PhysDiff: A Physically-Guided Diffusion Model for Multivariate Time Series Anomaly Detection
Authors: Long Li
Wencheng Zhang
Shi Yuan
Hongle Guo
Wanghu Chen
Venue: NeurIPS 2025 poster ()
Keywords: Time Series Anomaly Detection
Physically-Informed Diffusion
​​Physics-Guided Decomposition
Non-stationary Dynamics​
Primary area: deep_learning
TL;DR: A physically-guided diffusion model for multivariate time series anomaly detection is proposed.

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
Long Li Wencheng Zhang Shi Yuan Hongle Guo?, Wanghu Chen
Coollege of Computer Science & Engineering, Northwest Normal University School of Management, Northwest Normal University
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Abstract
Unsupervised anomaly detection of multivariate time series remains challenging in complex non-stationary dynamics, due the high false-positive rates and limited in- terpretability. We propose PhysDifff combining physics-guided guided decomposition with diffusion-based reconstruction, to address these issues. The physics-guide -rsiiie--gieed 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 diffu- sion modeling captures deviations from learned normal behavior, making anomalies more distinguishable. Notably, PhysDiff introduces an amplitude-sensitee per- mutation entropy criterion to adaptively determine the optimal decomposition depth, and automatically extract adaptive frequency components used as explicit physics-bus based constraints for the diffusion process. Furthermore, the proposed conditional diffusion network employs dual-path conditioning mechanism that integrates frequerry and low-frequency physical priors, dynamically regulat- ing the denoising process via novel time frequency energy routing mechanism. By weighting reconstruction errors across frequency bands, our method improves anomaly localization and enhances interpretability. Extensive experiments five benchmark datasets and two NeurIPS-T'S scenarios demonstrate that PhysDiff out- performs 18 state-of-the-art baselines, with average F1 score improvements both standard and challenging datasets. Experimental results validate the ad advantages of combining principled signal decomposition with diffusion-basse reconstruction for robust, interpretable anomaly detection in complex dynamic systems.
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 [.] 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 [I]. Reconstruction- based methods have demonstrated promising performance in detecting point anomalies, as these anomalies manifest 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 greater detection
'These authors contributed equally to this work. Corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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challenge. In real-world scenarios, time series data is predominantly -ottsttiiona,, 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- noise-adding and de- noising 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 gen- uine 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 decomposi- tion 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 se- ries 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 44. 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 enhanc- ing 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. Overem- phasis on low frequency information can obscure anomalies, whereas excessive incorporation of high frequency details risks model overfitting. These challenges highlight the need for more dynamic and physically-informee approach to decomposition and reconstruction.
Inspired by the concept of entropy in physics, we propose PhysDiff, a physically-guides diffusion model for multivariate time series anomaly detection that combines permutation entropy with an implitude-sensitivv weighting scheme based actual magnitude variations (6]. For Challenge 1, we dynamically adjust the decomposition depth during frequency-doman dommnnn analysis by leveraging entropy values that reflect the intrinsic physical characteristics of the data. Specifically, signal complexity is assessed using an anplitude-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 applitude-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 physically-guide diffusion model that effectively addresses non-stationarity challenges.
We introduce an ppplitudd-sensitivv permutation entropy guided decomposition mechanism that dynamically determines optimal decomposition depth.
We develop dual-path conditional diffusion framework with novel frequency- routing attention mechanism.
2

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Extensive experimental validation demonstrates that the method surpasses 18 tfae-oof-the-rr baselines, with improved average Fl 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 significant and widely recognized challenge. According to detection strategies, existing approaches can be categorized into forecasting-basee based, 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 non- stationary conditions where both the statistical properties and underlying patterns may evolve over time.
rccosttruction-baees distance- encoding-
Forecasting-base based methods 7 construct predictive models from historical data but struggle to distinguish anomalies from normal fluctuations due to inherent data instabilitty Distance-bar based meth- ods 8 [ compute similarities between subsequences yet typically fail to capture joint multivariate anomalies when they treat variables independently or use naive concatenation. Encoding- approaches [0] [1] transform subsequences into discrete symbols or probabilistic representations, but discretization can obscure fine-grained variations critical for detecting non-stationary fluctuations. [3] 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 tem- poral continuity and dynamic dependencies by treating time series as independent points, significantly hampering contextual anomaly detection.
Dnstribution-basee methods [12
Reconstruction-basse approaches [5] have emerged as particularly promising, employing encoder- decoder architectures to extract essential data representations with robust performance across various domains. 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 powerful framework for capturing complex distributions and temporal dependencies non-stationary time series [66]. However, current diffusion- 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 shortcomings often result in either underfitting (missing subtle anomalies) or overfitting (generating false positives), motivating development of PhysDiff as physically-ifformed approach that dynamically adapts to complex non-stationary behaviors in multivariate time series.
the integration of high and low frequency information. These
3 Methodology
3.1 Framework Overview
Given a multivariate time series:
X (m,,. [T) Txd
where denotes the sequence length and d is the feature dimensionality, our goal to identify anomalous patterns that deviate from typical behaviors. To facilitate local context modeling, employ sliding-window paradigm that partitions the original series into overlapping windows:
W [11, W2, WT-w-1)
where, each window W; E Boxd contains consecutive observations. The proposed Phys 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 [ our framework consists of three key modules: (1) Poysics-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- -nnsically-Ifformedd Diffusion Model. which employs conditional generative diffusion
3

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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
X
complexity X
Add&Norm
Threshold
complexity
Feed Forward
Complexity Adaptive Weighting
ASPE Complexity Arplltude-Sensitive
Add&Norm
Reverse Diffusion Process Iterative Denoising with Physical Guidance
Permutation Entropy
MatMul
Physics-Guided Routing Attention
Softmax
Physics
Guided
Frequancy Based Routing
Combined Attention Score
Forward Diffusion Process Progressive Naise Addition
High-Freen Frequency
Low- Frw-Freyyyncy
wWW
Routing Mechanism
Physics-Guided Feature Extraction
MatMul
Physical Attention Score
MAFD Decomposition
Embedding
Ø
@
P
Q
P
Data
Time
Poshoa
X
X
Embedding
Embedding
X
Input Time Series
x
X
X_
Figure 1: A physically-guider decomposition and diffusion reconstruction framework (PhysDiff). The framework contains three main modules: (1) Prysics-Guuded Feature Extraction; (2) Physically- Informed Diffusion Model; (3) Physics- -Driven Anomaly Detection.
3.2 Diffusion-Based Anomaly Detection Foundations
The forward process incrementally adds noise the original data according predefined schedule, transforming a clean signal into pure noise:
T
a (x Xo) II - 8xx-,,,81
=
where Br controls the noise schedule at each diffusion step. denoises the signal by learning a sequence of transformations:
The reverse process then iteratively
pe (x)T)) = p(XT) II pe (x:-1
For anomaly detection, we leverage this framework by training the model exclusively normal patterns. During inference, anomalous sequences will result in higher reconstruction errors as the model attempts 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, 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 novel Multi-channel Adaptive Fourier Decomposition (MAFD). Specifically, for multivariate

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MAFD recursively decomposes the signal using cross- channel shared Blaschke basis functions:
time series ((t) [z1 x22 (t))
( = ΣA A. Bn (t(
where the projection coefficients are defined as A, = [AI,n: , Ac.al, and (v(()) denotes the residual component. The basis function selection maximizes cross-channel energy convergence: maxa, E KG B.
A critical challenge in adaptive decomposition is determining the optimal stopping point that bal- ances signal structure preservation with anomaly distinguishability. To address this, we introduce an amplitude-sensitie permutation entropy (ASPE) measure ( that dynamically determines decompo- sition depth:
In(d!) Σ Wa P(T)
HASPE
where P(x) is the probability of ordering pattern and the weight ws 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-segmnniing the signal or burying anomaly-relen features in irrelevant high-frequee frequency bands.
emphasizes p(v
3.4 Physically-Informe 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:
po (x (2x) p(xx)
II t=1
po (x
(5)
where the physical context information 24 is defined as a composite feature vector:
= (t), HASPE(t), E(x))
(6)
Here, (t) and (t) represent figh-freeuency and lwwffrequency components obtained from MAFD decomposition, capturing rapid transient changes and long-term trends respectively; HASPE (t) is the amplitude-sensitiee permutation entropy value quantifying signal complexity; and E(x,) is the physics- based energy function measuring physical plausibility the current state. Notably, these components serve 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 amplituee-sensitiee weights are directly embedded into the basis selection process as follows:
AMAFD-ASPE Σ ||..
Σ
A A,
BIl + XHASPE
This composite loss not only refines the decomposition strategy but also increases sensitivity to abnormal patterns while preserving physical interpretabilityy
3.4.2 Frequency-Based Routing Attention
Notably, PhysDiff introduces cross-modal routing mechanism that enables dynamic interaction between high frequency and low frequency components during the diffusion process. Specifically, the
5

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input signal ((t) is first decomposed via MAFD into high- high-frequency and low- fow-rrquencyy components: (t) (t) + (t), where Xur captures rapid transients and X1F represents stable trends. These physical components then serve as conditioning information for the routing network:
2 = R(,, (H)()), XIF(())
(8)
It is important to note that the diffusion process operates the original time series X(t, while XHF(() and X(r(() guide the reconstruction through the routing attention mechanism.
In detail, the physically guided attention is computed by:
QK" + 9h QP 91 QP? V
Attention (Q, K, V, Ph, Pt) = softmax
Vdr.
with adaptive gating coefficients 9A, = a (QP)) and 9r a(PT)) This mechanism allows the model 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 energy function derived from instantaneous frequency analyses:
E(x,) >
(10)
where (-)) 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 follows:
where À represents the step size and n denotes Gaussian noise. Additionally, a frequency-adape (11) dynamic trend influence factor yt, w) a (10(1 - Vo.) ) Aw) modulates the reliance on physical components throughout the denoising trajectory, where A. cxp(-lw12/o 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.
x x XVx, (x,) + V2An
3.5 Training Strategy and Optimization
The entire framework is trained by minimizing a composite loss function:
Lioual Lain + 7 LMAFD-ASPE 1) {(x))
(12)
where the diffusion loss Lai Ex,x,, [Ile cx t, 22 ]l?] ensures accurate modeling of normal patterns, LMAFD-ASPE incorporates physical insights, and E(x,) enforces physical plausibility.
To improve robustness, incorporate controlled disturbance mechanism during training where W disturbed W; + 8- p. with 8 ~ U(0,1 1) as a random perturbation. Optimization uses the AdamW optimizer with early stopping mitigate overfitting.
3.6 Physics-Driven Anomaly Detection
For each sliding window, we define an anomaly score that fuses reconstruction fidelity with physical plausibility:
Score(t) a [((t) X(t)lle +(1-a) a)
DKL (PTTFD PPTTTDDDPPPrioo PI
(13)
Reconstruction Error
Time Frequency Distribution Divergence
where {(t) is the reconstruction of the observed window, PrIED represents the transient time- frequency distribution, and Parior denotes the prior distribution of normal patterns.
We employ extreme value theory through SPOT 18 to dynamically calibrate detection threshold by fitting generalized Pareto distribution to the tail of the anomaly score distribution, deriving the threshold as t, = /'score + k dscore This adaptive thresholding strategy accommodates data-specific characteristics, enabling robust anomaly detection across diverse time series domains.

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4 Experiments
4.1 Experimental Details
Datasets. We evaluated our approach using five widely recognized benchmark datasets: SMD 19]
MSL. SMAP 77, SWaT (20), and PSM (21). plus the NeurIPS-TS dataset- comprisinng Crediteard and GECCO subsets as detailed by Lai et al. (2021) [. 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-buu based Fl score is adopted our primary performance metric, which comprised precision and recall indicators extended from affiliation metrics [22]. Traditional point adjustment (PA) methods, where detecting 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 16]. The iffiliation-based F1 score offers 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 oftte-of-the-att anomaly detection approaches across six categories: (1) Forecasting methods (LSTM (7): (2) Reconstruction methods (PCA [23), AE 24],
DAGMM [25]. BeatGAN (T5). Omni [19], D3R [66); (3) Distance methods (OCSVM 26].
HBOS 13. LOF 9 DeepSVDD 12);; (4) Encoding methods (Anomaly Transformer (77.. DCdetec-
tor 11, SensitiveHUE: 28). MTAD-GAT 29), TEAD (T]); (5) Distribution methods (LODA BD));
(6) Tree methods (iForest (44 ). Additionally, we included an adversarial baseline marking anomalies at fixed intervals to provide non-informative temporal reference. For baselines with publicly avail- able 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 [6), 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-arr anomaly detec- tion methods across five real-world benchmark datasets, as shown in Table [ PhysDiff consistently
Table I: Performance comparison on five real-world anomaly detection datasets.
Dataset
SMI>
MSL
SMAP
SWaT
PSM
Metric
P
R
F1
P
R
F1
P
R
F1
P
R
I-
R
H-
OCSVM
66.98
73.75
72.
PCA
64.92
40.19
54.34
52.69
98.33
68.61
50.62
98.48
66.87
62.32
82.96
71.18
77.44
37:71
53.53
HBOS
56.28
63.11
62.17
59.25
$3.32
69.25
41.54
66.17
51.04
57.71
29.82
43.21
100.00
6.54
12.28
LOP
72.92
49.00
72.
47.92
60.72
$3.20
53.90
99.91
70,02
FFrrect
100.00
9.37
17.13
53.87
94.58
68.6 65.65
41.12
68.91
51.51
53.03
62. 80
62 03:03
100.00
3.35
6.48
LODA
59.02
18
62. 40
57.79
95.65
72.05
51.51
100.00
68.00
56.30
70.14
62.3 54
62.22
40.17
56.05
AE
69.22
98.48
$1.30
55.75
66
70.72
39.12
70.31
50.52
54,92
95.20
70.45
60.67
98.24
75.01
DAGMM
63.57
67.00
54.07
50.75
72.
70.50
LSTM
60.12
84.77
70.35
58.82
14.68
23.49
55.25
27.70
36.90
49.99
$2.11
62.15
57.06
95.92
71.55
BearGNN
74.11
81.64
77.69
58.74
98.94
71.30
$4,04
98.30
69.74
61.89
83.46
71.08
58.81
99.08
73.81
Oruii
79.09
$2.74
71.41
A.T.
100.00
3.19
6.19
51.04
95.
66.49
56.91
6/9
71.65
53.63
59.94
$7.59
52.01
82. 12
64.55
DCdetector
50.93
95.57
66.45
55.94
95.53
70.56
53.12
98.37
68.99
53.25
98.12
69.03
$4.72
$6.36
66.99
SemitiveHUE
60.34
90.13
72.29
55.92
98.95
71,46
53.63
98.37
69.42
58.91
91.71
71.74
56.15
98. 75
71.59
DeD
64.98
64.77
64,88
100.00
19.06
29.73
72.44
74.05
30.64
MTAD-GAT
85.90
67.69
75.71
54.96
94.93
69.81
39.05
93.99
55.08
65.90
77.51
71.23
79.90
60.14
68.63
TFAD
56.32
97.83
71.49
54.96
94.93
69.81
39.05
93.99
55 .8.
60.38
81.96
69.53
79.14
71.63
75.20
DER
64.87
97.93
78.04
56.45
71.81
$1.08
94,46
66.30
64.25 25
70.25
100.00
09.43
PhysDaff
71.03
96.00
81.65
62.75
92.66
74.83
61.36
86.81
73.91
60.00
92.04
72.64
66.09
84.47
74.16
outperforms all baseline models, most notably achieving an F1 score of 74.16% on PSM, outper- forming 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% % Fl score on SMAP, outperforming Anomaly Transformer by more than 2 percentage points, and 74.83% on MSL, exceeding LODA by 2.78
7

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percentage points, These gains can be attributed to our model's 's ability to effectively decompose time series into physically meaningful components using the proposed Multi-channel Adaptive Fourier Decomposition.
To validate PhysDiff'' performance in more challenging scenarios, conducted additional experi- ments on the NeurIPS-TS datasets featuring more complex anomaly patterns and higher- dimensional feature spaces. As shown Table 2 PhysDiff achieves Fl scores of 69. 69.44% and 57.92% on the Creditcard and GECCO datasets respectively, substantially outperforming previous best approaches. 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 dif-
Table 2: Performance comparison on the NeurlPS-TS datasets.
advantages of our physically-guide fusion approach. The adaptive decom- position mechanism effectively captures both high frequency transient patterns and low frequency trend components, enabling more accurate anomaly detection across different anomaly types. The incorpora- tion of umplitude-sensitive permutation en- tropy provides physically meaningful measure of signal complexity that helps distinguish between normal variations and actual anomalies. Our rrequency-adaptive
Dataset
Creditcard
GECCO
Metric
P
R
F1
P
R
F1
iForst AE
1.52
88.01
2.99
2.59
97.40
5.04
14.76
31.71
20.14
76.56
20.14
31.89
A.T.
55.86
57.55
56.69
54.77
57.21
55.96
DCdetector
49.86
67.67
57.41
38.30
59.70
46.60
D3R
56.90
69.61
62.62
59.00
36.61
45.18
PhysDiff
64.10
75.88
69.44
60.89
55.23
57.92
routing mechanism dynamically calibrates the contri- butions 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 pro- vided in Appendix E.3 This validates hypothesis that adaptive decomposition mechanisms are particularly beneficial for handling complex temporal patterns in non-stationary environments. These results collectively establish PhysDiff
state-of-the-ar 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 's contribution to our PhysDiff framework, we conducted comprehensive ablation studies across five datasets in Table 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. 23.13% decrease PSM while high frequency component removal resulted 11. 141 decrease on SMAP. Our information- thhoortic components also proved essential, as removing permutation entropy caused a 5.32% average decrease, while replacing amplitude-sensitivy permutation entropy with standard entropy resulted a 4.38% reduction.
Table 3: Results of ablation studies. Fl scores are reported, with higher values meaning better performance. The best scores are highlighted in bold.
w/o PE
w/o ASPE
w/o RA
w/o
w/o LF
w/o PG
w/o coms ()
w/o
dataset
PhysDiff
HF
coms( 16)
SMD 81.65
70.61
70.51
72.37
70.91
70.02
70.84
69.86
69.49
MSL
74.83
71.59
71.06
67.95
69.92
71.46
70.54
71.59
71.12
SMAP
73.91
70.76
70.51
66.40
62.77
67.27
65.03
65.70
68.78
SWaT
72.64
68.76
69.84
70.46
69.81
69.84
70.41
69.84
69.83
PSM
74.16
68.86
73.38
49.03
70.65
68.86
70.62
73.47
68,86
Average
75.44
70.12
71.06
65.24
68.81
69.49
69.49
70.09
69.62

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Regarding decomposition granularity, neither 8 nor 16 components achieved optimal performance, showing decreases of 5.35% and 5.82% respectively, demonstrating the importance of appropri- ate decomposition depth selection. Low frequency component removal caused 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 decompo- sition, and rnformation-theoreel
measures. Each component makes substantial contributions to the framework's s effectiveness across diverse real-world scenarios, validating architectural design choices and providing empirical evidence for the importance 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 with importance score 0.0998, Feature 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 exhibits a particularly distinctive pattern where fraudulent transactions display a wider, left-skewed distribution compared the sharp, concentrated peaks of normal transactions. This pattern provides strong discriminative signals for the detection model.
10 Important Fealures Fraud Detection
Anomaly Score Distribution
Normal
0.6
Fraud
Threshold:- 661337
Importatte Store
0.4
0.02
0.2
0.00
Restre
Feature
Feature
Feetave
Festure
Feature
fanture
Feeture
feature
featuee
-9
-6
Fealures
Anumaly
Figure 2: Top 10 features for fraud detection, with Feature (0.0998) contributing most signif- icantly.
Figure 3: Anomaly score distribution showing separation between normal transactions (below -6.1374) and fraudulent ones.
The anomaly detection timeline in Figure illustrates PhysDiff's performance across 1,000 sequential transactions, where the model successfully identifies two true fraud cases while generating several false alarms 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 a 69. 69.44% Fl score that significantly outperforms traditional fraud detection approaches.
Anomaly Detection Timeline (First 1000 Transactions))
Feature Correlation Matrix
Anomaly Score
Threshold: 6.1374
Anamely Skow
-6
True Fraud
False Alarm
-7
0.4
0.2
-8
0.0
-0.2
20:0
400
600
800
1000
Figure 4: Detection timeline across 1,000 trans- actions identifying true fraud cases (red x) and false alarms (orange).
Figure 5: Feature correlation matrix revealing inter-feature relationships that help identify sus- picious activities.

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The feature correlation matrix in Figure 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 distri- butions 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 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-sensiiee 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 Fl 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.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No. 62462059).
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/ supplemmnntll material
A
Notations
In this paper, employ various mathematical notations represent the components of our proposed PhysDiff framework. To facilitate the reader's understanding of the mathematical formulations presented throughout the paper, provide a comprehensive list of the key notations in Table These notations cover the multivariate time series representation, decomposition components, diffusion model parameters, and anomaly detection metrics. Understanding these notations essential for following the theoretical development and algorithmic implementations of our physically-guit approach time series anomaly detection. The reader encouraged to refer to this table when interpreting the equations and algorithms presented in subsequent sections.
Table Mathematical Notations Used PhysDiff
Symbol
Description
X
12.,,,..., 2TJ
Multivariate time series data with 7' sequence length and d di- mensions
RTxd
W
Sequence of overlapping windows with W2 E RRxdd
(W1, WW,WW,.... Wr-
((t)
Multivariate time series with channels
(t), x2((), ...., x2())...., xc((t]
(x(e
Blaschke basis functions in MAFD decomposition Projection coefficients in MAFD
An
[Ai,n . Ac, Ry(1)
Residual component after decomposition Amplitude-sensiitve permutation entropy Probability of permutation pattern ı
HASPE ((T))
Wx M(U"""
Weight for amplitude variations
XHF((). XIF(1)
High-frequency and low-frequency components
( (F0)
Forward diffusion process
Pe (e(ooTllz)
Reverse diffusion process
Noise schedule parameter
- Be
Single step noise parameter
ā [T aii
Cumulative noise parameter
Co
Noise prediction network
E(x:)
V (xx))11
Physics-based energy function Anomaly detection score Dynamic trend influence factor
(core(t)
(1) (10 vo.))
F 3.0 (3)
Generalized Pareto distribution for threshold calibration Adaptive anomaly threshold
Tt Mscore Sccore
MAAFD-ASP
Composite loss for decomposition Diffusion model loss
Cair
Leatl
Total framework loss
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 hierarchical manner, with each level of decomposition building upon the results obtained from the previous stage. The analytic signal of the e-th channel is expressed as:
N
Go (e5))
Σ n=1
Acn Bn (est) + Ro,N (eft)
(14)
20

---

where Ac, is the decomposition coefficient of the c-th channel at the n-th level, B,,ee*) denotes the shared basis function (modified Blaschke product), and (e.v(e")) represents the remainder term after N-level decomposition. The decomposition process recursively determines the coefficients Ac, and the basis functions (e(( at each level.
For a multivariate time series (t) = (t), (f). IC)() each channel xx(() performs the Hilbert transform to obtain the analytic signal:
Ga(e"t) (t) + jH{{x(tt)
(15)
where H(-} denotes the Hilbert transform. After removing the mean value of Ge(e)t), the initial residual is defined as:
Re,o(e = G.(e")
(16)
For each decomposition level n ........ N, the recursive procedure mainly consists of three steps:
Search for the optimal basis function parameter. Select the parameter a, E by maximizing the total energy across all channels:
an arg max
Σ [((Gelle") aED
(17)
1- T-
where C(a3(e
is the normalized evaluator function.
Compute the decomposition coefficients. After determining the optimal an the decomposition coefficient Ac, for each channel is calculated as:
Acn (Gaa(e"))
(18)
which represents the projection of the residual signal onto the basis function efa,]
Update the residual. The residual Re, (ett) is updated as: Re An,
Rega
(19)
where the basis function B, (e constructed as modified Blaschke product from the parameter sequence {a1, ......, Gn is defined by:
Bn(e)t) 11-lan est
ettt 0d āze
(20)
ane
d=1
B.2 Amplitude-Sensitivee Permutation Entropy
Amplitude-Sensiiee Permutation Entropy (ASPE) 6 extends traditional permutation entropy by incorporating amplitude information. The ASPE is defined
HASPE = In(d)) Σ w Lu)P(T)
(21)
where P(x)) is the probability of a given ordering pattern and the weight w, amplitude variations. This entropy measure effectively discriminates between typical variations and anomalies by integrating both ordering and amplitude information.
atv
emphasizes
For time series {xx ) we construct embedding vectors " d.T j ,, (d 1)7, where d is the embedding dimension and is the time delay. Each embedding vector is then assigned a permutation pattern π = (r1, T2.... ..,,. ra) based on the relative ordering of its elements, where:
[x Tj+rs......... ()) for
ijtnt < Tj+rar S
S Ijarer
(22)
The probability ((T) of each pattern calculated the frequency of occurrence normalized by the total number of embedding vectors.
21

---

C Noise diffusion proofs
C.1 Forward Process Derivation
The forward process in our diffusion model is defined as Markov chain that gradually adds noise to the input:
(f1r) II (( = 1,,
(23)
t=1
This can be rewritten in non-recursive form enable direct sampling from any timestep: q( (o) = (v (1 ō;)
(24)
where 1 - 3e and a III 0g This formulation allows us to express directly in terms of To:
It Va + v1 ūee, e~ c0011
(25)
C.2 Reverse Process Properties
The reverse process learns gradually denoise the signal:
T
po( p(xr)) II po (1
(26)
where 2, represents the physical guidance information. The conditional density po 21) is modeled as Gaussian:
Po( 24) = (Ho((a, t, 2), Mo (2,,,)))
(27)
The mean be incorporates our physics-guided information through the denoising network eg:
no(x,,t,21) Var vI ar
_ co(
(28)
The variance follows schedule that interpolates between fixed timestep-dependent values:
Eg E2 t) 0 (1 (1-aa)
(29)
Our physically-guided reverse process optimizes the evidence lower bound (ELBO):
Er, [||c - co (2+, f, =+]
LMAFD-ASPE + - +x+)(x))
(30)
where (xx)) = V ① (r:))l represents our physics- 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 period of 5 weeks, with 38 metries monitored for each machine. (2) The PSM (Pooled Server Metrics) dataset is signal dataset from eBay's '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 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 organized in Table 5
22

---

Algorithm Physically- Guideed Diffusion Process
Require: Time series Xg. diffusion steps T', noise schedule {at)es
Ensure: Reconstructed time series Xo
1: function
2: Pv. P. f- MAFD(Xx)
Multi-channel Adaptive Fourier Decomposition D High and low frequency components
3: return PA, Pt
4: end function
5: function FREQQUECCYRESPOSSFFUUNN
6: return exp(-Iw | /o²)
Frequency response function
7: end function
8: function FORWARDPROCESS(XO.
9:
Sample random noise
O4 II Oi
Cumulative noise level Add noise
11: X₁ Vox + V 02f6
12: return X
end function
14: function PIYSICALLGUDANNCESS,,,.P Pn, Pi)
15:
dr + II==100 Og (1(1) 1/TT
> Cumulative noise level High frequency influence decreases with time > Low frequency influence increases with time
17:
fi(t) t t/T
Compute routing attention with physical guidance
19:
Q, K, V t LinearProjectio(xx))
(1) + (t) QP)
High frequency gating
21:
91(1) o(i(() QP()
Low frequency gating
Attention softmax (0) (0-0P 00 VV
22:
vide
Apply frequency-adan adaatiive dynamic trend influence for each frequency band w in signal do
24:
25:
Aw + FrequencyRessposeeuun
26:
7(, + (10 (z) Aw)
27:
x; - (t,
(t,, (P + +P))
28:
end for
29: Predict noise using physical information
30:
60 <- NooieFstiimatioNeeoorrk., I,, Attention, PA, Pt)
31: return 60
32:
end function
33:
function REVERSEPROCESSS(y: Ph. Pt)
for i T,T I,.... do
35:
Eg f- PlysicalGGuidancexx... f, Ph, Pt)
36:
0g Di= 04
37:
P4 & Jau (X, V1-8c
1-a cp)
ift then
1-ā.-1 (1 Oit))
39:
σe f- I-as
(O,II)
41:
X + H4 + 012
42:
else
43:
Xn 1
end
45:
end for
46: return X
47: end function
48: // Main algorithm
49: Pt. Pt -ExtracPyyyscallCCmppennntts
50: xr: + ForwardProcess( T)
51: Xy & ReverseProcess/ PA. Pi)
23

---

Table 5: Details of benchmark datasets for evaluation. (AR:anomalyy ratio)
Dataset
Domain
Dimension
Training
Validation
Test (labeled)
A AR( %
SMD
Server Machine
38
566 724
141 681
708 420
4.2
MSL
Spacecraft
55
46 653
11 664
73 729
10.5
SMAP
Spacecraft
25
108 146
27
427 617
12.8
SWaT
Water treatment
31
396 000
000
449919
12.1
PSM
Server Machine
25
105 984
26
87 841
27.8
Creditcard
Finance
29
113922
481
142 404
0.17
GECCO
Water treatment
408
852
69 261
1.25
Table 6: Categorization and Characteristics of baselines.
Category
Method
Advantages
Limitation
Forecasting
Capable multi-step forecasting
Less sensitive abrupt
PCA
Easily interpretable
Only linear
AE
Relatively straightforward to implement
Prone to overfitting
DAGMM
Complex distribution modeling
Heavy memory
Reconstruction
BeaGAN
Realistic sequence modeling
Hard to train
Omni
Models uncertainty variable coupling
Gaussian assumptions
Catches diverse anomaly types
Co-training can misalign
OCSVM
Theoretically solid for small data
Poor in high dims
HOBS
Shange-sensitive
Window design critical
Distance
LOF
Detects local outliers
Scalability and high-dim issues
DeepSVDD
End-to-end optimization
Risk trivial zero-radius solution
A.T.
Captures complex dependencies
Requires parameter tuning
DCdetector
Pattern capture
Negative sampling critical
Encoding
SensitiveHUUE
High sensitivity to subtle anomalies
Weak on sparse anomalies
MTAD-GAT
Inter-variable correlation modeling
Prior knowledge dependent
TEAD
Dual-domain feature capture
Weak on non-stationary series
Distribution
LODA
Online learning capability
May lose information
Tree
Forset
Highly efficient
Struggles with very local anomalies
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
D.3
Metrics
In this study, 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:
P Prreession
1
Σ P Precissonn jES
(31)
[S|
where S' === nled n 1, * 0} represents the set of affiliation zones that have intersections with predictions, and LaddPPeeee Pprecision, the precision probability calculated within each affiliation zone. Similarly, affiliation recall measures the extent to which actual anomalies are correctly detected, defined
1 Σ Precatty
Precatl
(32)
1=1
where is the total number of actual anomalous events, and Precatt, is the recall probability calculated within each affiliation zone. To comprehensively evaluate algorithm performance, we use the Fl score based on affiliation metries, calculated as:
Fl Pprecision Precall
(33)
Pprecision + Precall
24

---

This parameter- purameter-free evaluation method partitions the timeline according to the relationship with the nearest actual anomaly and evaluates algorithm performance independently within each partition. 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
Model Architecture PhysDiff consists of three primary modules: Physics-Guided Feature Extrac- tion, Physically-Informee 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 spe- cialized routing attention mechanism: (ttention(Q, K, V. Ph, Pt) = softmax((QK" + 9h QP 9: //())//))) where gh = o o(QP)) and gi (QPP)) control high and low frequency compo- nents. We use a noise schedule with 3 values from 0.0001 to 0.02 over 1000 timesteps, window size of 64, and embedding dimension of 512.
Training Procedure We train Phys using AdamW optimizer with learning rate le-4 and weight decay le-4, with cosine annealing and warm restarts. Early stopping monitors validation loss with patience of epochs. For robustness, we apply controlled disturbance during training with random noise scaled by factor p. MAFD components are set 8 for SMD and SWaT, and 16 for SMAP. The parameter Aaspe 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 composite approach combining reconstruction error and model--pedictee 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 GTX 2080Ti with 22GB memory. Our implementation includes optimizations: (1) acCe-ecceler MAFD calculations using nvmath when available, (2) efficient time series embeddings with convolu- tional 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 s0eps=1000, window size=64, model dimension=512, foed-forward dimension=2048, at- tention dimension=64, head count=8, block count=2, dropout rate=0.2, batch size=8, and disturbance factor 1=10.0 Code is available at htt.://ooooooooooooooooocee/ee/e
E
Experiments Analysis
E.1 Analysis of Comparative Study Results
As demonstrated in Table T PhysDiff consistently outperforms all baseline models across five real- world benchmark datasets. The performance gains are particularly significant in datasets characterized by complex temporal dynamics. On the SMD (Server Machine Dataset), PhysDiff achieves an Fl score of 81.65%, outperforming the previous best model (AE at 81 81.30%) by 0.35 percentage points, demonstrating superior capability in detecting anomalies in server telemetry data with high- dimensional features. For the MSL (Mars Science Laboratory) dataset, PhysDiff attains an Fl score of 74.83%, exceeding LODA (72.05%) by 2.78 percentage points, showing enhanced ability 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% Fl 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% Fl score
versus DAGMM's 72.32%, demonstrating marginally better performance critical infrastructure
25

---

Table 7: Hyperparameters for PhysDiff
Category
Parameter
Value
Time Steps
1000
Beta Range
0.0001-0.02
Window Size
64
Model Dimension
512
Model Architecture
Foed-Forward Dimension
2048
Attention Dimension
64
Head Count
8
Block Count
Dropout Rate
0.2
Optimizer
AdamW
Learning Rate
0.0001
Batch Size
8
Training
Epochs
10
Early Stopping Patience
3
Disturbance Factor (p)
10.0
ASPE Weight (A)
0.05-0.1 (dataset dependent)
Components
8-16 (dataset dependent)
MAFD
Dictionary Distance (dist)
0.05
Maximum Magnitude
0.95
SPOT Purameter
0.01
Evaluation
Kernel Size for Smoothing
5
monitoring data. For PSM (Pooled Server Metries), PhysDiff scores 74.16%, which is competitive though slightly below AE's 75.01 %, showing strong performance on server performance data.
iForest
A.T.
D3R
DCdetector
PhysDiff
SMD
69.4
+2.8
Absalute f1 Score improvement
LODA
TRAD PSM -1.0
$6.0
PhycDin Score
SMAP
Fl Score (96
98.5
+2.3
Creditcard
+00.2
31.9
3.0
70
75
90
85
Busi Buseline Score 156)
Creditcard
GECCO
Figure 6: Performance comparison between PhysDiff and baseline methods across seven datasets. Points above the diagonal line indicate PhysDiff's improvement
Figure 7: Performance comparison on NeurIPS- TS datasets (Creditcard and GECCO). PhysDiff significantly outperforms all baseline methods, achieving 69.4% and 57.9% F1 scores Cred- itcard and GECCO datasets respectively, with particularly large improvements over methods like iFFrest, AE, A.T., DCdetector and D3R.
previous methods. The most substantial improvements are observed on Creditcard. The bubble size and color repre- sent the absolute Fl score improvement.
As shown in Figure 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% Fl score, substantially
26

---

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% Fl score versus Anomaly Transformer's s 55.96%, with 1.96 percentage point improvement on this water treatment dataset featuring complex anomaly patterns. Figure visually emphasizes this dominance, with PhysDifffs '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 peecalln--reaal space, indicating its ability to minimize both false positives and false negatives simultaneously. Notably, on the SMD dataset (Figure 9 [nd([O), PhysDiff positions itself in the optimal region with Fl score contours approaching 0.8, surrounded by a higher density of performance points, further validating its robust performance characteristics.
SMD
PSM
SWaT
100
Procision (8)
MhysDiff tAA DSK SQNAAN STRD-DATT
70
Phys
70
70
Megnaa PhynDirt
60
D3R
60
60
con
50-
50-
50-
40
40
40
40
50
60
70
80
90
100
40
50
60
70
80
90
100
40
50
60
70
80
90
100
Recall (96)
Recall 1%6)
Recall (36)
Figure 8: Rrecision-Recall analysis across SMD, PSM, and SWaT datasets. PhysDiff (marked with a star) achieves a better balance between precision and recall compared to competing methods, maintaining high scores on both metrics. Contour lines represent Fl score values.
Precision-Recall Density Analysis (SMAP Dataset)
Precision-Recall Density Analysis (MSL Dataset)
100
80
80
Precision (%)
Physoiff
Precision (%)
PhysDitr
OFaIGAL
NTKD GAT
40
BeatGAN MTAD-GAT
40
BeatGAN
TRAD
MIAD-GAT
TFAD
TEAD
D3R
01-02
D3R
PhysDiff
PhysDitf
0
20
80
100
60
100
Recall (%)
Recall (%6
Figure 9: Precision- density analysis for the SMAP dataset, showing the distribution of model performance in PR space. PhysDiff (red star) positions optimally with high density near F1=0.8, outperforming AE, BeatGAN, MTAD- GAT, TFAD, and D3R.
Figure 10: Precision-Recall density analysis for the MSL dataset, showing the distribution of model performance in PR space. PhysDiff (red star) positions optimally with high density near F1=0.8, outperforming AE, BeatGAN, MTAD- GAT, TFAD, and D3R.
These consistent performance gains across diverse domains validate the effectiveness of our physically- guided 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 's integration of physical priors into the diffusion process significantly enhances anomaly detection capabilities, particularly for complex, non-stationary time series data.
27

---

E.2
Computational Complexity Analysis
This section analyzes PhysDiff's computational complexity and provides empirical runtime compar- isons 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 --C-L-log L) where cross-channel sharing significantly reduces the coefficient compared to independent processing. The diffusion process requires ( D operations per timestep. Our routing attention mechanism adds O D) complexity that adapts dynamically based on signal characteristics. Here C' represents the number of input channels, denotes the input sequence length, Tanr the number of diffusion timesteps. D refers the hidden dimension size, and W indicates the attention window size.
E.2.2 Empirical Runtime Analysis
Table g presents runtime mcasurements per epoch on the SWaT dataset, demonstrating clear per- formance 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 SMD and 7.61 % on SMAP.
Table Runtime comparison across method categories on SWaT dataset
Method Category
Runtime Range (s)
Complexity Order
Examples
Linear Methods
0.02-0.13
O(TD)
PCA, LODA
Distance-based
0.25-0.30
O(T²)
LOF, IForest
Classical ML
1.17-1.48
O(T2D)
OCSVM. LSTM AE, DCdetector BeatGAN, TEAD D3R, PhysDiff
Neural Networks
3.70-6.76
O(TD²)
Advanced Models
9.31-19.77
O2)
Diffusion Models
78.5-81.4
O(Tar TD2)
The additional overhead in PhysDiff stems from physics-guided routing attention and adaptive decom- position mechanisms. These components provide interpretable frequency -ommain insights unavailable baseline diffusion methods. Our channel-shared MAFD strategy achieves significant memory optimization, reducing requirements by approximately 60% compared channel-independen decom- position approaches. This optimization transforms complexity from OCC T log 7') to O log T).
The efficiency-eeffommance performance ratio demonstrates that PhysDiff's computational investment is justified by superior anomaly detection capabilities and enhanced interpretability through physics-guided 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 non-stationary time series, conducted statistical char- acterization of dataset properties using stationarity tests and derived metrics. Table presents the quantitative analysis results across five benchmark datasets.
We employ four key indicators to characterize non-stationarity: KPSS Score measures trend station- arity using the KwiatkowskiiPhiin Shii-PPhllllpSSS- Paaiiiippsss--mmiidd Seaiips-SShhiit-Shhhhe test statistic [II, 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 [22
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-
28

---

Table Non-stationarity characteristics and PhysDiff performance improvements
Dataset
KPSS Score
Var. Instability
Trend Strength
Mean Shift
F1 Improvement
SMD
0.9870
0.3968
0.4402
0.4331
+0.35%
SWaT
0.9791
0.5763
0.2377
0.2485
+0.32%
MSL
0.9000
0.1611
0.0003
0.0088
+2.78%
SMAP 0,9000
0.0061
0,0055
0.0115
+2.26%
PSM
0.9094
0.0215
0.0077
0.0080
-0.85%
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 -0,55, validating our approach's specificity for complex temporal patterns rather than over- regularizing stationary data.
E.4 Analysis of Ablation Study Results
To evaluate the contribution individual components within the PhysDiff framework, we con- ducted comprehensive ablation studies across five datasets. configuration, while Figure 1I visualizes the performance degradation when removing specific components.
the Fl scores for each
Our ablation analysis reveals that the physical guidance (PG) mechanism is critically important, with its removal causing the largest average performance drop (-5.95% Fl 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 resulting in a -10.20% average F1 score decrease. Most notably, disabling RA caused a dramatic -25.1% % degradation ibrating the contributions of different frequency components during reconstruction, especially in datasets with complex multi-scale temporal dependencies. This suggests that the ability adaptively weight frequency components based on their relevance to the current context is essential for accurate anomaly detection.
crucial in certain contexts, with its removal
the PSM dataset, underscoring the critical importance of dynamically cal-
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 essential for detecting subtle anomalies that manifest as
captured by high frequency components rapid fluctuations. Similarly, excluding low frequency components (LF) resulted in a -5.95% decrease, with substantial effects SMD (-1 (.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-thoome measures incorporated PhysDiff also proved highly valuable. Removing permutation entropy (PE) caused -5.32% average decrease, while replacing amplitude-sensitive permutation entropy (ASPE) with standard entropy resulted -4.38% reduction. These results confirm that entropy-based measures effectively capture signal complexity and distinguish between normal variations and anomalies. The superior performance of ASPE over standard entropy validates approach of incorporating amplitude information when assessing time series complexity.
Decomposition granularity experiments showed that neither 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 characteristies of different time series.
29

---

25
ASPE
4.38
w/o RA
9.3
69
75
22
25.1
20
w/o comst81
5.35
Score DebreDeoon Decrease
coms(16)
5.82
w/o PG
10.8
4.3
11
we U
5.05
w/o coms(16)
12.2
3.7
5.1
28
5.3
6.63
coms(8)
wio RA
w/o
Score Decrease
W/o ASPE
11.1
3.8
Figure 11: Impact of removing individual com- ponents from PhysDiff, measured by F1 score decrease averaged across all datasets. Routing Attention (RA) has the largest impact (10.20% decrease), followed by High frequency compo- nents (HF, 6.63%), Physical Guidance (PG) and Low frequency components (L.F) (both 5.95%).
Figure 12: The dataset-specific impact of re- moving each component. Darker colors indicate larger FI score decreases. Removing Routing Attention (RA) severely impacts PSM (25, % decrease), while High frequency (HF) removal significantly affects SMAP (11.1% decrease).
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.
Dataset-Specific Impact Key Components
w/o RA
w/o coms(8)
HF
20
w/o ASPE
F1 Score Decrease
w/o PG
15
10
5
MSL
SMAP
Figure 13: Dataset-specific impact of removing key components. The bar chart highlights how different datasets rely on different aspects of PhysDiff: PSM depends heavily on Routing Attention, SMAP is sensitive High frequency components, while SMD benefits from all components more evenly.
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- -thooreti 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.
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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-guides 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 broader range of domains.
G
Limitations
Despite PhysDiff's superior performance in anomaly detection, it has notable limitations. The dif- fusion model architecture requires lengthy training times and substantial computational resources, limiting its application in resource-constrained environments. PhysDiff assumes continuous obser- vations throughout the time series, 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 Fl scores, anomaly detection evaluation should be more compre- hensive. Future work will address these limitations by employing model distillation techniques to compress the mullii-teep inference process, incorporating uncertainty-awee diffusion mechanisms for handling incomplete data, and designing an integrated evaluation framework combining detection latency, false alarm tolerance, and interpretability metries to provide more objective standards for time series anomaly detection.
MAFD decomposition requires consistent data points for
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Reviews:
[{"review_id": "MuSN4cLUdW", "reviewer": "Reviewer_gw7N", "summary": "The 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.", "questions": "1. 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?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strenghts\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.", "was_revised": true, "final_justification": "After careful consideration of the authors’ rebuttal and comments with other reviewers, I would give \"Accept\". The paper proposes a novel, effective method (PhysDiff) with strong results and clear writing.", "rebuttal": "> **Q1: 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?**\n\n**A1:** We sincerely appreciate the reviewer's valuable suggestion. Detailed explanations of MAFD, ASPE, and energy-guided Langevin dynamics, including their roles in anomaly detection, are provided in Appendix B and Section 3.4.3 of the paper.\n\nIn lines 426--434 (Section B.1): MAFD recursively decomposes multivariate signals into frequency components and a residual via shared Blaschke basis functions. At each level, the basis is chosen to maximize cross-channel energy convergence, ensuring extracted components capture common dynamics across variables. This disentangling of high-frequency oscillations and low-frequency trends enhances sensitivity to anomalies at different scales.\n\nIn lines 450--456 (Section B.2): To adaptively determine how deep this decomposition should proceed, ASPE extends permutation entropy by weighting each ordering pattern $\\pi$ with $\\omega_\\pi = \\sigma(y_{d,\\tau_\\pi})/\\mu(y_{d,\\tau_\\pi})$, integrating both ordinal patterns and amplitude variations. By monitoring the ASPE of residual components at each decomposition layer, the process stops when entropy indicates diminishing complexity, thus adaptively determining the optimal decomposition depth.\n\nIn line 189 (Section 3.4.3, Eq. 10): Furthermore, during the reverse (denoising) process, we enforce physical consistency via the energy function $E(x_t) = \\|\\nabla_x \\Phi(x_t)\\|^2$ and Langevin dynamics: $x_{t-1} \\leftarrow x_t - \\lambda \\nabla_{x_t} E(x_t) + \\sqrt{2\\lambda} n$, where $\\lambda$ is the step size and $n$ Gaussian noise. This biases sampling toward physically plausible states. A dynamic trend influence $\\gamma(t) = \\sigma(10(1-\\sqrt{\\bar{\\alpha}_t}))$ adjusts the strength of physical guidance over time, balancing trend preservation and fine-detail reconstruction to better distinguish anomalies.\n\n> **Q2: How does adaptive decomposition (MAFD+ASPE) compare to fixed high-low frequency decomposition in performance, and in which scenarios is adaptivity critical?**\n\n\n**A2:** The ablation studies (see in lines 272-290 and Table 3) clearly demonstrated the advantages of adaptive decomposition. Removing permutation entropy (PE) caused a 5.32% average F1 score decrease, while replacing ASPE with standard entropy resulted in a 4.38% performance reduction.\n\nMeanwhile, performance changes on domain-specific datasets highlight the critical role of adaptivity in certain scenarios. It is found that ASPE removal leads to a performance decrease of 3.77% and 3.4% on the spacecraft telemetry dataset MSL and SMAP, respectively. Similarly, it shows a 10.84% F1 score decrease after removing physical guidance on SMD server data. Particularly, there is a dramatic 25.13% performance drop on PSM dataset when routing attention was removed.\n\n\n> **Q3: How does PhysDiff perform in domains with weak physical constraints, and what adaptations could improve its versatility?**\n\n**A3:** We appreciate the reviewer's feedback. It is nesscessary to note that PhysDiff is primaryly designed for time series anomaly detection, where physical constraints are usually inherent in the data.\n\nWe also acknowledge that PhysDiff has limitations in domains with weak physical constraints, such as pure text sequences, social media metrics, and abstract financial derivative data, where the contribution of physical guidance components may decreases. Nevertheless, in scenarios with weak physical constraints, PhysDiff can still  improve the performance primarily depending on the intrinsic advantages of the diffusion model framework and multi-scale decomposition rather than physical prior knowledge.\n\nTo enhance PhysDiff's applicability in domains with weak physical constraints, we propose three key improvements: (1) Introducing a domain-adaptive weighting mechanism to automatically adjust the physical guidance weight η in Equation 11 based on signal complexity, reducing reliance on physical constraints when clear patterns are absent; (2) Employing a hybrid decomposition strategy that combines MAFD with domain-specific techniques like wavelet decomposition for financial data or empirical mode decomposition for text sequences; (3) Utilizing learnable physical priors to adaptively capture implicit structural patterns tailored to different domains, which can be implemented by replacing fixed physical energy functions with learnable neural networks.\n\nIn future work, we are to systematically evaluate PhysDiff's performance across domains with varying physical constraints, develop adaptive physical weight adjustment mechanisms, and explore domain-agnostic structured prior knowledge integration polices."}, {"review_id": "Fe61IjqGzq", "reviewer": "Reviewer_TJwG", "summary": "This 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.", "questions": "i) 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?", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "I am satisfied about authors' response during the rebuttal period and appreciate their effort in trying to address my concerns in a short period of time. Originally I already leaned to accept their paper so I would keep my rating, the questions are more for myself to give a rating that I am confident with.", "rebuttal": "> **Q1: Could the authors provide more details on the computational complexity and runtime of PhysDiff compared to other baselines? Any empirical comparisons would be appreciated.**\n\n**A1:** We sincerely appreciate the reviewer's valuable suggestion. We have analyzed the computational complexity of PhysDiff compared to the basedlines and conducted empirical runtime comparisons on the SWaT dataset. The runtime of different models can validate our theoretical complexity analysis. \n\nOur analysis reveals that PhysDiff's computational overhead mainly occurs during the diffusion-based reconstruction phase, while maintaining competitive efficiency compared to similar models. Runtime measurements per epoch on the SWaT dataset demonstrate clear performance stratification across method types, as shown in the following table. \n\n| Method Category | Runtime of Methods (seconds) | Runtime of Categories (seconds) | Complexity Order |\n|---|---|---|---|\n| **Linear Methods** | PCA (0.017), LODA (0.124) | 0.02-0.13 | $\\mathcal{O}(TD)$ |\n| **Distance-based** | DeepSVDD (0.256), LOF (0.289), IForest (0.301) | 0.25-0.30 | $\\mathcal{O}(T^2)$ |\n| **Classical ML** | OCSVM (1.173), LSTM (1.25), DAGMM (1.48) | 1.17-1.48 | $\\mathcal{O}(T^2D)$ |\n| **Neural Networks** | AE (3.701), DCdetector (4.81), MTAD-GAT (6.76) | 3.70-6.76 | $\\mathcal{O}(TD^2)$ |\n| **Advanced Models** | A.T. (9.31), BeatGAN (11.0), SensitiveHUE (16.26), TFAD (19.77) | 9.31-19.77 | $\\mathcal{O}(T^2D^2)$ |\n| **Diffusion Models** | D3R (78.52), **PhysDiff (81.43)** | 78.5-81.4 | $\\mathcal{O}(T_{diff} \\cdot TD^2)$ |\n\nPhysDiff's computational complexity is determined mainly by its three primary components. MAFD decomposition operates at $\\mathcal{O}(C \\cdot L \\cdot \\log L)$ where cross-channel sharing reduces the coefficient significantly compared to independent processing. The diffusion process requires $\\mathcal{O}(T_{diff} \\cdot D^2)$ operations per timestep, with our routing attention mechanism adding $\\mathcal{O}(W^2 \\cdot D)$ complexity that adapts dynamically based on signal characteristics. Here $C$ represents the number of input channels/features, $L$ 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.\n\nTo be noted that, though PhysDiff requiring 81.43 seconds compared to D3R's 78.52 seconds (3.7% overhead), this investment yields substantial performance returns. PhysDiff achieves 3.61% F1-score improvement on SMD and 7.61% on SMAP datasets, demonstrating favorable computational efficiency relative to performance gains. The additional overhead primarily stems from our physics-guided routing attention and adaptive decomposition mechanisms, which provide interpretable frequency-domain insights unavailable in baseline diffusion methods.\n\n> **Q2: Can PhysDiff handle missing values in multivariate time series data? If not directly, what preprocessing or modifications would be necessary?**\n\n**A2:** We appreciate the reviewer's insightful feedback. Our study specifically focuses on physically-guided diffusion for time series anomaly detection, which leads to limited consideration of missing value preprocessing in multivariate time series. Consequently, the proposed MAFD decomposition in PhysDiff method inherently assumes continuous observations. To some extent, missing values represent a common challenge in diffusion-based approaches for time series analysis. \n\nTraditionally, for sparse random value missing ($<10$\\%), linear interpolation can maintain frequency pattern continuity with an acceptable performance degradation. For block missing ($>15$\\%), it severely disrupts MAFD basis function estimation, especially when missing segments contain critical frequency components. Meanwhile, channel-specific missing makes cross-channel shared decomposition problematic because different channels' missing patterns break the consistency of energy convergence. In such situations, we recommend frequency-domain interpolation to reconstruct missing segments utilizing partial MAFD results, forward/backward filling for short gaps, and variational inference-based joint estimation methods for severe missing scenarios.\n\nFurthermore, in future work, we plan to integrate uncertainty-aware diffusion mechanisms that handle missing observations through masked attention and develop variational inference frameworks for jointly estimating missing values and anomaly scores. This will enable PhysDiff to work directly on incomplete data without preprocessing.\n\n\n> **Q3:How is the dynamic trend influence factor determined or learned during training? Is it interpretable or tunable?**\n\n**A3:** The dynamic trend influence factor $\\gamma(t)$ is explicitly defined as $\\gamma(t) = \\sigma(10(1-\\sqrt{\\bar{\\alpha}_t}))$, where $\\bar{\\alpha}_t = \\prod \\alpha_i$ represents cumulative noise parameters (see in lines 190--193).\n\n**Determination Mechanism of the Dynamic Trend Influence Factor:**\n\n1. **Automatic Calculation:** $\\gamma(t)$ is automatically computed based on diffusion timestep $t$, requiring no additional learnable parameters. This design avoids overfitting risks while maintaining model interpretability.\n\n2. **Physical Meaning:** When $t$ is small (low noise stage), $\\sqrt{\\bar{\\alpha}_t}$ approaches 1, while $\\gamma(t)$ approaches $\\sigma(0) \\approx 0.5$, providing moderate physical guidance. When $t$ is large (high noise stage), $\\sqrt{\\bar{\\alpha}_t}$ approaches 0, while $\\gamma(t)$ approaches $\\sigma(10) \\approx 1$, providing strong physical guidance.\n\n**Interpretability and Tunability:**\n\nAs shown in Algorithm 1 (lines 14--32), we implement frequency-adaptive dynamic regulation: $\\gamma(t,\\omega) = \\sigma(10(1 - \\sqrt{\\bar{\\alpha}\\_t}) \\cdot A\\_\\omega)$, where $A\\_\\omega = \\exp(-|\\omega|^2/\\sigma^2)$ is the frequency response function. This design allows different frequency bands to receive adaptive physical guidance strength at different diffusion stages, maintaining both interpretability and necessary flexibility."}, {"review_id": "8aBuOR4c3k", "reviewer": "Reviewer_rchn", "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.", "questions": "- What is the exact definition of physical context information zt mentioned in Line 167? Could the authors provide a detailed\n   explanation of how physical priors guide the diffusion model?\n- 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\n   computational costs and model performance?\n- Could the authors provide a more detailed analysis of performance\nvariations across different datasets? Are there differences in recall and precision? What are the underlying reasons for these discrepancies?", "limitations": "Yes.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- Empirical validation of the model's effectiveness;\n- A case study in real-world scenarios highlights the practical\n   significance of the model.\nWeaknesses:\n-  Some formulas lack detailed explanations and analysis;\n- Frequency-domain feature extraction may incur computational\n   overhead, particularly for multivariate time series;\n- Performance improvements are not significant across five real-world\ndatasets.", "was_revised": true, "final_justification": "According to the authors’ rebuttal and additional experiments, I increase my score. Thanks for the additional explanation.", "rebuttal": "> **Q1: What is the exact definition of physical context information $z_t$ mentioned in Line 167? Could the authors provide a detailed explanation of how physical priors guide the diffusion model?**\n\n**A1:** The physical context information $z_t$ is a composite feature vector containing multi-dimensional physical prior information derived from our feature extraction module: $z_t = \\lbrace P_h(t), P_l(t), \\mathrm{HASPE}(t), E(x_t) \\rbrace$.\n$P_h(t)$ represents high-frequency components obtained from MAFD decomposition, capturing rapid transient changes and noise characteristics of the system; $P_l(t)$ represents low-frequency components, reflecting long-term trends and fundamental dynamic patterns; $\\mathrm{HASPE}(t)$ is the amplitude-sensitive permutation entropy value at the current time step, quantifying signal complexity; $E(x_t)$ is the physics-based energy function value derived from instantaneous frequency analysis, measuring the physical reasonableness of the current state. These components together constitute a comprehensive description of time series physical characteristics.\n\nThe conditional diffusion process integrates physical priors through multiple pathways. In routing attention integration, physical components guide attention weight calculation as following:\n$\\mathrm{Attention}(Q, K, V, P_h, P_l) = \\mathrm{softmax}\\left(\\frac{QK^\\top + g_h \\cdot QP_h^\\top + g_l \\cdot QP_l^\\top}{\\sqrt{d_k}}\\right)V$,\nwhere gating coefficients $g_h = \\sigma(QP_h^\\top)$ and $g_l = \\sigma(QP_l^\\top)$ dynamically emphasize frequency components.\n\nIn energy regularization, the composite loss function\n$L_\\mathrm{total} = L_\\mathrm{diff} + \\gamma \\cdot L_\\mathrm{MAFD-ASPE} + \\eta \\cdot E(x_t)$\nensures physical plausibility through energy constraints.\n\nIn conditional denoising, the noise prediction network $\\epsilon_\\theta(x_t, t, z_t)$ explicitly conditions on physical context, enabling physics-informed reconstruction.\n\nThis multi-level physical guidance mechanism ensures that the diffusion process can not only reconstruct statistically normal patterns but also maintain physical consistency. In industrial sensor data, physical constraints prevent reconstructions that violate physical laws (such as energy conservation, causality, etc.); in spacecraft telemetry, physical priors help maintain the reasonableness of orbital mechanics and system dynamics; in water treatment systems, physical guidance ensures continuity of fluid dynamics and chemical processes.\n\n> **Q2: 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?**\n\n**A2:** Frequency-domain methods indeed introduce some computational overhead compared to time-domain methods in specific operations, but demonstrate overall efficiency advantages in the complete anomaly detection workflow.\n\nIn signal analysis operations, time-domain methods have complexity \\$\\mathcal{O}(T)\\$, while frequency-domain methods require \\$\\mathcal{O}(T \\log T)\\$ due to the use of FFT, with an overhead ratio of approximately 1.3×. The MAFD decomposition is frequency-domain-specific with complexity \\$\\mathcal{O}(C \\cdot L \\cdot \\log L)\\$, where \\$C\\$ is the number of channels and \\$L\\$ is the sequence length.\n\nHowever, in pattern recognition, frequency-domain methods show clear advantages. Time-domain correlation calculations typically have \\$\\mathcal{O}(T^2)\\$ complexity, whereas frequency-domain methods that leverage frequency features require only \\$\\mathcal{O}(T \\log T)\\$, yielding an overhead ratio of 0.6×. For the full anomaly detection workflow, time-domain methods have complexity \\$\\mathcal{O}(T^2 \\cdot D)\\$, while frequency-domain methods reduce this to \\$\\mathcal{O}(T \\cdot D \\cdot \\log T)\\$, showing asymptotic advantage especially for long sequences.\n\nThe channel-independent strategy maintains separate decompositions for each channel, with computational cost \\$\\mathcal{O}(C \\cdot T \\cdot \\log T)\\$, incurring high memory usage but achieving a 2.1% performance gain. In contrast, the channel-shared strategy adopts shared basis functions, reducing the cost to \\$\\mathcal{O}(T \\cdot \\log T)\\$ with significantly lower memory usage and is used as the baseline configuration. The hybrid strategy achieves a balance, with complexity \\$\\mathcal{O}(C^{0.5} \\cdot T \\cdot \\log T)\\$, yielding a +0.8% performance gain at moderate cost.\n\nDespite slightly increased overhead in individual operations, frequency-domain methods provide meaningful acceleration in key steps of anomaly detection. The energy concentration property of frequency representations makes anomaly patterns easier to isolate, thus reducing computation in searching and comparison stages. Although MAFD’s cross-channel sharing introduces an additional 15% decomposition cost, it eliminates redundancy by performing unified multivariate processing, resulting in 60% memory savings.\n\n> **Q3: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?**\n\n**A3:** We provide detailed explanations focusing on precision and recall differences across datasets, as shown in the following table.\n\n| Dataset | Metric | SMD | MSL | SMAP | SWaT | PSM |\n|---------|--------|-----|-----|------|------|-----|\n| **OCSVM** | P | 66.98 | 50.26 | 41.05 | 56.08 | 57.51 |\n| | R | 62.03 | 99.86 | 69.37 | 98.72 | 58.11 |\n| | F1 | 73.75 | 66.87 | 51.58 | 72.11 | 57.81 |\n| **PC2** | P | 64.92 | 52.69 | 50.62 | 62.32 | 77.44 |\n| | R | 40.19 | 98.33 | 98.48 | 82.96 | 37.71 |\n| | F1 | 54.34 | 68.61 | 66.87 | 71.18 | 53.53 |\n| **HBOS** | P | 56.28 | 59.25 | 41.54 | 57.71 | 100.00 |\n| | R | 63.11 | 83.32 | 66.17 | 29.82 | 6.54 |\n| | F1 | 62.17 | 69.25 | 51.04 | 43.21 | 12.28 |\n| **LOF** | P | 57.69 | 49.89 | 47.92 | 53.20 | 53.90 |\n| | R | 99.10 | 72.18 | 82.86 | 96.73 | 99.91 |\n| | F1 | 72.92 | 59.00 | 60.72 | 68.65 | 70.02 |\n| **IForest** | P | 100.00 | 53.87 | 41.12 | 53.03 | 100.00 |\n| | R | 9.37 | 94.58 | 68.91 | 62.80 | 4.35 |\n| | F1 | 17.13 | 68.65 | 51.51 | 62.03 | 6.48 |\n| **LODA** | P | 59.02 | 57.79 | 51.51 | 56.30 | 62.22 |\n| | R | 66.18 | 95.65 | 100.00 | 70.14 | 40.17 |\n| | F1 | 62.40 | 72.05 | 68.00 | 62.54 | 56.05 |\n| **AE** | P | 69.22 | 55.75 | 39.42 | 54.92 | 60.67 |\n| | R | 98.48 | 96.66 | 70.31 | 98.20 | 98.24 |\n| | F1 | 81.30 | 70.72 | 50.52 | 70.45 | 75.01 |\n| **DAGMM** | P | 63.57 | 54.07 | 50.75 | 59.42 | 68.22 |\n| | R | 70.83 | 92.11 | 96.38 | 92.36 | 70.50 |\n| | F1 | 67.00 | 68.14 | 66.49 | 72.32 | 69.34 |\n| **LSTM** | P | 60.12 | 58.82 | 55.25 | 49.99 | 57.06 |\n| | R | 84.77 | 14.68 | 27.70 | 82.11 | 95.92 |\n| | F1 | 70.35 | 23.49 | 36.90 | 62.15 | 71.55 |\n| **BeatGAN** | P | 74.11 | 55.74 | 54.04 | 61.89 | 58.81 |\n| | R | 81.64 | 98.94 | 98.30 | 83.46 | 99.08 |\n| | F1 | 77.69 | 71.30 | 69.74 | 71.08 | 73.81 |\n| **Omni** | P | 79.09 | 51.23 | 52.74 | 62.76 | 69.20 |\n| | R | 75.77 | 99.40 | 98.51 | 82.82 | 80.79 |\n| | F1 | 77.40 | 67.61 | 68.70 | 71.41 | 74.55 |\n| **A.T.** | P | 100.00 | 51.04 | 56.91 | 53.63 | 52.01 |\n| | R | 3.19 | 95.36 | 66.69 | 59.94 | 82.18 |\n| | F1 | 6.19 | 66.49 | 71.65 | 71.59 | 64.55 |\n| **DCDetector** | P | 50.93 | 55.94 | 53.12 | 53.25 | 54.72 |\n| | R | 95.57 | 95.53 | 98.37 | 98.12 | 86.36 |\n| | F1 | 66.45 | 70.56 | 68.99 | 69.03 | 66.99 |\n| **SensitiveHUE** | P | 60.34 | 55.92 | 53.63 | 58.91 | 56.15 |\n| | R | 90.13 | 98.95 | 98.37 | 91.71 | 98.75 |\n| | F1 | 72.29 | 71.46 | 69.42 | 71.74 | 71.59 |\n| **DeepSVDD** | P | 64.98 | 10.53 | 29.73 | 59.11 | 74.05 |\n| | R | 64.77 | 100.00 | 17.09 | 93.53 | 50.64 |\n| | F1 | 64.88 | 19.06 | 11.45 | 72.44 | 60.15 |\n| **MTAD-GAT** | P | 85.90 | 54.96 | 39.05 | 65.90 | 79.90 |\n| | R | 67.69 | 94.93 | 93.99 | 77.51 | 60.14 |\n| | F1 | 75.71 | 69.81 | 55.08 | 71.23 | 68.63 |\n| **TFAD** | P | 56.32 | 54.96 | 39.05 | 60.38 | 79.14 |\n| | R | 97.83 | 94.93 | 93.99 | 81.96 | 71.63 |\n| | F1 | 71.49 | 69.81 | 55.08 | 69.53 | 75.20 |\n| **D3R** | P | 64.87 | 56.45 | 51.08 | 64.25 | 53.17 |\n| | R | 97.93 | 95.55 | 94.46 | 77.50 | 100.00 |\n| | F1 | 78.04 | 71.81 | 66.30 | 70.25 | 69.43 |\n| **PhysDiff** | P | **71.03** | **62.75** | **64.36** | **60.00** | **66.09** |\n| | R | **96.00** | **92.66** | **86.81** | **92.04** | **84.47** |\n| | F1 | **81.65** | **74.83** | **73.91** | **72.64** | **74.16** |\n\nPhysDiff demonstrates a consistent pattern of high recall (84-96%) across most datasets while showing significant precision (60-71%) compared to other methods. In details, the best overall performance occurs on SMD due to clear anomaly patterns in server monitoring data where physical relationships are well-defined. SWaT exhibits a low precision (60.00%) despite maintaining high recall (92.04%) because water treatment systems involve complex interdependencies that can trigger false alarms in our physics-guided routing mechanism.\n\nOur MAFD excels at comprehensive anomaly detection across different frequency bands, explaining the consistently high recall performance. In well-structured domains like server monitoring (SMD), physical priors align effectively with actual patterns, yielding high precision. In complex interdependent systems like water treatment (SWaT), physics-guided assumptions may introduce noise, reducing precision while maintaining strong anomaly detection capabilities.\n\nWe adopt affiliation-based F1 score, which fundamentally differs from traditional point adjustment (PA) methods used in many original papers. Traditional PA methods detect any point within an anomalous segment as successful detection of the entire segment, potentially leading to overly optimistic evaluations, while affiliation F1 computes precision and recall based on average directed distance between predicted anomaly events and ground truth, considering spatio-temporal adjacency."}, {"review_id": "otNXylTnOk", "reviewer": "Reviewer_pK7y", "summary": "This 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.", "questions": "1. 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.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "## 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.", "was_revised": true, "final_justification": "During the process of rebuttal, I believe:\n\n1. The authors clearly illustrated the definition of optimal depth and verified its effectiveness for method performance, demonstrating that the cross-entropy-like loss function can approximately find the optimal depth.  \n2. The authors clearly compare the different experimental settings, which leads to the performance differences between the results reported in this paper and the original paper. I believe their explanations are reasonable.\n3. The authors compare the performance of their method's performance on non-stationary datasets and stationary ones, verifying that the method can deal with the non-stationary issue as claimed.\n\nI think the following concerns remains unsolved, but considering the novelty highlighted by reviewer TJwG and the soundness verified in the rebuttal process and highlighted by reviewer gW7N, I think the reasons to accept outweigh the reasons to reject.\n\n1. I still think that for deep-learning-based methods, the non-stationary issue is not a huge problem. Since many previous methods have been reported very good performances on the non-stationary datasets (according to the stationary indicators provided by the authors, SMD and SWaT are non-stationary datasets). I attached the reported F1 scores of previous methods on these datasets in the end.\n2.  Limited comparative analysis in case study.\n\n\n\n*Reported F1 scores of deep-learning-based anomaly detection methods on non-stationary datasets*\n\n|                        | SMD   | SWaT  |\n| ---------------------- | ----- | ----- |\n| AnomalyTransformer [1] | 92.33 | 94.07 |\n| PUAD [2]               | 95.68 | ---   |\n| DCdetector [3]         | 87.18 | 96.33 |\n| DAEMON [4]             | 96.2  | 92.9  |\n| TranAD [5]             | 96.05 | 81.51 |\n\n[1] Xu J, Wu H, Wang J, et al. Anomaly transformer: Time series anomaly detection with association discrepancy[J]. ICLR 2022.\n\n[2] Li, Yuxin, et al. \"Prototype-oriented unsupervised anomaly detection for multivariate time series.\" *International Conference on Machine Learning*. PMLR, 2023.\n\n[3] Yang, Yiyuan, et al. \"Dcdetector: Dual attention contrastive representation learning for time series anomaly detection.\" *Proceedings of the 29th ACM SIGKDD conference on knowledge discovery and data mining*. 2023.\n\n[4] Chen, Xuanhao, et al. \"Daemon: Unsupervised anomaly detection and interpretation for multivariate time series.\" *2021 IEEE 37th International Conference on Data Engineering (ICDE)*. IEEE, 2021.\n\n[5] Tuli, Shreshth, Giuliano Casale, and Nicholas R. Jennings. \"Tranad: Deep transformer networks for anomaly detection in multivariate time series data.\" *VLDB* (2022).", "rebuttal": "> **Q1: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\n**A1:** Thank you for the suggestion. We conducted quantitative analyses using statistical indicators to characterize the non-stationarity dataset properties, including the ADF test p-value, KPSS score, variance instability, trend strength, mean shift, and structural breaks. \n\n| Dataset | ADF p-value | KPSS Score | Variance Instability | Trend Strength | Mean Shift | Structural Breaks | F1 Improvement |\n|---------|-------------|------------|---------------------|----------------|------------|-------------------|----------------------------|\n| **SMD** | 0.0368 | **0.9870** | **0.3968** | **0.4402** | **0.4331** | 0.0600 | **+0.35%** |\n| **SWaT** | 0.1248 | **0.9791** | **0.5763** | **0.2377** | **0.2485** | **0.2000** | **+0.32%** |\n| **MSL** | 0.0000 | **0.9000** | **0.1611** | 0.0003 | 0.0088 | 0.0000 | **+2.78%** |\n| **SMAP** | 0.0000 | **0.9000** | 0.0061 | 0.0055 | 0.0115 | 0.0000 | **+2.26%** |\n| **PSM** | 0.0000 | 0.9094 | 0.0215 | 0.0077 | 0.0080 | 0.0000 | **-0.85%** |\n\nAs shown in the table, PhysDiff achieves consistent positive improvements across all datasets exhibiting significant non-stationary characteristics.  SMD and SWaT demonstrate the strongest non-stationary properties, with PhysDiff improving F1-scores by +0.35% and +0.32%, respectively. On datasets MSL and SMAP，which exhibit distinct non-stationary patterns with high KPSS scores，PhysDiff achieves significantly greater F1-score improvements of +2.78% and +2.26%, respectively.\n\nThe technical effectiveness stems from our MAFD mechanism, which preserves both high-frequency transients and low-frequency trends through cross-channel shared basis functions. The ASPE criterion adaptively determines decomposition depth based on signal complexity, ensuring optimal representation of non-stationary characteristics. The physics-guided routing attention mechanism dynamically balances frequency components based on their anomaly relevance, proving particularly effective when anomalies span multiple temporal scales in non-stationary environments. Only PSM, which exhibits minimal non-stationary characteristics, shows negative performance (-0.85%). This validates our approach's specificity for handling complex temporal patterns rather than over-regularizing simple, stationary data.\n\nThis quantitative analysis  demonstrates PhysDiff's effectiveness on non-stationary time series data.\n> **Q2: 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\n**A2:** Regarding the analysis of how decomposition depth affects model performance, we provide explanations from the following aspects:\n\nAccording to the MAFD method described in lines 151-157, decomposition depth directly affects signal frequency resolution. Shallow decomposition may fail to capture multi-scale patterns, while overly deep decomposition may cause overfitting. Our ASPE mechanism dynamically assesses residual component complexity to determine optimal stopping condition.\n\nThe ablation study shows that fixed use of 8 or 16 decomposition components resulted in average F1 score decreases of 5.35% and 5.82% respectively (lines 281-290, Table 3). This proves that fixed decomposition depths cannot adapt to different datasets' characteristics.\n\nAs shown in Formula 4, ASPE determines stopping conditions by monitoring entropy changes in residuals. When residual complexity no longer decreases significantly, continued decomposition introduces noise rather than useful information, which is the core reason for proposing our dynamic depth determination mechanism.\n\n> **Q3: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\n**A3:** In our paper, the term “optimal depth” refers to the number of decomposition layers in the MAFD (Multi-channel Adaptive Fourier Decomposition) process at which the Amplitude-Sensitive Permutation Entropy (ASPE) of the residual component stabilizes and stops decreasing significantly (lines 159–162).\n\nThis “optimality” is defined with respect to balancing two objectives:\n\n1. Preserving physically interpretable components (e.g., trends and dominant cycles);\n\n2. Enhancing the distinguishability of anomalous patterns across scales.\n\nIf the decomposition is too shallow, important structures such as high-frequency spikes or periodicities might remain entangled in the signal, making anomalies harder to detect. Conversely, if it is too deep, the process might over-segment the signal and bury anomaly-relevant features in noisy or irrelevant high-frequency bands. Therefore, the goal is to decompose just enough to isolate meaningful frequency components without degrading anomaly separability.\n\nWhy does optimizing ASPE help us achieve this? ASPE is an entropy-based complexity measure that accounts for both ordinal patterns and amplitude variation. If the ASPE continues to decrease, it indicates the residual still contains structure or meaningful patterns. When the ASPE reaches a plateau, the residual is likely dominated by noise or unstructured variability — further decomposition would no longer add value and might even harm detection performance.\n\nThus, optimizing ASPE is equivalent to maximizing the information gain from decomposition, and the stopping point defines the depth that best supports the objective of interpretable and effective anomaly detection.\n\n> **Q4: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\n**A4:** We provide detailed quantitative analysis of computational and memory efficiency. Our analysis reveals that while diffusion models require substantial computational resources, PhysDiff achieves competitive efficiency among comparable methods while delivering superior performance. \n\nThe efficiency comparison demonstrates clear trade-offs between computational cost and detection capability across method categories: \n\n| Efficiency Tier | Methods | Training Time (s) | Performance Trade-off |\n|---|---|---|---|\n| **Ultra-efficient** | PCA (0.017), LODA (0.124) | 0.02-0.12 |Low accuracy, limited capability |\n| **Efficient** | LOF (0.289), IForest (0.301), DeepSVDD (0.256) | 0.25-0.30 | Moderate accuracy, scalability issues |\n| **Moderate** | OCSVM (1.173), LSTM (1.25), DAGMM (1.48), AE (3.701) | 1.17-3.70| Good accuracy, limited complexity handling |\n| **Resource-intensive** | DCdetector (4.81), MTAD-GAT (6.76), A.T. (9.31), BeatGAN (11.0) | 4.81-11.0 | High accuracy, complex patterns |\n| **High-end** | SensitiveHUE (16.26), TFAD (19.77), D3R (78.52), **PhysDiff (81.43)** | 16.3-81.4 | Superior accuracy, interpretability |\n\nPhysDiff's memory consumption of 81.43s represents only an 3.7% increase over D3R (78.52s), while providing substantial performance improvements and interpretability benefits. The additional memory overhead primarily accommodates decomposed frequency components and physical context information $z_t$, which enable our physics-guided reconstruction capabilities. \n\nOur channel-shared MAFD strategy achieves significant memory optimization, reducing requirements by approximately 60% compared to channel-independent decomposition approaches. This optimization transforms complexity from $\\mathcal{O}(C \\cdot T \\cdot \\log T)$ to $\\mathcal{O}(T \\cdot \\log T)$, providing favorable scaling properties for high-dimensional multivariate time series. The 3.7% computational overhead compared to D3R (81.43s vs 78.52s) yields disproportionate performance benefits: 3.61% F1-score improvement on SMD and 7.61% on SMAP datasets. \n\nThis efficiency-performance ratio demonstrates that PhysDiff's additional computational investment is well-justified, particularly considering the interpretable frequency-domain insights unavailable in baseline diffusion methods. \n\n> **Q5: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\n**A5:** We understand the reviewer's concern about baseline performance differences. The key differences arise from several aspects:\n\n**Evaluation Metrics Differences (lines 218–226):**\nWe adopt the affiliation-based F1 score as our primary performance metric instead of traditional point-adjustment (PA) methods. Traditional PA-based metrics tend to produce overly optimistic evaluations by inflating true positives and suppressing false negatives. In contrast, the affiliation-based F1 score provides a more rigorous evaluation by computing the average directed distance between predicted anomaly events and ground truth.\n\n**Experimental Setup Consistency (lines 232–235):**\nAll experiments were conducted under consistent settings using official implementations and recommended hyperparameters. We evaluated 18 state-of-the-art baseline methods spanning six major categories.\n\n**Dataset-Specific Challenges (lines 212–218):**\nOur evaluation is conducted on widely recognized benchmark datasets, including SMD, MSL, SMAP, SWaT, and PSM, along with  the NeurIPS-TS dataset. These datasets cover diverse domains (e.g., IT systems, spacecraft, water treatment, finance).\n\n**Implementation Environment:**\nWe detail our experimental environment in Appendix D.4, supporting full reproducibilit and eliminating ambiguit regarding implementation-related discrepancies."}]

Author rebuttal:

Meta-review: This paper presented an approach for diffusion-based time-series anomaly detection. It proposes to enhance the diffusion model using a on both a specific multi-variate spectral signal decomposition and especially tailored routing network and energy guidance. All reviewers were enthusiastic about this paper. The AC is concerned by the high complexity of the approach and the limited analysis and justification regarding non-obvious choices. This is compounded by the application to time-series AD which is known to be plagued with evaluation issues. However, as the reviewers all recommended acceptance, nearly all without reservation, the AC also recommends acceptance.
Decision: Accept (poster)