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arxiv: 2606.11138 · v1 · pith:6XH44RGZnew · submitted 2026-06-09 · 💻 cs.LG · cs.NA· math.NA

First-Order Trajectory Matching: Fast Ensemble Predictions of Chaotic, Turbulent, Stochastic Systems

Shreya Jha , Timo Schorlepp , Nicholas Geissler , Jules Berman , Benjamin Peherstorfer This is my paper

Pith reviewed 2026-06-27 13:46 UTC · model grok-4.3

classification 💻 cs.LG cs.NAmath.NA
keywords First-Order Trajectory Matchingensemble predictionstochastic dynamical systemsprobability currentsurrogate modelingchaotic systemsturbulent flowstrajectory matching
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The pith

First-order trajectory matching learns probability current velocity directly from data to match ensemble averages in stochastic systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces First-Order Trajectory Matching (FTM) to learn the first-order local transport of probability mass from trajectories of stochastic systems. By matching the symmetric first-order motion of trajectories, FTM recovers the probability current velocity whose flow preserves time marginals. This yields ensemble-average predictions while also tracking fluxes, circulations, and barrier-crossing currents, all without estimating drift, diffusion, or scores. A stability analysis separates discretization error from sampling variance, and experiments on dynamical systems and PDEs show the method runs at deterministic-rollout cost.

Core claim

FTM matches the symmetric first-order motion of trajectories to learn the probability current velocity, whose flow preserves time marginals to match ensemble averages, while capturing current-like trajectory quantities such as fluxes, circulations, and barrier-crossing currents; the method learns this velocity directly from trajectories and avoids drift, diffusion, and score estimation.

What carries the argument

Symmetric first-order motion matching, which learns the probability current velocity from trajectory data to preserve time marginals.

If this is right

  • FTM supplies trajectory-aware ensemble predictions for chaotic, turbulent, and stochastic systems.
  • Predictions are obtained at low deterministic-rollout cost after training.
  • The one-step simulation-free loss remains stable when temporal resolution and sample size are balanced.
  • FTM captures fluxes, circulations, and barrier-crossing currents in addition to marginal statistics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may scale to high-dimensional real-world systems where full Monte Carlo ensembles remain prohibitive.
  • FTM outputs could serve as cheap initial conditions or constraints for hybrid physics-ML models.
  • Systems with known analytic probability currents offer direct tests of whether first-order matching recovers the exact velocity field.

Load-bearing premise

Matching the symmetric first-order motion of trajectories is sufficient to recover the probability current velocity whose flow preserves time marginals.

What would settle it

A direct comparison on a stochastic system where FTM ensemble averages diverge from independent Monte Carlo ground truth when higher-order trajectory effects dominate.

Figures

Figures reproduced from arXiv: 2606.11138 by Benjamin Peherstorfer, Jules Berman, Nicholas Geissler, Shreya Jha, Timo Schorlepp.

Figure 1
Figure 1. Figure 1: Statistically stationary process where probability mass circulates. A learned op￾erator model leads to trajectories that spiral inward, collapsing to the mean. Population dynamics (marginal matching) match time marginals but with zero velocity (for minimum kinetic energy) and so miss the rotation. Conditional gener￾ative modeling such as conditional flow matching reproduces the mass transport but typically… view at source ↗
Figure 2
Figure 2. Figure 2: The one-step FTM loss is stable in the practically relevant time-resolution/sample￾size balanced regime: longer chunks sizes τ are necessary only at very small time-step sizes h to reduce variance (a)–(b), but otherwise add substantial cost per training step and provide little accuracy gain at realistic resolutions (here for Duffing oscillator example) (c). FTM trains robustly (d). Path-dependent QoI infer… view at source ↗
Figure 3
Figure 3. Figure 3: Duffing oscillator (left): FTM matches the evolving ensemble and generates physically meaningful trajectories. Stochastic Burgers (right): FTM is robust to increasing noise levels (σ). forcing is applied to the oscillator’s acceleration equation. 2. Chaotic Rayleigh–Bénard convection. This is a stochastically forced nine-mode model of Rayleigh–Bénard convection introduced in [50]. The system is of dimensio… view at source ↗
Figure 4
Figure 4. Figure 4: FTM predicts ensembles with accurate energy and enstrophy statistics of solution trajectories of the stochastic Burgers (left panel) and stochastically forced turbu￾lence (right panel) example. (See [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: On stochastically forced turbulence, FTM produces accurate rollouts with one NFE per physical time step, while ARDM uses up to 100 inner sampling steps in our experiments (left). FTM also matches the energy spectrum decay well (right). FTM scales to stochastic PDEs with one NFE per physical time step We next demonstrate in Figures 4 and 5 and [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Statistically stationary “two-lane process” defined in Example [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FTM predicts ensembles with accurate energy and enstrophy statistics of stochastic Burgers (left panel) and stochastically forced turbulence (right panel) solution trajectories. the QoI is computed (Appendices D.1 and D.2) over all test trajectories. The reported number is the average relative error. The number in brackets is the standard deviation of the Monte Carlo estimator [PITH_FULL_IMAGE:figures/ful… view at source ↗
read the original abstract

We introduce First-Order Trajectory Matching (FTM), a surrogate-modeling method that learns the first-order local transport of probability mass from trajectories of stochastic systems. By matching the symmetric first-order motion of trajectories, FTM learns the probability current velocity, whose flow preserves time marginals to match ensemble averages, while also capturing current-like trajectory quantities such as fluxes, circulations, and barrier-crossing currents. FTM learns the current velocity directly from trajectories, avoiding drift, diffusion, and score estimation. Our stability analysis separates discretization error from sampling variance and shows that the one-step simulation-free FTM loss is stable when temporal resolution and sample size are properly balanced. Across stochastic dynamical systems and PDE examples, we empirically demonstrate that FTM provides trajectory-aware ensemble predictions at low, deterministic-rollout cost.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces First-Order Trajectory Matching (FTM), a surrogate-modeling approach that learns the probability current velocity directly from trajectories of stochastic systems by matching symmetric first-order motions. The central claim is that the resulting deterministic flow preserves time marginals to match ensemble averages, while also capturing fluxes and circulations, at low rollout cost without estimating drift, diffusion, or scores. A stability analysis separates discretization error from sampling variance for the one-step simulation-free loss, and empirical results are shown on stochastic dynamical systems and PDEs.

Significance. If the central claim holds, FTM offers a computationally efficient alternative for ensemble predictions in chaotic, turbulent, and stochastic regimes by replacing stochastic rollouts with deterministic integration of a learned velocity field. The separation of error sources in the stability analysis and the direct use of trajectory data without auxiliary estimations are positive features that could impact surrogate modeling for high-dimensional systems.

major comments (2)
  1. [Abstract] Abstract (FTM definition paragraph): the assertion that matching symmetric first-order trajectory motions yields a velocity field v whose deterministic flow exactly satisfies the continuity equation for the time-evolving marginals p(t) is stated without an explicit weak-form derivation (e.g., equivalence of the loss to ∫(v−J/p)·∇ϕ p dx=0 for test functions ϕ). This equivalence is load-bearing for the claim that deterministic rollouts match ensemble averages over long horizons.
  2. [Stability analysis] Stability analysis: the separation of discretization error from sampling variance is presented for the one-step loss, but it is unclear whether the analysis extends to the preservation of marginals under iterated deterministic integration in turbulent regimes where higher-order correlations may affect long-horizon behavior.
minor comments (2)
  1. [Abstract] The abstract mentions empirical demonstrations but does not name the specific stochastic systems or PDE examples; adding one sentence would improve clarity.
  2. Notation for the probability current velocity and the symmetric first-order motion should be introduced with a short equation in the main text for readers unfamiliar with the continuity-equation context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their detailed review and constructive feedback. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (FTM definition paragraph): the assertion that matching symmetric first-order trajectory motions yields a velocity field v whose deterministic flow exactly satisfies the continuity equation for the time-evolving marginals p(t) is stated without an explicit weak-form derivation (e.g., equivalence of the loss to ∫(v−J/p)·∇ϕ p dx=0 for test functions ϕ). This equivalence is load-bearing for the claim that deterministic rollouts match ensemble averages over long horizons.

    Authors: We agree that an explicit reference to the weak-form equivalence would strengthen the abstract. The main text establishes that the FTM loss is equivalent to a weak-form minimization aligning the learned velocity with the probability current (via integration against test functions), which ensures the deterministic flow satisfies the continuity equation for the marginals. In the revised manuscript we will add a concise statement of this equivalence to the abstract. revision: yes

  2. Referee: [Stability analysis] Stability analysis: the separation of discretization error from sampling variance is presented for the one-step loss, but it is unclear whether the analysis extends to the preservation of marginals under iterated deterministic integration in turbulent regimes where higher-order correlations may affect long-horizon behavior.

    Authors: The stability analysis deliberately targets the one-step loss to isolate discretization error from sampling variance. Preservation of marginals under iterated integration follows directly from the velocity approximating the probability current, whose flow satisfies the continuity equation by construction; higher-order correlations are not required for this first-order transport property. Our empirical results on long-horizon predictions in chaotic and turbulent regimes provide supporting evidence. We will expand the discussion section to clarify the scope of the analysis and its relation to multi-step behavior. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The abstract and description present FTM as learning the probability current velocity via one-step trajectory matching, with stability analysis separating discretization and sampling effects, and empirical validation on systems. No quoted equations or self-citations reduce the central claim (that matching symmetric first-order motions yields a flow preserving marginals) to a definitional identity, fitted input renamed as prediction, or load-bearing self-citation chain. The method is described as simulation-free and directly from trajectories without the target result presupposed in the loss. This meets the default expectation of a non-circular paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that first-order symmetric motion matching recovers the probability current velocity without needing higher-order terms or explicit estimation of other quantities; no free parameters or invented entities are identifiable from the abstract.

axioms (1)
  • domain assumption Matching the symmetric first-order motion of trajectories learns the probability current velocity whose flow preserves time marginals.
    Stated in the abstract as the core of FTM.

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Reference graph

Works this paper leans on

75 extracted references · 2 linked inside Pith

  1. [1]

    Albergo, N

    M. Albergo, N. M. Boffi, and E. Vanden-Eijnden. Stochastic interpolants: A unifying framework for flows and diffusions.Journal of Machine Learning Research, 26(209):1–80, 2025

    2025

  2. [2]

    M. S. Albergo and E. Vanden-Eijnden. Building normalizing flows with stochastic interpolants.arXiv preprint arXiv:2209.15571, 2022

    Pith/arXiv arXiv 2022

  3. [3]

    B. D. Anderson. Reverse-time diffusion equation models.Stochastic Processes and their Applications, 12(3):313–326, 1982

    1982

  4. [4]

    Bartosh, D

    G. Bartosh, D. Vetrov, and C. A. Naesseth. SDE Matching: Scalable and Simulation- Free Training of Latent Stochastic Differential Equations. InProceedings of the 42nd International Conference on Machine Learning, volume 267 ofProceedings of Machine Learning Research, pages 3054–3070. PMLR, 2025

    2025

  5. [5]

    Benton, G

    J. Benton, G. Deligiannidis, and A. Doucet. Error bounds for flow matching methods. arXiv preprint arXiv:2305.16860, 2023

    arXiv 2023

  6. [6]

    Berman, T

    J. Berman, T. Blickhan, and B. Peherstorfer. Parametric Model Reduction of Mean-Field and Stochastic Systems via Higher-Order Action Matching. InAdvances in Neural Information Processing Systems, 2024. 14

    2024

  7. [7]

    Blickhan, J

    T. Blickhan, J. Berman, A. Stuart, and B. Peherstorfer. DICE: Discrete inverse continuity equation for learning population dynamics, 2025

    2025

  8. [8]

    N. M. Boffi, M. S. Albergo, and E. Vanden-Eijnden. How to build a consistency model: Learning flow maps via self-distillation. InThe Thirty-ninth Annual Conference on Neural Information Processing Systems, 2025

    2025

  9. [9]

    N. M. Boffi and E. Vanden-Eijnden. Probability flow solution of the Fokker–Planck equation.Machine Learning: Science and Technology, 4(3):035012, jul 2023

    2023

  10. [10]

    N. M. Boffi and E. Vanden-Eijnden. Deep learning probability flows and entropy production rates in active matter.Proceedings of the National Academy of Sciences, 121(25):e2318106121, 2024

    2024

  11. [11]

    N. M. Boffi and E. Vanden-Eijnden. Model-Free Learning of Probability Flows: Eluci- dating the Nonequilibrium Dynamics of Flocking.Phys. Rev. Lett., 135:238301, Dec 2025

    2025

  12. [12]

    Boninsegna, F

    L. Boninsegna, F. Nüske, and C. Clementi. Sparse learning of stochastic dynamical equations.The Journal of chemical physics, 148(24), 2018

    2018

  13. [13]

    S. L. Brunton, B. R. Noack, andP. Koumoutsakos. Machine Learning forFluid Mechanics. Annual Review of Fluid Mechanics, 52(Volume 52, 2020):477–508, 2020

    2020

  14. [14]

    ProximalOptimalTransport Modeling of Population Dynamics.arXiv preprint arXiv:2106.06345, 2022

    C.Bunne, L.Meng-Papaxanthos, A.Krause, andM.Cuturi. ProximalOptimalTransport Modeling of Population Dynamics.arXiv preprint arXiv:2106.06345, 2022

    arXiv 2022

  15. [15]

    S. Chen, S. Chewi, H. Lee, Y. Li, J. Lu, and A. Salim. The probability flow ODE is provably fast.Advances in Neural Information Processing Systems, 36:68552–68575, 2023

    2023

  16. [16]

    Y. Chen, M. Goldstein, M. Hua, M. S. Albergo, N. M. Boffi, and E. Vanden-Eijnden. Probabilistic Forecasting with Stochastic Interpolants and Föllmer Processes. InPro- ceedings of the 41st International Conference on Machine Learning, volume 235 of Proceedings of Machine Learning Research, pages 6728–6756. PMLR, 21–27 Jul 2024

    2024

  17. [17]

    Chetrite and K

    R. Chetrite and K. Gawędzki. Eulerian and Lagrangian pictures of non-equilibrium diffusions.Journal of Statistical Physics, 137(5):890–916, 2009

    2009

  18. [18]

    Comte and V

    F. Comte and V. Genon-Catalot. Nonparametric drift estimation for i.i.d. paths of stochastic differential equations.The Annals of Statistics, 48(6):3336 – 3365, 2020

    2020

  19. [19]

    Course and P

    K. Course and P. B. Nair. Amortized Reparametrization: Efficient and Scalable Variational Inference for Latent SDEs. InAdvances in Neural Information Processing Systems, volume 36, 2023

    2023

  20. [20]

    Das and S

    B. Das and S. K. Manikandan. Localizing entropy production along non-equilibrium trajectories.arXiv preprint arXiv:2503.20427, 2025

    Pith/arXiv arXiv 2025

  21. [21]

    M. Deng, H. Li, T. Li, Y. Du, and K. He. Generative Modeling via Drifting, 2026. 15

    2026

  22. [22]

    Dresdner, D

    G. Dresdner, D. Kochkov, P. Norgaard, L. Zepeda-Núñez, J. A. Smith, M. P. Brenner, and S. Hoyer. Learning to correct spectral methods for simulating turbulent flows. 2022

    2022

  23. [23]

    Dridi, L

    N. Dridi, L. Drumetz, and R. Fablet. Learning stochastic dynamical systems with neural networks mimicking the Euler–Maruyama scheme. In2021 29th European Signal Processing Conference (EUSIPCO), pages 1990–1994, Dublin, Ireland, Aug. 2021. IEEE

    1990

  24. [24]

    Frishman and P

    A. Frishman and P. Ronceray. Learning Force Fields from Stochastic Trajectories.Phys. Rev. X, 10:021009, Apr 2020

    2020

  25. [25]

    Fukumizu, T

    K. Fukumizu, T. Suzuki, N. Isobe, K. Oko, and M. Koyama. Flow matching achieves almost minimax optimal convergence.arXiv preprint arXiv:2405.20879, 2024

    arXiv 2024

  26. [26]

    Z. Geng, M. Deng, X. Bai, J. Z. Kolter, and K. He. Mean Flows for One-step Generative Modeling. InThe Thirty-ninth Annual Conference on Neural Information Processing Systems, 2025

    2025

  27. [27]

    U. G. Haussmann and E. Pardoux. Time Reversal of Diffusions.The Annals of Probability, 14(4):1188–1205, Oct 1986

    1986

  28. [28]

    J. Ho, A. Jain, and P. Abbeel. Denoising diffusion probabilistic models. InAdvances in Neural Information Processing Systems, volume 33, 2020

    2020

  29. [29]

    Jiang, P

    R. Jiang, P. Y. Lu, E. Orlova, and R. Willett. Training neural operators to preserve invariant measures of chaotic attractors. InThirty-seventh Conference on Neural Information Processing Systems, 2023

    2023

  30. [30]

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang. Physics-informed machine learning.Nature Reviews Physics, 3(6):422–440, Jun 2021

    2021

  31. [31]

    Kidger, J

    P. Kidger, J. Foster, X. Li, and T. Lyons. Efficient and Accurate Gradients for Neural SDEs. InAdvances in Neural Information Processing Systems, volume 34, pages 18747– 18761, 2021

    2021

  32. [32]

    P. E. Kloeden and E. Platen.Numerical Solution of Stochastic Differential Equations. Stochastic Modelling and Applied Probability. Springer, Berlin, Heidelberg, 1 edition,

  33. [33]

    Springer-Verlag Berlin Heidelberg (1992); eBook ISBN 978-3-662-12616-5 (2013 reprint)

    1992

  34. [34]

    Kochkov, J

    D. Kochkov, J. A. Smith, A. Alieva, Q. Wang, M. P. Brenner, and S. Hoyer. Machine learning–accelerated computational fluid dynamics.Proceedings of the National Academy of Sciences, 118(21), 2021

    2021

  35. [35]

    Kohl, L.-W

    G. Kohl, L.-W. Chen, and N. Thuerey. Benchmarking autoregressive conditional diffusion models for turbulent flow simulation.Neural Networks, 199:108641, 2026

    2026

  36. [36]

    J. Li, J. M. Horowitz, T. R. Gingrich, and N. Fakhri. Quantifying dissipation using fluctuating currents.Nature communications, 10(1):1666, 2019. 16

    2019

  37. [37]

    Li, T.-K

    X. Li, T.-K. L. Wong, R. T. Q. Chen, and D. Duvenaud. Scalable Gradients for Stochastic Differential Equations. InProceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, volume 108 ofProceedings of Machine Learning Research, pages 3870–3882. PMLR, 2020

    2020

  38. [38]

    Z. Li, N. Kovachki, K. Azizzadenesheli, B. liu, K. Bhattacharya, A. Stuart, and A. Anand- kumar. Fourier Neural Operator for Parametric Partial Differential Equations. In International Conference on Learning Representations, 2021

    2021

  39. [39]

    Lipman, R

    Y. Lipman, R. T. Q. Chen, H. Ben-Hamu, M. Nickel, and M. Le. Flow matching for gen- erative modeling. InThe Eleventh International Conference on Learning Representations, 2023

    2023

  40. [40]

    L. Lu, P. Jin, G. Pang, Z. Zhang, and G. E. Karniadakis. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators.Nature Machine Intelligence, 3(3):218–229, Mar 2021

    2021

  41. [41]

    J. Lyu, K. J. Ray, and J. P. Crutchfield. Learning Stochastic Thermodynamics Directly from Correlation and Trajectory-Fluctuation Currents.arXiv, 2504.19007, 2025

    arXiv 2025

  42. [42]

    Maoutsa, S

    D. Maoutsa, S. Reich, and M. Opper. Interacting Particle Solutions of Fokker–Planck Equations Through Gradient–Log–Density Estimation.Entropy, 22(8), 2020

    2020

  43. [43]

    Molinaro, S

    R. Molinaro, S. Lanthaler, B. Raonić, T. Rohner, V. Armegioiu, S. Simonis, D. Grund, Y. Ramic, Z. Y. Wan, F. Sha, S. Mishra, and L. Zepeda-Núñez. Generative AI for fast and accurate statistical computation of fluids. 2025

    2025

  44. [44]

    K. Na, J. Lee, S.-Y. Yun, and S. Lim. Probability-flow ODE in infinite-dimensional function spaces.arXiv preprint arXiv:2503.10219, 2025

    arXiv 2025

  45. [45]

    Neklyudov, R

    K. Neklyudov, R. Brekelmans, D. Severo, and A. Makhzani. Action Matching: Learning Stochastic Dynamics from Samples. InProceedings of the 40th International Conference on Machine Learning, volume 202 ofProceedings of Machine Learning Research, pages 25858–25889. PMLR, 2023

    2023

  46. [46]

    Nelson.Dynamical theories of Brownian motion

    E. Nelson.Dynamical theories of Brownian motion. Princeton University Press, 1967

    1967

  47. [47]

    J. N. Nielsen, H. Madsen, and P. C. Young. Parameter estimation in stochastic differential equations: an overview.Annual Reviews in Control, 24:83–94, 2000

    2000

  48. [48]

    Otness, A

    K. Otness, A. Gjoka, J. Bruna, D. Panozzo, B. Peherstorfer, T. Schneider, and D. Zorin. An extensible benchmark suite for learning to simulate physical systems. InThirty-fifth Conference on Neural Information Processing Systems Datasets and Benchmarks Track (Round 1), 2021

    2021

  49. [49]

    G. A. Pavliotis.Stochastic Processes and Applications: Diffusion Processes, the Fokker- Planck and Langevin Equations, volume 60 ofTexts in Applied Mathematics. Springer New York, New York, NY, 2014. 17

    2014

  50. [50]

    On certain relations between classical statistics and quantum mechanics

    L. Peliti and P. Muratore-Ginanneschi. R. Fürth’s 1933 paper “On certain relations between classical statistics and quantum mechanics” [“Über einige Beziehungen zwischen klassischer Statistik und Quantenmechanik”, Zeitschrift für Physik, 81 143–162].The European Physical Journal H, 48(1):4, May 2023

    1933

  51. [51]

    Reiterer, C

    P. Reiterer, C. Lainscsek, F. Schürrer, C. Letellier, and J. Maquet. A nine-dimensional Lorenz system to study high-dimensional chaos.Journal of Physics A: Mathematical and General, 31(34):7121, aug 1998

    1998

  52. [52]

    Rey-Bellet and K

    L. Rey-Bellet and K. Spiliopoulos. Irreversible Langevin samplers and variance reduction: a large deviations approach.Nonlinearity, 28(7):2081–2103, 2015

    2081

  53. [53]

    Ryder, A

    T. Ryder, A. Golightly, A. S. McGough, and D. Prangle. Black-Box Variational Inference for Stochastic Differential Equations. InProceedings of the 35th International Conference on Machine Learning, volume 80 ofProceedings of Machine Learning Research, pages 4423–4432. PMLR, 2018

    2018

  54. [54]

    Sanderse, P

    B. Sanderse, P. Stinis, R. Maulik, and S. E. Ahmed. Scientific machine learning for closure models in multiscale problems: A review.Foundations of Data Science, 7(1):298–337, 2025

    2025

  55. [55]

    U. Seifert. Stochastic thermodynamics, fluctuation theorems and molecular machines. Reports on progress in physics, 75(12):126001, 2012

    2012

  56. [56]

    Y. Song, P. Dhariwal, M. Chen, and I. Sutskever. Consistency models.International conference on machine learning, 2023

    2023

  57. [57]

    Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole. Score- Based Generative Modeling through Stochastic Differential Equations. InInternational Conference on Learning Representations, 2021

    2021

  58. [58]

    Stachenfeld, D

    K. Stachenfeld, D. B. Fielding, D. Kochkov, M. Cranmer, T. Pfaff, J. Godwin, C. Cui, S. Ho, P. Battaglia, and A. Sanchez-Gonzalez. Learned Coarse Models for Efficient Turbulence Simulation. InInternational Conference on Learning Representations (ICLR 2021), 2021

    2021

  59. [59]

    Terpin, N

    A. Terpin, N. Lanzetti, M. Gadea, and F. Dörfler. Learning Diffusion at Lightspeed. In Advances in Neural Information Processing Systems, 2024

    2024

  60. [60]

    A. Tong, J. Huang, G. Wolf, D. van Dijk, and S. Krishnaswamy. TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics.arXiv preprint arXiv:2002.04461, 2020

    arXiv 2002

  61. [61]

    Tsimpos, D

    P. Tsimpos, D. Sharp, and Y. Marzouk. One-Shot Generative Flows: Existence and Obstructions, 2026

    2026

  62. [62]

    Tzen and M

    B. Tzen and M. Raginsky. Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit.arXiv preprint arXiv:1905.09883, 2019. 18

    arXiv 1905

  63. [63]

    Zhang, Y

    X. Zhang, Y. Pu, Y. Kawamura, A. Loza, Y. Bengio, D. L. Shung, and A. Tong. Trajectory Flow Matching with Applications to Clinical Time Series Modeling. In Advances in Neural Information Processing Systems, volume 37, pages 107198–107224, 2024

    2024

  64. [64]

    L. Zhou, S. Ermon, and J. Song. Inductive Moment Matching.arXiv preprint arXiv:2503.07565, 2025. Appendix A. Illustrative examples, and gauge freedom for global mass transport We first specify the SDE toy example shown in Figure 1 in the main text via Example 1. Example1.We consider a statistically stationary, rotating Ornstein–Uhlenbeck process in d= 2di...

    arXiv 2025

  65. [65]

    This probability current corresponds to sample paths of(18) circulating in the same direction, while preserving their time marginal distribution

    truth:We show the constant marginal distributionρ(t) ≡ρ ∞ of the process onR2 as a heatmap, with white arrows indicating the direction of the analytically known probability current velocityv(x) = ΩRx from (7), which circulates counterclockwise forΩ > 0. This probability current corresponds to sample paths of(18) circulating in the same direction, while pr...

  66. [66]

    Learning only the drift, and solving the ODEd dt ˆx(t) = b(ˆx(t))at inference, leads to trajectories that spiral inwards and collapse onto the mean0ofρ∞ at large t

    operator learning:The target for operator learning is the drift vector fieldb in (18) via (4). Learning only the drift, and solving the ODEd dt ˆx(t) = b(ˆx(t))at inference, leads to trajectories that spiral inwards and collapse onto the mean0ofρ∞ at large t. Hence, this does not preserve the correct time marginals. 19

  67. [67]

    population dynamics:The main goal of population dynamics is to match the correct time marginals, which are stationary here. These methods typically assume that onlyunpairedsamples from X(t)at different times t are available, without trajectory information (which, as we argue in the present work, is actually available in many problems and useful to train w...

  68. [68]

    In the figure, we hence show sample paths of the SDE(18) generated using the Euler–Maruyama method

    conditional flow matching:Conditional flow matching methods are capable of reproducing the correct transition law ofX(t + h) |X (t)of the SDE(18). In the figure, we hence show sample paths of the SDE(18) generated using the Euler–Maruyama method. By construction, these will have the correct time marginalsρ(t) ≡ρ ∞, and circulate counterclockwise according...

  69. [69]

    orthogonal component

    FTM:Here, we integrate the ODE d dt ˆx(t) = v(t,ˆx(t)), which preserves the time marginals, can be done at low computational cost, and the resulting pure circulation at constant radius mimics the behavior of true sample paths of (18). In order to illustrate and contextualize these points further, it is useful to provide a slightly more abstract view on Ex...

  70. [70]

    1 τ Z t+τ t ∥vθ(s, X(s))∥2 2ds 2# | {z } ≤2V 4 + 8Et∼U([0,T−τ]),ω∼P

    the current velocityvis a gradient field, v(t, x) =−∇U(t, x)− σ(t)2 2 ∇logρ(t, x) =−∇ U(t, x) + σ(t)2 2 logρ(t, x) . 2.vis the unique minimizer of the kinetic energy functional K(u) := Z T 0 Z Rd ρ(t, x)∥u(t, x)∥2 2 dxdt, among all velocity fieldsusatisfying the continuity equation ∂tρ+∇ ·(ρu) = 0. Hence, in this setting, the probability current velocityv...

  71. [71]

    Using the Cauchy-Schwarz inequality for integrals and the uniform bounds∥b(t, x)∥2 ≤ bmax and∥v θ(t, x)∥2 ≤V, we get Eω " 1 τ Z t+τ t ⟨vθ(s, Xω(s)), b(s, Xω(s))⟩ds 2# ≤V 2b2 max

  72. [72]

    Using the Itô isometry and the boundΣ(t) =A(t)A(t)⊤ ⪯σ maxId, we have Eω " 1 τ Z t+τ t ⟨vθ(s, Xω(s)), A(s)dWω(s)⟩ 2# = 1 τ 2 Eω Z t+τ t ⟨vθ(s, Xω(s)),Σ(s)v θ(s, Xω(s))⟩ds ≤ 1 τ 2 Eω Z t+τ t σmax∥vθ(s, Xω(s))∥2 2ds ≤ σmaxV 2 τ

  73. [73]

    C.3 Variance of the one-step FTM loss ash↓0, and relation to symmetric differences In this section, we discuss the one-step FTM objective(16) from the main text

    Lastly, using the assumptionCdiv = supt,x 1 2 |Tr[Σ(t)∇v θ(t, x)]|<∞, we have Eω " 1 2τ Z t+τ t Tr [Σ(s)∇vθ(s, Xω(s))] ds 2# ≤C 2 div , which completes the proof since all of these bounds are independent oft∼ U ([0, T−τ ]). C.3 Variance of the one-step FTM loss ash↓0, and relation to symmetric differences In this section, we discuss the one-step FTM objec...

  74. [74]

    For fixedθ∈R p, the velocity fieldvθ is bounded in the sense that there existsV > 0 such that ∥vθ(t, x)∥2 ≤Vfor allt∈[0, T], x∈R d

  75. [75]

    We formulate this as an assumption here for brevity, but such a bound can be derived by using standard weak Taylor arguments for the forward and reverse process [32]

    The symmetric increment (6) is first-order accurate in the sense ∥vh(t, x)−v(t, x)∥ 2 ≤Ch ,(25) for a constantC > 0and all t∈ [h, T−h ], x∈R d. We formulate this as an assumption here for brevity, but such a bound can be derived by using standard weak Taylor arguments for the forward and reverse process [32]. We have the following upper bound on the mean-...