Wavelet Flow Matching for Time Series

Lucas Poinsignon; Jorge da Silva Goncalves; Samuel Ruiperez-Campillo; Julia E. Vogt

cs.LGSep 30, 2026

Wavelet Flow Matching for Time Series

Authors: Lucas Poinsignon, Jorge da Silva Goncalves, Samuel Ruiperez-Campillo, Julia E. Vogt

Organizations: Dept. of Computer Science ETH Zurich, Switzerland

Abstract

Synthetic time series are increasingly used for data augmentation, privacy-preserving data sharing, and downstream model development, yet faithfully reproducing both multi-scale temporal structure and cross-channel dependencies remains challenging. We study multivariate time-series generation through flow matching in the wavelet domain. By operating on multilevel discrete wavelet coefficients rather than directly in the time domain, the model represents coarse structure and progressively finer details at separate scales. Their naturally different variances further induce an implicit coarse-to-fine generative process without requiring an explicit multi-scale schedule. Since the transform acts independently on each channel, we pair it with a channel-token transformer whose attention directly models cross-channel dependencies. Across seven benchmark datasets and four sequence lengths, our method is best or tied on a majority of dataset-metric combinations, with the largest and most consistent improvements in Context-FID and discriminative score.

Figures & tables

Context-FID ( $\downarrow$ )
	ETTh1	ETTh2	Stocks	Exchange	EEG	Energy	MuJoCo
Diffusion-TS	0.137 $\pm$ .012	0.056 $\pm$ .006	0.175 $\pm$ .022	0.047 $\pm$ .005	0.018 $\pm$ .004	0.090 $\pm$ .011	0.016 $\pm$ .002
SigDiffusion	2.473 $\pm$ .227	1.162 $\pm$ .051	3.285 $\pm$ .504	1.629 $\pm$ .152	0.020 $\pm$ .003	4.241 $\pm$ .372	2.681 $\pm$ .263
FourierDiffusion	0.025 $\pm$ .003	0.026 $\pm$ .001	0.039 $\pm$ .005	0.080 $\pm$ .021	0.015 $\pm$ .002	0.217 $\pm$ .015	0.062 $\pm$ .006
WaveletDiff	0.026 $\pm$ .002	0.031 $\pm$ .002	0.020 $\pm$ .002	0.009 $\pm$ .000	0.008 $\pm$ .001	0.482 $\pm$ .042	1.180 $\pm$ .095
FlowTS	0.025 $\pm$ .001	0.012 $\pm$ .001	0.017 $\pm$ .005	0.009 $\pm$ .001	0.005 $\pm$ .000	0.042 $\pm$ .004	0.012 $\pm$ .001

Table 1: Unconditional generation on short sequences (

T=24

). Mean

\pm

std over three independent end-to-end training runs with five evaluations each. Bold : best; underlined : second best.

Figure 4: Mean rank of each method over the seven datasets as a function of the window length

T

(1 = best; tied methods receive the average of their ranks).

Panel of Figure 5 — Figure 5: t-SNE visualization and probability distributions on Energy. Red is for real data, and blue for generated data.

Figure 6: Effect of removing the implicit coarse-to-fine progression at

T=24

. Bars show the ratio of the score under natural coefficient scaling to that under per-level standardization. Values below

1

favor natural scaling and values above

1

favor standardization; the dashed line denotes equal performance.

Variant	ETTh1	ETTh2	Stocks	Exchange	EEG	Energy	MuJoCo
(a) Without $e_{f}$	1.488 $\pm$ 0.099	0.977 $\pm$ 0.168	1.763 $\pm$ 0.321	0.938 $\pm$ 0.072	0.011 $\pm$ 0.001	2.617 $\pm$ 0.145	1.390 $\pm$ 0.139
(b) Std. levels	0.021 $\pm$ 0.002	0.014 $\pm$ 0.001	0.018 $\pm$ 0.004	0.010 $\pm$ 0.001	0.010 $\pm$ 0.001	0.019 $\pm$ 0.002	0.008 $\pm$ 0.000
(c) Time domain	0.006 $\pm$ 0.001	0.005 $\pm$ 0.001	0.014 $\pm$ 0.001	0.007 $\pm$ 0.000	0.014 $\pm$ 0.001	0.012 $\pm$ 0.000	0.005 $\pm$ 0.000
(d) Uniform $t$	0.007 $\pm$ 0.000	0.008 $\pm$ 0.001	0.013 $\pm$ 0.001	0.005 $\pm$ 0.000	0.014 $\pm$ 0.002	0.016 $\pm$ 0.001	0.010 $\pm$ 0.001
Full model	0.005 $\pm$ 0.001	0.004 $\pm$ 0.000	0.005 $\pm$ 0.002	0.004 $\pm$ 0.001	0.013 $\pm$ 0.002	0.010 $\pm$ 0.003	0.006 $\pm$ 0.000

Table 2: Context-FID (

\downarrow

) for one-factor-at-a-time ablations at

T=24

. The full model uses db2 coefficients, natural level scaling, logit-normal time sampling, and channel-identity embeddings

e_{f}

. Variants (a)–(d) respectively remove

e_{f}

, standardize the wavelet levels, replace the wavelet transform with the identity, and sample

t

uniformly. Results are mean

\pm

standard deviation. Best values are in bold and second-best values are underlined.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Component	Formula	Parameters	Share
MLP, all blocks	$B\,(8d_{\mathrm{m}}^{2}+5d_{\mathrm{m}})$	$4{,}204{,}544$	$42.99\%$
AdaLN-Zero, all blocks	$B\,(6d_{\mathrm{m}}^{2}+6d_{\mathrm{m}})$	$3{,}158{,}016$	$32.29\%$
Channel attention, all blocks	$B\,(4d_{\mathrm{m}}^{2}+4d_{\mathrm{m}})$	$2{,}105{,}344$	$21.53\%$
Time embedding	$8d_{\mathrm{t}}^{2}+5d_{\mathrm{t}}$	$131{,}712$	$1.35\%$
Output modulation	$2d_{\mathrm{m}}^{2}+2d_{\mathrm{m}}$	$131{,}584$	$1.35\%$
Conditioning projection	$d_{\mathrm{m}}d_{\mathrm{t}}+d_{\mathrm{m}}$	$33{,}024$	$0.34\%$

Appendix

Table 3: Trainable parameters of the default configuration on ETTh1 at

T=24

(

d_{\mathrm{m}}=256

B=8

d_{\mathrm{t}}=128

, MLP ratio

4

J=3

D=T=24

coefficients per channel,

F=7

Dataset	Type	Length	Channels $F$	Source
ETTh1	Electricity transformer	$17{,}420$	$7$	Zhou et al. [2021]
ETTh2	Electricity transformer	$17{,}420$	$7$	Zhou et al. [2021]
Stocks	Daily Google prices	$3{,}685$	$6$	Yoon et al. [2019]
Exchange	Daily exchange rates	$7{,}588$	$8$	Lai et al. [2018]
EEG	14-electrode recording	$14{,}980$	$14$	Roesler [2013]
Energy	Appliance energy	$19{,}735$	$28$	Candanedo et al. [2017]

Appendix

Table 4: Datasets used in our experiments. Length is the number of time steps in the raw series; for MuJoCo,

10{,}000

trajectories of length

T

are simulated.

Model	Sampler	NFE per sample
Diffusion-TS	DDPM / DDIM	100–1000 $\dagger$
SigDiffusion	Tsit5 (128 steps)	768
FourierDiffusion	VP-SDE (1000 steps)	1000
WaveletDiff	DDPM (1000 steps)	1000
FlowTS	Euler (100 steps)	100
Ours	Euler (100 steps)	100

Appendix

Table 5: Network function evaluations (NFE) per generated sample at

T=24

\dagger

Diffusion-TS uses 500 evaluations on ETTh1, ETTh2, Stocks, and Exchange, 100 on EEG, and 1000 on Energy and MuJoCo.

	$a_{3}$		$d_{3}$		$d_{2}$		$d_{1}$
Dataset	$\sigma$	$t$	$\sigma$	$t$	$\sigma$	$t$	$\sigma$	$t$
ETTh1	2.57	0.28	0.90	0.53	0.42	0.70	0.24	0.81
ETTh2	2.76	0.27	0.40	0.71	0.24	0.81	0.17	0.85
Stocks	2.75	0.27	0.30	0.77	0.23	0.81	0.17	0.85
Exchange	2.82	0.26	0.12	0.89	0.07	0.93	0.04	0.96
EEG	1.41	0.41	0.95	0.51	0.93	0.52	0.92	0.52

Appendix

Table 6: Per-level coefficient standard deviation

\sigma

and SNR crossing time

t=1/(1+\sigma)

T=24

, under the db2 wavelet. Coarser levels have larger

\sigma

and therefore cross earlier.

Figure 7: Level-wise SNR along the linear probability path on the other six datasets. Coarse levels cross

\mathrm{SNR}=1

earlier than fine levels.

Context-FID ( $\downarrow$ )
	ETTh1	ETTh2	Stocks	Exchange	EEG	Energy	MuJoCo
Diffusion-TS	0.187 $\pm$ .006	0.073 $\pm$ .004	0.238 $\pm$ .050	0.048 $\pm$ .005	0.024 $\pm$ .002	0.106 $\pm$ .019	0.021 $\pm$ .002
SigDiffusion	2.972 $\pm$ .112	1.413 $\pm$ .181	4.086 $\pm$ .784	1.702 $\pm$ .101	0.032 $\pm$ .004	4.730 $\pm$ .547	3.117 $\pm$ .150
FourierDiffusion	0.039 $\pm$ .005	0.034 $\pm$ .004	0.055 $\pm$ .010	0.058 $\pm$ .009	0.022 $\pm$ .001	0.296 $\pm$ .018	0.099 $\pm$ .012
WaveletDiff	0.059 $\pm$ .005	0.074 $\pm$ .005	0.024 $\pm$ .002	0.007 $\pm$ .001	0.011 $\pm$ .002	0.541 $\pm$ .022	1.269 $\pm$ .149
FlowTS	0.030 $\pm$ .002	0.013 $\pm$ .001	0.019 $\pm$ .003	0.010 $\pm$ .001	0.008 $\pm$ .001	0.061 $\pm$ .004	0.026 $\pm$ .001

Appendix

Table 7: Unconditional generation at

T=32

. Mean

\pm

std over five recomputations of each metric on the same generated set. Bold : best; underlined : second best.

Context-FID ( $\downarrow$ )
	ETTh1	ETTh2	Stocks	Exchange	EEG	Energy	MuJoCo
Diffusion-TS	0.267 $\pm$ .023	0.112 $\pm$ .010	0.354 $\pm$ .080	0.056 $\pm$ .003	0.058 $\pm$ .005	0.085 $\pm$ .007	0.035 $\pm$ .002
SigDiffusion	5.948 $\pm$ .465	1.581 $\pm$ .174	3.851 $\pm$ .744	1.986 $\pm$ .125	0.056 $\pm$ .004	6.403 $\pm$ .271	3.621 $\pm$ .181
FourierDiffusion	0.089 $\pm$ .005	0.068 $\pm$ .005	0.111 $\pm$ .012	0.081 $\pm$ .008	0.045 $\pm$ .005	0.446 $\pm$ .030	0.167 $\pm$ .009
WaveletDiff	0.104 $\pm$ .008	0.072 $\pm$ .003	0.059 $\pm$ .011	0.151 $\pm$ .015	0.025 $\pm$ .003	0.438 $\pm$ .013	0.347 $\pm$ .021
FlowTS	0.052 $\pm$ .003	0.027 $\pm$ .002	0.038 $\pm$ .005	0.016 $\pm$ .001	0.015 $\pm$ .003	0.172 $\pm$ .016	0.093 $\pm$ .005

Appendix

Table 8: Unconditional generation at

T=64

. Same conventions as .

Context-FID ( $\downarrow$ )
	ETTh1	ETTh2	Stocks	Exchange	EEG	Energy	MuJoCo
Diffusion-TS	0.994 $\pm$ .025	0.146 $\pm$ .018	0.611 $\pm$ .068	0.059 $\pm$ .004	0.139 $\pm$ .023	0.074 $\pm$ .008	0.073 $\pm$ .004
SigDiffusion	11.619 $\pm$ .739	2.087 $\pm$ .284	4.743 $\pm$ .381	2.005 $\pm$ .267	0.130 $\pm$ .012	10.471 $\pm$ .272	3.738 $\pm$ .269
FourierDiffusion	0.319 $\pm$ .020	0.200 $\pm$ .013	0.272 $\pm$ .023	0.235 $\pm$ .040	0.086 $\pm$ .014	0.837 $\pm$ .034	0.224 $\pm$ .021
WaveletDiff	0.158 $\pm$ .023	0.088 $\pm$ .006	0.112 $\pm$ .010	0.154 $\pm$ .015	0.059 $\pm$ .005	0.501 $\pm$ .037	0.277 $\pm$ .018
FlowTS	0.097 $\pm$ .003	0.058 $\pm$ .004	0.063 $\pm$ .010	0.035 $\pm$ .004	0.027 $\pm$ .003	0.260 $\pm$ .021	0.156 $\pm$ .009

Appendix

Table 9: Unconditional generation at

T=128

. Same conventions as .

Panel of Figure 8 — Figure 8: Evolution of each metric with the window length $T$ .

Panel of Figure 9 — Figure 9: t-SNE embeddings of real (red) and generated (blue) windows for all methods and datasets at $T=24$ .

Panel of Figure 10 — Figure 10: Probability distributions of real (red) and generated (blue) data for all methods and datasets at $T=24$ .

Context-FID ( $\downarrow$ )
Variant	ETTh1	ETTh2	Stocks	Exchange	EEG	Energy	MuJoCo
(a) Without $e_{f}$	1.488 $\pm$ 0.099	0.977 $\pm$ 0.168	1.763 $\pm$ 0.321	0.938 $\pm$ 0.072	0.011 $\pm$ 0.001	2.617 $\pm$ 0.145	1.390 $\pm$ 0.139
(b) Std. levels	0.021 $\pm$ 0.002	0.014 $\pm$ 0.001	0.018 $\pm$ 0.004	0.010 $\pm$ 0.001	0.010 $\pm$ 0.001	0.019 $\pm$ 0.002	0.008 $\pm$ 0.000
(c) Time domain	0.006 $\pm$ 0.001	0.005 $\pm$ 0.001	0.014 $\pm$ 0.001	0.007 $\pm$ 0.000	0.014 $\pm$ 0.001	0.012 $\pm$ 0.000	0.005 $\pm$ 0.000
(d) Uniform $t$	0.007 $\pm$ 0.000	0.008 $\pm$ 0.001	0.013 $\pm$ 0.001	0.005 $\pm$ 0.000	0.014 $\pm$ 0.002	0.016 $\pm$ 0.001	0.010 $\pm$ 0.001
Full model	0.005 $\pm$ 0.001	0.004 $\pm$ 0.000	0.005 $\pm$ 0.002	0.004 $\pm$ 0.001	0.013 $\pm$ 0.002	0.010 $\pm$ 0.003	0.006 $\pm$ 0.000

Appendix

Table 10: One-factor-at-a-time ablations at

T=24

. The full model uses db2 coefficients, natural level scaling, logit-normal time sampling, and channel-identity embeddings

e_{f}

. Variants (a)–(d) respectively remove

e_{f}

, standardize the wavelet levels, replace the wavelet transform with the identity, and sample

t

uniformly. Results are mean

\pm

standard deviation; Best values are in bold and second-best values are underlined. The shaded rows show the full model.

Context-FID ( $\downarrow$ )
	Time shift $\alpha$
Dataset	$0.33$	$0.5$	$1.0^{\dagger}$	$2.0$	$3.0$	$5.0$
ETTh1	0.007 $\pm$ 0.000	0.008 $\pm$ 0.000	0.005 $\pm$ 0.001	0.011 $\pm$ 0.001	0.010 $\pm$ 0.000	0.021 $\pm$ 0.001
ETTh2	0.008 $\pm$ 0.000	0.005 $\pm$ 0.000	0.004 $\pm$ 0.000	0.004 $\pm$ 0.000	0.005 $\pm$ 0.000	0.007 $\pm$ 0.000
Stocks	0.008 $\pm$ 0.001	0.011 $\pm$ 0.001	0.005 $\pm$ 0.002	0.006 $\pm$ 0.001	0.010 $\pm$ 0.001	0.023 $\pm$ 0.003
Exchange	0.005 $\pm$ 0.001	0.005 $\pm$ 0.000	0.004 $\pm$ 0.001	0.003 $\pm$ 0.000	0.007 $\pm$ 0.000	0.008 $\pm$ 0.001

Appendix

Table 11: Effect of the sampling time shift

\alpha

T=24

, with

N=100

Euler steps, all from the same trained model. † Default setting. Results are mean

\pm

std; the best value in each row is in bold.

Context-FID ( $\downarrow$ )
	Euler steps $N$
Dataset	$25$	$50$	$100^{\dagger}$	$200$	$400$
ETTh1	0.019 $\pm$ 0.002	0.011 $\pm$ 0.001	0.005 $\pm$ 0.001	0.007 $\pm$ 0.000	0.008 $\pm$ 0.001
ETTh2	0.009 $\pm$ 0.001	0.005 $\pm$ 0.001	0.004 $\pm$ 0.000	0.005 $\pm$ 0.000	0.009 $\pm$ 0.000
Stocks	0.021 $\pm$ 0.003	0.004 $\pm$ 0.001	0.005 $\pm$ 0.002	0.003 $\pm$ 0.000	0.010 $\pm$ 0.001
Exchange	0.009 $\pm$ 0.000	0.003 $\pm$ 0.000	0.004 $\pm$ 0.001	0.005 $\pm$ 0.000	0.006 $\pm$ 0.000

Appendix

Table 12: Effect of the number of Euler steps

N

T=24

, with time shift

\alpha=1

, all from the same trained model. † Default setting. Results are mean

\pm

std; the best value in each row is in bold.

Context-FID ( $\downarrow$ )
Transform	ETTh1	ETTh2	Stocks	Exchange	EEG	Energy	MuJoCo
db2	0.005 $\pm$ 0.001	0.004 $\pm$ 0.000	0.005 $\pm$ 0.002	0.004 $\pm$ 0.001	0.013 $\pm$ 0.002	0.010 $\pm$ 0.003	0.006 $\pm$ 0.000
coif1	0.005 $\pm$ 0.000	0.003 $\pm$ 0.000	0.009 $\pm$ 0.001	0.009 $\pm$ 0.001	0.014 $\pm$ 0.001	0.008 $\pm$ 0.000	0.007 $\pm$ 0.001
bior2.2	0.006 $\pm$ 0.000	0.005 $\pm$ 0.000	0.006 $\pm$ 0.000	0.004 $\pm$ 0.001	0.014 $\pm$ 0.001	0.013 $\pm$ 0.001	0.006 $\pm$ 0.001
rbio2.2	0.004 $\pm$ 0.000	0.005 $\pm$ 0.000	0.019 $\pm$ 0.001	0.006 $\pm$ 0.001	0.010 $\pm$ 0.002	0.013 $\pm$ 0.001	0.006 $\pm$ 0.000
Learned	0.004 $\pm$ 0.000	0.003 $\pm$ 0.000	0.008 $\pm$ 0.001	0.008 $\pm$ 0.000	0.011 $\pm$ 0.002	0.009 $\pm$ 0.001	0.005 $\pm$ 0.000

Appendix

Table 13: Effect of the wavelet transform at

T=24

. Results are mean

\pm

std; the best transform for each dataset and metric is in bold.

Figure 11: Deviation of the lattice angles from their db2 initialization during training at $T=24$ .

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Vector quantization (VQ) with autoregressive (AR) token modeling is a widely adopted and highly competitive paradigm for time-series generation. However, such models are fundamentally limited by exposure bias: during inference, errors can accumulate across sequential predictions, leading to pronounced quality degradation in long-horizon generation. To address this, we propose SDFlow (

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