MercerFlow: Flow Matching in a Kernel-Induced Latent Space for Probabilistic Forecasting

Ilya Kuleshov; Egor Serov; Alexey Zaytsev

cs.LGOct 5, 2026

MercerFlow: Flow Matching in a Kernel-Induced Latent Space for Probabilistic Forecasting

Authors: Ilya Kuleshov, Egor Serov, Alexey Zaytsev

Organizations: Applied AI Institute Moscow, Russia

Abstract

Recent work has shown that probabilistic flow matching for time series forecasting benefits from a data-matched prior. The resulting prior introduces local correlations, which a sequential architecture usually absorbs: a recurrent neural network (RNN), a structured state-space model (S4), or a Transformer. However, such a backbone costs GPU memory and time per epoch. A cheaper alternative is MLP-based latent-space flow matching: embed the time series via an invertible map to a single latent vector and learn the flow there, so a tabular MLP can treat the series as a set of features. The relationship between the prior and the choice of linear latent map is understudied in conditional flow matching (CFM) forecasting, yet we found it strongly affects performance. Fixed transforms such as Fourier or discrete cosine (DCT) are only well-conditioned for Ornstein--Uhlenbeck priors, while a principal-component (PCA) map fit to the data is a strong but training-set-dependent reference sensitive to train--test shift. Instead, we propose to use the Mercer eigenbasis of the prior kernel: it diagonalises the centred covariance exactly, decouples from training data, and adapts to non-stationary and periodic priors. On five GluonTS benchmarks (ETTh1, ETTh2, Weather, Electricity, Traffic) under a shared protocol with TSFlow, the resulting MLP matches or beats it on CRPS at about $4.7\times$ less training memory and $3.5\times$ -- $4.4\times$ less time per epoch.

Figures & tables

Figure 1: Overview of MercerFlow . The joint context and horizon window is mapped into the decorrelated Mercer eigenbasis of the GP prior. A lightweight MLP learns conditional flow matching directly in this latent space, enabling fast and scalable probabilistic forecasting without sequential backbones.

	Ours	Linear Codec Baselines				Sequential
Metric	Mercer	Fourier	DCT	PCA	Raw	TSFlow (S4)
Mean CRPS rank ( $\downarrow$ )	$\mathbf{1.00}$	$1.20$	$1.40$	$1.80$	$3.40$	$5.80$
# CRPS wins ( $\uparrow$ )	$\mathbf{5/5}$	$4/5$	$4/5$	$3/5$	$2/5$	$0/5$
Mean MAE rank ( $\downarrow$ )	$\mathbf{1.00}$	$1.60$	$1.40$	$2.00$	$3.20$	$5.00$
# MAE wins ( $\uparrow$ )	$\mathbf{5/5}$	$3/5$	$4/5$	$3/5$	$2/5$	$1/5$

Table 1: Predictive performance summary across all five benchmarks (ETTh1, ETTh2, Weather, Electricity, Traffic). Rank

=1+

number of methods strictly outperforming the model under a paired two-sided

t

-test on shared seeds (

p<0.2

, lower score is better); a win denotes rank

1

Metric / Architecture	ETTh1	ETTh2	Weather	Electricity	Traffic
MLP Time (s, $\downarrow$ )	0.67	0.69	1.80	0.93	0.68
S4 Time (s, $\downarrow$ )	2.50	2.40	7.70	4.10	2.60
MLP VRAM (GB, $\downarrow$ )	1.5	1.5	1.5	1.5	1.5
S4 VRAM (GB, $\downarrow$ )	7.1	7.1	7.1	7.1	7.1

Table 2: Computational efficiency benchmark at batch size 512. Comparison of per-epoch training time (seconds,

\downarrow

) and peak GPU memory (GB,

\downarrow

Figure 2: Traffic decorrelation on conditional-prior draws $\mathbf{c}_{\mathrm{0}}=\Phi^{\top}[\mathbf{y}_{\mathrm{p}};\mathbf{y}_{\mathrm{0}}]$ for the joint window. Each panel displays $|DAD|$ for the leading $64$ modes, with condition number $\kappa(DAD)$ and test CRPS ( $\eta=10^{-3}$ ) shown above each plot. Aligning the codec basis with the prior kernel maintains low off-diagonal correlation and preserves downstream performance.

Appendix figures & tables19 assets

Supplementary material from the paper’s appendix.

Appendix

Codec	ETTh1	ETTh2
Mercer (Ours)	$0.2255\pm 0.0007$	$0.145\pm 0.002$
Fourier	$0.2240\pm 0.0020$	$0.145\pm 0.003$
DCT	$0.2260\pm 0.0010$	$0.145\pm 0.002$
PCA	$0.2248\pm 0.0004$	$0.145\pm 0.002$
Raw	$0.2240\pm 0.0010$	$0.145\pm 0.003$

Appendix

Table 3: SGD ablation on ETTh1 and ETTh2 (Test CRPS,

\downarrow

) for Post-AdaLN Nesterov SGD (momentum

0.9

, learning rate

10^{-3}

). Evaluated at

20

NFE under the identical joint window (

L=384

) and OU length scales as the AdamW runs.

Codec	First-layer Gram Matrix $A$	Jacobi Condition Number Bound $\kappa(DAD)$
Identity (Raw)	$T_{n}(f)+\varepsilon I_{n}$	$\kappa(A)$ , increases with length scale $\ell$
Mercer (Ours)	$\Lambda=\operatorname{diag}(\lambda_{1},\dots,\lambda_{n})$	$1$ (exact decorrelation)
Fourier	$\Phi_{F}^{\top}T_{n}(f)\,\Phi_{F}+\varepsilon I_{n}$	$1+\mathcal{O}\left(1/\min_{\omega}f(\omega)\right)$
DCT	$\Phi_{D}^{\top}T_{n}(f)\,\Phi_{D}+\varepsilon I_{n}$	$1+\mathcal{O}\left(1/\min_{\omega}f(\omega)\right)$
PCA	Empirical data covariance	Data-dependent (not aligned with prior kernel)

Appendix

Table 4: Theoretical first-layer Gram matrix

A=\mathbb{E}[\mathbf{x}\mathbf{x}^{\top}]

and corresponding Jacobi condition number

\kappa(DAD)

under the centred Ornstein–Uhlenbeck prior

\mathbf{y}\sim\mathcal{N}(\mathbf{0},K)

on a window of length

n=L

, with regularizing jitter

\varepsilon>0

Dataset	Raw	Fourier	Mercer	DCT	PCA
ETTh1	$1.72\times 10^{5}$	$268.50$	$\mathbf{1.00}$	$1.92$	$8.15$
ETTh2	$8.16\times 10^{4}$	$189.71$	$\mathbf{1.00}$	$2.07$	$46.98$
Weather	$2.80\times 10^{4}$	$119.74$	$\mathbf{1.00}$	$2.12$	$13.46$
Electricity	$2.23\times 10^{3}$	$41.63$	$\mathbf{1.00}$	$2.08$	$43.93$
Traffic	$195.79$	$13.44$	$\mathbf{1.00}$	$1.90$	$44.68$

Appendix

Table 5: Empirical Jacobi condition number

\kappa(\mathrm{DAD})

under the centred prior

y\sim\mathcal{N}(0,K)

on the joint grid (joint window

192

192

; last channel; scaler fit on the first

70\%

). This is the population the proposition’s identity is proved against: Mercer hits

\kappa(\mathrm{DAD})=1

on every dataset, with jitter only, by construction. The training input is not this population; Table 12 reports the same quantity under the conditional draw

\mathbf{c}_{\mathrm{0}}=\Phi^{\top}[\mathbf{y}_{\mathrm{p}};\mathbf{y}_{\mathrm{0}}]

Dataset	OU	RBF	Periodic
ETTh1	$\mathbf{-1.231}$	$-0.970$	$0.062$
ETTh2	$-0.836$	$\mathbf{-1.184}$	$0.608$
Traffic	$0.465$	$0.226$	$\mathbf{0.035}$
Electricity	$0.228$	$\mathbf{0.209}$	$0.969$
Weather	$\mathbf{-0.587}$	$-0.372$	$1.203$

Appendix

Table 6: Mean negative log density of candidate Gaussian process priors on the validation slice. Lower is better (

\downarrow

). The periodic entry for Traffic uses period

p=168

, length scale

\ell=0.1

, and jitter

\varepsilon=0.05

; all other entries represent the optimal fit across

400

TPE trials. Bold indicates the best score in each row.

Dataset	Kernel	$\ell$	Period $p$	Jitter $\varepsilon$
ETTh1	OU	$479$	—	$4.4\times 10^{-4}$
ETTh1	RBF	$7.41$	—	$2.8\times 10^{-3}$
ETTh1	Periodic	$1.98$	$276$	$0.062$
ETTh2	OU	$183$	—	$1.0\times 10^{-5}$
ETTh2	RBF	$4.58$	—	$7.2\times 10^{-4}$
ETTh2	Periodic	$2.77$	$24$	$0.23$

Appendix

Table 7: Continuous optimal parameters of the validation likelihood fits from Table 6 . Length scale

\ell

, period

p

(periodic kernel only), and regularizing diagonal jitter

\varepsilon

. The jitter search interval is

\varepsilon\in[10^{-5},1.0]

; the OU fits for ETTh2, Traffic, Electricity, and Weather saturate at the lower bound

1.0\times 10^{-5}

Dataset	Length Scale $\ell$	Runtime Jitter $\varepsilon$
ETTh1	$336$	$10^{-4}$
ETTh2	$192$	$10^{-4}$
Weather	$96$	$10^{-4}$
Electricity	$24$	$10^{-4}$
Traffic	$7$	$10^{-4}$

Appendix

Table 8: Baseline OU length scale

\ell

and runtime diagonal jitter

\varepsilon

used in primary benchmark runs, shared across Mercer, Fourier, DCT, PCA, and Raw.

Prior	Mercer	Fourier	Raw	DCT	PCA
OU	$687$	$891$	$8.54\times 10^{3}$	$712$	$\mathbf{477}$
Periodic	$1.18\times 10^{3}$	$188$	$1.62\times 10^{4}$	$2.45\times 10^{4}$	$\mathbf{23.0}$

Appendix

Table 9: Traffic first-layer Jacobi condition number

\kappa(DAD)

under conditional prior samples

c_{0}=\Phi^{\top}[y_{p},y_{0}]

: stationary OU prior (

\ell=7

\varepsilon=10^{-4}

) versus validation-optimal periodic prior (period

p=168

\ell=0.1

\varepsilon=0.05

). Evaluated on the joint window (

L_{p}=192,L_{f}=192

L=384

), last channel, random seed

42

, standard scaler fit on the first

70\%

. Bold indicates the best (lowest) condition number in each row.

Prior	Learning Rate $\eta$	Mercer	Fourier	Raw	DCT	PCA
OU	$10^{-3}$	$\mathbf{0.0875\pm 0.0009}$	$\mathbf{0.0871\pm 0.0015}$	$0.0884\pm 0.0003$	$0.0878\pm 0.0011$	$0.0883\pm 0.0018$
OU	$10^{-1}$	$\mathbf{0.1065\pm 0.0036}$	$0.1080\pm 0.0086$	$0.2462\pm 0.0717$	$0.1116\pm 0.0026$	$\mathbf{0.1065\pm 0.0041}$
Periodic	$10^{-3}$	$\mathbf{0.0821\pm 0.0006}$	$0.0827\pm 0.0011$	$0.0839\pm 0.0016$	$0.0840\pm 0.0007$	$0.0828\pm 0.0007$
Periodic	$10^{-1}$	$0.0951\pm 0.0130$	$0.7773\pm 1.0350$	$1.0198\pm 0.1078$	$0.1641\pm 0.0688$	$\mathbf{0.0855\pm 0.0032}$

Appendix

Table 10: Traffic probabilistic forecasting performance (Test CRPS,

20

NFE,

\downarrow

) under the stationary OU prior vs. the validation-optimal periodic prior (

p=168

\ell=0.1

\varepsilon=0.05

). Under the periodic prior, MercerFlow dynamically adapts and achieves top performance, whereas fixed harmonic codecs (DCT, Fourier) suffer from severe condition degradation and numerical instability under learning rate stress (

\eta=10^{-1}

). Bold indicates the best score in each row.

	ETTh2	Weather	Electricity	ETTh1	Traffic
OU NLL	$-0.836$	$-0.587$	$0.228$	$-1.231$	$0.465$
Residual	$0.33$	$0.44$	$1.00$	$1.44$	$1.46$

Appendix

Table 11: Dataset hardness from the frozen OU prior. Residual is the mean squared error of the conditional OU mean on the held-out last

15\%

, after the training standard scaler, averaged over channels; length scale from Table 8 . OU NLL is the validation score from Table 6 ; white noise on unit-scaled series scores about

1.42

. Easy: residual

<1/2

; hard: residual

\geq 1

Dataset	Raw	Fourier	Mercer	DCT	PCA
ETTh1	$(2.08/3.25)\,$ 1e5	$539$ / $471$	$34.7$ / $14.8$	$34.7$ / $14.6$	$26.8$ / $15.1$
ETTh2	$(3.45/6.21)\,$ 1e5	$1770$ / $2970$	$155$ / $248$	$154$ / $260$	$121$ / $198$
Weather	$(4.20/1.11)\,$ 1e4	$295$ / $272$	$13.5$ / $4.5$	$14.3$ / $4.6$	$17.6$ / $15.9$
Electricity	$(5.85/5.22)\,$ 1e3	$234$ / $394$	$60.2$ / $171$	$59.6$ / $176$	$115$ / $285$
Traffic	$(8.60/1.10)\,$ 1e3	$922$ / $1140$	$683$ / $925$	$709$ / $936$	$481$ / $1290$

Appendix

Table 12: Empirical Jacobi condition number

\kappa(\mathrm{DAD})

under the conditional prior draws

\mathbf{c}_{0}=\Phi^{\top}[\mathbf{y}_{\mathrm{p}};\mathbf{y}_{0}]

on the training split and on the held-out last

15\%

, standard-scaled;

D=\mathrm{diag}(A)^{-1/2}

with

A

the empirical second moment of

c_{0}

. Each cell gives train/test. Scaler fit on the first

70\%

; the PCA basis is fit on data windows, not on prior draws. Raw values share a single exponent per row.

$\kappa(\mathrm{DAD})$	Overall ( $n{=}5$ )	Hard ( $n{=}3$ )	Easy ( $n{=}2$ )
Test	$0.44$	$0.79$	$-0.08$
Train	$0.38$	$0.68$	$-0.08$

Appendix

Table 13: Mean across datasets of within-dataset Spearman

\rho

between

\kappa(\mathrm{DAD})

under the conditional prior (Table 12 ) and CRPS, over the five MLP codecs; TSFlow excluded. Each per-dataset

\rho

ranks the five codecs on the two axes within that dataset; the column is the mean of those

\rho

’s across the listed datasets.

n

is the number of datasets averaged.

Method	ETTh1	ETTh2	Weather	Electricity	Traffic
DLinear ( Zeng et al., 2023 )	$23.06\pm 0.01$	$16.02\pm 0.01$	$6.26\pm 0.02$	$5.57\pm 0.00$	$10.75\pm 0.01$
PatchTST ( Nie et al., 2022 )	$23.02\pm 0.61$	$16.50\pm 0.47$	$5.99\pm 0.58$	$6.70\pm 0.20$	$12.45\pm 0.30$
DMamba ( Chen and Sun, 2026 )	$25.38\pm 0.89$	$16.00\pm 0.41$	$2.56\pm 0.02$	$5.48\pm 0.10$	$10.42\pm 0.24$
TSFlow (S4) ( Kollovieh et al., 2025 )	$19.1\pm 0.2$ (5)	$15.4\pm 0.3$ (6)	$2.18\pm 0.08$ (6)	$7.4\pm 0.3$ (6)	$23.0\pm 9.0$ (6)
Raw	$19.2\pm 0.2$ (5)	13.1 $\pm$ 0.2 (1)	$1.82\pm 0.03$ (1)	$4.23\pm 0.06$ (5)	$8.9\pm 0.1$ (5)
Fourier	$19.0\pm 0.3$ (1)	$13.2\pm 0.3$ (1)	$1.82\pm 0.03$ (1)	$4.11\pm 0.04$ (2)	8.70 $\pm$ 0.09 (1)

Appendix

Table 14: Comprehensive long-horizon probabilistic forecasting performance (Test CRPS,

10^{-2}

\downarrow

). Top: deterministic baselines; bottom: conditional flow matching models. Parenthetical values denote paired two-sided

t

-test rank (

p<0.2

) among flow matching models on shared random seeds (

42

–

48

). Bold : best; second best .

Method	ETTh1	ETTh2	Weather	Electricity	Traffic
DLinear ( Zeng et al., 2023 )	$2.12\pm 0.01$	$4.91\pm 0.01$	$27.56\pm 0.08$	$187.2\pm 0.2$	$4.00\pm 0.01$
PatchTST ( Nie et al., 2022 )	2.11 $\pm$ 0.06	$5.06\pm 0.15$	$26.38\pm 2.54$	$225.4\pm 6.8$	$5.00\pm 0.01$
DMamba ( Chen and Sun, 2026 )	$2.33\pm 0.08$	4.91 $\pm$ 0.12	$11.26\pm 0.10$	$184.2\pm 3.5$	$4.00\pm 0.01$
TSFlow (S4) ( Kollovieh et al., 2025 )	$2.25\pm 0.03$ (1)	$6.0\pm 0.1$ (6)	$11.6\pm 0.3$ (6)	$320\pm 10$ (6)	$11.0\pm 5.0$ (6)
Raw	$2.31\pm 0.02$ (6)	$5.13\pm 0.09$ (1)	10.0 $\pm$ 0.2 (1)	$181\pm 3$ (5)	$4.04\pm 0.08$ (3)
Fourier	$2.27\pm 0.03$ (3)	$5.1\pm 0.1$ (1)	$10.1\pm 0.2$ (1)	$176\pm 2$ (2)	3.95 $\pm$ 0.04 (1)

Appendix

Table 15: Point forecasting performance (Test MAE,

\downarrow

, Traffic in

10^{-3}

). Parenthetical values denote paired two-sided

t

-test rank (

p<0.2

) among flow matching models on shared random seeds (

42

–

48

). Bold : best; second best .

CRPS ( $\times 10^{-2}$ , $\downarrow$ )
$L_{\mathrm{f}}$	Method	ETTh1	ETTh2	Weather	Electricity	Traffic †
336	Raw	17.77 $\pm$ 0.23	13.07 $\pm$ 0.33	3.98 $\pm$ 0.05	4.43 $\pm$ 0.09	10.36 $\pm$ 0.20
	Fourier	18.07 $\pm$ 0.49	12.91 $\pm$ 0.19	3.82 $\pm$ 0.02	4.51 $\pm$ 0.10	10.41 $\pm$ 0.15
	DCT	17.81 $\pm$ 0.67	12.65 $\pm$ 0.31	3.82 $\pm$ 0.02	4.49 $\pm$ 0.09	10.45 $\pm$ 0.15
	PCA	17.93 $\pm$ 0.49	12.51 $\pm$ 0.30	3.83 $\pm$ 0.03	4.49 $\pm$ 0.08	10.29 $\pm$ 0.16
	MercerFlow	17.82 $\pm$ 0.49	12.66 $\pm$ 0.36	3.82 $\pm$ 0.03	4.52 $\pm$ 0.10	10.41 $\pm$ 0.21

Appendix

Table 16: Sensitivity to extended forecasting horizons (

L_{\mathrm{p}}=192

). Bold: best; underline: second best.

Dataset	Mercer (Ours)	Fourier	DCT	PCA	Raw
ETTh1	$17.99\pm 0.44$	$18.17\pm 0.55$	$18.51\pm 0.33$	$\mathbf{17.81\pm 0.00}$	$122.42\pm 80.10$
ETTh2	$\mathbf{13.05\pm 0.56}$	$13.27\pm 0.06$	$14.00\pm 1.13$	$\mathbf{12.92\pm 0.20}$	$15.18\pm 0.69$
Weather	$\mathbf{1.70\pm 0.01}$	$\mathbf{1.71\pm 0.01}$	$\mathbf{1.71\pm 0.01}$	$\mathbf{1.70\pm 0.01}$	$\mathbf{1.71\pm 0.06}$
Electricity	$\mathbf{4.34\pm 0.06}$	$\mathbf{4.33\pm 0.06}$	$\mathbf{4.34\pm 0.04}$	$\mathbf{4.39\pm 0.05}$	$6.53\pm 0.35$
Traffic	$\mathbf{10.65\pm 0.36}$	$\mathbf{10.80\pm 0.86}$	$11.16\pm 0.26$	$\mathbf{10.65\pm 0.41}$	$24.62\pm 7.17$

Appendix

Table 17: Test CRPS (

\times 10^{-2}

\downarrow

) at AdamW learning rate

\eta=10^{-1}

(default is

10^{-3}

), mean

\pm

std over seeds

42

–

44

. Evaluated at matched mid-run EMA checkpoints

E^{*}

. Bold: within one std of the best.

Figure 3: Training loss trajectories on ETTh1 and ETTh2 at an elevated learning rate (

\eta=10^{-2}

). MercerFlow (solid blue) converges faster and strictly monotonically. In contrast, the unrotated baseline ( RawFlow , dashed orange) suffers from optimization friction and visible instability (e.g., the loss spike around epoch

60

on ETTh1), demonstrating the empirical benefits of first-layer preconditioning.

Codec / arm	$\kappa(DAD)$ prior	$\kappa(DAD)$ interp.	Test CRPS ( $\downarrow$ )	SGD gap (%)
Mercer ( $\ell=32$ )	$1.00$	$98.5$	$\mathbf{0.6063\pm 0.0025}$	$0.00$
Fourier ( $\ell=32$ )	$24.90$	$654.0$	$\mathbf{0.6064\pm 0.0023}$	$0.00$
DCT ( $\ell=32$ )	$1.82$	$105.0$	$\mathbf{0.6062\pm 0.0024}$	$0.00$
Raw ( $\ell=32$ )	$1{,}501.0$	$3{,}515.0$	$\mathbf{0.6069\pm 0.0030}$	$0.00$
Mercer ( $\ell=320$ )	$1.71$	$192.0$	$0.6125\pm 0.0057$	—

Appendix

Table 18: Synthetic OU comparison (context/horizon

32/32

, data length scale

\ell=32

). Matched codecs share the OU prior at

\ell=32

; the wrong-

\ell

arm uses a Mercer/GP model at

\ell=320

. AdaLN MLP (

64

hidden,

2

layers), AdamW

\eta=10^{-3}

2{,}000

steps, seeds

0

–

2

(mean

\pm

std).

\kappa(DAD)

prior is computed on the population covariance

\Phi^{\top}K\Phi

; interpolant

\kappa(DAD)

is empirical on conditional CFM states

c_{t}

. SGD control: Nesterov momentum

0.9

, matched compute budget; gaps are

100\times(\mathrm{CRPS}/\mathrm{CRPS}_{\mathrm{Raw}}-1)

(mean over seeds). Bold indicates values within one standard deviation of the best.

		NFE (Euler Steps)
Dataset	Method	4	10	20	50	128
ETTh1	MercerDiff	$19.03\pm 0.07$	$1.04\pm 0.06$	$0.56\pm 0.07$	$0.51\pm 0.06$	$0.47\pm 0.03$
ETTh1	MercerFlow	$\mathbf{0.200\pm 0.017}$	$\mathbf{0.199\pm 0.017}$	$\mathbf{0.199\pm 0.017}$	—	—
ETTh2	MercerDiff	$7.78\pm 0.02$	$0.47\pm 0.01$	$0.211\pm 0.005$	$0.176\pm 0.005$	$0.177\pm 0.003$
ETTh2	MercerFlow	$\mathbf{0.137\pm 0.005}$	$\mathbf{0.136\pm 0.006}$	$\mathbf{0.136\pm 0.007}$	—	—

Appendix

Table 19: Inference sampling efficiency: Test CRPS (

\downarrow

, mean

\pm

std) across varying Numbers of Function Evaluations (NFE) at horizon

L_{\mathrm{f}}=192

. Due to straight probability paths, MercerFlow saturates within

4

–

10

steps, whereas diffusion ( MercerDiff ) suffers from severe discretization error. Bold: best per column.

			Diagonal Jitter $\varepsilon$
Metric	Dataset	Codec	$10^{-4}$	$10^{-3}$	$10^{-2}$	$10^{-1}$	$1.0$
$\kappa(DAD)$ ( $\mathbf{c}_{t}$ , $\downarrow$ )	ETTh1	Mercer	23.55	30.64	29.37	47.03	334.67
$\kappa(DAD)$ ( $\mathbf{c}_{t}$ , $\downarrow$ )	ETTh1	Raw	$2.01\times 10^{5}$	$3.02\times 10^{5}$	$1.85\times 10^{5}$	$1.66\times 10^{5}$	$1.44\times 10^{5}$
	ETTh2	Mercer	62.14	74.75	95.27	524.26	$\mathbf{3.64\times 10^{3}}$
	ETTh2	Raw	$4.07\times 10^{5}$	$1.22\times 10^{6}$	$1.20\times 10^{6}$	$1.17\times 10^{6}$	$1.01\times 10^{6}$
CRPS ( $\downarrow$ )	ETTh1	Mercer	0.188	0.186	0.187	0.195	0.255

Appendix

Table 20: Sensitivity to GP prior diagonal jitter

\varepsilon

on ETTh1 and ETTh2 (

L=384

\kappa(DAD)

is evaluated on conditional interpolants

\mathbf{c}_{t}

. Standard regularisation (

\varepsilon\leq 10^{-2}

) preserves well-conditioned representations; extreme jitter (

\varepsilon\geq 0.1

) collapses the prior toward white noise (

K\approx I_{L}

), degrading predictive accuracy. Bold: best per column within each dataset block.

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Multivariate Time Series Forecasting Flow Matching Flow-Based Generative Modeling

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Generative modeling has shown strong promise for multivariate time mseries (MTS) forecasting, especially scale to high-dimensional settings. Diffusion-based methods achieve competitive performance but typically require many sampling steps at inference. VAE-based non-iterative forecasting frameworks have therefore emerged as an efficient alternative. Within this line of work, vector quantization (VQ) enables controllable latent space modeling by mapping multivariate series into compact discrete representations. Existing VQ-based forecasting methods, however, typically rely on autoregressive (AR) token generation, which suffers from exposure bias and training-inference mismatch. Flow matching provides an efficient non-autoregressive alternative for latent forecasting, but existing formulations usually initialize transport from a generic Gaussian prior. We instead observe that the trained VQ codebook already captures representative latent prototypes and can thus serve as a more informative prior for flow matching. Based on this insight, we propose ProtoFlow, a forecasting framework that combines vector-quantized autoencoding with Prototype-prior Flow matching. Our method first maps multivariate sequences into a discrete latent space, then constructs a structured prior from the learned codebook, and finally learns a DiT-based rectified flow to transport samples from this prior to future latent representations conditioned on historical observations. By replacing generic noise initialization with a learned prototype prior, ProtoFlow avoids the rollout mismatch of AR token prediction and promotes faster training convergence. Extensive experiments on benchmark datasets show that it consistently achieves superior forecasting performance with efficient inference.

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