Flow matching generates samples by gradually transforming noise into data. In practice, using a finite number of sampling steps introduces a numerical error that depends on the chosen schedule. We study this dependence for Gaussian targets and the explicit midpoint sampling method, using the exact flow field. We measure sampling error by the squared Wasserstein distance between the target distribution and the final distribution produced by the midpoint sampler. We show that the standard conditional optimal transport (CondOT) schedule cancels the leading midpoint error and improves the general convergence bound, even when the sampling steps are unequally spaced. On a uniform grid of S sampling steps, we fix the signal schedule at αt=t and prove the existence of scalar noise schedules βt that approach the CondOT noise schedule 1−t at rate 1/S and yield exact Gaussian sampling for every sufficiently large S. Controlled Gaussian experiments illustrate the convergence rates and exact calibration.
Figures & tables
Target
r
κ
Unnormalized covariance eigenvalues w
A
2
4
(1,1,1,4,4,4)
B
6
4
(4j/5)j=05
C
2
104
(1,1,1,104,104,104)
D
6
104
(104j/5)j=05
Table 1: Four Gaussian targets. The vectors w are normalized by ( 20 ).
Figure 1: Fixed schedules for Target D. The four alternatives approach S−4 squared Wasserstein decay, while CondOT achieves S−6 . Dashed lines indicate the asymptotic rates.
Figure 2: First-order correction for Target D. The explicit choice θ=K/S improves CondOT’s S−6 decay to S−8 . Dashed lines indicate these rates.
S
S8JS(βK/S)
JS(βθS) (numerical)
256
1.8642×107
<10−170
512
1.8610×107
<10−170
1024
1.8602×107
<10−170
2048
1.8601×107
<10−170
Table 2: Calibration for Target D using the proof’s iteration. The last column reports numerical squared Wasserstein loss in 90-digit arithmetic.
Figure 3: Calibrated schedule corrections at S=256 . Top: modest spectral spread ( κ=4 ). Bottom: broad spectral spread ( κ=104 ). Left: two distinct eigenvalues. Right: six distinct eigenvalues. The horizontal line is CondOT, and all panels share the same vertical scale. The smaller corrections for Targets A and B are magnified.
Appendix figures & tables6 assets
Supplementary material from the paper’s appendix.
Appendix
Target
S
JSCondOT
JSK/S
Final JS
Updates
A
64
2.07×10−12
3.24×10−16
<10−170
47
A
128
3.23×10−14
1.25×10−18
<10−170
39
A
256
5.05×10−16
4.87×10−21
<10−170
34
A
512
7.89×10−18
1.90×10−23
<10−170
29
A
1024
1.23×10−19
7.40×10−26
<10−170
26
B
64
1.91×10−12
1.34×10−14
<10−170
72
Appendix
Table 3: High-precision proof iteration across targets. JSCondOT and JSK/S denote the baseline and first-order losses. The “Final JS ” column reports the loss at the last iterate. Successful runs achieve JS<10−170 . An asterisk marks runs that reached the 600-update limit without satisfying the stopping criterion. A dash marks a run stopped after a nonpositive midpoint factor; “updates” counts applications of the proof iteration.
Target
(1−t)2
1−t2
cos(πt/2)
(1−t)(1+0.1t)
CondOT
K/S
A
4.0000
4.0000
4.0000
4.0001
6.0000
8.0000
B
4.0000
4.0000
4.0000
4.0001
6.0000
7.9997
C
4.0000
4.0370
4.0043
4.0478
6.0000
7.9959
D
4.0000
4.0057
4.0011
4.0077
6.0000
8.0000
Appendix
Table 4: Observed decay exponents
Figure 4: Fixed-schedule convergence for all four targets. Each panel repeats the comparison in Figure 1 ; the dashed lines indicate S−4 and S−6 .
Figure 5: CondOT and the first-order correction for all four targets. The dashed lines indicate S−6 and S−8 . The vertical limits vary between panels.
Target
θ1
θ2
θ3
θ4
θ5
θ6
A
−0.00160120
0.00153963
–
–
–
–
B
0.00578858
0.02958382
−0.38062679
1.13618161
−1.39374372
0.60501822
C
−0.04198104
−0.14789871
–
–
–
–
D
−0.45092399
−1.70655900
26.80927083
−73.25730884
76.48650974
−27.84974968
Appendix
Table 5: Calibrated coefficients at S=256 , rounded to eight decimal places.
Target
S
Updates
Maximum scale error
JS
Solver status
A
64
4
1.33×10−15
5.83×10−30
Converged
A
128
3
1.16×10−15
6.65×10−30
Converged
A
256
2
9.67×10−16
2.82×10−30
Converged
A
512
2
7.36×10−17
2.47×10−32
Converged
A
1024
1
3.12×10−16
5.08×10−31
Converged
B
64
7
7.76×10−16
1.07×10−30
Converged
Appendix
Table 6: Proof iteration computed in binary64 and reevaluated at 90 digits without coefficient refinement. The scale error is maxk∣D1,kM−1/λk∣ ; JS includes eigenvalue multiplicities. Capped runs report the final iterate’s errors after 1000 updates. Dashes indicate a nonpositive midpoint factor.
Flow-matching schedules affect both sampling dynamics and the variance of the regression target. For centered commuting Gaussians, we show that a direction-dependent schedule decomposes into two independent design choices: a variance path, which fully determines the intermediate laws and probability flow, and a factorization, which leaves this flow unchanged while controlling irreducible regression variance. On the sampling side, we analyze finite-step Euler accuracy and derive a necessary drift bound for exact N -step sampling, connecting the geodesic and the logarithmic path. On the training side, for any fixed path, we derive closed-form factorizations that either minimize time-averaged regression variance or make it constant along the path.
Flow matching models generate samples by numerically integrating a learned velocity field, with each integration step requiring a neural network evaluation. Fast generation therefore requires using a small fixed evaluation budget effectively: the key question is not only how to integrate the flow, but where the sampler should spend its steps. We propose SharpEuler, a training-free sampler that profiles a pretrained model offline by estimating where the learned velocity field changes most rapidly along calibration trajectories. This finite-difference estimate defines a solver-aware sharpness profile, which is smoothed and converted by a quantile transform into a timestep grid for any desired inference budget. At test time, sampling remains ordinary Euler integration with the same number of model evaluations as a uniform schedule. We justify SharpEuler using three principles: a numerical principle identifying trajectory acceleration as the leading source of Euler discretization error, a variational principle deriving sharpness-based power-law timestep densities, and a statistical guarantee showing that the finite-sample calibrated sampler is stable at the terminal distribution level. Our experiments show that SharpEuler improves sample quality at fixed budgets, reducing inter-mode leakage and increasing mode coverage.
Aditi Gupta, Soon Hoe Lim, Annan Yu +1
Lawrence Berkeley National Laboratory · International Computer Science Institute · 1Lawrence Berkeley National Laboratory, 2International Computer Science Institute +4
Functional flow matching is posed on distributions of functions but implemented from finitely many coefficients or point values. Under scattered or adaptive refinement, the resulting conditioning sigma-algebras need not be nested, so martingale convergence does not justify the sensor limit. We prove strong L2 convergence of finite conditional velocity targets for every strongly consistent sequence of finite-rank reconstructions, with quantitative bounds for orthogonal projections and a point-sensor extension through a regularity space. For learned flows, coupling directly to a population superposition path yields an end-to-end Wasserstein bound without assuming uniqueness of the population finite-dimensional ODE. We verify sensor-independent constants for a normalized quadrature neural operator, including globally Lipschitz activations through an explicit magnitude recurrence. A noncommuting trace-class Gaussian example gives boundary multiplier 0 under projected restriction and 0.72 under exact conditioning. A spatial regularity--cubature certificate closes the operator-realization term, a Bernstein argument gives a O(n−1) excess-risk term for fixed model dimension and envelopes, and an exactly realizable clipped Gaussian scaling specialization yields an explicit end-to-end rate.
Lennon J. Shikhman
School of Computer Science, College of Computing Georgia Institute of Technology Atlanta, Georgia 30332, USA