Title: PhysDiff: A Physically-Guided Diffusion Model for Multivariate Time Series Anomaly Detection
Venue: NeurIPS 2025
Accepted: yes
Keywords: Time Series Anomaly Detection

Physically-Informed Diffusion

​​Physics-Guided Decomposition

Non-stationary Dynamics​
Abstract: Unsupervised anomaly detection of multivariate time series remains challenging in complex nonstationary dynamics, due to the high false-positive rates and limited interpretability. We propose PhysDiff, combining physics-guided decomposition with diffusion-based reconstruction, to address these issues. The physics-guided signal decomposition is introduced to disentangle overlapping dynamics by isolating high frequency oscillations and low frequency trends, which can reduce interference and provide meaningful physical priors. The reconstruction through conditional diffusion modeling captures deviations from learned normal behavior, making anomalies more distinguishable. Notably, PhysDiff introduces an amplitude-sensitive permutation entropy criterion to adaptively determine the optimal decomposition depth, and automatically extract adaptive frequency components used as explicit physics-based constraints for the diffusion process. Furthermore, the proposed conditional diffusion network employs a dual-path conditioning mechanism that integrates high-frequency and low-frequency physical priors, dynamically regulating the denoising process via a novel time frequency energy routing mechanism. By weighting reconstruction errors across frequency bands, our method improves anomaly localization and enhances interpretability. Extensive experiments on five benchmark datasets and two NeurIPS-TS scenarios demonstrate that PhysDiff outperforms 18 state-of-the-art baselines, with average F1-score improvements on both standard and challenging datasets. Experimental results validate the advantages of combining principled signal decomposition with diffusion-based reconstruction for robust, interpretable anomaly detection in complex dynamic systems.

PhysDiff: A Physically-Guided Diffusion Model for Multivariate Time Series Anomaly Detection

Abstract

Unsupervised anomaly detection of multivariate time series remains challenging in complex non-stationary dynamics, due to the high false-positive rates and limited interpretability. We propose PhysDiff, combining physics-guided decomposition with diffusion-based reconstruction, to address these issues. The physics-guided signal decomposition is introduced to disentangle overlapping dynamics by isolating high frequency oscillations and low frequency trends, which can reduce interference and provide meaningful physical priors. The reconstruction through conditional diffusion modeling captures deviations from learned normal behavior, making anomalies more distinguishable. Notably, PhysDiff introduces an amplitude-sensitive permutation entropy criterion to adaptively determine the optimal decomposition depth, and automatically extract adaptive frequency components used as explicit physics-based constraints for the diffusion process. Furthermore, the proposed conditional diffusion network employs a dual-path conditioning mechanism that integrates high-frequency and low-frequency physical priors, dynamically regulating the denoising process via a novel time frequency energy routing mechanism. By weighting reconstruction errors across frequency bands, our method improves anomaly localization and enhances interpretability. Extensive experiments on five benchmark datasets and two NeurIPS-TS scenarios demonstrate that PhysDiff outperforms 18 state-of-the-art baselines, with average F1 score improvements on both standard and challenging datasets. Experimental results validate the advantages of combining principled signal decomposition with diffusion-based reconstruction for robust, interpretable anomaly detection in complex dynamic systems.

1 Introduction

Multivariate time series anomaly detection is a critical task across various industries, including industrial manufacturing, financial risk management, and healthcare. Anomalies in time series data are typically categorized into point anomalies and pattern anomalies [1]. Point anomalies can be further divided into contextual and global anomalies, while pattern anomalies are classified into seasonal, shapelet, and trend anomalies based on behavior-driven taxonomies [1]. Reconstructionbased methods have demonstrated promising performance in detecting point anomalies, as these anomalies manifest as individual data points exhibiting significant deviations from the expected probability distribution [2]. However, pattern anomalies, which often consist of subtle structural changes over longer periods and remain within the range of normal values, pose a greater detection

challenge. In real-world scenarios, time series data is predominantly non-stationary, and genuine anomalies frequently manifest as pattern anomalies. Although considerable progress has been made, most existing research focuses primarily on stationary datasets. As a result, when point anomalies are obscured, the effectiveness of conventional methods declines significantly.

Diffusion models have emerged as a powerful approach for time series anomaly detection due to their unique ability to model complex data distributions through a progressive denoising process. Unlike traditional reconstruction methods based on simple encoder-decoder architectures, diffusion models demonstrate a stronger ability to model complex patterns and temporal dependencies in non-stationary time series data. This advantage stems from their iterative noise-adding and denoising procedures [3]. This progressive approach makes diffusion models particularly effective at reconstructing normal patterns while amplifying the reconstruction error for anomalous segments. Furthermore, the probabilistic nature of diffusion models enables them to quantify uncertainty in predictions, providing valuable information for distinguishing between normal variations and genuine anomalies. By learning the distribution of normal time series behaviors, diffusion models can generate counterfactual normal versions of input sequences, making them ideal for detecting both point anomalies and subtle pattern anomalies that conventional methods often miss in non-stationary environments. To address non-stationarity problem, conditional diffusion models enhance generation capabilities through contextual information integration. These models rely on conditional decomposition to identify dynamic changes between signals and use constrained reconstruction to regulate the denoising process accordingly.

Nevertheless, two challenges remain in applying diffusion models to time series anomaly detection. Challenge 1: When extracting features through conditional decomposition, transforming time series to the frequency domain makes pattern anomalies easier to detect [2], but existing frequency domain decomposition methods rely on manually preset decomposition levels or energy thresholds to determine termination conditions [4]. Such static criteria are ill-suited for complex, real-world non-stationary data, where adaptive decomposition is necessary. Challenge 2: In the process of constrained sequence reconstruction, the reasonable use of input conditions for denoising and enhancing model interpretability is rarely mentioned. The original sequence, when decomposed, produces multiple detail coefficient sequences [5], representing signal information at various frequency levels. High frequency components typically capture transient, non-stationary changes with high uncertainty and entropy, while low frequency components reflect more stable trends with lower entropy. Overemphasis on low frequency information can obscure anomalies, whereas excessive incorporation of high frequency details risks model overfitting. These challenges highlight the need for a more dynamic and physically-informed approach to decomposition and reconstruction.

Inspired by the concept of entropy in physics, we propose PhysDiff, a physically-guided diffusion model for multivariate time series anomaly detection that combines permutation entropy with an amplitude-sensitive weighting scheme based on actual magnitude variations [6]. For Challenge 1, we dynamically adjust the decomposition depth during frequency-domain analysis by leveraging entropy values that reflect the intrinsic physical characteristics of the data. Specifically, signal complexity is assessed using an amplitude-weighted entropy measure, which effectively captures both the average magnitude and the variability of the sequence. By monitoring the entropy of residual components after each decomposition layer, the stopping condition becomes data-driven rather than heuristic. For Challenge 2, we propose an innovative reconstruction strategy based on a conditional diffusion model, in which amplitude-weighted entropy acts as a dynamic constraint during the generative reconstruction process. This approach leverages the controllability of conditional diffusion models to dynamically regulate the reconstruction of each frequency component.

The main contributions of this article are as follows:

We propose a physically-guided diffusion model that effectively addresses non-stationarity challenges.

We introduce an amplitude-sensitive permutation entropy guided decomposition mechanism that dynamically determines optimal decomposition depth.

We develop a dual-path conditional diffusion framework with a novel frequency-based routing attention mechanism.

Extensive experimental validation demonstrates that the method surpasses 18 state-of-the-art baselines, with improved average F1 score on both standard benchmarks and challenging NeurIPS-TS datasets.

2 Related Work

Anomaly detection of multivariate time series, particularly on large-scale non-stationary datasets, remains a significant and widely recognized challenge. According to detection strategies, existing approaches can be categorized into forecasting-based, reconstruction-based, distance-based, encodingbased, distribution-based, and tree-based ones. They offer distinct advantages but also face various challenges when addressing the complexities of real-world time series data, especially under nonstationary conditions where both the statistical properties and underlying patterns may evolve over time.

Forecasting-based methods [7] construct predictive models from historical data but struggle to distinguish anomalies from normal fluctuations due to inherent data instability. Distance-based methods [8, 9] compute similarities between subsequences yet typically fail to capture joint multivariate anomalies when they treat variables independently or use naive concatenation. Encoding-based approaches [10, 11] transform subsequences into discrete symbols or probabilistic representations, but discretization can obscure fine-grained variations critical for detecting non-stationary fluctuations. Distribution-based methods [12, 13] assume specific statistical distributions, rendering them sensitive to extreme values while neglecting temporal dependencies that are essential for modeling complex multivariate structures. Tree-based methods [14] segment data using isolation trees but overlook temporal continuity and dynamic dependencies by treating time series as independent points, significantly hampering contextual anomaly detection.

Reconstruction-based approaches [15] have emerged as particularly promising, employing encoderdecoder architectures to extract essential data representations with robust performance across various domains. A significant advancement in this category came with the introduction of Denoising Diffusion Probabilistic Models [3], which reformulate the conventional encoding-decoding paradigm into progressive diffusion and reconstruction processes. These models operate by gradually adding noise to data and learning to reverse this process, providing a powerful framework for capturing complex distributions and temporal dependencies in non-stationary time series [16]. However, current diffusion-based methods still face two critical limitations: rigid decomposition processes with static parameters that cannot adapt to evolving time series characteristics, and reconstruction strategies lacking fine-grained control over the integration of high and low frequency information. These shortcomings often result in either underfitting (missing subtle anomalies) or overfitting (generating false positives), motivating our development of PhysDiff as a physically-informed approach that dynamically adapts to complex non-stationary behaviors in multivariate time series.

3 Methodology

3.1 Framework Overview

Given a multivariate time series:

where T denotes the sequence length and d is the feature dimensionality, our goal is to identify anomalous patterns that deviate from typical behaviors. To facilitate local context modeling, we employ a sliding-window paradigm that partitions the original series into overlapping windows:

where, each window W i ∈ R w × d contains w consecutive observations. The proposed PhysDiff framework leverages diffusion models as a foundation for anomaly detection due to their exceptional ability to learn complex data distributions and reconstruct normal patterns. As depicted in Figure 1, our framework consists of three key modules: (1) Physics-Guided Feature Extraction , where we extract interpretable features via adaptive multi-scale signal decomposition, capturing both transient dynamics and long-term trends that serve as physical priors for conditioning the subsequent diffusion process; (2) Physically-Informed Diffusion Model , which employs a conditional generative diffusion

process incorporating these physical priors to robustly learn the distribution of normal patterns; and (3) Anomaly Detection Scoring Module , where anomalies are detected by contrasting reconstructed signals against observed data, with mechanisms sensitive to both point anomalies and sequence-level pattern deviations.

Nx

Xhigh

complexity

X1ow

Add&Norm

complexity

Feed Forward

ComplexityAdaptiveWeighting

ASPE Complexity

X

resichual

Add&Norm

Amplitude-Sensitive

ReverseDiffusionProcess

↑

Permutation Entropy

IterativeDenoisingwithPhysicalGuidance

MatMul

↑

Physics-Guided Routing Attention

Softmax

Physics

Guided

Frequency-BasedRoutingNetwork

Combined Attention Score

Forward DiffusionProcess

High-Frequency

Low-Frequency

ProgressiveNoiseAddition

Routing Mechanism

Physics-GuidedFeatureExtraction

MatMul

Physical Attention Score

MAFD Decomposition

Embedding

Data

Time

Position

K

P

P

Xda

Xtime

↑X

Embedding

Embedding

Input Time Series

X

X1ow

3.2 Diffusion-Based Anomaly Detection Foundations

The forward process incrementally adds noise to the original data according to a predefined schedule, transforming a clean signal into pure noise:

where β t controls the noise schedule at each diffusion step. The reverse process then iteratively denoises the signal by learning a sequence of transformations:

For anomaly detection, we leverage this framework by training the model exclusively on normal patterns. During inference, anomalous sequences will result in higher reconstruction errors as the model attempts to normalize them according to learned patterns. This approach is particularly effective for detecting subtle pattern anomalies in non-stationary environments.

3.3 Physics-Guided Feature Extraction

To enhance the diffusion model's ability to capture complex time series dynamics, a physics-guided feature extraction method is proposed. This method adaptively decomposes multivariate signals into interpretable components and quantitatively assesses signal complexity. These extracted physical features serve as conditioning information to guide the subsequent diffusion-based reconstruction, enabling the model to distinguish normal variations from genuine anomalies.

Our approach generalizes adaptive Fourier decomposition principles [17] to the multivariate scenario via a novel Multi-channel Adaptive Fourier Decomposition (MAFD). Specifically, for a multivariate

time series X ( t ) = [ x 1 ( t ) , x 2 ( t ) , . . . , x C ( t )] , MAFD recursively decomposes the signal using crosschannel shared Blaschke basis functions:

where the projection coefficients are defined as A n = [ A 1 ,n , A 2 ,n , . . . , A C,n ] , and R N ( t ) denotes the residual component. The basis function selection maximizes cross-channel energy convergence: max a n ∑ C c =1 |⟨ G c , B n ⟩| 2 .

A critical challenge in adaptive decomposition is determining the optimal stopping point that balances signal structure preservation with anomaly distinguishability. To address this, we introduce an amplitude-sensitive permutation entropy (ASPE) measure [6] that dynamically determines decomposition depth:

where P ( π ) is the probability of ordering pattern π , and the weight ω π = σ ( y d,τ π ) µ ( y d,τ π ) emphasizes amplitude variations. ASPE effectively discriminates between meaningful signal structure and noise by monitoring the complexity of residual components at each decomposition layer. When ASPE stabilizes and stops decreasing significantly, this indicates that the residual is dominated by unstructured variability rather than meaningful patterns. This data-driven stopping criterion ensures that decomposition proceeds just enough to isolate physically interpretable frequency components without over-segmenting the signal or burying anomaly-relevant features in irrelevant high-frequency bands.

3.4 Physically-Informed Diffusion Model

3.4.1 Conditional Diffusion with Physical Priors

In PhysDiff, the basic diffusion framework is extended by conditioning the reverse process on physical priors extracted from the data as:

where the physical context information z t is defined as a composite feature vector:

Here, P h ( t ) and P l ( t ) represent high-frequency and low-frequency components obtained from MAFD decomposition, capturing rapid transient changes and long-term trends respectively; H ASPE ( t ) is the amplitude-sensitive permutation entropy value quantifying signal complexity; and E ( x t ) is the physics-based energy function measuring physical plausibility of the current state. Notably, these components serve as conditioning features that guide the denoising process, rather than being the targets of diffusion themselves. This conditioning mechanism allows the model to incorporate domain knowledge about time series dynamics, significantly improving its ability to distinguish between normal variations and anomalous patterns.

Furthermore, to integrate MAFD and ASPE into the conditioning mechanism, the amplitude-sensitive weights are directly embedded into the basis selection process as follows:

This composite loss not only refines the decomposition strategy but also increases sensitivity to abnormal patterns while preserving physical interpretability.

3.4.2 Frequency-Based Routing Attention

Notably, PhysDiff introduces a cross-modal routing mechanism that enables dynamic interaction between high frequency and low frequency components during the diffusion process. Specifically, the

input signal X ( t ) is first decomposed via MAFD into high-frequency and low-frequency components: X ( t ) = X HF ( t ) + X LF ( t ) , where X HF captures rapid transients and X LF represents stable trends. These physical components then serve as conditioning information for the routing network:

It is important to note that the diffusion process operates on the original time series X ( t ) , while X HF ( t ) and X LF ( t ) guide the reconstruction through the routing attention mechanism.

In detail, the physically guided attention is computed by:

with adaptive gating coefficients g h = σ ( Q P T h ) and g l = σ ( Q P T l ) . This mechanism allows the model to dynamically emphasize either rapid transient changes or stable trend information according to the contextual requirements at each diffusion step.

3.4.3 Physical Consistency through Energy Guidance

To ensure that generated samples adhere to underlying physical principles, we introduce an energy function derived from instantaneous frequency analyses:

where Φ( · ) represents the physical potential field learned from normal patterns. This energy term regularizes the denoising process by penalizing physically implausible states.

Moreover, physical consistency is enhanced by integrating Langevin dynamics into the sampling procedure as follows: √

where λ represents the step size and n denotes Gaussian noise. Additionally, a frequency-adaptive dynamic trend influence factor γ ( t, ω ) = σ (10(1 - √ ¯ α t ) · A ω ) modulates the reliance on physical components throughout the denoising trajectory, where A ω = exp( -| ω | 2 /σ 2 ) is the frequency response function. This design allows different frequency bands to receive adaptive physical guidance strength at different diffusion stages, ensuring stronger physical guidance when needed most while allowing the model to capture fine-grained details that may not be explicitly encoded in the physical priors.

3.5 Training Strategy and Optimization

The entire framework is trained by minimizing a composite loss function:

where the diffusion loss L diff = E t, x 0 ,ϵ [ ∥ ϵ -ϵ θ ( x t , t, z t ) ∥ 2 ] ensures accurate modeling of normal patterns, L MAFD-ASPE incorporates physical insights, and E ( x t ) enforces physical plausibility.

To improve robustness, we incorporate a controlled disturbance mechanism during training where W disturbed i = W i + δ · p , with δ ∼ U (0 , 1) as a random perturbation. Optimization uses the AdamW optimizer with early stopping to mitigate overfitting.

3.6 Physics-Driven Anomaly Detection

For each sliding window, we define an anomaly score that fuses reconstruction fidelity with physical plausibility:

where ˆ X ( t ) is the reconstruction of the observed window, P TTFD represents the transient timefrequency distribution, and P prior denotes the prior distribution of normal patterns.

We employ extreme value theory through SPOT [18] to dynamically calibrate the detection threshold by fitting a generalized Pareto distribution to the tail of the anomaly score distribution, deriving the threshold as τ t = µ score + k · σ score. This adaptive thresholding strategy accommodates data-specific characteristics, enabling robust anomaly detection across diverse time series domains.

4 Experiments

4.1 Experimental Details

Datasets. We evaluated our approach using five widely recognized benchmark datasets: SMD [19], MSL[7], SMAP [7], SWaT [20], and PSM [21], plus the NeurIPS-TS dataset-comprising Creditcard and GECCO subsets as detailed by Lai et al. (2021) [1]. Data labeled as normal were partitioned with 80% allocated for training and 20% for validation, ensuring the model is properly optimized on typical behavior. These datasets represent diverse domains including spacecraft telemetry, water treatment systems, and financial transactions, providing a comprehensive evaluation landscape for anomaly detection methods.

Metrics. The affiliation-based F1 score is adopted as our primary performance metric, which comprised precision and recall indicators extended from affiliation metrics [22]. Traditional point adjustment (PA) methods, where detecting a single point within an anomalous segment counts as successful detection of the entire segment, can lead to overly optimistic evaluations by inflating true positives and suppressing false negatives [11, 16]. The affiliation-based F1 score offers a more rigorous evaluation by computing precision and recall based on the average directed distance between predicted anomaly events and ground truth, accounting for both spatial and temporal adjacency. All results are reported as percentages, with best performances in bold and second-best underlined.

Baselines. Our method is compared against 18 state-of-the-art anomaly detection approaches across six categories: (1) Forecasting methods (LSTM [7]); (2) Reconstruction methods (PCA [23], AE [24], DAGMM[25], BeatGAN [15], OmniAnomaly [19], D3R [16]); (3) Distance methods (OCSVM [26], HBOS[13], LOF [9], DeepSVDD [12]); (4) Encoding methods (Anomaly Transformer [27], DCdetector [11], SensitiveHUE [28], MTAD-GAT [29], TFAD [10]); (5) Distribution methods (LODA [30]); (6) Tree methods (iForest [14]). Additionally, we included an adversarial baseline marking anomalies at fixed intervals to provide a non-informative temporal reference. For baselines with publicly available implementations, we re-ran experiments using official code and recommended hyperparameters. For methods where identical results on the same datasets with the same evaluation protocol were reported in recent work [11, 16], we directly cite those results to ensure methodological consistency and fair comparison.

4.2 Comparative Study

Our comprehensive comparative analysis evaluates PhysDiff over 18 state-of-the-art anomaly detection methods across five real-world benchmark datasets, as shown in Table 1. PhysDiff consistently

outperforms all baseline models, most notably achieving an F1 score of 74.16% on PSM, outperforming AE by over 7 percentage points, and 81.65% on SMD, surpassing the previous best AE performance of 81.30%. On datasets characterized by complex temporal dynamics, the improvements are particularly significant. PhysDiff achieves 73.91% F1 score on SMAP, outperforming Anomaly Transformer by more than 2 percentage points, and 74.83% on MSL, exceeding LODA by 2.78

percentage points. These gains can be attributed to our model's ability to effectively decompose time series into physically meaningful components using the proposed Multi-channel Adaptive Fourier Decomposition.

To validate PhysDiff's performance in more challenging scenarios, we conducted additional experiments on the NeurIPS-TS datasets featuring more complex anomaly patterns and higher-dimensional feature spaces. As shown in Table 2, PhysDiff achieves F1 scores of 69.44% and 57.92% on the Creditcard and GECCO datasets respectively, substantially outperforming previous best approaches. On Creditcard, PhysDiff surpasses D3R by 6.82 percentage points, while on GECCO it exceeds Anomaly Transformer by 1.96 percentage points. These significant improvements demonstrate our approach's superior ability to detect anomalies in complex real-world scenarios.

The consistent performance gains across all seven datasets highlight several key advantages of our physically-guided diffusion approach. The adaptive decomposition mechanism effectively captures both high frequency transient patterns and low frequency trend components, enabling more accurate anomaly detection across different anomaly types. The incorporation of amplitude-sensitive permutation entropy provides a physically meaningful measure of signal complexity that helps distinguish between normal variations and

actual anomalies. Our frequency-adaptive routing mechanism dynamically calibrates the contributions of different frequency components during the diffusion process, leading to more accurate reconstruction of normal patterns. Quantitative analysis confirms PhysDiff's superior effectiveness on non-stationary datasets compared to stationary ones. Datasets exhibiting strong non-stationarity characteristics achieve larger performance improvements, with detailed statistical validation provided in Appendix E.3. This validates our hypothesis that adaptive decomposition mechanisms are particularly beneficial for handling complex temporal patterns in non-stationary environments. These results collectively establish PhysDiff as a new state-of-the-art in multivariate time series anomaly detection, offering both superior performance and enhanced interpretability through its physics-guided approach.

4.3 Ablation Study

To evaluate each component's contribution to our PhysDiff framework, we conducted comprehensive ablation studies across five datasets in Table 3. The physical guidance mechanism proved most critical, with its removal causing the largest performance drop of 5.95% average F1 score, confirming our hypothesis that incorporating physical constraints substantially improves anomaly detection accuracy. Similarly, disabling routing attention and high frequency components led to significant performance degradation of 10.20% and 6.63% respectively, with particularly severe impacts on specific datasets: routing attention removal caused 25.13% decrease on PSM while high frequency component removal resulted in 11.14% decrease on SMAP. Our information-theoretic components also proved essential, as removing permutation entropy caused a 5.32% average decrease, while replacing amplitude-sensitive permutation entropy with standard entropy resulted in a 4.38% reduction.

Regarding decomposition granularity, neither 8 nor 16 components achieved optimal performance, showing decreases of 5.35% and 5.82% respectively, demonstrating the importance of appropriate decomposition depth selection. Low frequency component removal caused a 5.95% decrease, with varying impacts across datasets. These findings collectively confirm that PhysDiff's superior performance stems from the synergistic integration of physical insights, adaptive frequency decomposition, and information-theoretic measures. Each component makes substantial contributions to the framework's effectiveness across diverse real-world scenarios, validating our architectural design choices and providing empirical evidence for the importance of incorporating domain knowledge into anomaly detection systems.

4.4 Case Study: Fraud Detection in Financial Transactions

The effectiveness of PhysDiff in real-world scenarios is demonstrated through its application to financial transaction fraud detection. Our analysis of the feature importance rankings in Figure 2 reveals that Feature 1 with importance score 0.0998, Feature 15 with 0.0677, and Feature 22 with 0.0650 contribute most significantly to fraud identification, collectively accounting for over 23% of the model's discriminative power. As shown in Figure 3, Feature 1 exhibits a particularly distinctive pattern where fraudulent transactions display a wider, left-skewed distribution compared to the sharp, concentrated peaks of normal transactions. This pattern provides strong discriminative signals for the detection model.

Feature 1

Feature 15

Feature 22

Feature 11

Feature 19

Feature 14

Feature 25

Feature 3

Feature 20

Feature 24

Features

0.00

0.02

0.04

0.06

0.08

0.10

Importance Score (abs correlation)

Top 10 Important Features for Fraud Detection

9

8

7

6

5

4

3

Anomaly Score

0.0

0.2

0.4

0.6

Density

Anomaly Score Distribution

Normal

Fraud

Threshold: -6.1374

The anomaly detection timeline in Figure 4 illustrates PhysDiff's performance across 1,000 sequential transactions, where the model successfully identifies two true fraud cases while generating several false alarms at high anomaly score peaks. The anomaly score distribution shows clear separation between normal and fraudulent transactions, with most normal transactions generating scores below the determined threshold of -6.1374, while fraudulent transactions frequently produce higher scores. This separation enables the model to achieve 64.10% precision and 75.88% recall, resulting in a 69.44% F1 score that significantly outperforms traditional fraud detection approaches.

0

200

400

600

800

1000

Transaction Index

9

8

7

6

5

Anomaly Score

Anomaly Detection Timeline (First 1000 Transactions)

Anomaly Score

Threshold: -6.1374

True Fraud

Feature Correlation Matrix

0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

The feature correlation matrix in Figure 5 and distribution analyses reveal that PhysDiff effectively captures complex inter-feature relationships that would be missed by univariate detection methods. Features with lower individual importance rankings show distinct separation patterns in their distributions that complement the primary features. This multi-scale decomposition enables PhysDiff to distinguish between natural transaction variability and truly suspicious patterns, making it particularly effective at identifying complex fraud schemes that evolve gradually across multiple transactions. This represents a significant advantage over conventional threshold-based detection systems.

5 Conclusion

We proposed PhysDiff , a physically-guided decomposition and diffusion framework for anomaly detection in non-stationary multivariate time series. By combining amplitude-sensitive permutation entropy for optimized decomposition with dual-path conditional diffusion, our approach effectively models both abrupt failures and gradual degradations while maintaining interpretability. Experiments show that PhysDiff achieves improved average F1 score on both standard and NeurIPS-TS datasets, outperforming state-of-the-art methods. Its effectiveness across domains from industrial systems to financial fraud detection validates the integration of physical priors into deep generative models. Future work includes improving computational efficiency, incorporating privacy mechanisms, and extending the framework to handle multimodal data with varying sampling rates and missing values.

References

Kwei-Herng Lai, Daochen Zha, Junjie Xu, Yue Zhao, Guanchu Wang, and Xia Hu. Revisiting time series outlier detection: Definitions and benchmarks. In Thirty-fifth conference on neural information processing systems datasets and benchmarks track (round 1) , 2021.

Xingjian Wu, Xiangfei Qiu, Zhengyu Li, Yihang Wang, Jilin Hu, Chenjuan Guo, Hui Xiong, and Bin Yang. Catch: Channel-aware multivariate time series anomaly detection via frequency patching. arXiv preprint arXiv:2410.12261 , 2024.

Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in neural information processing systems , 33:6840-6851, 2020.

Han Yu, Peikun Guo, and Akane Sano. Adawavenet: Adaptive wavelet network for time series analysis. arXiv preprint arXiv:2405.11124 , 2024.

Md Mahmuddun Nabi Murad, Mehmet Aktukmak, and Yasin Yilmaz. Wpmixer: Efficient multi-resolution mixing for long-term time series forecasting. In Proceedings of the AAAI Conference on Artificial Intelligence , volume 39, pages 19581-19588, 2025.

Jun Huang, Huijuan Dong, Na Li, Yizhou Li, Jing Zhu, Xiaowei Li, and Bin Hu. Amplitudesensitive permutation entropy: A novel complexity measure incorporating amplitude variation for physiological time series. Chaos: An Interdisciplinary Journal of Nonlinear Science , 35(3):033113, 2025.

Kyle Hundman, Valentino Constantinou, Christopher Laporte, Ian Colwell, and Tom Soderstrom. Detecting spacecraft anomalies using lstms and nonparametric dynamic thresholding. In Proceedings of the 24th ACM SIGKDD international conference on knowledge discovery & data mining , pages 387-395, 2018.

Bernhard Schölkopf, John C Platt, John Shawe-Taylor, Alex J Smola, and Robert C Williamson. Estimating the support of a high-dimensional distribution. Neural computation , 13(7):14431471, 2001.

Markus M Breunig, Hans-Peter Kriegel, Raymond T Ng, and Jörg Sander. Lof: identifying density-based local outliers. In Proceedings of the 2000 ACM SIGMOD international conference on Management of data , pages 93-104, 2000.

Chaoli Zhang, Tian Zhou, Qingsong Wen, and Liang Sun. Tfad: A decomposition time series anomaly detection architecture with time-frequency analysis. In Proceedings of the 31st ACM international conference on information & knowledge management , pages 2497-2507, 2022.

Yiyuan Yang, Chaoli Zhang, Tian Zhou, Qingsong Wen, and Liang Sun. Dcdetector: Dual attention contrastive representation learning for time series anomaly detection. In Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining , pages 3033-3045, 2023.

Lukas Ruff, Robert Vandermeulen, Nico Goernitz, Lucas Deecke, Shoaib Ahmed Siddiqui, Alexander Binder, Emmanuel Müller, and Marius Kloft. Deep one-class classification. In International conference on machine learning , pages 4393-4402. PMLR, 2018.

Markus Goldstein and Andreas Dengel. Histogram-based outlier score (hbos): A fast unsupervised anomaly detection algorithm. KI-2012: poster and demo track , 1:59-63, 2012.

Fei Tony Liu, Kai Ming Ting, and Zhi-Hua Zhou. Isolation forest. In 2008 eighth ieee international conference on data mining , pages 413-422. IEEE, 2008.

Bin Zhou, Shenghua Liu, Bryan Hooi, Xueqi Cheng, and Jing Ye. Beatgan: Anomalous rhythm detection using adversarially generated time series. In IJCAI , volume 2019, pages 4433-4439, 2019.

Chengsen Wang, Zirui Zhuang, Qi Qi, Jingyu Wang, Xingyu Wang, Haifeng Sun, and Jianxin Liao. Drift doesn't matter: dynamic decomposition with diffusion reconstruction for unstable multivariate time series anomaly detection. Advances in Neural Information Processing Systems , 36:10758-10774, 2023.

Ze Wang, ChiMan Wong, Agostinho Rosa, Tao Qian, and Feng Wan. Adaptive fourier decomposition for multi-channel signal analysis. IEEE Transactions on Signal Processing , 70:903-918, Jan 2022.

Alban Siffer, Pierre-Alain Fouque, Alexandre Termier, and Christine Largouet. Anomaly detection in streams with extreme value theory. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining , Aug 2017.

Ya Su, Youjian Zhao, Chenhao Niu, Rong Liu, Wei Sun, and Dan Pei. Robust anomaly detection for multivariate time series through stochastic recurrent neural network. In Proceedings of the 25th ACM SIGKDD international conference on knowledge discovery & data mining , pages 2828-2837, 2019.

Aditya P Mathur and Nils Ole Tippenhauer. Swat: A water treatment testbed for research and training on ics security. In 2016 international workshop on cyber-physical systems for smart water networks (CySWater) , pages 31-36. IEEE, 2016.

Ahmed Abdulaal, Zhuanghua Liu, and Tomer Lancewicki. Practical approach to asynchronous multivariate time series anomaly detection and localization. In Proceedings of the 27th ACM SIGKDD conference on knowledge discovery & data mining , pages 2485-2494, 2021.

Alexis Huet, Jose Manuel Navarro, and Dario Rossi. Local evaluation of time series anomaly detection algorithms. In Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining , pages 635-645, 2022.

Mei-Ling Shyu, Shu-Ching Chen, Kanoksri Sarinnapakorn, and LiWu Chang. A novel anomaly detection scheme based on principal component classifier. In Proceedings of the IEEE foundations and new directions of data mining workshop , pages 172-179. IEEE Press Piscataway, NJ, USA, 2003.

Mayu Sakurada and Takehisa Yairi. Anomaly detection using autoencoders with nonlinear dimensionality reduction. In Proceedings of the MLSDA 2014 2nd workshop on machine learning for sensory data analysis , pages 4-11, 2014.

Bo Zong, Qi Song, Martin Renqiang Min, Wei Cheng, Cristian Lumezanu, Dae-ki Cho, and Haifeng Chen. Deep autoencoding gaussian mixture model for unsupervised anomaly detection. In 6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30 - May 3, 2018, Conference Track Proceedings . OpenReview.net, 2018.

Bernhard Schölkopf, Robert C Williamson, Alex Smola, John Shawe-Taylor, and John Platt. Support vector method for novelty detection. Advances in neural information processing systems , 12, 1999.

Jiehui Xu, Haixu Wu, Jianmin Wang, and Mingsheng Long. Anomaly transformer: Time series anomaly detection with association discrepancy. arXiv preprint arXiv:2110.02642 , 2021.

Yuye Feng, Wei Zhang, Yao Fu, Weihao Jiang, Jiang Zhu, and Wenqi Ren. Sensitivehue: Multivariate time series anomaly detection by enhancing the sensitivity to normal patterns. In Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining , pages 782-793, 2024.

Hang Zhao, Yujing Wang, Juanyong Duan, Congrui Huang, Defu Cao, Yunhai Tong, Bixiong Xu, Jing Bai, Jie Tong, and Qi Zhang. Multivariate time-series anomaly detection via graph attention network. In 2020 IEEE international conference on data mining (ICDM) , pages 841-850. IEEE, 2020.

Tomáš Pevn` y. Loda: Lightweight on-line detector of anomalies. Machine Learning , 102:275304, 2016.

Denis Kwiatkowski, Peter CB Phillips, Peter Schmidt, and Yongcheol Shin. Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? Journal of Econometrics , 54(1-3):159-178, 1992.

Said E Said and David A Dickey. Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika , 71(3):599-607, 1984.

NeurIPS Paper Checklist

The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit.

Please read the checklist guidelines carefully for information on how to answer these questions. For each question in the checklist:

You should answer [Yes] , [No] , or [NA] .

[NA] means either that the question is Not Applicable for that particular paper or the relevant information is Not Available.

Please provide a short (1-2 sentence) justification right after your answer (even for NA).

The checklist answers are an integral part of your paper submission. They are visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will be asked to also include it (after eventual revisions) with the final version of your paper, and its final version will be published with the paper.

The reviewers of your paper will be asked to use the checklist as one of the factors in their evaluation. While "[Yes] " is generally preferable to "[No] ", it is perfectly acceptable to answer "[No] " provided a proper justification is given (e.g., "error bars are not reported because it would be too computationally expensive" or "we were unable to find the license for the dataset we used"). In general, answering "[No] " or "[NA] " is not grounds for rejection. While the questions are phrased in a binary way, we acknowledge that the true answer is often more nuanced, so please just use your best judgment and write a justification to elaborate. All supporting evidence can appear either in the main paper or the supplemental material, provided in appendix. If you answer [Yes] to a question, in the justification please point to the section(s) where related material for the question can be found.

IMPORTANT, please:

Delete this instruction block, but keep the section heading 'NeurIPS Paper Checklist" ,

Keep the checklist subsection headings, questions/answers and guidelines below.

Do not modify the questions and only use the provided macros for your answers .

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: The abstract and introduction focus on the challenges of the multivariate time series anomaly detection and the advantages and contributions of our proposed method to address the challenges.

Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.

It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: The diffusion model architecture requires lengthy training times and substantial computational resources. In future work, we plan to employ model distillation techniques to compress the model, making it more lightweight.

Guidelines:

The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.

The authors are encouraged to create a separate "Limitations" section in their paper.

The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.

The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.

The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.

The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.

If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.

While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

3. Theory assumptions and proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]

Justification: The paper provides complete proofs for each module and method proposed.

Guidelines:

The answer NA means that the paper does not include theoretical results.

All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced.

All assumptions should be clearly stated or referenced in the statement of any theorems.

The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.

Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.

Theorems and Lemmas that the proof relies upon should be properly referenced.

4. Experimental result reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [Yes]

Justification: The new method proposed in the paper is introduced in detail in the implementation details, and the method introduced in the paper can be replicated through each complete module.

Guidelines:

The answer NA means that the paper does not include experiments.

If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.

If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.

Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.

While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example

(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.

(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.

(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).

(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.

5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [Yes]

Justification: The datasets used in this paper are public, and the code used for the experiments will be made publicly available once it is organized, to ensure that the experiments can be reproduced by users.

Guidelines:

The answer NA means that paper does not include experiments requiring code.

Please see the NeurIPS code and data submission guidelines ( https://nips.cc/ public/guides/CodeSubmissionPolicy ) for more details.

While we encourage the release of code and data, we understand that this might not be possible, so 'No' is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).

The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines ( https: //nips.cc/public/guides/CodeSubmissionPolicy ) for more details.

The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.

The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.

At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).

Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.

6. Experimental setting/details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [Yes]

Justification: Detailed training and testing information is provided in the appendix of the article.

Guidelines:

The answer NA means that the paper does not include experiments.

The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.

The full details can be provided either with the code, in appendix, or as supplemental material.

7. Experiment statistical significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [Yes]

Justification: The paper accurately presents error bars for the execution speed benchmark. Guidelines:

The answer NA means that the paper does not include experiments.

The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).

The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)

The assumptions made should be given (e.g., Normally distributed errors).

It should be clear whether the error bar is the standard deviation or the standard error of the mean.

It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.

For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).

If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.

8. Experiments compute resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes]

Justification: The paper provides a detailed introduction to the environment in which the experiments were conducted, and the code also includes detailed information on the environmental setup.

Guidelines:

The answer NA means that the paper does not include experiments.

The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.

The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.

The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).

9. Code of ethics

Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines ?

Answer: [Yes]

Justification: The research in this paper adheres to the NeurIPS Code of Ethics, including references to datasets and other models.

Guidelines:

The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.

If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.

The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).

10. Broader impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [Yes]

Justification: The paper interweaves an introduction to the role of time series anomaly detection in time series analysis and provides a prospect on how the proposed methods may impact the field and society.

Guidelines:

The answer NA means that there is no societal impact of the work performed.

If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.

Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.

The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.

The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.

If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).

11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA]

Justification: The method proposed in this paper is suitable for time series anomaly detection and does not pose a high risk of misuse.

Guidelines:

The answer NA means that the paper poses no such risks.

Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.

Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.

We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.

12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [Yes]

Justification: The datasets and models used in the paper are all clearly cited in the references, including the data sources and author information.

Guidelines:

The answer NA means that the paper does not use existing assets.

The authors should cite the original paper that produced the code package or dataset.

The authors should state which version of the asset is used and, if possible, include a URL.

The name of the license (e.g., CC-BY 4.0) should be included for each asset.

For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.

If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.

For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.

If this information is not available online, the authors are encouraged to reach out to the asset's creators.

13. New assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [Yes]

Justification: The experimental code will be made openly accessible, along with the necessary documents to facilitate reproducibility of the experimental results and utilization of the code for future work.

Guidelines:

The answer NA means that the paper does not release new assets.

Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.

The paper should discuss whether and how consent was obtained from people whose asset is used.

At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.

14. Crowdsourcing and research with human subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [No]

Justification: The paper does not involve the issues mentioned above.

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.

According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.

15. Institutional review board (IRB) approvals or equivalent for research with human subjects

Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

Answer: [NA]

Justification: The experiments in this paper are based on public datasets and do not involve the above issues.

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.

We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.

For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.

16. Declaration of LLM usage

Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.

Answer: [No]

Justification: This research does not utilize LLMs as a component of the core methodology. LLMs were only used for assistance with writing and editing the manuscript, which does not impact the scientific rigor or originality of our research methods.

Guidelines:

The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.

Please refer to our LLM policy ( https://neurips.cc/Conferences/2025/LLM ) for what should or should not be described.

Appendix / supplemental material

A Notations

In this paper, we employ various mathematical notations to represent the components of our proposed PhysDiff framework. To facilitate the reader's understanding of the mathematical formulations presented throughout the paper, we provide a comprehensive list of the key notations in Table 4. These notations cover the multivariate time series representation, decomposition components, diffusion model parameters, and anomaly detection metrics. Understanding these notations is essential for following the theoretical development and algorithmic implementations of our physically-guided approach to time series anomaly detection. The reader is encouraged to refer to this table when interpreting the equations and algorithms presented in subsequent sections.

B MAFD and ASPE

B.1 Multi-channel Adaptive Fourier Decomposition

The Multi-channel Adaptive Fourier Decomposition (MAFD) [17] extends traditional adaptive Fourier decomposition to multivariate scenarios. MAFD recursively decomposes multi-channel signals in a hierarchical manner, with each level of decomposition building upon the results obtained from the previous stage. The analytic signal of the c -th channel is expressed as:

where A c,n is the decomposition coefficient of the c -th channel at the n -th level, B n ( e jt ) denotes the shared basis function (modified Blaschke product), and R c,N ( e jt ) represents the remainder term after N -level decomposition. The decomposition process recursively determines the coefficients A c,n and the basis functions B n ( e jt ) at each level.

For a multivariate time series X ( t ) = [ x 1 ( t ) , x 2 ( t ) , . . . , x C ( t )] , each channel x c ( t ) performs the Hilbert transform to obtain the analytic signal:

where H{·} denotes the Hilbert transform. After removing the mean value of G c ( e jt ) , the initial residual is defined as:

For each decomposition level n = 1 , 2 , . . . , N , the recursive procedure mainly consists of three steps:

Search for the optimal basis function parameter. Select the parameter a n ∈ D by maximizing the total energy across all channels:

where e { a } ( e jt ) = √ 1 -| a | 2 1 -¯ ae jt is the normalized evaluator function.

Compute the decomposition coefficients. After determining the optimal a n , the decomposition coefficient A c,n for each channel is calculated as:

which represents the projection of the residual signal onto the basis function e { a n } .

Update the residual. The residual R c,n ( e jt ) is updated as:

where the basis function B n ( e jt ) , constructed as a modified Blaschke product from the parameter sequence { a 1 , a 2 , . . . , a n } , is defined by:

B.2 Amplitude-Sensitive Permutation Entropy

Amplitude-Sensitive Permutation Entropy (ASPE) [6] extends traditional permutation entropy by incorporating amplitude information. The ASPE is defined as:

where P ( π ) is the probability of a given ordering pattern π , and the weight ω π = σ ( y d,τ π ) µ ( y d,τ π ) emphasizes amplitude variations. This entropy measure effectively discriminates between typical variations and anomalies by integrating both ordering and amplitude information.

For a time series { x t } T t =1 , we construct embedding vectors y d,τ j = [ x j , x j + τ , . . . , x j +( d -1) τ ] for j = 1 , 2 , . . . , T -( d -1) τ , where d is the embedding dimension and τ is the time delay. Each embedding vector is then assigned a permutation pattern π = ( r 1 , r 2 , . . . , r d ) based on the relative ordering of its elements, where:

The probability P ( π ) of each pattern is calculated as the frequency of occurrence normalized by the total number of embedding vectors.

C Noise diffusion proofs

C.1 Forward Process Derivation

The forward process in our diffusion model is defined as a Markov chain that gradually adds noise to the input:

This can be rewritten in a non-recursive form to enable direct sampling from any timestep:

where α t = 1 -β t and ¯ α t = ∏ t i =1 α i . This formulation allows us to express x t directly in terms of x 0 : √ √

C.2 Reverse Process Properties

The reverse process learns to gradually denoise the signal:

where z t represents the physical guidance information. The conditional density p θ ( x t -1 | x t , z t ) is modeled as a Gaussian:

The mean µ θ incorporates our physics-guided information through the denoising network ϵ θ :

The variance follows a schedule that interpolates between fixed timestep-dependent values:

Our physically-guided reverse process optimizes the evidence lower bound (ELBO):

where E ( x t ) = ∥∇ x Φ( x t ) ∥ 2 represents our physics-based energy constraint.

D Detailed experimental settings

D.1 Datasets

To prove the effectiveness of our method, we evaluated six real-world datasets: (1) SMD (Server Machine Dataset) collects data from 28 servers over a period of 5 weeks, with 38 metrics monitored for each machine. (2) The PSM (Pooled Server Metrics) dataset is a signal dataset from eBay's IT systems, consisting of 26 dimensions. (3) The MSL (Mars Science Laboratory rover) and SMAP (Soil Moisture Active Passive satellite) are NASA spacecraft telemetry datasets that contain multivariate data captured from multiple telemetry channels across various entities. The MSL dataset comprises 27 entities with 55 telemetry channels per entity, while the SMAP dataset contains 55 entities with 25 telemetry channels per entity. (4) The SWaT (Secure Water Treatment) was collected from 51 sensors in a critical infrastructure system under continuous operational conditions. (5) The NeurIPS-TS (NeurIPS 2021 Time Series Benchmark) introduces a collection of five time series anomaly detection scenarios, systematically classified through behavior-driven taxonomy into distinct categories: point-global, pattern-contextual, pattern-shapelet, pattern-seasonal, and pattern-trend. Detailed specifications regarding the datasets are organized in Table 5.

Algorithm 1 Physically-Guided Diffusion Process

D.2 Baselines

To understand the effectiveness of the proposed method, we extensively compare our model with 18 baselines. Further details concerning the baselines are shown in Table 6.

D.3 Metrics

In this study, we employ Precision (P), Recall (R) and F1 score (F1) based on affiliation metrics [22] as our primary evaluation metrics. Unlike traditional sample-based metrics, affiliation metrics consider the unique characteristics of time series data, including temporal adjacency and event duration. Affiliation precision measures the degree of association between predicted anomalies and actual anomalies, defined as:

̸

where S = { j ∈ [1 , n ]; pred ∩ I j = ∅} represents the set of affiliation zones that have intersections with predictions, and P precision j is the precision probability calculated within each affiliation zone. Similarly, affiliation recall measures the extent to which actual anomalies are correctly detected, defined as:

where n is the total number of actual anomalous events, and P recall j is the recall probability calculated within each affiliation zone. To comprehensively evaluate algorithm performance, we use the F1 score based on affiliation metrics, calculated as:

This parameter-free evaluation method partitions the timeline according to the relationship with the nearest actual anomaly and evaluates algorithm performance independently within each partition. It not only maintains interpretability in physical terms but also resists adversarial prediction strategies, providing more objective and fair assessment results for time series anomaly detection algorithms.

D.4 Implementation

In our experiments, we implement PhysDiff with careful attention to model architecture, training procedure, and anomaly detection strategies, with all key hyperparameters summarized in Table 7.

Model Architecture PhysDiff consists of three primary modules: Physics-Guided Feature Extraction, Physically-Informed Diffusion Model, and Anomaly Detection Scoring. We implement the feature extractor with MAFD that adaptively decomposes multivariate signals, paired with ASPE to assess signal complexity. The core diffusion model integrates physical guidance through a specialized routing attention mechanism: Attention ( Q,K,V,P h , P l ) = softmax (( QK T + g h · QP T h + g l · QP T l ) / √ d k ) V , where g h = σ ( QP T h ) and g l = σ ( QP T l ) control high and low frequency components. We use a noise schedule with β values from 0.0001 to 0.02 over 1000 timesteps, window size of 64, and embedding dimension of 512.

Training Procedure We train PhysDiff using AdamW optimizer with learning rate 1e-4 and weight decay 1e-4, with cosine annealing and warm restarts. Early stopping monitors validation loss with patience of 3 epochs. For robustness, we apply controlled disturbance during training with random noise scaled by factor p . MAFD components are set to 8 for SMD and SWaT, and 16 for SMAP. The parameter λ aspe modulates signal complexity influence with dataset-specific base values: 0.05 for SMAP, 0.02 for SMD, and 0.1 for other datasets.

Anomaly Detection For scoring anomalies, we implement a composite approach combining reconstruction error and model-predicted scores with dataset-specific weightings: SMAP (0.4/0.6), SMD (0.3/0.7), SWaT (0.3/0.7), and others (0.5/0.5). Dynamic thresholding uses SPOT algorithm, fitting a Generalized Pareto Distribution to anomaly score tails. Initial thresholds are determined by percentile-based approach (96th for SMAP, 94th for SMD, 95th for others).

Implementation Environment Experiments were conducted using PyTorch 2.1.2 on NVIDIA GTX2080Ti with 22GB memory. Our implementation includes optimizations: (1) CUDA-accelerated MAFD calculations using nvmath when available, (2) efficient time series embeddings with convolutional layers of kernel size 1, (3) optimized batch matrix multiplications for routing attention, and (4) reconstruction head with two-layer MLP, GELU activation, and dropout rate 0.2. Key hyperparameters include: time steps=1000, window size=64, model dimension=512, feed-forward dimension=2048, attention dimension=64, head count=8, block count=2, dropout rate=0.2, batch size=8, and disturbance factor p =10.0. Code is available at https://anonymous.4open.science/r/PhysDiff-4726.

E Experiments Analysis

E.1 Analysis of Comparative Study Results

As demonstrated in Table 1, PhysDiff consistently outperforms all baseline models across five realworld benchmark datasets. The performance gains are particularly significant in datasets characterized by complex temporal dynamics. On the SMD (Server Machine Dataset), PhysDiff achieves an F1 score of 81.65%, outperforming the previous best model (AE at 81.30%) by 0.35 percentage points, demonstrating superior capability in detecting anomalies in server telemetry data with highdimensional features. For the MSL (Mars Science Laboratory) dataset, PhysDiff attains an F1 score of 74.83%, exceeding LODA (72.05%) by 2.78 percentage points, showing enhanced ability to identify anomalies in spacecraft telemetry with complex non-stationary patterns. The advantage extends to the SMAP (Soil Moisture Active Passive) dataset, where PhysDiff reaches 73.91% F1 score, surpassing Anomaly Transformer (71.65%) by 2.26 percentage points, highlighting its effectiveness on satellite telemetry data. On SWaT (Secure Water Treatment), PhysDiff obtains 72.64% F1 score versus DAGMM's 72.32%, demonstrating marginally better performance on critical infrastructure

monitoring data. For PSM (Pooled Server Metrics), PhysDiff scores 74.16%, which is competitive though slightly below AE's 75.01%, showing strong performance on server performance data.

Figure 7: Performance comparison on NeurIPSTS datasets (Creditcard and GECCO). PhysDiff significantly outperforms all baseline methods, achieving 69.4% and 57.9% F1 scores on Creditcard and GECCO datasets respectively, with particularly large improvements over methods like iForest, AE, A.T., DCdetector and D3R.

As shown in Figure 6, PhysDiff consistently positions above the diagonal line, indicating superior performance compared to the previous best baselines across all datasets. The most substantial improvements are observed on the Creditcard dataset (+6.8 percentage points) and MSL dataset (+2.8 percentage points). This visual representation clearly illustrates the magnitude of improvement across different application domains, with larger bubbles and warmer colors representing greater performance gains.

The performance advantage is even more pronounced on the more challenging NeurIPS-TS datasets (Table 2 and Figure 7). On the Creditcard dataset, PhysDiff achieves 69.44% F1 score, substantially

outperforming the previous best model D3R (62.62%) by 6.82 percentage points, demonstrating exceptional capability in detecting fraudulent financial transactions. Similarly for GECCO, PhysDiff reaches 57.92% F1 score versus Anomaly Transformer's 55.96%, with a 1.96 percentage point improvement on this water treatment dataset featuring complex anomaly patterns. Figure 7 visually emphasizes this dominance, with PhysDiff's bars clearly exceeding those of all baseline methods.

The precision-recall analysis illustrated in Figure 8 reveals that PhysDiff achieves a more balanced trade-off between precision and recall compared to competing methods. The plots for SMD, PSM, and SWaT datasets show PhysDiff (marked with a star) consistently positioned near the optimal regions of the precision-recall space, indicating its ability to minimize both false positives and false negatives simultaneously. Notably, on the SMD dataset (Figure 9 and 10), PhysDiff positions itself in the optimal region with F1 score contours approaching 0.8, surrounded by a higher density of performance points, further validating its robust performance characteristics.

These consistent performance gains across diverse domains validate the effectiveness of our physicallyguided approach, which effectively captures both transient high frequency patterns and stable low frequency trends through adaptive decomposition mechanisms. The superior results across multiple metrics and datasets suggest that PhysDiff's integration of physical priors into the diffusion process significantly enhances anomaly detection capabilities, particularly for complex, non-stationary time series data.

E.2 Computational Complexity Analysis

This section analyzes PhysDiff's computational complexity and provides empirical runtime comparisons to address scalability concerns. PhysDiff incurs higher computational costs than traditional methods but achieves competitive efficiency among comparable approaches while delivering superior performance.

E.2.1 Theoretical Complexity

PhysDiff's computational complexity consists of three primary components. MAFD decomposition operates at O ( C · L · log L ) where cross-channel sharing significantly reduces the coefficient compared to independent processing. The diffusion process requires O ( T diff · D 2 ) operations per timestep. Our routing attention mechanism adds O ( W 2 · D ) complexity that adapts dynamically based on signal characteristics. Here C represents the number of input channels, L denotes the input sequence length, T diff is the number of diffusion timesteps, D refers to the hidden dimension size, and W indicates the attention window size.

E.2.2 Empirical Runtime Analysis

Table 8 presents runtime measurements per epoch on the SWaT dataset, demonstrating clear performance stratification across method categories. PhysDiff requires 81.43 seconds compared to D3R's 78.52 seconds, representing only a 3.7% computational overhead while providing substantial performance returns: 3.61% F1 score improvement on SMD and 7.61% on SMAP.

The additional overhead in PhysDiff stems from physics-guided routing attention and adaptive decomposition mechanisms. These components provide interpretable frequency-domain insights unavailable in baseline diffusion methods. Our channel-shared MAFD strategy achieves significant memory optimization, reducing requirements by approximately 60% compared to channel-independent decomposition approaches. This optimization transforms complexity from O ( C · T · log T ) to O ( T · log T ) .

The efficiency-performance ratio demonstrates that PhysDiff's computational investment is justified by superior anomaly detection capabilities and enhanced interpretability through physics-guided decomposition. The scalable design ensures practical applicability for real-world time series anomaly detection scenarios.

E.3 Non-Stationarity Effectiveness Analysis

To validate PhysDiff's effectiveness on non-stationary time series, we conducted statistical characterization of dataset properties using stationarity tests and derived metrics. Table 9 presents the quantitative analysis results across five benchmark datasets.

We employ four key indicators to characterize non-stationarity: KPSS Score measures trend stationarity using the Kwiatkowski-Phillips-Schmidt-Shin test statistic [31], with higher values indicating stronger non-stationarity; Variance Instability quantifies temporal variance changes; Trend Strength measures the prominence of linear trend components; and Mean Shift captures the degree of mean level changes over time [32].

As shown in Table 9, PhysDiff achieves consistent positive improvements across datasets exhibiting significant non-stationary characteristics. Datasets with high KPSS scores and variance instability demonstrate larger performance gains, with SMD and SWaT showing +0.35% and +0.32% improve-

ments respectively. MSL and SMAP achieve even larger improvements of +2.78% and +2.26% respectively, despite different non-stationarity patterns.

Our adaptive decomposition mechanism effectively captures non-stationary characteristics through MAFD's cross-channel shared basis functions and ASPE's complexity-based stopping criterion. Only PSM, exhibiting minimal non-stationary characteristics, shows negative performance change of -0.85%, validating our approach's specificity for complex temporal patterns rather than overregularizing stationary data.

E.4 Analysis of Ablation Study Results

To evaluate the contribution of individual components within the PhysDiff framework, we conducted comprehensive ablation studies across five datasets. Table 3 presents the F1 scores for each configuration, while Figure 11 visualizes the performance degradation when removing specific components.

Our ablation analysis reveals that the physical guidance (PG) mechanism is critically important, with its removal causing the largest average performance drop (-5.95% F1 score). The impact is particularly severe on SMAP (-8.9%) and SMD (-10.8%) datasets, confirming our hypothesis that incorporating physical constraints substantially improves anomaly detection accuracy by providing meaningful priors for the diffusion model. This validates our core design principle of integrating domain knowledge directly into the generative process.

The routing attention (RA) mechanism proves even more crucial in certain contexts, with its removal resulting in a -10.20% average F1 score decrease. Most notably, disabling RA caused a dramatic -25.1% degradation on the PSM dataset, underscoring the critical importance of dynamically calibrating the contributions of different frequency components during reconstruction, especially in datasets with complex multi-scale temporal dependencies. This suggests that the ability to adaptively weight frequency components based on their relevance to the current context is essential for accurate anomaly detection.

Frequency components analysis reveals significant insights about their relative importance. Removing high frequency components (HF) led to a -6.63% average performance reduction, with the most significant impact on SMAP (-11.1%) and SMD (-10.7%). This demonstrates that transient patterns captured by high frequency components are essential for detecting subtle anomalies that manifest as rapid fluctuations. Similarly, excluding low frequency components (LF) resulted in a -5.95% decrease, with substantial effects on SMD (-11.6%) and SMAP (-6.6%), highlighting the importance of trend information in establishing normal baseline patterns. These results confirm that both frequency ranges contain complementary information necessary for comprehensive anomaly detection.

The information-theoretic measures incorporated in PhysDiff also proved highly valuable. Removing permutation entropy (PE) caused a -5.32% average decrease, while replacing amplitude-sensitive permutation entropy (ASPE) with standard entropy resulted in a -4.38% reduction. These results confirm that our entropy-based measures effectively capture signal complexity and distinguish between normal variations and anomalies. The superior performance of ASPE over standard entropy validates our approach of incorporating amplitude information when assessing time series complexity.

Decomposition granularity experiments showed that neither 8 nor 16 components achieved optimal performance (-5.35% and -5.82% respectively), indicating that appropriate decomposition depth is critical and dataset-dependent. This highlights the importance of adaptive decomposition strategies that can adjust to the specific characteristics of different time series.

As illustrated in Figure 12, the impact of component removal varies significantly across datasets. The heatmap visualization reveals distinct patterns of dependency, with darker cells indicating larger performance decreases when specific components are removed from particular datasets. Figure 13 further emphasizes these dataset-specific dependencies, showing that PSM relies heavily on routing attention while SMD benefits from all components more uniformly. This heterogeneity in component importance across datasets suggests that each time series domain has unique characteristics that require different aspects of the PhysDiff framework.

SMD

MSL

SMAP

SWaT

PSM

0

5

10

15

20

25

F1 Score Decrease

Dataset-Specific Impact of Key Components

w/o RA

w/o coms(8)

w/o HF

w/o ASPE

w/o PG

These comprehensive ablation findings collectively validate our architectural design choices and demonstrate that PhysDiff's superior performance stems from the synergistic integration of physical insights, adaptive frequency decomposition, and information-theoretic measures. Each component makes substantial contributions to the framework's effectiveness across diverse real-world scenarios, with their relative importance varying based on dataset characteristics. The ablation results provide empirical evidence for the importance of incorporating domain knowledge into anomaly detection systems, particularly for handling complex non-stationary time series data.

F Broader Impacts

In recent years, time series anomaly detection technology demonstrates significant value in industrial manufacturing, finance, and healthcare, enabling early identification of equipment failures, fraudulent transactions, and abnormal physiological indicators, thereby reducing potential risks and losses. Our model improves detection accuracy and result interpretability through physics-guided decomposition methods. However, when processing sensitive data (such as medical records or financial information), our framework currently lacks data anonymization mechanisms. In future research, we plan to incorporate technologies like differential privacy to effectively protect user privacy while maintaining detection performance, enabling safe application of this technology across a broader range of domains.

G Limitations

Despite PhysDiff's superior performance in anomaly detection, it has notable limitations. The diffusion model architecture requires lengthy training times and substantial computational resources, limiting its application in resource-constrained environments. PhysDiff assumes continuous observations throughout the time series, as our MAFD decomposition requires consistent data points for effective basis function estimation. Significant missing values disrupt the decomposition process, particularly when missing segments contain critical frequency components, and channel-specific missing patterns break the consistency of cross-channel energy convergence. Additionally, while we validated effectiveness through F1 scores, anomaly detection evaluation should be more comprehensive. Future work will address these limitations by employing model distillation techniques to compress the multi-step inference process, incorporating uncertainty-aware diffusion mechanisms for handling incomplete data, and designing an integrated evaluation framework combining detection latency, false alarm tolerance, and interpretability metrics to provide more objective standards for time series anomaly detection.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper argues that most existing time series anomaly detection methods have been developed and evaluated primarily on stable datasets, and thus tend to underperform when encountering pattern anomalies in unstable time series. To address this limitation, the authors propose a frequency-domain diffusion model aimed at improving detection performance on unstable data.\n\nThe core contributions of the paper include: (1) a permutation entropy-guided decomposition mechanism, which determines the optimal decomposition depth in the frequency domain; and (2) a frequency-based routing attention mechanism, designed to differentiate attention between high-frequency and low-frequency components during anomaly detection.\n\nSTRENGTHS AND WEAKNESSES:\n## Strength\n\n1. The paper clearly articulates the specific problem it aims to address and presents the motivation behind the proposed methodology in a well-structured manner.\n2. Extensive experiments are conducted across multiple commonly used datasets, with comprehensive comparisons against a wide range of baseline methods.\n\n## Weakness\n\n**Inappropriate characterization of prior work:**\n The authors claim that many existing time series anomaly detection methods are primarily evaluated on stationary datasets and thus perform poorly on non-stationary data. However, many widely used benchmark datasets such as **MSL**, **SMAP**, and **PSM** are known to be **non-stationary** in nature. Moreover, numerous existing methods have demonstrated strong performance on these datasets. For instance, AnomalyTransformer achieves F1 scores *93% (MSL)*, *96% (SMAP)*, and *97% (PSM)*. These results contradict the paper’s assertion and suggest the need for a more accurate review of prior work.\n\n**Unclear necessity of addressing the proposed challenge:**\n One of the main challenges the paper aims to address is how to determine the “optimal depth” in frequency domain decomposition. The paper does not provide evidence to demonstrate that different decomposition depths lead to significantly different detection performances. As such, it remains unclear whether this is a challenge that requires explicit resolution.\n\n**Unclear notion: optimal depth:**\nAs one of the main contributions declared in this paper, the notion of what is the *optimal depth* of frequency decomposition is not clear. Optimal for what? (e.g. optimal depth for F1 score, for efficiency, etc.?) \n\n**Significant deviation from baseline performance:**\n The reported F1 scores for several baselines in the paper deviate substantially from those reported in their original papers. For example, DCdetector originally reports F1 scores of *87% (SMD)*, *96% (MSL)*, *97% (SMAP)*, *96% (SWaT)*, and *97% (PSM)*, whereas in the current paper, the scores are only *66%*, *70%*, *68%*, *69%*, and *66%*, respectively. I recommend the authors to clarify whether the discrepancies arise from differences in experimental setup, metric computation, or implementation details to verify the fairness of comparison.\n\n**Limited comparative analysis in the case study:**\n The paper includes a case study intended to showcase the effectiveness of the proposed method in a real-world application. However, the case study only presents results for the proposed method without comparing it to any baseline methods. As a result, the case study provides limited insight into the relative advantages or practical value of the proposed approach.\n\nQUESTIONS:\n1. I recommend conducting a detailed analysis comparing the proposed method’s performance on non-stationary versus stationary datasets, to demonstrate that it indeed improves the performance on non-stationary time series as claimed.\n\n2. Could you please analyze how the decomposition depth in the frequency domain affects the model’s performance. This would help justify the necessity of finding an optimal depth.\n\n3. What is the definition of “optimal depth” — optimal with respect to which objective? Moreover, please explain why optimizing permutation entropy leads to a decomposition depth that is optimal for this specific goal.\n\n4. As the paper mentions that diffusion models incur higher computational and training costs, could you please provide a quantitative comparison of time and memory efficiency between the proposed method and baseline methods.\n\n5. Could you please explain the large performance gaps between the baseline results reported in this paper and those in the original papers. Clarifying potential differences in implementation, evaluation metrics, or experimental setup would strengthen the credibility of the comparisons.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-23"}

comments: SUMMARY:
The paper proposes a diffusion-based reconstruction model for time series anomaly detection guided by frequency-domain features. By leveraging high- and low-frequency features derived from frequency decomposition as physical priors, the method better distinguishes anomalous patterns during the reconstruction process. Experimental results demonstrate the effectiveness of the approach.

STRENGTHS AND WEAKNESSES:
Strengths:
- Empirical validation of the model's effectiveness;
- A case study in real-world scenarios highlights the practical
   significance of the model.
Weaknesses:
-  Some formulas lack detailed explanations and analysis;
- Frequency-domain feature extraction may incur computational
   overhead, particularly for multivariate time series;
- Performance improvements are not significant across five real-world
datasets.

QUESTIONS:
- What is the exact definition of physical context information zt mentioned in Line 167? Could the authors provide a detailed
   explanation of how physical priors guide the diffusion model?
- Has the computational overhead of the frequency-domain approach compared to the time-domain approach been quantified? How do channel-independent and channel-shared strategies affect
   computational costs and model performance?
- Could the authors provide a more detailed analysis of performance
variations across different datasets? Are there differences in recall and precision? What are the underlying reasons for these discrepancies?

LIMITATIONS:
Yes.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-01

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a physically-guided diffusion framework for anomaly detection in multivariate non-stationary time series (PhysDiff). The authors first propose a Multi-channel Adaptive Fourier Decomposition (MAFD) method to extract high- and low-frequency components, with decomposition depth adaptively determined using Amplitude-Sensitive Permutation Entropy (ASPE). These physics-inspired features are then used as priors in a dual-path conditional diffusion model, which integrates a frequency-based routing attention mechanism to guide denoising. Anomaly scores are computed by combining reconstruction error with time-frequency divergence. Experiments on seven datasets demonstrate that PhysDiff outperforms 18 baseline methods in F1 score with strong interpretability and generalization.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\ni) The integration of MAFD, ASPE, and conditional diffusion modeling is both novel and clearly articulated.\nii) The use of a physically-guided diffusion model to address non-stationary multivariate time series is innovative and well-justified.\niii) The experimental evaluation is extensive, involving 7 benchmark datasets and 18 baseline methods, and shows that PhysDiff consistently outperforms existing approaches.\niv) A comprehensive ablation study supports the design choices and highlights the contribution of each model component.\n\nWeaknesses\ni) The paper lacks discussion on model complexity and computational overhead of PhysDiff, particularly in comparison to the baselines.\nii) The methodology section could benefit from additional visualizations to help readers more intuitively follow each component and processing step.\n\nQUESTIONS:\ni) Could the authors provide more details on the computational complexity and runtime of PhysDiff compared to other baselines? Any empirical comparisons would be appreciated.\nii) Can PhysDiff handle missing values in multivariate time series data? If not directly, what preprocessing or modifications would be necessary?\niii) How is the dynamic trend influence factor determined or learned during training? Is it interpretable or tunable?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes **PhysDiff**, a physically-guided diffusion model for anomaly detection in non-stationary multivariate time series, addressing high false positive rates and low interpretability. PhysDiff integrates physics-guided signal decomposition with conditional diffusion reconstruction. The model introduces adaptive frequency extraction and a time-frequency energy routing mechanism to enhance detection accuracy and interpretability. Experiments on five benchmark datasets and NeurIPS-TS scenarios show PhysDiff significantly outperforms 18 baseline algorithms, demonstrating robustness and interpretability in complex dynamic systems.\n\nSTRENGTHS AND WEAKNESSES:\nStrenghts\n1. PhysDiff effectively incorporates physical principles through Multi-channel Adaptive Fourier Decomposition (MAFD) and Amplitude-Sensitive Permutation Entropy (ASPE), enabling adaptive signal decomposition that captures cross-channel dynamics and enhances interpretability in non-stationary time series.\n2. It uses frequency domain analysis to separate high- and low-frequency components, reducing interference and providing physical priors, while the conditional diffusion model distinguishes anomalies via reconstruction errors. \n3. The model demonstrates superior anomaly detection performance, achieving significant F1-score improvements over 18 baselines across five benchmark datasets, showcasing its generalizability and effectiveness in complex scenarios.\n4. The ablation experiment and comparative experiment are very comprehensive.\n5. This paper is well written and easy to follow.\n\nWeaknesses\n1. The paper provides limited introductory explanation of the mathematical foundations underpinning Physics-Guided Feature Extraction (e.g., MAFD and ASPE) and Physical Consistency through Energy Guidance (e.g., Langevin dynamics and energy functions), making it challenging for readers without strong mathematical backgrounds to fully grasp their implementation and purpose.\n2. The paper lacks a thorough analysis of key hyperparameters (e.g., $\\lambda$ in the composite loss or $\\gamma(t)$ in energy guidance), which could impact reproducibility and robustness across diverse datasets.\n3. The literature reviews can be further improved by incorporating more recent studies.\n\nQUESTIONS:\n1. Can the authors provide a more accessible explanation of how MAFD, ASPE, and energy-guided Langevin dynamics are implemented and their specific roles in anomaly detection?\n2. How does adaptive decomposition (MAFD+ASPE) compare to fixed high-low frequency decomposition in performance, and in which scenarios is adaptivity critical?\n3. How does PhysDiff perform in domains with weak physical constraints, and what adaptations could improve its versatility?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-07-04"}