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.
Lawrence Berkeley National Laboratory · International Computer Science Institute · 1Lawrence Berkeley National Laboratory, 2International Computer Science Institute +4