Is Flow Matching Just Trajectory Replay for Sequential Data?

Soon Hoe Lim; Shizheng Lin; Michael W. Mahoney; N. Benjamin Erichson

stat.MLFeb 9, 2026

Is Flow Matching Just Trajectory Replay for Sequential Data?

Authors: Soon Hoe Lim, Shizheng Lin, Michael W. Mahoney, N. Benjamin Erichson

Organizations: Department of Mathematics, KTH Royal Institute of Technology · Nordita, KTH Royal Institute of Technology and Stockholm University · Department of Statistics, University of California at Berkeley · International Computer Science Institute · Lawrence Berkeley National Laboratory

Abstract

Flow matching (FM) is increasingly used in scientific domains for time series generation and forecasting, where data often arise from underlying dynamical systems. However, it is not well-understood whether it learns transferable dynamical structure or simply performs an effective ''trajectory replay''. We study this question by deriving the optimal velocity field targeted by the empirical FM objective on sequential data. For the Gaussian conditional paths commonly used in practice, we show that the resulting sampler is an ODE whose dynamics describe a nonparametric, memory-augmented continuous-time dynamical system. The driving velocity field admits a closed-form expression as a weighted mixture of instantaneous velocities determined by observed transitions. This characterization provides an interpretable view of neural FM models as parametric estimation of an underlying nonparametric target and naturally leads to FreeFM: a closed-form, training-free realization of this target. FreeFM provides a transparent reference for studying sequential FM while yielding effective probabilistic forecasts directly from historical transitions.

Figures & tables

Figure 1: What Dynamical System Are FM Forecasters Actually Sampling From? For sequential transition data, the optimal empirical FM velocity field induces a nonparametric, memory-based ODE that can be evaluated directly from historical transitions. For the Gaussian bridge considered in this work, the same field admits two equivalent representations: $\hat{v}(t,z)=G_{t}z+h(t,z;\mathcal{D}_{M})$ (see Eq. ( 14 )) and a decomposition into transition replay plus a path-induced score correction. Starting from a local Gaussian initialization around the current state $x_{\tau}$ , integration of this field gives a next-state sample, and repeated application gives a multi-step forecast. Propagating multiple particles yields a predictive ensemble.

Panel of Figure 2 — Figure 2: Conditional Forecast. (a) Examples of forecast trajectories generated by FreeFM and baseline models for the Aizawa attractor. Each trajectory originates from a different initial condition. (b) sMAPE and VPT of forecast results from FreeFM and baseline models. Shaded regions indicate the mean $\pm 0.3$ standard deviation across the 135 dynamical systems, each with 20 trajectories originating from randomly sampled initial conditions.

Panel of Figure 3 — Figure 3: Probabilistic Forecast for Lorenz-63. Forecasts generated by FreeFM and the trained VanillaFM baseline. The upper plots show ensemble means with pointwise standard deviation bands. The lower plot shows the corresponding two-standard-deviation tubes around the ensemble means. Ensemble statistics are computed from 50 Monte Carlo forecasts.

Panel of Figure 4 — Figure 4: Probabilistic Forecast for Dysts. (a) sMAPE of forecasts generated by FreeFM and the trained VanillaFM baseline. (b) CRPS of the corresponding probabilistic forecasts. Shaded bands indicate the mean $\pm 0.3$ standard deviation across systems. Results are aggregated over 135 dynamical systems, with 20 randomly sampled initial conditions per system and 50 Monte Carlo forecasts per initial condition.

Panel of Figure 5 — Figure 5: Attractor Reconstruction. The upper plot shows boxplots across systems of the RMSE between the correlation dimensions estimated from the predicted and ground truth trajectories (white diamonds mark the mean). The lower plot shows the KL divergence between the true and reconstructed attractors for FreeFM and the baseline models; shaded bands indicate the mean $\pm 0.3$ standard deviation across systems. Results are aggregated over 135 dynamical systems, each with 20 randomly sampled initial conditions. For FreeFM and VanillaFM, 50 Monte Carlo forecasts are generated per initial condition.

Appendix figures & tables23 assets

Supplementary material from the paper’s appendix.

Appendix

Method family	Local	Nonparametric	Cont-time	Probabilistic	Neural
EDM ( Sugihara and May, 1990 )	✔	✔	$\times$	$\sim$	$\times$
KAF ( Zhao and Giannakis, 2016 )	✔	✔	$\times$	$\sim$	$\times$
KCE	✔	✔	×	$\sim$	×
Diffusion map ( Coifman and Lafon, 2006 )	✔	✔	×	×	×
CNF/FM models ( Lim et al., 2025 )	×	×	✔	✔	✔
Retrieval-based models ( Tire et al., 2024 )	✔	$\sim$	×	$\sim$	$\sim$

Appendix

Table 1: Comparison of FreeFM with related model families. For a fixed empirical transition distribution and conditional probability path, the full-memory FreeFM velocity field is the unrestricted

L^{2}

minimizer of the corresponding CFM objective. A neural model trained on that same objective approximates this field subject to function class and optimization error. FreeFM combines local retrieval, nonparametric transition modeling, continuous-time flow-based forecasting, and training-free predictive ensemble generation. Here,

\sim

indicates that a property may be present depending on the specific method instantiation.

Panel of Figure 6 — Figure 6: Computational Cost. (a) FLOPs Comparison during training phase. (b) FLOPs Comparison during inference phase. All the FLOPs are computed among $6220$ transition pairs.

Systems	dense	top- $R$	VanillaFM	TiDE	LSTM	Trans.	N-BEATS	Linear	ESN
	FreeFM
All	225.4	128.4	90.7	14.1	61.4	177.0	294.4	0.3	0.4
$d=3$	227.3	129.4	90.6	14.1	61.4	177.0	291.7	0.3	0.4
$d=4$	212.0	118.8	90.0	14.1	61.2	176.8	295.9	0.3	0.4
$d=5$	214.8	125.0	91.2	14.3	61.2	176.5	310.3	0.4	0.4
$d=6$	221.3	121.9	88.5	14.0	61.4	176.3	303.8	0.4	0.4

Appendix

Table 2: CPU-only mean runtime, in seconds, grouped by state dimension.

Figure 7: Hénon–Heiles System.

Figure 8: Sprott G System.

Figure 9: Isothermal Chemical Process.

Figure 10: Jerk Circuit Oscillator.

Figure 11: Forced Brusselator System.

Panel of Figure 12 — Figure 12: Ablation Study. (a) Ablation study for ODE integration steps: $[30,50,100]$ . (b) Ablation study for three ODE solvers: $[\text{Euler},\text{Runge\textendash Kutta},\text{Exponential Euler}]$ . (c) Ablation study for truncation model, truncation number $R=256$ . Shaded regions in (a)-(c) indicate the mean $\pm 0.3$ standard deviation across the 135 dynamical systems, each with 20 trajectories originating from randomly sampled initial conditions.

Model	Aus. E. ( $d=5$ )	Exchange ( $d=8$ )	Bitcoin ( $d=18$ )	Solar ( $d=137$ )	Elec. ( $d=321$ )	Traffic ( $d=862$ )
FreeFM	$\underline{0.0248\pm 0.0025}$	$\underline{0.0085\pm 0.0002}$	$\underline{5.1263\pm 0.0202}$	$\underline{0.0004\pm 0.0003}$	$\underline{1.0900\pm 0.0009}$	$0.1090\pm 0.0003$
VanillaFM	$0.0343\pm 0.0079$	$1.4640\pm 0.3683$	$12.5831\pm 1.2431$	$0.0055\pm 0.0006$	$1.8024\pm 0.1189$	$\underline{0.0764\pm 0.0017}$
LSTM	$0.0611\pm 0.0362$	$1.1282\pm 0.1050$	$11.4140\pm 0.7479$	$0.0016\pm 0.0015$	$1.7666\pm 0.3259$	$0.9175\pm 0.3364$
NBEATS	$0.0620\pm 0.0128$	$0.9466\pm 0.2604$	$13.0375\pm 4.1430$	$0.0090\pm 0.0121$	$1.9242\pm 0.0523$	$0.0766\pm 0.0051$
TiDE	$0.0402\pm 0.0120$	$0.3209\pm 0.2017$	$7.5923\pm 2.5347$	$0.0091\pm 0.0105$	$1.4096\pm 0.0681$	$0.0819\pm 0.0267$
Transformer	$0.0680\pm 0.0295$	$1.3253\pm 0.0697$	$14.3725\pm 0.8058$	$0.0010\pm 0.0007$	$1.3642\pm 0.1202$	$0.1124\pm 0.0408$

Appendix

Table 3: Forecasting performance in terms of MSE on real-world datasets for horizon 5.

Figure 13: Real-World Trajectory Forecasting Examples (Horizon 5). Predicted trajectories on the Traffic dataset. We show four coordinates forecast by FreeFM and the baseline models over a prediction horizon of five time steps. Shaded regions indicate the 10–90th percentile interval, computed at each future time step from 50 Monte Carlo samples drawn from each model.

Model	Aus. E. ( $d=5$ )	Exchange ( $d=8$ )	Bitcoin ( $d=18$ )	Solar ( $d=137$ )	Elec. ( $d=321$ )	Traffic ( $d=862$ )
FreeFM	$0.5600\pm 0.0682$	$\underline{0.4706\pm 0.0024}$	$\underline{10.5514\pm 1.0760}$	$0.8675\pm 0.7373$	$2.1937\pm 0.0592$	$0.3087\pm 0.0123$
VanillaFM	$\underline{0.3017\pm 0.1273}$	$13.6606\pm 1.5202$	$15.5929\pm 2.1633$	$\underline{0.1497\pm 0.1485}$	$\underline{0.8512\pm 0.0575}$	$\underline{0.2238\pm 0.0189}$
LSTM	$0.5842\pm 0.3684$	$10.7276\pm 3.1165$	$19.5922\pm 2.0108$	$0.5650\pm 0.7364$	$2.0271\pm 0.7627$	$1.5010\pm 0.7156$
NBEATS	$0.5358\pm 0.1991$	$9.6630\pm 1.5681$	$11.1885\pm 3.4449$	$0.6536\pm 0.7858$	–	$0.2298\pm 0.0239$
TiDE	$0.8860\pm 0.5535$	$10.3629\pm 7.4060$	$27.6160\pm 35.7030$	$0.1664\pm 0.2195$	$1.0334\pm 0.1614$	$0.4046\pm 0.2179$
Transformer	$0.5309\pm 0.3648$	$7.5731\pm 0.9196$	$18.4517\pm 3.3422$	$1.1303\pm 2.1847$	$1.1184\pm 0.2967$	$0.9943\pm 0.7531$

Appendix

Table 5: Forecasting performance in terms of MSE on real-world datasets for horizon

50

Figure 14: Real-World Trajectory Forecasting Examples (Horizon 50). Predicted trajectories on the Traffic dataset. We show four coordinates forecast by FreeFM and the baseline models over a prediction horizon of fifty time steps. Shaded regions indicate the 10–90th percentile interval, computed at each future time step from 50 Monte Carlo samples drawn from each model.

Dataset	FreeFM	VanillaFM	TiDE	LSTM	Trans.	N-BEATS
Aus. Elec.	82.2	580.0	12.6	25.3	171.5	333.5
Exchange	57.2	528.0	11.3	22.4	139.1	317.2
Bitcoin	74.1	552.2	11.6	23.2	140.3	320.4
Solar	125.3	843.2	14.8	30.2	155.7	943.9
Electricity	229.7	1263.5	17.3	41.0	155.7	1730.8
Traffic	574.6	4814.0	32.3	87.6	156.4	5223.5

Appendix

Table 7: CPU-only mean runtime over five seeds, in seconds. FreeFM and VanillaFM totals include model selection.

$\sigma$	sMAPE (%) $\downarrow$	VPT $\uparrow$	MSE $\downarrow$	CRPS $\downarrow$
0.001	42.02 $\pm$ 0.27	0.730 $\pm$ 0.008	2.8521 $\pm$ 0.0436	0.3417 $\pm$ 0.0023
0.005	42.12 $\pm$ 0.28	0.733 $\pm$ 0.007	2.8988 $\pm$ 0.0535	0.3423 $\pm$ 0.0025
0.010	42.37 $\pm$ 0.27	0.728 $\pm$ 0.007	2.9214 $\pm$ 0.0611	0.3456 $\pm$ 0.0025
0.020	43.51 $\pm$ 0.26	0.707 $\pm$ 0.007	3.0479 $\pm$ 0.0373	0.3561 $\pm$ 0.0014
0.050	53.88 $\pm$ 0.29	0.614 $\pm$ 0.005	8.0008 $\pm$ 0.1510	0.5295 $\pm$ 0.0038
0.100	105.86 $\pm$ 0.35	0.404 $\pm$ 0.005	$(1.1834\pm 0.0130)\times 10^{5}$	22.4631 $\pm$ 0.1591

Appendix

Table 8: Sensitivity to

\sigma

on Dysts with

\sigma_{\min}=0.05

fixed and

H=300

. Mean

\pm

standard error over five seeds. Lower is better for sMAPE, MSE, and CRPS; higher is better for VPT.

$\sigma$	Aus. Elec.	Exchange	Bitcoin	Solar	Electricity	Traffic
	$(d=5)$	$(d=8)$	$(d=18)$	$(d=137)$	$(d=321)$	$(d=862)$
0.001	0.0295 $\pm$ 0.0021	0.0084 $\pm$ 0.0004	5.1512 $\pm$ 0.0170	0.0006 $\pm$ 0.0005	1.0829 $\pm$ 0.0008	0.1084 $\pm$ 0.0004
0.005	0.0295 $\pm$ 0.0021	0.0084 $\pm$ 0.0004	5.1423 $\pm$ 0.0175	0.0006 $\pm$ 0.0005	1.0829 $\pm$ 0.0008	0.1085 $\pm$ 0.0005
0.010	0.0295 $\pm$ 0.0021	0.0084 $\pm$ 0.0004	5.1132 $\pm$ 0.0167	0.0006 $\pm$ 0.0005	1.0832 $\pm$ 0.0008	0.1090 $\pm$ 0.0006
0.050	0.0273 $\pm$ 0.0019	0.0099 $\pm$ 0.0004	4.6111 $\pm$ 0.0139	0.0006 $\pm$ 0.0005	1.0899 $\pm$ 0.0007	0.1226 $\pm$ 0.0003
0.100	0.0235 $\pm$ 0.0016	0.0167 $\pm$ 0.0005	4.6349 $\pm$ 0.0146	0.0006 $\pm$ 0.0006	1.1020 $\pm$ 0.0008	0.1321 $\pm$ 0.0002

Appendix

Table 9: Sensitivity of MSE to

\sigma

with

\sigma_{\min}

fixed for the short-horizon real-world tasks. Lower is better.

$\sigma$	Aus. Elec.	Exchange	Bitcoin	Solar	Electricity	Traffic
	$(d=5)$	$(d=8)$	$(d=18)$	$(d=137)$	$(d=321)$	$(d=862)$
0.001	0.0931 $\pm$ 0.0051	0.0528 $\pm$ 0.0012	1.6828 $\pm$ 0.0031	0.0131 $\pm$ 0.0007	0.7526 $\pm$ 0.0003	0.1846 $\pm$ 0.0009
0.005	0.0930 $\pm$ 0.0051	0.0529 $\pm$ 0.0012	1.6816 $\pm$ 0.0034	0.0131 $\pm$ 0.0007	0.7526 $\pm$ 0.0003	0.1850 $\pm$ 0.0010
0.010	0.0930 $\pm$ 0.0051	0.0530 $\pm$ 0.0012	1.6780 $\pm$ 0.0028	0.0132 $\pm$ 0.0007	0.7527 $\pm$ 0.0003	0.1860 $\pm$ 0.0010
0.050	0.0884 $\pm$ 0.0047	0.0589 $\pm$ 0.0013	1.6318 $\pm$ 0.0027	0.0134 $\pm$ 0.0007	0.7550 $\pm$ 0.0003	0.2002 $\pm$ 0.0002
0.100	0.0804 $\pm$ 0.0035	0.0787 $\pm$ 0.0014	1.7000 $\pm$ 0.0046	0.0139 $\pm$ 0.0007	0.7594 $\pm$ 0.0003	0.2102 $\pm$ 0.0001

Appendix

Table 10: Sensitivity of CRPS to

\sigma

with

\sigma_{\min}

fixed for the short-horizon real-world tasks. Lower is better.

	MSE $\downarrow$
Dataset	$0$	$0.25$	$0.50$	$0.75$	$1$
Electricity	1.082710 $\pm$ 0.001030	1.084720 $\pm$ 0.001040	1.086580 $\pm$ 0.001070	1.088400 $\pm$ 0.001070	1.089880 $\pm$ 0.001020
Exchange Rate	0.008701 $\pm$ 0.000462	0.008715 $\pm$ 0.000462	0.008728 $\pm$ 0.000463	0.008742 $\pm$ 0.000463	0.008755 $\pm$ 0.000463
Solar Energy	0.000268 $\pm$ 0.000098	0.000292 $\pm$ 0.000128	0.000294 $\pm$ 0.000129	0.000294 $\pm$ 0.000129	0.000303 $\pm$ 0.000127
Traffic	0.108113 $\pm$ 0.000520	0.108292 $\pm$ 0.000428	0.108414 $\pm$ 0.000376	0.108606 $\pm$ 0.000347	0.108796 $\pm$ 0.000339
Bitcoin	5.155820 $\pm$ 0.013000	5.147300 $\pm$ 0.012800	5.136480 $\pm$ 0.012100	5.127580 $\pm$ 0.012200	5.118460 $\pm$ 0.012200

Appendix

Table 11: Correction-strength ablation on the real-world tasks for the short-horizon (

H=5

) setting. Only the coefficient

\lambda_{\mathrm{corr}}

is varied, with

\lambda_{\mathrm{corr}}=0

corresponding to replay only and

\lambda_{\mathrm{corr}}=1

recovering the original FreeFM field.

	CRPS $\downarrow$
Dataset	$0$	$0.25$	$0.50$	$0.75$	$1$
Electricity	0.752643 $\pm$ 0.000450	0.753326 $\pm$ 0.000438	0.753950 $\pm$ 0.000455	0.754562 $\pm$ 0.000473	0.755057 $\pm$ 0.000454
Exchange Rate	0.054103 $\pm$ 0.001580	0.054160 $\pm$ 0.001580	0.054218 $\pm$ 0.001580	0.054276 $\pm$ 0.001580	0.054334 $\pm$ 0.001580
Solar Energy	0.012725 $\pm$ 0.000144	0.012762 $\pm$ 0.000175	0.012764 $\pm$ 0.000175	0.012766 $\pm$ 0.000175	0.012780 $\pm$ 0.000153
Traffic	0.183996 $\pm$ 0.001220	0.184448 $\pm$ 0.001120	0.184820 $\pm$ 0.001080	0.185137 $\pm$ 0.000981	0.185689 $\pm$ 0.001030
Bitcoin	1.683860 $\pm$ 0.001710	1.682560 $\pm$ 0.001810	1.681220 $\pm$ 0.001600	1.680110 $\pm$ 0.001720	1.678980 $\pm$ 0.001480

Appendix

Table 12: Correction-strength ablation on the real-world tasks for the short-horizon (

H=5

) setting. Only the coefficient

\lambda_{\mathrm{corr}}

is varied, with

\lambda_{\mathrm{corr}}=0

corresponding to replay only and

\lambda_{\mathrm{corr}}=1

recovering the original FreeFM field.

Metric	FreeFM	TFM
sMAPE (%) $\downarrow$	23.52 $\pm$ 0.02	76.42 $\pm$ 0.79
VPT $\uparrow$	1.192 $\pm$ 0.003	0.370 $\pm$ 0.012
MSE $\downarrow$	1.4397 $\pm$ 0.0082	6.6982 $\pm$ 0.4929
CRPS $\downarrow$	0.2238 $\pm$ 0.0004	1.1108 $\pm$ 0.0283

Appendix

Table 13: Comparison with TFM on Dysts (

H=300

). Values are mean

\pm

standard error over five seeds.

	sMAPE (%) $\downarrow$		CRPS $\downarrow$		MSE $\downarrow$
Dataset	FreeFM	TFM	FreeFM	TFM	FreeFM	TFM
Aus. Elec.	34.08 $\pm$ 0.09	33.88 $\pm$ 1.23	0.1211 $\pm$ 0.0006	0.1633 $\pm$ 0.0115	0.0415 $\pm$ 0.0002	0.0582 $\pm$ 0.0054
Exchange	9.16 $\pm$ 0.05	9.37 $\pm$ 0.68	0.0685 $\pm$ 0.0006	0.0712 $\pm$ 0.0054	0.0120 $\pm$ 0.0002	0.0134 $\pm$ 0.0018
Bitcoin	61.66 $\pm$ 0.17	71.53 $\pm$ 3.17	0.7821 $\pm$ 0.0005	0.8940 $\pm$ 0.0569	1.6334 $\pm$ 0.0016	2.0112 $\pm$ 0.2558
Solar E	8.82 $\pm$ 2.14	5.54 $\pm$ 0.45	0.0194 $\pm$ 0.0027	0.0241 $\pm$ 0.0022	0.0060 $\pm$ 0.0026	0.0017 $\pm$ 0.0002
Elec.	119.99 $\pm$ 0.03	98.31 $\pm$ 0.84	1.5216 $\pm$ 0.0003	0.7252 $\pm$ 0.0089	4.1381 $\pm$ 0.0008	0.9880 $\pm$ 0.0217

Appendix

Table 14: Short-horizon comparison with TFM on the real-world tasks. Values are mean

\pm

standard error over five seeds. Bold indicates the best result for each metric and dataset.

Metric	FreeFM	KCE	KAF
MSE $\downarrow$	1.7092 $\pm$ 0.0201	2.9089 $\pm$ 0.0225	5.3139
sMAPE (%) $\downarrow$	26.845 $\pm$ 0.029	26.795 $\pm$ 0.123	39.564
VPT $\uparrow$	1.1522 $\pm$ 0.0068	1.0059 $\pm$ 0.0094	0.8278

Appendix

Table 15: Nonparametric comparison on the Dysts systems for which all three evaluations are available. FreeFM–KCE is the controlled retrieval comparison; KAF is a complementary dynamics-adapted benchmark.

Dataset	FreeFM	KCE	KAF
Horizon 5
Electricity	1.0900	1.5439	1.3538
Exchange	0.0085	1.1770	3.2789
Solar	0.000361	$2.0\times 10^{-7}$	0.006683
Traffic	0.1090	0.0744	0.0709
Bitcoin	5.1263	14.8034	14.0718

Appendix

Table 16: MSE comparison with nonparametric baselines on the real-world tasks. KCE is the controlled retrieval baseline; KAF is a complementary dynamics-adapted benchmark.

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Jul 13, 2026J/K move · Enter open · S save

Velocity Scheduled Flow Matching

Vitalii Bondar

Cherkasy State Technological University Cherkasy, Ukraine

Flow Matching Conditional Flow Matching Few-Step Diffusion Sampling Flow-Based Generative Modeling

Open paper AskPDF arXiv

Flow matching trains a neural network to regress the conditional velocity along a linear interpolant between noise and data, and the number of network evaluations~(NFE) sets the cost of sampling. The straight-line interpolant carries an implicit choice: the sample moves at constant speed throughout the trajectory. We relax this choice and introduce Velocity Scheduled Flow Matching~(VSFM), which replaces the conditional target

x_1 - x_0

with

v(t)(x_1 - x_0)

for any nonnegative profile

v:[0,1]\to\mathbb{R}_{\geq 0}

satisfying

\int_0^1 v\,dt = 1

. We study six polynomial profiles drawn from motion planning. The first use of VSFM is at inference time: a pretrained linear flow-matching model can be sampled under any admissible profile by integrating its ODE on a non-uniform

τ

-schedule, with no retraining and no additional computation; on CIFAR-10 this lowers FID by up to

19.8\%

. Training from scratch under a braking profile gives a further reduction of

17.4\%

4

~NFE. Both gains follow from the local truncation error of the Euler integrator on the induced grid.

Abstract

Explore similar work

Velocity Scheduled Flow Matching

PrismFlow: Residual Dynamics for Flow Matching in Time-Series Generation

Smoother Flow Matching via Contrastive Trajectory Repulsion