ProtoFlow: Prototype-Guided Flow Matching for Multivariate Time Series Forecasting

Shibo Feng; Wanjin Feng; Yang Qiu; Deheng Ye; Peilin Zhao; Chunyan Miao

cs.AIOct 1, 2026

ProtoFlow: Prototype-Guided Flow Matching for Multivariate Time Series Forecasting

Authors: Shibo Feng, Wanjin Feng, Yang Qiu, Deheng Ye, Peilin Zhao, Chunyan Miao

Organizations: Nanyang Technological University · Tsinghua University · Shanghai Jiao Tong University

Abstract

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.

Figures & tables

Figure 1: Illustration of our ProtoFlow. Stage 1 discretizes the future target sequence into compact latent tokens via similarity-based VQ latent modeling. Stage 2 compares three latent-generation paradigms under the same tokenized space: (a) Autoregressive rollout, which suffers from training–inference mismatch; (b) Gaussian-initialized flow matching, which starts transport from an unstructured prior; and (c) The proposed ProtoFlow, which initializes conditional rectified flow from a learned codebook prior for latent flow matching generation. The future time features $\text{y}^{c}$ is omitted here.

	Electricity		Solar		Traffic		Taxi		Wikipedia
Horizon+Method	$\text{CRPS}_{\text{sum}}$	$\text{NRMSE}_{\text{sum}}$	$\text{CRPS}_{\text{sum}}$	$\text{NRMSE}_{\text{sum}}$	$\text{CRPS}_{\text{sum}}$	$\text{NRMSE}_{\text{sum}}$	$\text{CRPS}_{\text{sum}}$	$\text{NRMSE}_{\text{sum}}$	$\text{CRPS}_{\text{sum}}$	$\text{NRMSE}_{\text{sum}}$
48 Steps
$\bullet~$ TimeGrad ( 2021 )	0.031	0.044	0.324	0.736	0.043	0.072	0.252	0.334	0.082	0.105
$\bullet~$ CSDI ( 2021 )	0.029	0.050	0.328	0.742	-	-	-	-	-	-
$\bullet~$ TSDiff ( 2023 )	0.036	0.057	0.324	0.731	0.057	0.070	0.243	0.330	0.079	0.091
$\bullet~$ MG-TSD ( 2024 )	0.024	0.038	0.328	0.737	0.042	0.067	0.217	0.327	0.071	0.096

Table 1: The detailed results of ProtoFlow on the 48 and 96 prediction horizons. ProtoFlow-G: the variant of ProtoFlow that uses a standard Gaussian prior. - marks out-of-memory failures, Best and second-best forecasting results are bolded and underlined, respectively.

Method	Electricity		Solar		Traffic		Taxi		Wikipedia		Avg.
Method	C-S	NM-S	C-S	NM-S	C-S	NM-S	C-S	NM-S	C-S	NM-S	C-S	NM-S
VAE + AR	0.421	0.544	1.154	2.618	1.005	1.511	0.337	0.448	0.130	0.169	0.609	1.058
VAE + FM	0.129	0.213	0.643	1.324	0.323	0.564	0.227	0.293	0.117	0.128	0.288	0.504
VQ + AR	0.037	0.054	0.347	0.833	0.066	0.084	0.205	0.322	0.108	0.121	0.153	0.283
ProtoFlow-G	0.019	0.037	0.305	0.717	0.036	0.058	0.159	0.287	0.096	0.114	0.123	0.243
ProtoFlow	0.015	0.022	0.284	0.638	0.034	0.054	0.157	0.281	0.092	0.101	0.116	0.219

Table 2: The comparison results of 96 prediction horizons with baselines regarding

\text{CRPS}_{\text{sum}}

(C-S) and

\text{NRMSE}_{\text{sum}}

(NM-S). AR: Auto-regressive Transformer and FM: Flow Matching .

Figure 2: Comparison of training convergence ( a ), inference efficiency, training memory ( b ), and forecasting performance ( c-d ) on Electricity, Traffic, and Solar with a prediction length of 96.

Method	Electricity		Solar		Traffic		Taxi		Wikipedia		Avg.
Method	C-S	NM-S	C-S	NM-S	C-S	NM-S	C-S	NM-S	C-S	NM-S	C-S	NM-S
ProtoFlow-G	0.019	0.037	0.305	0.717	0.036	0.058	0.159	0.287	0.096	0.114	0.123	0.243
ProtoFlow-uniform	0.018	0.030	0.356	0.769	0.036	0.056	0.157	0.284	0.099	0.108	0.133	0.249
ProtoFlow	0.015	0.022	0.284	0.638	0.034	0.054	0.157	0.281	0.092	0.101	0.116	0.219

Table 3: The forecasting performance with frequency-weighted prototype (ProtoFlow), uniform-weight prototype (ProtoFlow-uniform), and isotropic Gaussian priors (ProtoFlow-G) at a prediction length of 96.

Figure 3: Effects of codebook size on ProtoFlow at the 96-step prediction length.

	Electricity			Solar			Traffic
	Frequency	Uniform	Stage1 MSE	Frequency	Uniform	Stage1 MSE	Frequency	Uniform	Stage1 MSE
Coodebook size
64	0.039	0.044	7.76e-4	1.035	0.962	7.70e-4	0.113	0.105	5.82e-4
128	0.030	0.032	7.13e-4	0.774	0.761	6.75e-4	0.062	0.066	5.23e-4
256	0.022	0.030	6.02e-4	0.638	0.769	6.13e-4	0.054	0.056	4.78e-4
512	0.031	0.033	6.01e-4	0.993	0.758	6.18e-4	0.056	0.057	4.69e-4

Table 4: Effects of codebook size on Stage 1 reconstruction MSE and

\text{NRMSE}_{\text{sum}}

under frequency and uniform weighting at a 96-step prediction length.

Figure 4: Effect of ODE sampling steps on 96-step probabilistic forecasting performance and

W_{2}

transport distance for ProtoFlow and ProtoFlow-G.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

DATASET	Dimension	Domain	Freq	Total Time Steps	Pred Length	Rolling Windows
SOLAR	137	$\mathbb{R}^{+}$	Hourly	7,009	$\{48,96\}$	7
ELECTRICITY	370	$\mathbb{R}^{+}$	Hourly	5,790	$\{48,96\}$	7
TRAFFIC	963	(0,1)	Hourly	10,413	$\{48,96\}$	7
TAXI	1214	$\mathbb{N}$	30-Min	1,488	$\{48,96\}$	57
WIKIPEDIA	2000	$\mathbb{N}$	Daily	792	$\{48,96\}$	5

Appendix

Table 5: The dataset properties used in our experiments are summarized as follows.

Datasets	Codebook Size	Codebook dim	Enc/Dec Layers	$\lambda$	$\beta$
Solar	256	128	2	0.99	0.1
Electricity	256	256	2	0.99	0.1
Traffic	256	256	2	0.99	0.1
Taxi	256	512	2	0.99	0.05
Wikipedia	256	512	2	0.99	0.05

Appendix

Table 6: Detailed hyperparameters of Stage 1.

Datasets	Hidden dim	$\text{DiT}_{\theta}$ layers	Batch size	Steps	lr
Solar	128	4	64	3	5e-4
Electricity	256	4	64	3	5e-4
Traffic	256	5	64	3	3e-4
Taxi	512	5	64	3	3e-4
Wikipedia	512	4	64	5	3e-4

Appendix

Table 7: Detailed hyperparameters of Stage 2.

Layer	Function	Descriptions
1	Convolution	input channel= $d$ , output channel=D, kernel size=3, stride=1, padding=1
2	ReLU	nn.ReLU()
3	Convolution	input channel=D, output channel=D, kernel size=4, stride=2, padding=1
4	ResNet	input channel=D, depth=3, dilation growth rate=3
5	ReLU	nn.ReLU()
6	Convolution	input channel=D, output channel=D, kernel size=4, stride=2, padding=1

Appendix

Table 8: The detailed architecture of the time serie tokenizer’s encoder.

Layer	Function	Descriptions
1	Convolution	input channel= $d_{c}$ , output channel=D, kernel size=3, stride=1, padding=1
2	ReLU	nn.ReLU()
3	ResNet	input channel=D, depth=3, dilation growth rate=3
4	ReLU	nn.ReLU()
5	Upsample	nn.Upsample()
6	Convolution	input channel=D, output channel=D, kernel size=3, stride=1, padding=1

Appendix

Table 9: The detailed architecture of the time serie tokenizer’s decoder.

Figure 5: Illustration of model architecture of $\text{DiT}_{\theta}$ of Stage2.

\mathcal{L}_{\mathrm{FM}}\leftarrow\frac{1}{|\mathcal{B}|}\sum_{b\in\mathcal{B}}\left\|\mathbf{v}_{\theta}(\mathbf{x}_{s,b}^{t_{b}},t_{b}\mid\mathbf{c}_{b})-(\mathbf{x}_{s,b}^{1}-\mathbf{x}_{s,b}^{0})\right\|_{F}^{2}

Appendix

Algorithm 2 ProtoFlow Stage 2: Codebook-Prior Flow Matching

Figure 6: ProtoFlow forecasts with a prediction horizon of 96. Rows show Electricity (370 dimensions), Traffic (963 dimensions), Solar (137 dimensions), Taxi (1214 dimensions), and Wikipedia (2000 dimensions) from top to bottom, with four examples per dataset.

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Open paper AskPDF arXiv

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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imilarity-

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riven

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Matching), a non-autoregressive framework that operates entirely in the frozen VQ latent space and enables parallel sequence generation via flow matching. We tackle three key challenges in making this transition: (1) eliminating exposure bias by replacing step-wise token prediction with a global transport map; (2) mitigating the high-dimensionality of VQ token spaces via a low-rank manifold decomposition with a learned anchor prior over the latent manifold; and (3) incorporating discrete supervision into continuous transport dynamics by introducing a categorical posterior over codebook indices within a variational flow-matching formulation. Extensive experiments show that SDFlow achieves state-of-the-art performance, improving Discriminative Score and substantially reducing Context-FID, particularly for challenging long-sequence generation. Moreover, SDFlow provides significant inference speedups over autoregressive baselines, offering both high fidelity and computational efficiency. Code is available at https://anonymous.4open.science/r/SDFlow-D6F3/

Abstract

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