Curly Flow Matching for Learning Non-gradient Field Dynamics

AI summary

Overview

Research area: Machine learning for dynamical systems, generative modeling, and trajectory inference — specifically optimal transport, flow matching, and Schrödinger bridge methods applied to scientific data.

Technical level: Intermediate to Advanced. The paper assumes familiarity with continuous normalizing flows, conditional flow matching, stochastic differential equations, and the Schrödinger bridge problem with entropic optimal transport.

Scope: The paper introduces Curly Flow Matching (Curly-FM), a method that learns non-gradient (curl-containing, periodic) field dynamics by solving a Schrödinger bridge problem with a non-zero drift reference process constructed from inferred velocities in addition to population snapshot data, and evaluates it on synthetic, ocean-current, computational fluid dynamics, and single-cell trajectory inference tasks. (arXiv:2510.26645v1 [cs.LG], 30 Oct 2025; code at https://github.com/kpetrovicc/curly-flow-matching.git)

What This Paper Is About

Trajectory inference methods in the natural sciences typically rely on a least-action assumption, which restricts the learned dynamics to gradient fields and produces trajectories that minimize an energy functional between two probability measures. Many real systems, such as cell cycles in single-cell RNA data, instead exhibit non-gradient, periodic behavior that such methods fundamentally cannot represent. The paper's goal is to learn these non-gradient dynamics by modifying the Schrödinger bridge problem to use a reference process with non-zero drift.

Key Contributions

1. Curly Flow Matching (Curly-FM): A new approach for learning non-gradient field dynamics by designing and solving a Schrödinger bridge problem with a non-zero drift reference process, in contrast to the typical zero-drift reference processes used in methods such as Diffusion Schrödinger Bridges.
2. Use of inferred velocities as drift: The reference process is constructed from inferred velocities in addition to population snapshot data, letting the method exploit approximate velocity information (for example, RNA-velocity) that is not used by zero-drift flow matching approaches.
3. A two-stage, simulation-free algorithm: Stage one learns a neural path interpolant by regressing against the constructed reference drift; stage two learns a generative process as a mixture of conditional bridges built using optimal-transport-based couplings that minimize the length of the neural path interpolant's velocity field.
4. Empirical validation across scientific domains: The method is demonstrated on single cells, computational fluid dynamics, and ocean currents with approximate velocities, showing trajectories that better match both the reference process and the population marginals.

Main Findings

• Synthetic circles task: On source and target distributions arranged as asymmetric circles with a circular reference velocity field of constant rotational speed, prior flow-matching methods with a zero reference field produce straight paths between source and target and fail to capture cycling patterns. Curly-FM learns the cyclical behavior.
• Ocean currents (Gulf of Mexico): Using 1 km bathymetry resolution from HYCOM with 111 particles per time-point, Curly-FM achieves the best results for the majority of reported metrics across left-out time points. EMD values for Curly-FM were 0.019 ± 0.003 (t2), 0.045 ± 0.005 (t4), 0.027 ± 0.001 (t6), and 0.030 ± 0.006 (t8), outperforming SBIRR (0.073 ± 0.020, 0.072 ± 0.012, 0.120 ± 0.029, 0.094 ± 0.023) on EMD. MFM had lower cosine distance at t2 (0.179 ± 0.010 vs. Curly-FM's 0.231 ± 0.004) but Curly-FM was lower at t4, t6, and t8.
• Computational cost: Curly-FM achieves its ocean-current results in minutes, compared with 4 hrs for the simulation-based SBIRR.
• Cell cycle in human fibroblasts: Curly-FM is reported as the only method able to learn the cell cycle. Its cosine distance to the reference field was 0.295 ± 0.040 (d=2), 0.300 ± 0.058 (d=10), and 0.249 ± 0.024 (d=20), substantially lower than CFM, OT-CFM, and TrajectoryNet. Its 𝒲₂ values were higher than OT-CFM's (1.199 ± 0.177 vs. 0.248 ± 0.030 at d=2), which the authors attribute to the objective emphasizing velocity alignment over exactly matching endpoint marginals.
• Mouse erythroid development: Using 9,815 erythroid cells split into three temporal snapshots with the central marginal withheld, Curly-FM outperformed OT-CFM at reconstructing the underlying RNA-velocity field and cell trajectories in the majority of selected dimensions. At d=2 it had the lowest cosine distance (0.009 ± 0.000) and lowest L₂ (1.663 ± 0.293); MFM retained lower 𝒲₂ (0.269 ± 0.004 vs. Curly-FM's 0.369 ± 0.090), which the authors associate with stronger adherence to the underlying manifold.
• Computational fluid mechanics: The paper states that Curly-FM is evaluated on a particle-based PDE dataset generated by a Lagrangian solver, but the truncated content provided here does not report the quantitative results for this experiment.

Methodology in Plain English

The starting point is the Schrödinger bridge problem: find the path measure that stays as close as possible (in KL divergence) to a chosen reference process while still transporting one observed distribution of particles to another at the endpoints. Classical flow and bridge matching methods use a zero-drift reference process, effectively a Brownian motion, and the resulting optimal solution follows gradient-field dynamics — the trajectories minimize kinetic energy and cannot curl.

Curly-FM changes the reference process so that it has a non-zero drift, which is built from approximate velocity information collected alongside the population snapshots. Learning to follow that drift allows the model to reproduce cyclic, non-gradient motion while still being anchored to the observed marginal distributions.

The method solves this in two stages:

1. Neural path interpolant. The bridge mean is parameterized as a straight interpolation between the two endpoints plus a learned time-dependent correction term. The correction network is trained so that the time derivative of this mean matches the reference drift, which is estimated from data using a nearest-neighbor kernel weighting. This is done with automatic differentiation and no simulation.
2. Marginal flow and score matching. A transport plan between the endpoints is estimated by solving an optimal transport problem whose cost measures how far the neural path interpolant's velocity deviates from the reference drift over the trajectory; this cost is estimated stochastically with a small number of samples. Conditional bridges constructed with that plan are then used to train the generative vector field via flow matching, together with a score-matching term when the diffusion coefficient is positive. The reference drift is detached from the gradient of this second-stage loss.

Because the reference process is not constrained to match the endpoint marginals, the learned velocity field is not equal to the reference drift — the model must trade off matching the reference dynamics against transporting mass correctly between the observed time points.

Why This Matters

Impact on research. The paper argues that the field's default least-action, gradient-field assumption is a structural limitation, not just an accuracy issue, because it makes certain classes of real dynamics unrepresentable. By showing that a Schrödinger bridge with non-zero drift can be solved with a simulation-free algorithm, the work extends flow matching beyond population-level modeling and toward modeling known periodic behavior in physical systems. It also offers a faster alternative to simulation-based methods such as TrajectoryNet and SBIRR, which the authors describe as much slower and numerically unstable.

Real-world applications.

• Single-cell biology: Recovering realistic cell state trajectories under cell-cycle and differentiation dynamics, using RNA-velocity as the approximate velocity signal.
• Oceanography: Modeling ocean current transport, demonstrated on Gulf of Mexico HYCOM data, where vortex and periodic structure matter.
• Fluid mechanics: Learning Lagrangian particle dynamics from PDE simulations, where fluid flow is inherently rotational.
• Developmental and disease biology: The introduction frames trajectory inference as an upstream step for finding regulators of biological processes in development or disease.

Industry relevance. Domains that rely on transport and trajectory modeling with cyclical or rotational structure — climate and ocean modeling, fluid simulation, and computational biology pipelines — could benefit from a simulation-free training procedure that is cheaper than simulation-based alternatives while respecting known velocity information.

Future Directions

• Making the diffusion coefficient learnable. The paper fixes g_t (or tailors it to be small) and notes that GSBM learns it instead, but that this is computationally expensive. Extending Curly-FM to learn g_t is left open.
• Better optimal transport coupling. The paper states that the optimal plan is intractable and that a biased minibatch approximation is used instead; improving this approximation is an open practical question.
• Completing and reporting the computational fluid mechanics evaluation. The provided content truncates before the Lagrangian particle results, so the full quantitative outcome of that experiment is not reported here.
• Reducing the trade-off between velocity fidelity and marginal matching. The results show Curly-FM achieving much better reference-drift alignment than baselines but sometimes worse 𝒲₂ distance to the left-out marginal, which the authors attribute to its objective. Closing this gap is a natural next step.

Target Audience

Machine learning researchers working on generative models, flow matching, and Schrödinger bridges; computational biologists doing single-cell trajectory inference with RNA-velocity; and scientific computing practitioners in fluid dynamics and oceanography who need to learn dynamics from population-level observations under known approximate velocity fields. Readers should be comfortable with stochastic differential equations, continuous normalizing flows, and optimal transport concepts.

Authors’ 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