HCFM: Helmholtz Component Flow Matching for Time Series Anomaly Detection

Shyam Sundar Saravanan; Tie Luo; Sanjay Madria

cs.LGOct 18, 2026

HCFM: Helmholtz Component Flow Matching for Time Series Anomaly Detection

Authors: Shyam Sundar Saravanan, Tie Luo, Sanjay Madria

Organizations: Department of Computer Science, Missouri University of Science and Technology, Rolla, MO, USA · Department of ECE, University of Kentucky, Lexington, KY, USA

Abstract

Flow Matching (FM) is an efficient simulation-free approach for training flow-based generative models, making it a promising foundation for scalable time-series anomaly detection (TSAD). However, standard FM learns a monolithic vector field in which nominal transport and local density-changing behavior are entangled. This creates a central challenge for TSAD; local expansion and contraction carry crucial anomaly-sensitive information, yet isolating this signal requires expensive Jacobian-trace estimation over the entire field. We propose Helmholtz Component Flow Matching (HCFM), an unsupervised TSAD framework that parameterizes the vector field prediction network with separate transport, compression, and residual heads. The transport head is constrained to be divergence-free, so divergence estimation is restricted to the compression head, while a lightweight residual head absorbs the remaining mismatch. This structured parameterization improves scalability and detection performance by separating volume-preserving nominal transport from density-changing compression, yielding cleaner anomaly signals from distinct deviations of nominal behavior. Experiments on large-scale univariate and multivariate TSAD benchmarks show that HCFM consistently improves detection accuracy while maintaining efficient inference compared with 16 baselines. The code is available at https://github.com/TrustAIoT/HCFM__time-series-anomaly/

Figures & tables

Fig. 1: Motivating illustration of HCFM. (a) In standard Flow Matching, a monolithic predicted field entangles nominal transport with local density-changing behavior, producing an entangled divergence signal that bends otherwise clean transport paths. (b) The transport head in HCFM is constrained to be divergence-free, encouraging volume-preserving nominal movement without local expansion or contraction. (c) HCFM predicts a single final vector field through additive transport, compression, and residual heads,

v_{\theta}=v_{T}+v_{C}+v_{R}

. This structured parameterization preserves a clean transport backbone while localizing density-changing behavior to the compression component and using the residual head to absorb residual mismatch.

Fig. 2: Architecture of Helmholtz Component Flow Matching (HCFM). HCFM trains a data-space Flow Matching model over interpolated time-series windows and replaces a monolithic vector-field predictor with additive transport, compression, and residual heads. The transport head captures divergence-free nominal movement, the compression head provides localized density-changing evidence through component-specific divergence, and the residual head absorbs remaining mismatch while being kept lightweight. Training supervises the combined final field and uses residual-energy and orthogonality regularization to shape the internal component allocation. At inference, HCFM combines base-energy and compression-divergence for anomaly detection and localization.

Fig. 3: Parameter and divergence-trace target allocation for HCFM. (a) HCFM is matched to a monolithic predictor with the same total parameter budget. Most capacity is assigned to transport and compression, while the residual head remains lightweight. (b) HCFM differentiates only the compression component for divergence scoring, reducing the traced target to $0.376\times$ of the full-field monolithic target in this configuration.

Operation	Kernel / Stride	Units / Shape	Activation
Shared Input
Input $x_{t}$	–	$B\times C\times W$	–
Timestep embedding	–	$d_{\mathrm{emb}}$	SiLU
Transport Head
Low-rank skew transport	–	$v_{T}(x_{t},t)\in\mathbb{R}^{C\times W}$	–
Compression Head

TABLE I: HCFM structured vector-field network and training and inference hyperparameters.

Property	Dataset
Property	UCR	SMAP	MSL	GECCO	SMD	PSM	SWAN
# Sub-datasets	250	55	27	1	28	1	1
# Dimensions ( $C$ )	1	25	55	9	38	25	38
# Train size	5,302,449	140,825	58,317	69,260	708,405	132,481	60,000
# Test size	14,051,317	444,035	73,729	69,261	708,420	87,841	60,000
AR (%)	0.225	13.13	10.72	1.05	4.16	27.76	32.60

TABLE II: Statistics of the datasets used in our experiments. AR denotes anomaly ratio.

Model	Best F1 $\uparrow$	Precision $\uparrow$	VUS-PR $\uparrow$	AUROC $\uparrow$	AUPRC/AP $\uparrow$	FP@95R $\downarrow$	Hundman F1 $\uparrow$
TadGAN [ 12 ]	0.131	0.092	0.332	0.641	0.118	0.617	0.154
DENSE-AE [ 36 ]	0.159	0.136	0.352	0.662	0.147	0.584	0.181
LSTM-DT [ 36 ]	0.320	0.289	0.442	0.731	0.306	0.462	0.342
DONUT [ 30 ]	0.351	0.325	0.458	0.746	0.337	0.439	0.371
LSTM-AE [ 36 ]	0.355	0.301	0.472	0.754	0.351	0.427	0.384
MERLIN [ 31 ]	0.414	0.402	0.503	0.781	0.419	0.386	0.432

TABLE III: Aggregate UCR results over 250 datasets.

Method	GECCO		SWAN		MSL		SMAP		PSM		SMD
Method	VUS-PR $\uparrow$	Best F1 $\uparrow$	VUS-PR $\uparrow$	Best F1 $\uparrow$	VUS-PR $\uparrow$	Best F1 $\uparrow$	VUS-PR $\uparrow$	Best F1 $\uparrow$	VUS-PR $\uparrow$	Best F1 $\uparrow$	VUS-PR $\uparrow$	Best F1 $\uparrow$
OmniAnomaly	0.612	0.661	0.746	0.790	0.628	0.678	0.578	0.635	0.703	0.742	0.677	0.725
LSTM-AE	0.637	0.684	0.768	0.813	0.650	0.700	0.606	0.662	0.724	0.761	0.693	0.740
DCDetector	0.681	0.728	0.805	0.846	0.698	0.745	0.648	0.701	0.756	0.789	0.733	0.780
USAD	0.701	0.746	0.825	0.859	0.720	0.761	0.678	0.725	0.772	0.804	0.748	0.792
TranAD	0.724	0.768	0.839	0.878	0.749	0.789	0.698	0.748	0.791	0.821	0.784	0.808

TABLE IV: Multivariate TSAD results.

Method	AUROC $\uparrow$	AUPRC/AP $\uparrow$	FP@95R $\downarrow$	Hundman F1 $\uparrow$
OmniAnomaly	0.781	0.521	0.774	0.705
LSTM-AE	0.796	0.548	0.751	0.727
DCDetector	0.823	0.592	0.704	0.782
USAD	0.831	0.611	0.681	0.798
TranAD	0.846	0.637	0.646	0.821
CARLA	0.858	0.653	0.623	0.835

TABLE V: Aggregate multivariate TSAD results across GECCO, SWAN, MSL, SMAP, PSM, and SMD.

Method	Training time (min)	Latency (ms/window)	Rel. time
LSTM-AE	8.6	1.74	$0.44\times$
Vanilla FM	14.2	2.05	$0.52\times$
HCFM	22.8	3.92	$\mathbf{1.00\times}$
FDM	31.5	6.03	$1.54\times$
TranAD	24.7	7.64	$1.95\times$
SGFM	36.9	8.13	$2.07\times$

TABLE VI: Training and inference-time comparison against representative TSAD baselines. Training time is reported as wall-clock minutes, while latency is reported as average milliseconds per window. Relative inference time is normalized to HCFM.

Channels $C$	Training time (min)	Latency (ms/window)
1	22.80 ( $1.00\times$ )	3.92 ( $1.00\times$ )
8	23.92 ( $1.05\times$ )	3.98 ( $1.01\times$ )
16	26.52 ( $1.16\times$ )	4.03 ( $1.03\times$ )
32	34.06 ( $1.49\times$ )	3.98 ( $1.01\times$ )
64	49.04 ( $2.15\times$ )	4.06 ( $1.04\times$ )

TABLE VII: HCFM dimensional scalability with

W=64

. Values in parentheses are relative to

C=1

Fig. 4: Qualitative examples illustrating complementary anomaly evidence in HCFM. The shaded region denotes the ground-truth anomaly interval. Each column shows the raw test signal, reverse-flow base energy, and compression-divergence score. (a) On UCR_45, base energy provides weak anomaly separation, whereas compression divergence sharply localizes the anomalous interval. (b) On UCR_21, the anomaly is more clearly captured by base energy than by compression divergence. Together, these examples motivate combining endpoint mismatch evidence with localized compression-divergence evidence.

Scoring method	Divergence target	AUPRC	FP@95R	VUS-PR	Best F1	Rel. time	Rel. memory
Vanilla FM: base energy	None	0.566	0.270	0.589	0.562	$1.00\times$	$1.00\times$
Vanilla FM: base $+$ full divergence	Full field $v_{\theta}$	0.578	0.260	0.598	0.574	$2.71\times$	$2.18\times$
FDM: base $+$ full divergence	Full field $v_{\theta}$	0.594	0.248	0.606	0.588	$2.94\times$	$2.31\times$
HCFM: base only	None	0.682	0.206	0.651	0.671	$1.18\times$	$1.08\times$
HCFM: compression divergence only	Compression head $v_{C}$	0.735	0.180	0.684	0.724	$1.86\times$	$1.42\times$
HCFM: base $+$ compression divergence	Compression head $v_{C}$	0.757	0.171	0.694	0.742	$1.91\times$	$1.46\times$

TABLE VIII: Ablation of divergence source and scoring efficiency. Runtime and memory are reported relative to Jacobian-free vanilla FM base-energy scoring.

Variant	Field parameterization	Score used	AUPRC	FP@95R	VUS-PR	Best F1	Hundman F1
Vanilla FM	$v_{\theta}$	Base energy	0.566	0.270	0.589	0.562	0.586
Vanilla FM $+$ divergence	$v_{\theta}$	Base $+\nabla\!\cdot v_{\theta}$	0.578	0.260	0.598	0.574	0.598
No transport head	$v_{C}+\gamma_{R}v_{R}$	Base $+\nabla\!\cdot v_{C}$	0.724	0.185	0.676	0.713	0.743
No compression head	$v_{T}+\gamma_{R}v_{R}$	Base energy	0.647	0.226	0.632	0.636	0.665
No residual head	$v_{T}+v_{C}$	Base $+\nabla\!\cdot v_{C}$	0.743	0.178	0.686	0.728	0.759
Full HCFM	$v_{T}+v_{C}+\gamma_{R}v_{R}$	Base $+\nabla\!\cdot v_{C}$	0.757	0.171	0.694	0.742	0.776

TABLE IX: Ablation of the HCFM structured field decomposition. All variants use matched preprocessing, training budget, and reverse-integration settings.

$\gamma_{R}$	$\lambda_{R}$	AUPRC	FP@95R $\downarrow$	Best F1	$\rho_{\mathrm{div}}\downarrow$
$1.00$	$0$	0.686	0.232	0.668	1.721
$1.00$	$5{\times}10^{-4}$	0.715	0.207	0.697	1.084
$0.50$	$0$	0.723	0.199	0.705	0.923
$0.50$	$5{\times}10^{-4}$	0.741	0.181	0.727	0.514
$0.25$	$0$	0.731	0.193	0.713	0.682
$0.25$	$5{\times}10^{-4}$	0.757	0.171	0.742	0.287

TABLE X: Residual-scale and residual-regularization ablation for HCFM. All variants use the same transport and compression heads with

\lambda_{\mathrm{ortho}}=10

. Lower

\rho_{\mathrm{div}}

indicates less residual-head absorption of density-changing behavior.

$\lambda_{\mathrm{ortho}}$	Best F1	AUPRC	AUROC	VUS-PR	FP@95R $\downarrow$
$0$	0.704	0.721	0.912	0.671	0.194
$1$	0.724	0.739	0.917	0.684	0.183
$2$	0.732	0.748	0.920	0.689	0.177
$10$	0.742	0.757	0.922	0.694	0.171
$20$	0.736	0.751	0.919	0.690	0.174

TABLE XI: Orthogonality-weight ablation for HCFM.

Time-Series Anomaly Detection

Time-Series Anomaly Detection identifies unusual patterns or events in sequential data, from sensor readings and network traffic to health and industrial telemetry.

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VAN-AD: Visual Masked Autoencoder with Normalizing Flow For Time Series Anomaly Detection

PengYu Chen, Shang Wan, Xiaohou Shi, Yuan Chang+2

School of Computer Science (National Pilot Software Engineering School), Beijing University of Posts and Telecommunications, Beijing 100876, China · China Telecom Research Institute Beijing, China · Department of Computer Science, Missouri University of Science and Technology, Rolla, MO 65409 USA

Masked Autoencoders Vision Foundation Model Adaptation Reconstruction-Based Anomaly Detection Time-Series Anomaly Detection

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Time series anomaly detection (TSAD) is essential for maintaining the reliability and security of IoT-enabled service systems. Existing methods require training one specific model for each dataset, which exhibits limited generalization capability across different target datasets, hindering anomaly detection performance in various scenarios with scarce training data. To address this limitation, foundation models have emerged as a promising direction. However, existing approaches either repurpose large language models (LLMs) or construct largescale time series datasets to develop general anomaly detection foundation models, and still face challenges caused by severe cross-modal gaps or in-domain heterogeneity. In this paper, we investigate the applicability of large-scale vision models to TSAD. Specifically, we adapt a visual Masked Autoencoder (MAE) pretrained on ImageNet to the TSAD task. However, directly transferring MAE to TSAD introduces two key challenges: overgeneralization and limited local perception. To address these challenges, we propose VAN-AD, a novel MAE-based framework for TSAD. To alleviate the over-generalization issue, we design an Adaptive Distribution Mapping Module (ADMM), which maps the reconstruction results before and after MAE into a unified statistical space to amplify discrepancies caused by abnormal patterns. To overcome the limitation of local perception, we further develop a Normalizing Flow Module (NFM), which combines MAE with normalizing flow to estimate the probability density of the current window under the global distribution. Extensive experiments on nine real-world datasets demonstrate that VAN-AD consistently outperforms existing state-of-the-art methods across multiple evaluation metrics.We make our code and datasets available at https://github.com/PenyChen/VAN-AD.

Abstract

Time-Series Anomaly Detection

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