2025]. For low noise levels the convergence rate is significantly slower.

H Supplementary Results

H.1 Additional Synthetic Experiments

CURLY-FM robustness . To further assess CURLY-FM robustness, we designed a toy experiment with an analytical Schrödinger Bridge solution and non-gradient dynamics based on [Tong et al., 2024b, De Bortoli et al., 2021b, Shi et al., 2024]. We bridge two Gaussians in the presence of a spiral reference field across various dimensions d and stochasticity levels σ = g t = 0 . Specifically, we initialize two Gaussians in 20 dimensions centered at µ 0 = [ -0 . 1 , 0 , 0 , . . . , 0] and µ 1 = [0 . 1 , 0 , 0 , . . . , 0] with standard deviations σ 0 = σ 1 = [1 , . . . , 1] . We also define a ground truth transport field, which unlike the standard OT field, has an additional rotational component. We note that this has equivalent marginal probability distributions, but also rotates around the origin in the second and third dimensions. This allows us to test how well our method works in a simple toy setting.

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We compare against the closed-form solution and OT-CFM, measuring KL divergence, Wasserstein distance, and the cosine distance to the ideal rotational angle. Results in table 14 confirm CURLY-FM performs best in low-dimensional, low-stochasticity settings, achieving high cosine similarity.

Comparison to baselines in synthetic setting . We further compare CURLY-FM robustness in synthetic setting to SF2M and SB-CFM baselines. We find that CURLY-FM outperforms both SF2M and SB-CFM on stochasticity σ = 0 . 1 on cosine distance. In case of σ = 0 . 1 , CURLY-FM outperforms SB-CFM, and achieves similar cosine distance with SF2M while providing better W 2 metrics.

H.2 Computational Efficiency

We provide comparison across DM-SB, Vanilla-SB, TrajectoryNet and SBIRR baselines in terms of computational efficiency evaluated on the ocean currents experiments (shown in table 15). We further report results for the 2D and 10D settings for TrajectoryNet and CURLY-FM in table 16. We find that TrajectoryNet is considerably slower than CURLY-FM and is not a scalable approach. This is evident in that training TrajectoryNet takes, on average, 11 times longer than CURLY-FM in the 2D case, and 17 times longer in the 10D case. Moreover, as dimensionality increases from 2D to 10D, CURLY-FM incurs an increase in computational cost by 1.7x, whereas TrajectoryNet incurs

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H.3 Further Baseline Comparisons

H.3.1 Generalized Schrödinger Bridge Matching

In this section we provide a more extensive comparison of CURLY-FM to Generalized Schrödinger Bridge Matching (GSBM) [Liu et al., 2024].

Experimental Set-up . We make two modifications to improve GSBM in our setting for the fairest comparison to CURLY-FM: We use our loss instead of GSBM loss in equation 6a to control splines and to incorporate the signal from a reference drift directly. We fix σ to the target σ as we assume a constant σ and this is the optimum for each bridge. We keep the iterative algorithm as found in the original GSBM paper. We follow the formulation in Tong et al. [2024b] which separately learns the deterministic flow and stochastic score functions, which then can be added together to calculate the stochastic drift or used separately to integrate deterministically. We note that we do this so that we can experiment with different stochasticity and simulate forward or backwards in time.

Number of control points . We initially set the number of control points to 2. We provide an additional ablation over the number of control points in GSBM, noting that the default number in GSBM is 15. We also find 15 provides the best tradeoff. We find that more or fewer control points do not improve performance. Otherwise we keep the same hyperparameters as CURLY-FM in terms of iterations, kernels, batch size, learning rate, models, etc. for fair comparison. Table 17 compares CURLY-FM to GSBM on the Ocean currents dataset with varying number of control points

Onloss used in baseline comparison . We clarify that the loss (6a) in GSBM cannot take into account a non-zero reference drift directly. Loss 6a only considers potential functions V t ( x t ) and the kinetic energy of u t , and therefore is theoretically problematic in our setting. We therefore use CURLY-FM loss function, which we believe is a more fair comparison. For completeness we further include a GSBMbaseline with loss 6a found in the GSBM paper. We find that CURLY-FM is able to outperform both variations of GSBM on our oceans dataset across the three main quantitative metrics of interest. We further find our modification of GSBM's loss allows it to better match the non-zero reference drift as evidenced by a lower cosine distance, while also performing slightly better in W 1 distance.

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H.3.2 Metric Flow Matching

For completeness of our analysis, we additionally report results for Metric Flow Matching (MFM) [Kapu´ sniak et al., 2024]. MFM introduces a geometric bias by enforcing interpolations that remain close to the underlying data manifold, effectively learning smooth geodesic paths that reflect intrinsic geometric structure. The two-stage learning strategy in CURLY-FM is directly adapted from MFM replacing the manifold-constrained interpolations with regression against the reference non-gradient dynamics. In other words, CURLY-FM can be viewed as introducing an alternative inductive bias on trajectories: whereas MFM constrains paths to lie on the manifold defined by data, CURLY-FM enforces a bias on velocities , aiming to learn reference-consistent vector fields.

Since the core algorithmic structure of CURLY-FM (Algorithm 1) mirrors that of MFM, the two formulations are in fact compatible and can be combined-leveraging manifold-constrained interpolants together with velocity-based biases. We leave this promising direction for future work.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors proposed a generalization of flow matching method for trajectory inference without assuming a gradient field reference drift. The main idea is by decomposing Schrodinger bridge's solution into mixture of endpoint couplings and conditional bridges, and construct them in a simulation free manner separately. The authors tested their method in several different setups and seen their proposal out perform existing baselines.\n\nSTRENGTHS AND WEAKNESSES:\n## Strength\n1) The algorithm is simulation free and performs well.\n2) Can handle none potential dynamics and leverage velocity information. \n\n## Weakness \n1) I am not fully convinced that some of the model choice is compatible with Schrodinger bridge formulation esp. when the diffusion term is different. See questions for details. \n2) The notations is a bit hard to follow -- I understand this is partly due to Schrodinger bridge problems involving many different SDEs.\n\nQUESTIONS:\n1) In eq (9), with general diffusion term $g_t$, should the KL divergence being $E_{P}\\int_t g_t^{-1}||v||_2^2$? Similar question holds for the unnumbered equation below eq. (10).\n\n2) I might miss understood it, as I understood the authors modeled $\\mathbb{Q}$ using a mixture of Brownian bridges, with potentially different diffusion terms and $\\mathbb{P}$ to have $g_t$ as diffusion term. Will the KL divergence between $\\mathbb{P}$ and $\\mathbb{Q}$ constructed this way being finite? KL divergence between SDEs not sharing diffusion terms might be infinite since they are not absolute continuous to each other. \n\n3) The authors applied their method in a RNA velocity dataset treating RNA velocity as similar velocity in the SDE model. I am not sure to what extend they are connected. RNA velocity take advantage of splicing happens after transcription and use the ratio to infer whether genes are being transcribed more or less. These velocity would depends on each gene's splicing rate and mRNA's degradation rate. I wonder how they author think about these? In the ocean example one might indeed have some sparse velocity measurements but need interpolation. I encourage the authors to make some comments on how good the velocity measurements needs to be.\n\nLIMITATIONS:\nYes mostly. But I encourage the authors to provide some comments on how velocity measurements were taken in each experiments and assumptions on them.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the authors propose a numerical methodology to solve the standard dynamic Schrödinger Bridge (SB) problem ($\\min KL(P|Q):P_0=\\pi_0, P_1=\\pi_1$), in the particular case where the reference process $Q$ has non-zero (or at least non-linear) drift. This is motivated by the fact that various real-world dynamics exhibit non gradient, periodic behavior, which cannot be modeled by simple reference dynamics. In theory, the solution to this problem writes as the mixture of the induced entropic OT plan and the reference \"bridge\" process $Q_{|0,1}$ (i.e., the reference process pinned at terminal times 0 and 1). The goal of the method is to approximate the drift of this solution by a neural network to be able to generate dynamics afterwards. A popular and simulation-free approach to compute the SB problem is bridge matching, as proposed in [1], which relies on two requirements wrt $Q$ : (i) being able to sample from the conditional distribution $Q_{t|0,1}$ at any time $t$ and (ii) having a tractable expression of the drift of $Q_{|0,1}$ (which is Markovian). In the current setting, these two requirements are not satisfied.\n\nTo circumvent this issue, the authors propose to approximate the reference bridge by a neural interpolant (i.e., an interpolant that relies on a neural network) before applying the standard bridge matching technique, in a similar way to [2]. This ends up in a two stage procedure coined \"Curly Flow Matching\", which requires two neural networks (one to approximate the reference bridge, one to approximate the drift of the SB solution). Then, they present the results of numerical experiments on various types of dynamics (single-cell trajectory, fluid dynamics...) and compare their method to classic flow matching and mini-batch OT-based flow matching. \n\n[1] Diffusion Schrödinger Bridge Matching, Shi et al., Neurips 2024. \n\n[2] Generalized Schrödinger Bridge Matching, Liu et al. ICLR 2024.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**:\n- the motivation of tackling the SB problem with non-zero drift is well explained\n- the range of the numerical experiments is interesting and based on real-world systems\n\n**Weaknesses**:\n- in my opinion, the mathematical explanation of the Curly-FM algorithm (Section 3.2) lacks crucial details on the manner the losses are defined (I could not find any additional details in the appendix), which makes the reading very hard. While the authors claim to solve the SB problem (which is stochastic by nature, due to the consideration of non-zero volatilities), the proposed method seems to only solve the deterministic-limit case (i.e. when the volatilities go to zero)... In particular, I have serious doubts about (i) the loss to approximate the reference bridge and (ii) the way the OT plan is approximated (see section \"Questions\") to solve an arbitrary SB problem with non-linear drift.\n-  in [1], the authors propose a very close approach to Curly-FM in a general SB setting, where the reference process does not have a trivial drift too (i.e., (1) approximation of the reference bridge and (2) application of bridge matching). The key advantage in comparison with the current method is that the reference bridge is approximated by a Gaussian Stochastic interpolant (which has since been extended to interpolants with Gaussian mixtures) which does not rely on any neural parameterization (rather a spline parameterization). This method should be compared with in the numerical experiments, especially via the trade-off expressivity/computational budget.\n\n[1] Generalized Schrödinger Bridge Matching, Liu et al. ICLR 2024.\n\nQUESTIONS:\n**Questions**\n- from what I understand, the first stage of Curly-FM consists in approximating the marginal distribution of the reference bridge (i.e. $\\mathbb{Q}_{t|0,1}$) by a deterministic neural interpolant. Since $\\sigma_t$ (the volatility of the reference process) is not zero, I have the feeling that there is a mismatch between what needs to be approximated (a probability distribution) and the approximation itself (a Dirac mass). In this case, the resulting SB solution is not correct, since it is not stochastic. Do I miss something ? In my opinion, a more adapted parameterization of the marginal distribution of the reference bridge would be a Gaussian distribution whose mean is given by the proposed deterministic interpolant with parameterized variance. Of course, this would change the matching losses (since the drift of the corresponding bridge process would not simply given by the time derivate of the Gaussian mean).\n- I do not understand how the loss of Alg. 1 is built : putting aside my previous point (ie considering the deterministic case only), the objective function $\\mathbb{E}[\\|\\partial_t X_t,\\eta-f_t (X_{t, \\eta})\\|^2]$, where the expectation is taken wrt $(X_0, X_1, X_{t, \\eta})$ and $f_t$ is exactly the drift of the reference process, is perfectly legitimate in my opinion to approximate the reference bridge. In the stochastic case, this would correspond to minimize wrt $\\eta$ the objective $KL(\\mathbb{Q^\\eta}|\\mathbb{Q})$, where $\\mathbb{Q^\\eta}=\\int \\mathbb{Q^\\eta}(\\cdot|x_0, x_1)d\\pi(x_0,x_1)$,  as also proposed in [1] for the general SB setting. Why justifies the use of kernels here ?\n- I do not understand how the cost of the OT plan can be deduced. Regarding the formulation of the SB problem, the optimal plan $\\pi^\\star$ is given by $\\min KL(\\pi|Q_{0,1})$ (with the right marginals), where $Q_{0,1}$ is the joint distribution of the reference process at times 0 and 1 (which is not tractable). Even when using the approximated bridge (defined via the neural interpolant), this joint distribution remains untractable and an additional entropy term is missing in the loss. How do you obtain this OT cost ? How do you explain that you obtain a non-regularized OT cost, while the SB problem naturally leads to an entropy-regularized OT problem when reducing to the couple (0,1) ? \n- do you consider different values for $\\sigma_t$ and $g_t$ ? How these values are set in the experiments ? How does the algorithm behave when changing these values ?\n- the tradeoff between directly computing the OT plan (for example with Sinkhorn algorithm or mini-batch OT) + applying bridge matching VS doing an iterative refinement of bridge matching loss, by taking the current outputs of the model (see [1, 2]) has not been obviously solved in the past years: have you tried to turn Curly-FM into an iterative algorithm ?\n\n\n**Suggestions**\n- I would suggest the authors to write \"non-linear drift\" rather than \"non-zero drift\", which is more accurate since the DSBM approach also provides expressions for the first case, which is more general (see Appendix B.2)\n- I would recommend the authors to put more highlight on the mathematical formulation they want to solve, in particular on the choice of the volatilities $\\sigma_t$ and $g_t$ : is it any positive couple ? is the same value for both volatilities ? do you always consider the zero-limit ?\n- I would also suggest the authors to rewrite Section 3.2 in a clearer manner (for example, explicitly define two parts : (i) learning the bridge and (ii) learning the drift of the SB solution) to include the computation steps that justify the way the algorithm is built\n\n[1] Generalized Schrödinger Bridge Matching, Liu et al. ICLR 2024.\n\n[2] Schrödinger Bridge Flow for Unpaired Data Translation. De Bortoli et al, 2024\n\n**Update**: these questions have been addressed by the authors in the discussion and imply changes in the manuscript.\n\nLIMITATIONS:\nI think that the paper lacks detailed ablation studies on the proposed algorithm to evaluate its robustness (toy setting with increasing dimension, various values of volatility...). For example, it would be interesting to have an experiment, where the reference bridge can be analytically computed (i.e linear reference SDE) and compared with the output of Algorithm 1. This would enable to have some empirical guarantee on the first stage of Curly-FM.\n\n**Update**: these questions have been addressed by the authors in the discussion and imply changes in the manuscript.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 1, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nAn approach is presented that allows learning flows that include drifts rather than only gradient-field dynamics. The authors first learn an interpolant (neural path) and then fit a velocity field based on the coupling provided by the interpolant.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n+ Learning non-gradient field dynamics is indeed an important problem and little work in this direction has been done. So this work is timely and useful.\n\n+ The proposed two-stage approach of first learning a neural path interpolant and then fitting a velocity field is straightforward but reasonable and sound. The approach is well described.\n\n+ There are extensive numerical experiments that demonstrate that the approach captures drift dynamics whereas standard conditional flow matching fails\n\nWeaknesses:\n- There are extra training costs incurred by the additional steps (e.g., the OT step) that can be high. No discussion about it is provided.  \n\n- There is a limitation by first fitting the neural path interpolant, which is not addressed well. For example, if the initial neural path is poorly fitted, how will it affect the second step? The authors do mention briefly in the Conclusion that the approach hinges on inferring the reference field well, but this requires more discussion.\n\nQUESTIONS:\n- It is unclear how the trade off between matching the drift and marginals is handled. What knob needs to be turned in Algorithm 1 and 2 to trade off these two learning objectives? A more detailed discussion should be added because this question comes up several times in the paper (e.g., Intro on page 3 top and line 186-187)\n\n- There are other approaches than simply pairwise connecting marginals with OT such as action matching by [Neklyudov et al., 2024]. These approaches incur often lower training costs than solving many OT problems. Can this be used as a first stage in the presented approach as well to avoid some of the limitations of pairwise interpolants?\n\n- Minor questions: The authors state that TrajectoryNet is much slower and unstable (line 183). Can the authors quantify this for one of the examples?\n\nLIMITATIONS:\nYes, the authors have addressed in the Conclusion the limitation of first learning the reference field, which is the main limitation of this approach.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces Curly Flow Matching (Curly-FM), a simulation-free method for learning non-gradient, possibly periodic dynamics from snapshot data and approximate velocity measurements.  The authors cast trajectory inference as a Schrödinger-bridge problem whose reference process has non-zero drift built from the given velocities.  Because (most of the) existing diffusion/flow-matching techniques assume zero drift, the paper proposes a new method to account for non-zero drift via a two-stage approximation: (1) find a neural path interpolant such that its instantaneous velocity matches the reference drift, and (2) marginal flow matching to learn a drift that marginally modifies the neural path interpolant to make sure that the induced stochastic differential equation satisfies the marginal constraints at the initial and final times. The authors show promising results on empirical and real data experiments with rotational flows, improving over state-of-the-art baselines.\n\nSTRENGTHS AND WEAKNESSES:\n**Quality.** The submission is generally technically sound. It does not contain any formal propositions or main theoretical results, but most of the claims are well supported by the experimental findings. The authors do a good job discussing important background ideas, although in some parts it would be helpful to include either more intuitive explanations or more precise references (see the *Questions* section below for details).\n\n**Clarity.** I find most of the paper to be clearly written. I appreciate that the authors clearly motivate their work and set up the problem before presenting potential solutions. However, the paper becomes harder to follow in Section 3.1, where additional details and intuition would be very helpful for the reader to fully understand the method and be able to reproduce it. I will provide more specific feedback in the *Questions* section, but in my view, this is the first major weakness of the paper.\n\n**Significance.** The results have the potential to be impactful for the community. I believe that some researchers will find the experimental contributions appealing and may be motivated to pursue theoretical developments based on them. Additionally, practitioners working with similar data may be interested in applying this method. That said, the paper would benefit from an analysis of how crucial it is to accurately approximate — or have access to — the velocity field in order to obtain good results, since this data may not be readily available in all practical settings. More broadly, I think the paper should include a more thorough discussion of the role of the velocity field in the proposed approach.\n\n**Originality.** The paper presents a new method that clearly differs from most of the prior works cited. However, the authors repeatedly claim (e.g., Lines 43 and 151) that existing methods can only model gradient field dynamics, which is not accurate in general. For example, the work by Shen et al. (2024), which the authors use in one of their experiments, supports more general dynamics, The two works by Bartosh et al. (2024 and 2025) are also relevant in this context—they are briefly mentioned in Section 5 (line 284), but the authors do not provide any intuition about how these methods differ from their own (beyond a short comment about the “simulation-free” aspect) and do not compare against them experimentally. I view this as the second major weakness of the paper. If the proposed method is very similar to Bartosh’s work and its main advantage is being simulation-free, then I would expect some discussion or analysis of computational speed or efficiency. Without such analysis, it is difficult to justify a new method that does not clearly differentiate itself from recent work on a very similar task. Overall, I believe the work is original and constitutes an interesting contribution, but the discussion of related work should be expanded and the motivation for the proposed approach strengthened.\n\nQUESTIONS:\nIn this section, I will get in more details of the comments provided in the strengths and weaknesses section above, as well as provide some additional minor comments and questions. \n\n**Major Comments and Questions**\n\n- When I first encounter $\\eta$, it is not clear what this represents and why we need it. I think it would be great if the notation is setup and clearly described why we need this before using it. Similarly (and possibly relatedly) I am not sure I follow how $Q_{\\eta}$ (line 158) is actually constructed. I get it that is parametrized as a mixture of Brownian bridges, but above you are saying that these bridges need not be Brownian (Line 154). Is this making a modeling/approximation choice? I think it would be much clearer if the mixture were explicitly written out, with each component described both formally and intuitively.\n- After line 159, $f*_{\\eta}$ seems to correspond to the minimum value of the integral. Is that correct? Or is it instead meant to refer to one of the $f_{t,\\eta}$ terms? I am also not sure that the star notation is used anywhere later in the paper, so I don't follow why we need this.\n- What is the role of $\\kappa$, and how is it chosen in practice? Is it the same for each experiment? Why is it introduced below line 162? Similarly, what is the precise function of the neural path interpolant? How does it relate to model drift and the approximated velocity field? My understanding is that with the neural path interpolant I am trying to make the path $x_{t,\\eta}$ curly, and I am doing it by finding the best interpolant that matches a reference velocity field. And then once I found the interpolant I can use it in the derivative of the path to approximate the drift in the original SDE (equation (3)) with the loss after Line 172.\n- Overall, I find Section 3.1 quite difficult to follow, as it lacks both formal clarity and intuitive explanation. A clear breakdown of the two main steps, along with an explanation of what each quantity represents (both formally and intuitively), would greatly improve readability. I understand the high-level two-step procedure from the introduction, but I find it challenging to connect that to the formalism presented in this section.\n- More broadly, I think I am confused by how the reference velocity field is obtained. Is it computed by deriving the velocity along the stochastic path interpolant and then fitting a neural network to that data, using the loss after equation 162? Or are the velocities approximated using finite differences outside the optimization loop and then used in a regression setup? Is approximating the velocity field necessary at all? Could the method be adapted to scenarios where the true velocity field is known? And how does the accuracy of this approximation affect the performance or theoretical guarantees of the method? I know these are a lot of questions, but I am asking because I think this is the crucial innovation of the method and I am wondering whether it can be spelled out better. I am also asking because for most of these data there is no notion of observed trajectory for a particle, since the particles are unpaired, so I am not sure if this approach is impacted by this or not.\n- Lines 265–267: What is meant here? Could you provide a citation or further explanation? This relates to my broader confusion about the role of the velocity field (as noted above). It’s also not clear whether the method assumes access to RNA velocity fields for this experiment, as seems implied in Line 230.\n- How does the method generalize to more than two time points? In the ocean current experiment, there are more than two time points.  But in the method part the discussion is focused on just two end points at time 0 and 1. But from the experiment results it doesn't look like you're learning the drift piecewise between each pair of points (since the connections look smooth enough), so what is exactly going on? Does the need to approximate multiple velocity fields introduce cumulative bias?\n\n---\n\n**Minor Comments and Questions**\n\n- I really like the inclusion of Figure 1 on the first page! Could you increase the font size for the legend labels? Also, it would be helpful to spell out “OT-CFM” and “CFM” in the figure caption, since the acronyms are not introduced in the main text at that point. Many readers will look at the figure together with the abstract and will not yet know what those terms mean.\n- Line 43: I suggest adding a “Most of” qualifier here, since, as noted above and in the extended related work section, there are existing methods that can model non-gradient dynamics.\n- Line 68: The meaning of this sentence is unclear — could you rephrase or clarify?\n- Lines 140–141: A citation is needed.\n- Equation (9): Unless this particular characterization of the Schrödinger Bridge problem is used later in the paper, it may not be necessary to include it here.\n- Line 146: The phrase “We approximate...” makes it sound like these are the algorithms used in the current paper, which is not true. A more general phrasing would be preferable.\n- Lines 152–153: Is there a double negation issue here? It may be worth rephrasing for clarity.\n- Line 170: Is Equation (10) the correct reference? If so, what exactly do you mean by “total cost” in this context?\n- It would be helpful to explain how boldface is used in the tables. Currently, it seems that only the proposed method is bolded when it achieves the best performance, while other methods are not highlighted even when they perform best. This could be misleading unless explicitly stated.\n- Not a big deal, but note that Figure 5 currently appears before Figures 3 and 4.\n\nMy score is mostly driven by the points raised in the main comments and weaknesses sections. I will be very happy to revise my score if the authors can clarify my confusion and doubts.\n\nLIMITATIONS:\nYes, the authors discuss some limitations of their work in the discussion section. In particular, they briefly note that the accuracy of the inference process depends on how well the reference family can be reconstructed. As mentioned above, I believe it would be very valuable to explore this more formally — for example, by investigating whether certain assumptions on the data or on the quality of the derivative approximation could guarantee a desired level of inference accuracy.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}