Point-wise Anomaly Detection via Fold-bifurcation ODE
Authors: Sheo yon Jhin
Noseong Park
Venue: ICLR 2026 Poster (2026)
Keywords: time-series
anomly detection
Primary area: learning on time series and dynamical systems
TL;DR: point-wise anomaly detection via fold-bifurcation

Abstract: Anomaly detection in time series is essential for applications from industrial monitoring to financial risk management. Recent methods --- including forecasting error models, representation learning, augmentation, and weak-label learning --- have achieved strong results for specific anomaly types such as sudden point or gradual collective anomalies. While many prior works report window-level metrics that may mask errors, several recent methods evaluate at the point level as well. Our goal is to use a stricter point-wise protocol to make masking effects explicit. We introduce FOLD (Point-wise Anomaly Detection via fold-bifurcation), a framework that reframes detection as tracking a system’s proximity to a critical transition. FOLD extracts stress signals from a forecasting model and integrates them with a fold-bifurcation inspired ODE to produce the risk state, flagging anomalies once it crosses a threshold calibrated on normal data. This requires no anomaly labels and no additional detector training, enabling a parameter-free and efficient detection process. By modeling anomalies as stress accumulation toward a tipping point, FOLD naturally aligns with point-wise detection, providing a unifying and interpretable perspective that complements type-specific methods. Experiments on 40 benchmarks against 34 state-of-the-art baselines show that FOLD achieves competitive or superior performance, with particular strength under strict point-wise evaluation.

Published as a conference paper at ICLR 2026
POINT-WISE ANOMALY DETECTION BIFURCATION ODE
VIA FOLD-
Sheo Yon Jhin, Noseong Park
KAIST
Daejeon, South Korea sheoyon.jhin @ kaistac.kr, noseong kckr
ABSTRACT
Anomaly detection in time series is essential for applications from industrial mon- itoring to financial risk management. Recent methods including forecasting error models, representation learning, augmentation, and weak-label learning - have achieved strong results for specific anomaly types such as sudden point gradual collective anomalies. While many prior works report window-level met- ries that may mask errors, several recent methods evaluate at the point level well. Our goal to use stricter point-wise protocol to make masking effects ex- plicit. We introduce FOLD (Point-wise Anomaly Detection via fold- bidubbifurcation) a framework that reframes detection as tracking a system's proximity to a criti- cal transition. FOLD extracts stress signals from forecasting model and inte- grates them with oold-bifurcation inspired ODE to produce the risk state, flag- ging anomalies once it crosses a threshold calibrated on normal data. This requires no anomaly labels and no additional detector training, enabling parameter- free and efficient detection process. By modeling anomalies as stress accumulation toward tipping point, FOLD naturally aligns with point-wise detection, provid- ing unifying and interpretable perspective that complements type-specific meth- 40 benchmarks against 34 state-of-the-art baselines show that FOLD achieves competitive or superior performance, with particular strength under striet point- wise evaluation.
ods. Experiments
Prediction-based Anomaly Delection
FOLD: point-uise Anomaly Detection
INTRODUCTION
Inpat
Signal Modeling
Sanah fres fareodinn modii
Atnernat Parts Detectel
Anomaly detection is a fundamental problem with broad impact in domains such indus- trial diagnostics, predictive maintenance, and risk management, where the ability to foresee failures is critical for enabling proactive inter-
Forecasting
Fild Sittrrataaaa laqpred OIE
vention (Chevtchenko et al. 2023 Rodríguez
al. 2023).
strew tostaic
Capturms Strees
manaitee via fals iiturcation
Recent methods are largely dominated by two paradigms (Paparrizos et al.) 2025): prediction- based approaches, which monitor forecasting reconstruction errors (Tuli al. 2022 Xu eet al., 202T) and distance-based approaches, which rely on representation learning and em- bedding similarity (Wang et al.) 2025 Deng & Hooi 202T). However, these paradigms share common limitation, they primaily capture sud- den stress, i.c., sharp deviations at individual timesteps. Prediction-bazs based methods detect in- stantaneous error spikes, while distance-basee based models flag sudden embedding shifts. Even when extended over longer horizons, they re- main sensitive momentary fluctuations rather
Anomaly Scare
Gased
(stroes dggaal <ta Sute-kiburcationn dymaesicn)
Figure 1: Contrasting conventional prediction- based anomaly detection with our proposed FOLD. While prediction-based models capture only instantaneous error spikes (sudden stress), FOLD extracts sensitivity- nals and integrates them via fold-bifureation dy- namies, modeling how stress accumulates into state transitions. This enables accurate and inter- pretable point-wise anomaly detection.
stress sig-

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Published as a conference paper at ICLR 2026
than modeling how stresses accumulate over time. This limitation is often masked by window-level evaluation, where detections are counted correct if they fall anywhere within an anomaly window.
Under stricter point-wise anomaly detection, which demands precise localization at every timestep, performance degrades substantially (Wang et al. 2024] Paparrizos et al. 2025 Wang et al. 2025), highlighting the importance of point-wise evaluation more faithful and challenging criterion for real-world anomaly detection. Many real-world failures can arise from the accumulation of stress that drives a system toward a critical transition. Importandy, our formulation captures both grad- ual build-up and short, abrupt spikes within the same dynamical framework. We draw inspiration from biduriiiurattion dynamics, classical theory dynamical systems that explains how gradual external pressure can drive system toward an abrupt transition from normal to failed states. In its canonical form, fold-bifurcation assumes fixed control parameter r, which represents exter- nal pressure, and studies how stable and unstable equilibria appear or disappear simply, the system remains stable until the equilibria collide and vanish, at which point sudden collapse occurs.
changes. Put
Adapting this principle, we reinterpret the control parameter as a time-varying stress signal S(t) extracted from forecasting model. By integrating these stress signals with a fold-bifurcation- inspired ODE, we calculate the risk state z(t). which captures how small stresses compound into a tipping point, i.e., critical threshold beyond which the system abruptly shifts from normal to failed behavior. An anomaly is flagged when z(() crosses a threshold calibrated solely on normal data.
Building this perspective, propose FOLD (Fold-bifurcation based Anomaly Detection), framework that reframes detection as modeling stress accumulation, unifying the detection of both sudden deviations and gradual drifts. A distinguishing feature of FOLD is that it can be instantiated directly from an already trained forecasting model without any additional detector training.
We highlight the following contributions of this work:
1. To our knowledge, we introduce FOLD the first anomaly detection framework that lever- ages fold-bifurcation inspired dynamics for point-wise anomaly detection, requiring no anomaly labels and no additional detector training.
2. We provide a principled formulation of anomaly detection as stress-signal driven modeling, where stress signals are integrated through folu-bifurcatton ODE to capture how gradual pressures can accumulate and trigger sudden tipping point transitions.
3. We conduct extensive experiments on 40 benchmarks against 34 state-of-the-art baselines, demonstrating that FOLD achieves superior performance in both threshold-depende (e.g., Point-wise FI) and prreshold-ineppeddeer (c.g.. VUS-PR) metrics. This validates the robustness and practical value of our framework under strict point-wise evaluation pro- tocols.
2 RELATED WORK
2.1 TIME-SERIES ANOMALY DETECTION
Recent advances in time-series anomaly detection can be broadly categorized into two dominant paradigms (Paparrizos et al. (025): prediction- based methods, which monitor forecasting or recon- struction errors (Tuli al. [2022] Su et al. 2019; Zhang et al. 2019), and distance-based methods, which rely on representation learning and embedding similarity (Xu et al. 2021 Deng & Hooi, 2021 Wang et al., 2025). While effective under conventional benchmarks, both paradigms share key limitation: they primarily capture sudden stress, i.e., sharp deviations at individual timesteps. Prediction-based approaches flag error spikes, while distance-based methods detect sudden shifts in representation space. Even when extended to longer horizons, they remain sensitive momentary fluctuations rather than modeling how stresses accumulate over time.
This limitation is often masked by window-level evaluation, where detection is considered correct falls anywhere within an anomaly window. However, under the stricter point-wise anomaly detection setting, which requires precise localization at each timestep, these methods degrade sig- nificantly (Wang et al.) 22025) 2024). This explains why many prior approaches report strong results on lend metrics but fail to generalize under point-wise evaluation (Paparrizos al. 2025
2

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Published as a conference paper at ICLR 2026
Beyond deep neural methods, classical approaches such Isolation Forest (Liu et al. 2008 and one- class SVMs (Schölkopf et al., (001)) remain widely used in practice, while dynamical change-point detection methods emphasize carly-warning indicators in noisy systems. Early-wammn also drawn on bifurcation theory, for example by fitting autoregressive models to estimate Jacobian eigenvalues and track critical slowing down in multivariate settings (Williamson & Lenton 2015). These approaches highlight complementary perspectives, but they are not directly optimized for strict point-wise anomaly detection in complex multivariate data.
studies have
Building on these perspectives, FOLD leverages forecasting-derived sensitivity and uncertainty sig- nals time-varying stress input a aalubbifurattion ODE, enabling point-wise anomaly detection through risk-state thresholding. This avoids training an additional detector and instead operates di- rectly on top of pre-trained forecaster.
2.2 UNCERTAINTY ESTIMATION IN NEURAL TIME-SERIES MODELS
Quantifying predictive uncertainty has become increasingly important in time-series modeling, es- pecially for safety-critical domains where abnormal behavior must be distinguished from normal variability. Practical approaches include Monte Carlo Dropout (Gal & Ghahramani] (2016), which can be interpreted as Bayesian approximation, and deep ensembles (Laakshminarayanan et al.) 2017). which provide scalable estimates of both epistemic and aleatoric uncertainty, These methods not only improve the reliability of point forecasts but also create signals that can be exploited for detecting abnormal unstable regimes.
Uncertainty has since been directly leveraged in anomaly detection. (Li et al. 2018) used generative models to capture predictive variance in multivariate time series, while (Wiessner et al. 2024) pro- posed explicit uncertainty-aware detectors, highlighting that high variance is often associated with abnormal or unstable regimes. Beyond machine learning, evidence from complex systems further supports this connection: (Scheffer et al., (209)) showed that variance systematically increases as system approaches critical transition, suggesting that rising uncertainty itself can be interpreted as a form of accumulated stress.
In FOLD, we adopt this perspective and treat predictive uncertainty itself stress indicator, com- bining it with sensitivity signals, extending prior variance-basee -bsen approaches into unified stress formulation that anticipates anomalies.
2.3 BIFURCATION THEORY AND DYNAMICAL SYSTEMS PERSPECTIVE
Bifurcation theory studies how systems can remain stable under gradual change until critical point (or tipping point) reached, after which they suddenly shift qualitatively different state. A simple example is the fold-bifurcation, where the system has two equilibria (one stable, one unsta- ble) that move closer together as external pressure increases. At a critical threshold, these equilibria collide and vanish, causing the system to abruptly lose stability. Mathematically, this behavior is captured by the canonical equation:
dz(t) = -r-
dt
where z(t) is the system state and r is a control parameter. Such tipping phenomena have been widely studied in domains such ecology and climate science (Scheffer) 2009 [enton al., 2008). To capture resilience in real systems, this canonical form is often extended with decay term z(t):
dz(t) di
z(()² z(t).
While other bifurcation types (e.g., Hopf or pitchfork) describe oscillatory symmetry-breaking transitions, the bidubiiuurttion is particularly suited for anomaly detection because it directly cap- tures the gradual erosion of stability followed by abrupt collapse. Related carly-warning studies have exploited this property by fitting autoregressive models and tracking eigenvalue changes to detect critical slowing down in noisy multivariate systems (Williamson & Lenton 2015).
3

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Published as a conference paper at ICLR 2026
Parameterized Stress Signal Modeling
Forecasting
Y
Model
6- D(F [ar((F
Var(Y)
seruitivity
651 I...,
Freezed
I(O)
Forecasting
Y::
Viz
Fold- Bifurcation inspired ODE
MC Dropout
dx(tt)
aS(() (t)
(z(t)²
yz(()
dt
Xaut
Figure 2: Overall architecture of FOLD.
Building on this adapted form, we reinterpret as parameterized time-varying stress signal S(t)
and obtain our bold-bifurcation inspired dynamics:
dz((t) = aS(() - Bz(())² -
(3)
dt
where a, B, are fixed coefficients. This formulation preserves the key intuition stress gradu- ally accumulates, resilience erodes, and a sudden transition occurs while adapting it for anomaly detection. Unlike carly-warning approaches that monitor eigenvalue trends, FOLD instantiates fold-inspired dynamical mechanism driven by forecasting-derived stress signals to localize anoma- lies at the point level.
PROPOSED METHOD
3.1 OVERALL WORKFI.OW
The workflow of FOLD consists of the following three main steps (cf. Figure 2h:
1. A forecasting model fe is first trained on normal sequences using mean squared error (MSE) loss. After training, fo is frozen to serve fixed backbone (pink box).
2. For a test sequence, patches are masked and fed into the frozen fo with MC dropout. Sensi- tivity and uncertainty are combined to form time- marying stress signal S(t) (yellow box).
3. The stress signal S(t) is injected into bidubiffurattion inspired ODE to evolve the risk trajectory z(t). Anomalies are flagged once z(t) leaves its stable basin (gray box).
3.2 FORECASTING MODEL AND STRESS SIGNAL SCORING
Let X [xi........ zr. E R be an input time window of length L with d features. We begin with a forecasting model Jo RLxd HR trained solely on normal data using mean squared error (MSE) loss. The model is implemented with dropout layers that it can later provide Monte Carlo (MC) estimates predictive uncertainty. After training, fe is frozen for all subsequent steps.
Perturbed prediction K,- The input sequence = [.r..... ør. is divided into N patches [P.), For each patch P., we mask out (i.e., replace with mask token or zero) the corresponding segment and feed the modified sequence XxP, into the frozen forecasting model so. V, = (o(). (cf. Figure
Sensitivity term To measure how much the forecast depends given patch, compute distance between the baseline prediction ý fo(X) and the perturbed prediction Vú: D(R Y), where D(.-)) -) is distance metric (e.g., MSE, MAE, and cosine similarity). The rationale is to capture how strongly each local patch influences future predictions - patches whose removal causes large deviations Y are assigned higher sensitivity. We provide an ablation in Appendix E. to show the effect of different distance metrics.
4

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Published as a conference paper at ICLR 2026
Uncertainty term. To caputre how unstable the model becomes under local perturbations, estimate predictive variance via Monte Carlo dropout. For each input, we run TMC stochastic for- ward passes: Var(Y,)) - Var(Y) The variance difference measures the additional epistemic and aleatoric uncertainity introduced when the patch is masked. The of MC dropout is motivated by Bayesian approximations in neural networks (Lakshminarayanan et al. 2017). as discussed in Section 2.2
Stress Signal.
Combining the two components yields the patch-level stress score:
4 8. D(Y Y + A
- -
G Rexd
sensitivity term
uncertainty term
where 8, A 0 control the relative importance of sensitivity and uncertainty. Ablation studies in Section confirm that both terms are complementary: omitting either degrades performance.
Sequence-level stress signal. Finally, we aggregate patch-level scores into varying signal aligned with the original input:
S(t) = I(O)
Σ EEZ())
Esr S(t) E Rd, t = 1, I....,,.
(5)
where Z(t) is the set patches covering index This produce stress signal S E 4xd with the same temporal and feature dimensions as the input sequence, serving as the data-driven analogue of external pressure in the aid-bifurcation dynamics.
3.3 -ILL-IIFURCATION INSPIRED ODE MODELING
Real-world systems often exhibit tipping-poii point behavior: they appear stable while absorbing small stresses, but once a critical threshold crossed, they abruptly transition to a failed state. This phenomenon is mathematically captured by fold-bifurcation, where the canonical equation
dz(() dt
=tt
shows how gradual changes in a control parameter r can eliminate equilibria and precipitate sudden collapse. Among canonical bifurcations, we adopt the fold (saddle-node) because our targets abrupt losses of stability under monotone external pressure rather than oscillatory onsets (Hopf) or symmetry breaking (pitchfork). The fold directly captures equilibrium annihilation under gradually increasing pressure, which matches point-wise anomaly onset in practice.
FOLD builds directly on this intuition. Consistent with the stress-centric view in Section 2.3 reinterpret the fixed parameter r as the time-varying stress signal S(t) derived from the forecasting model. Conceptually, just increasing r leads collapse in the canonical form, a rising stress signal S() acts dynamic external force that drives the system state toward instability. This yields our bolu-bifurcation inspired dynamics of the risk state z(l) Rd:
dz(tt = aS(() -z(t)² -z((),
(7)
dt
where the dynamics are computed feature-wise over dimensions, resulting in full trajectory z E Lxd across the window. To obtain a unified anomaly indicator for detection, we aggregate these feature-wise trajectories into sesee risk score Zays R.
Interpretation coefficients. In this Eq. 7 Stt) replaces the constant control parameter r and acts data-driven analogue of external pressure, injected feature-wise from the forecasting model. The coefficient a > scales stress injection, ? > provides resilience by pulling the system back toward stability when stress subsides, and 8 0 induces nonlinear escalation, where accu- mulated risk amplifies disproportionately near tipping point. Together, these terms instantiate a aod-bbifurcatio inspired mechanism of stress accumulation leading critical transitions.
5

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Published as a conference paper at ICLR 2026
Numerical solution. Since Eq. has closed-form solution, we compute z(t) using standard adaptive ODE solvers (e.g.. Lange-Kuttall These solvers provide stable integration under time- varying stress signals, ensuring that anomaly detection reflects the dynamics of the model rather than artifacts of discretization.
Free anomaly detection. The coefficients (a, 3, act as fixed hyperparameters, selected based on data statistics rather than learned from anomaly labels. Anomalies are flagged once z(t) exits its stable basin under the influence of S(t). This realizes the principle of parameter-free detection: given a forecasting model, anomalies are identified directly through principled dynamical modeling without any detector-specific optimization. Furthermore, when integrated with pre-trained founda- tion models (e.g.. Chronos) demonstrated in experiments, this framework extends to fully training-fii free (zero-shot) anomaly detection pipeline, eliminating the need for backbone training as well.
3.4 ANOMALY CRITERION
To derive a stability boundary, we simulate Eq. 7 on normal training data. Because the forecasting model fo involves stochasticity (e.g., dropout), we generate multiple risk trajectories zıt) (or Zays (t) for multivariate settings) by varying the random seed. For each normal risk states z(t), we then record its maximum value, yielding
Marai = {max
X
E
train
(8)
Following the fold-bifurcation intuition, an anomaly should only be declared once the risk state exceeds the typical maxima observed in normal regimes. To capture this, we define the threshold as a high quantile of the normal maxima with a small multiplicative margin:
Zhr (1 p) Quantile,,MMaai
p 0.95-0.99, P 0.05.
The quantile term ensures robustness against rare fluctuations, while the margin reflects the bi- furcation principle that the system must not merely touch but clearly surpass the stability boundary before being considered anomalous. Conceptually, we interpret this calibrated threshold Zehr as the empirical boundary of the stable basin. Therefore, the event z(t) Zeh- syss (t) > Zehr for mul- tivariate settings) serves as the operational criterion for determining that the system has overcome its resilience and crossed the tipping point.
Point-wise decision rule. wise anomaly mask:
For test sequence X with trajectory Zaya (t), then produce point-
(()) (2sys Zun), 1= L.
(10)
All metrics (precision, recall, and F1) are computed at the timestep level. This rule provides a statis- tically grounded and label-free boundary for point- wise anomaly detection, as illustrated in Figure To further assess robustness of the calibrated threshold Zur. we conduct an ablation study under E-
contamination, with results reported in Section 5 and Appendix E.4 (Note: While this threshold
enables binary decisions, we primarily evaluate performance using the threshold-independeen VUS-
metric in Section 4 demonstrate the model's intrinsic robustness.)
EXPERIMENTS
We conduct a comprehensive evaluation on the TSB-AD benchmark (Liu & Paparrizos 2024). a widely recognized leaderboard comprising 40 curated datasets (1,070 time series in total). To ensure a rigorous and unbiased assessment, we adopt VUS-PR (Volume Under the Surface of Precision-Recall curve) (Paparrizos et al. 2022) as our primary evaluation metric. Unlike stan- dard Fl-scores, VUS-PR is threshold-independenn effectively eliminating potential biases arising from specific threshold selections. For completeness, we also provide threshold- deneeddend results (e.g.. Point-wise F1). This stricter protocol replaces window-leved scoring common prior work. See Appendix B for dataset, hyperparameter, and evaluation details.

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Published as a conference paper at ICLR 2026
Table 1: VUS-PR score (Higher is better) averaged over all time series for each dataset for univariate anomaly detection. The best VUS-PR are shown in bold and the second-best in italic Implementation details about FOLD(Chronos) are in Appendix B.5
RANK
2005
5ND
2016
SAHOO
Methed
0.47
11.11
11:2
8:94
0.27
1187
1,30
1.801
0.34
0.28
2,55
TSP (PT
0.49
0.51
0.67
0.63
0.01
0.23
0.51
0.54
0.66
0/08
0.07
0.74
0.56
1.10
0:44
1.101
0,68
0.75
0.81
(0:1
4.65
Sub- PCA
0.25
0.42
0399
0.25
0.55
0.01
0.36
0.51
0.44
0.91
0.88
0.10
0.52
01.55
0.52
0.30
0.84
03.5
0L54
0.12
11.99
0.14
13,39
KSkapsAD
0.25
0.64
0.33
0.09
883
0.02
0.55
0.57
0U34
0.55
0.19
10.9
0L56
01.15
0.43
5075
0..11
0.75
8:38
11,0
0.55
13,90
0.23
0.31
0.34
0.10
0U29
0.16
0.61
0.44
D382
0.37
0.15
Series20insh
ASSMENTOFT (FT)
0.38
0.51
0.33
0.58
0445
0.00
0.33
0.5
0.30
0.73
0.00
0.67
0.06
0.75
6.23
0.06
5.81
0.5c
0.58
0.8
0.25
14.00
AGMMNNT
0.90
0153
0.41
0.37
0.34
0.71
mm
D/68
0.6
0.
6.0
0.9
0.58
0.49
0.71
14.91
USAD
0112
0.15
0.38
0.71
0.8
0.66
0.51
018
0.00
15.55
SN-KNN
0.24
0101
0:47
0.10
035
8.2
0.36
0.51
0.39
0.31
070
0.31
037
0.11
0.56
0.30
1150
0A1
8:77
1130
6.31
15.87
Atarafbotiz
0.96
0.0
0.50
0.00
11.9
0.30
0.18
0.67
0.2
8:13
672
0-17
13
0.36
6111
0.92
0.26
0.91
0.43
CNS
0.32
0.40
0.36
042
0.01
0.15
0.3
0.10
0.73
0.0
6:08
0.00
0.34
0.55
0.2
0.68
091
180
0.54
045
0.24
0.5
13.50
LSTMAD
0.33
015
0.73
0.30
0.35
066
0.12
0.82
0.18
027
0.5
0.07
025
0.1
0.67
0.85
180
047
010
01.3
0.45
16.47
023
0.38
0.07
0.50
0.10
000
608
1.0
0.64
0.07
0.61
0.36
[Fues
098
0.36
0.67
0.38
0.34
0.00
0.10
9.9
0.22
0.59
0.5
0.08
630
0.5
0.3g
009
0.50
0.59
6.99
0.53
000
0.34
0.44
17.21
0.12
8.16
0.33
0.30
0.31
0.00
0.10
0.5
0.19
035
0.00
0.07
0.6
0.5
0.3
0.0
0.4
0.56
0.44
083
0.34
0.10
19.30
Lhrma
0.00
0.31
0.18
1.56
0.34
0153
0.14
958
180
0260
017
0.30
21.66
TimeeNe::
0.00
0.39
0.33
0.22
0024
800
031
0.30
105
Dos
616
0038
835
611
0076
091
0,99
010
11.77
0.2
0.18
0189
0.36
0.25
089
000
022
0.33
012
644
0.9
0116
0.31
0.38
802
0.5
0A3
019
0.3
0.7
0145
1137
1.20
035
0.34
0
110
015
0.0
0.00
505
0.55
010
0.08
21.31
031
0.6
0.25
0.11
034
0.36
0.35
0.32
0.5
012
014
0.5
0.00
0.1
011
826
0.2
0.53
0.2
063
0.27
19:80
0088
SA-MCD
0.66
0.31
5.66
0.1
0.06
0.55
0.35
0.38
23.05
S.b HBOS
0.1
0.05
0.45
0.05
069
0.0
0.17
0.25
6.30
0:33
0.6
0.13
8:0
0.55
0.24
6.12
030
0.5
0/4
0.14
001
0.00
31.91
0.3
8.06
0.78
0.00
0.31
001
0.14
0.78
0.56
0.36
11
0.1
0106
0.11
016
0.20
0.09
0.71
0.2
0/5
0.18
063
0.3
32.00
Dani
005
0.68
0.:0
0.30
9.23
0.61
18
0.09
0.16
0.29
0.47
077
041
0m1
R156
6.12
23.91
LOF
0.06
0:15
0.20
0.12
0.25
0.00
0.26
0.5
0.17
0.35
0.16
0009
0.11
0.13
0.15
0.12
0075
091
049
010
R69
0.37
26.44
Juunaay Trandonee
0.55
0.07
0.06
0.99
0.14
0.23
0.29
0.09
0.18
007
005
0.46
0/L
0.07
21.36
Table 2: VUS-PR score (Higher is better) averaged over all time series for each dataset for multivariate anomaly detection.
OPPORTUNTIY
Credit Cand
Avg. RANK
CATS22
Dophoed
Esathlnn
GEOCO
GHL
Genesis
LTDB
MITTE
MSL
PSM
SMAP
SMD
SVDR
SWHE TNO
Method
FOLD (ours)
0.23
0.19
0.39
0.93
0.08
0.07
0.28
0.37
0.09
0.50
0.82
0.19
0.45
0.46
0.39
0.50
0.97
3.11
TSPalse (PT)
0.07
0.00
0..35
0.91
0.18
0.001
0.02
0.57
0.14
0.21
0.07
0.14
0.32
0.36
11.47
0.14
0.93
9.88
TSPulse (2S)
0.05
0.00
0.35
0.89
0.17
0.01
0.01
0.36
0.07
0.20
0.07
0.14
0.30
0.35
0.38
0.13
0.93
12.17
CNN
0.08
0.02
0.21
0.68
0.03
0.02
0.10
0.35
0.19
0.35
0.19
0.41
1.00
7.52
OmniAnoomlly
0.02
0.34
0.84
0.02
0.07
0.11
0.12
0.15
0.81
PCA
0.12
0.10
0.13
0,95
0,20
0.01
0.02
0.21
0.07
0.15
0.30
0.16
0.09
0.56
0.11
0.45
1,00
9.11
LSTMAD
0.04
0.02
0.31
0.82
0.00
0.06
0.04
0.30
0.09
0.22
0.17
0.24
0.16
0.33
0.15
0.16
0.99
9.41
USAD
0.04
0.02
0.34
0.84
0.02
0.00
0.41
0.12
0.23
0.18
0.19
0.11
D.16
0.32
0.15
081
11
AutoEncoder
0.06
0.05
0.13
0.05
0.05
0.22
0.06
1.00
KMeansAD
0.12
0.02
0.30
0.37
0.06
0.03
0.06
0.06
0.20
0.16
0,86
8.58
CBLOF
0.06
0.03
0.10
0.86
0.03
0.02
0.02
0.20
0.04
0.21
0.14
0.19
0.14
0.22
0.07
0.29
1.00
13.11
MCD
0.13
0.06
0.14
0,80
0.03
0.01
0.06
0.21
0.04
0.25
0.17
0.26
0.30
0.26
0.07
0.54
1.00
10.11
OCSVM
0.08
0.02
0.06
0.33
0.04
0.04
0.08
0.20
0.04
0.22
0.12
0.19
0.12
0.28
0.06
0.44
0.81
12.64
Donut
0.02
0.66
0.75
RobasPCA
0.02
0.06
0.77
0.02
0.03
0.07
0.08
0.12
1,00
FITS
0.13
0.02
0.33
0.63
0.03
0.01
0.10
0.23
0.05
0.17
0.05
0.13
0.08
0.17
0.10
0.15
0.78
15.41
OFA
0.13
0.02
0.31
0.58
0.04
0.01
0.22
0.29
0.06
0.14
0.05
0.17
0.08
D.17
0.12
0.12
0.77
14.52
FIF
0.06
0:02
0.15
0.41
0.04
0.02
0.06
0.19
0.04
0.18
0.10
0.18
0.13
0.32
0:07
0.32
0.39
14
COPOD
0.05
0.05
0.11
0.40
0.04
0.03
0.08
0.19
0.07
0.31
0.99
12.52
IForest
0.05
0.03
0.13
0.35
0.04
0.05
0.07
0.93
53.52
HBOS
0.05
0.04
0.15
0.32
0.04
0.04
0.08
0.21
0.04
0.33
0.17
0.17
0.09
0.25
0.07
0.30
0.83
12.88
TimesNet
0.07
0.02
0.27
0.12
0.03
0.01
0.02
0.27
0.07
0.17
0.06
0.14
0.09
0.14
0.11
0.14
0.79
15.88
KNN
0.07
0.02
0.25
0.33
0.11
0.01
0.44
0.19
0.04
018
0.06
0.12
0.12
0.30
0.06
0.11
0.78
1682
TraADD
0.02
0.10
0.02
12.05
LOF
0.05
0.02
0.11
0.16
0.13
0.01
0.04
0:10
0.15
0.09
0.16
0.06
0.15
0.79
17.52
AnemalyTannffmer Transformer
Anemaly
0.03
0.02
0.07
0.10
0.02
0.03
0.01
0.21
0.05
0.12
0.07
0.21
0.06
0.07
0.08
0.18
0.77
19
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Published as a conference paper at ICLR 2026
025
Thrabeed
Thershold
0144
Statt Blak atato
Mallin MM
Thur Azasnads
Fradicted Anoraat
042
Tras Ansenaly
Producted Anceuity
12.30
62.0
Time
SMAP
(b) WADI
Figure 3: Illustration of threshold calibration three datasets. For each dataset, the top panel shows the trajectories of risk states sye (t) on normal training data, with the red dashed line indicating the chosen peeceneecettiee threshold Zur- The bottom panel shows the risk state trajectories test data, where anomalies are flagged once Zsya(1) exceeds this threshold.
M mmmmi
Malabbryly wW MaN MM
" Wpmny
mm
(a) SMD
(b) NAB
(c)
UCR
SMAP
Figure 4: Visualization of anomaly detection results 4 benchmark datasets. Additional visualiza-
tions are provided in Appendix
4.1 EXPERIMENTAL RESULTS
We conducted a comprehensive evaluation on the full TSB-AD benchmark suite (40 datasets) using the threshold-indepeendent VUS-PR metric. Additional experimental results are provided in Ap- pendix C
Univariate Performance and Foundation Model Synergy. FOLD achieves the best average rank of 3,8 3,86 across 23 datasets, significantly outperforming the runner-up TSPulse (Avg Rank 8.65). This validates our mechanism's 's dual capability: high-magnitude stress S(t) instantly overcomes damping
to detect abrupt spikes (e.g.. NAB), while the nonlinear term - =B2² integrates gradual accumula- tion (e.g., SMAP) to trigger tipping points. Furthermore, integrating FOL.D with Chronos (Ansari) al. 2024) in a zero-shot setting yields an even superior rank of 2.95. This demonstrates that our framework acts as a universal detection mechanism, effectively transforming probabilistic outputs whether from lightweight backbones or foundation models into robust external force that drives dynamical state transitions without additional detector training.
Multivariate Performance. In the multivariate track, FOLD maintains the top position with an average rank of 3.11, surpassing deep learning baselines such as CNN (Avg Rank 7.52). While deep learning methods often struggle with noise amplification in high-dimensional spaces, FOLD leverages the forecasting backbone to encode inter-variable dependencies into the stress signal. By aggregating these feature-wiss risk dynamics into a system-level score, FOLD effectively filters out isolated channel noise while amplifying synchronized stress events. This results in superior scalability and stability on complex, highly correlated systems like SWaT and OPPORTUNITY, where maintaining low false positives is crucial.
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Published as a conference paper at ICLR 2026
Overall Robustness. The analysis highlights key distinction: while statistical baselines (e.g., Sub-PCA) rely on rigid linearity assumptions - performing well only on simple stationary datasets FOLD's 's dynamical formulation naturally adapts nonlinear transitions actoss varying domains. Furthermore, compared to foundation model-based methods like MOMENT or TSPulse, FOLD achieves higher consistency without the computational burden of extensive pre-training fine- tuning. This suggests that principled dynamical mechanism can be more effective and efficient than purely data-driven scale capturing fundamental anomaly characteristics.
Tondo POLR
Piadee daeiah 2015
(a) SMAP(FOLD)
(b) NAB(FOLD)
(c) UCR(FOLD)
Figure 5: Visualization of uncertainty assessments to demonstrate reliability and confidence. The blue line represents the input data, and the shaded region indicates the uncertainty band (derived from MC dropout). Note that the uncertainty band remains narrow during normal states (indicating high confidence) but significantly widens at the onset of anomalies, triggering the risk accumula- tion mechanism. This visually confirms that FOLD's detections driven by model-intrinsic risk assessment rather than random fluctuations.
Visualizations and Uncertainty. Figure 4 demonstrates FOLD's detection performance across diverse failure modes. The model accurately tracks gradual stress accumulation (e.g., SMAP) as well abrupt spikes (e.g., NAB, UCR), effectively triggering alerts when the risk state (1)) Zays for multivariate setting) crosses the threshold. To further validate reliability, Figure 5 explicitly visualizes the predictive uncertainty bands (shaded regions derived from MC dropout). As shown, these bands remain narrow during normal states (indicating high confidence) but significantly widen the onset of anomalies. This visual evidence confirms that FOLD's detections are driven by model-intrinsic increase in risk and uncertainty, rather than random fluctuations, ensuring robust decision-making.
ABLATION AND SENSITIVITY STUDIES
design the stress signal by combining two terms: sensitivity, which reflects how fragile forecasts are to local perturbations, and uncertainty, which captures how unstable the model becomes under such perturbations. Both have been independently validated in prior work as meaningful indicators of anomalous behavior. ablation study confirming their complementary roles. Removing the uncertainty term leads to excessive false positives, as the model reacts strongly to transient fluctuations. Con- versely, removing the sensitivity sharp deviations. Only when combined do the two signals yield a balanced and robust quantifica- tion of stress, enabling the full FOLD model achieve the highest Fl-scores across benchmarks. This result substantiates our formulation: anomalies are best captured when both local fragility and systemic instability are jointly considered as drivers of stress accumulation.
Impact of stress signal components. As motivated in Section 3.2
causes under-detection, since uncertainty alone cannot capture
Calibration robustness. Since our anomaly criterion in Section relies on a calibrated threshold Zha de- rived from normal data, it is important to verify that the method remains stable when calibration data are partially contaminated. We probe the stability of the calibrated threshold Zehr against e-contamination. where an fraction of normal calibration windows placed by anomalous ones. For each E {0, 3, 10] recompute Zi,, and report the relative shift AZ (Ze, Zine)/ / Zuhr and the resulting pisnt-wise Fl on the test (no re-tuning of p, p).
Table 3: Effect of calibration contamina- tion threshold stability and detection performance on SMAP (S-1).
(%)
Znr
AZ (%)
F1
AFI (pp)
0.5982
0.8095
0.6105
+2.05
0.7918
-2.19
0.7075
+18.3
0.5366
-33.7
10
0.7204
+20.4
0.3832
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Published as a conference paper at ICLR 2026
1a.p--1.y--000 fa/t=00,-=0.00
a 0, 0.0 y.9 0.000)
Teus Asomaayy
300
1000
1300
2000
2000
3000
3300
0.4
0.75
Asomally
0.50
MAN
0.2
0:26
Dataneds
SMD
1300
Time Steps
2000
3500
Figure 6: Sensitivity of FOLD to ODE parameters a, 8, 7. (left) Fl-scores across benchmarks for two representative settings. (right) Example risk trajectories under different values, showing that B = 1.0 yields more informative dynamics.
Table 4: Ablation study on stress signal components.
NAB
UCR
SMAP
P
F1
P
R
F1
0.2367
FOLD (wlo Uncertaintt)
0.2083
0.0781
0.1136
0,3417
0.4116
0.1761
0.1879
0.0942
0.0655
Sensitivity) POLD (Full Model) 0.3519
0.3491
0.3532
0.6875
0.4245
0.4267
0.4058
0.2542
0.6820
0.7059
0.6013
Even a 10%. FOLD retains an Fl of 0.3832 on SMAP (S-1), dhich,despite the drop from clean
calibration, remains substantially higher than strong baselines on the same machine (e.g.. NRDetec- tor 0.2032, TranAD 0.0080).Small contaminations (s 3%) shift Zuee by only few percent and lead to modest Fl changes, consistent with the high-quantile stability we exploit.
Sensitivity to ODE parameters. We further analyze the role of the coefficients a, B,, y the fold- bifurcation inspired ODE (Eq.
7 Figure 6 compares two representative settings: (a 1.0, = 1.0, 0.001) versus (a I.0, B 0.0.y 0.001). The results highlight the importance of the nonlinear escalation term 2: when = O, the risk state z(t) grows monotonically with accumulated stress, essentially acting as simple integrator; in contrast, 3 1.0 introduces nonlin- ear suppression that yields more structured and interpretable trajectory. allowing z(() (or Zays (t) for multivariate settings) to capture meaningful rises and falls around anomalous intervals. Across benchmarks (left), = 1.0 consistently leads to higher Fl-scores, and the temporal plots (right) confirm that the dynamics with B 1.0 produce more discriminative risk states. This demonstrates that FOLD's performance does not hinge delicate parameter tuning, but that incorporating non- linear escalation (3 = 1.0) is key to realizing the full benefit of bidurrittr dynamics. (In Appendix E.2
we report various sensitivity studies a, 7.)
CONCLUSION
We presented FOLD, hadd-bffurcation-insieed framework for point-wise anomaly detection. Instead of relying on reconstruction or prediction errors, FOLD models how sensitivity- and uncertainty-based stress signals accumulate through a simple ODE to trigger a tipping-point tran- sition. Once a forecasting model is trained on normal data, no anomaly labels or additional detec- tor training are required, enabling fully label-free detection. This design provides interpretability grounded in dynamical-systee results under strict point-wiss wise evaluation across benchmarks against 34 sfate-of-the Beyond accuracy, FOLD yields a transparent continuous risk trajectory via the risk state z((), unify- ing both sudden spikes and gradual drifts within a single mechanism. In future work, we will explore lata-adaptive parameterization of the ODE using only normal data and extend FOLD streaming scenarios with drift-aware calibration.
theory, minimal computational overhead, and consistently strong baselines.
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Ethics statement. This work adheres to the ICLR Code of Ethics. Our study only publicly available, non-personal time-series benchmarks (e.g., SMAP, MSL, SWaT, WADI, SMD, NAB, UCR) under their respective licenses; no human subjects, personally identifiable information, or protected attributes are involved, and no re-identification is attempted. Data are used solely for research and are processed in accordance with the datasets' terms of use. Because anomaly detection can have dual-use risks (c.g., surveillance or misuse in operational settings), we limit our release to research artifacts (code, configs, and scripts) and provide guidance avoid deployment on sensitive data without appropriate consent, legal basis, and security review. We disclose no conflicts of interest or external sponsorship that could unduly influence the results.
Reproducibility Statement. To ensure the reproducibility and completeness of this paper, make our code available at /ht /ttttuu.//ithbb... com/ sheoyon jhin/FOLD We give details experimental protocol in the Appendix B
The Use of Large Language Models (LLMs). We used ChatGPT as a writing assistant for pol- ishing language and checking notation. No research ideas or content generation were conducted by LLMs.
ACKNOWLEDGMENTS
This work was supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT), Development of Financial and Economic Digital Twin Platform based on Al and Data under project number No.2022-0-00857 (34%), and Developing Sustainable Collaborative Multi-modal Lifelong Learning Framework un- der project number NO.RS-2022-IDD 13 (33%), and Samsung Research Funding & Incubation Center of Samsung Electronics under project number SRFC-IT2402-08 (33%).
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Table 5: Notations used in FOLD.
Symbol
Description
Symbol
Description
X E Rixd
Input sequence
H
Forecast horizon
fp
Forecasting model
(x(X)
Baseline prediction
P
i-th patch
X P.
Input with P. masked Distance function
Perturbed prediction
-(.,))
6,
Stress weights
TMC
MC dropout passes
Stress score for P:
I(t)
Patches covering t
S(t)
Stress signal t
z(t)
Risk trajectory
8,
ODE coefficients Calibration windows
Munin P Zthr
Maxima set from normal
Quantile level
Margin for threshold
Risk threshold
((t)
Point-wise decision
Zaya
System-level risk score
A
ALGORITHM
Algorithm 1: FOLD Calibration (pormal-only)
Input: Deormal So, (P,, (,, L,P
Output: Zihr
Mwain + Ø;
// Step
Train forecasting model on normal data
train fo on Dnomal.
// Step 2: Iterate through each normal training sequence
for X E Daammat do
Y fo(X);
// Obtain baseline prediction
Compute patch-wise perturbations and stress signals for P. X do
// step
(o()
6 D' 8 Y) > Var( Var(Ý)); -Vr())
// Step 4: Aggregate patch scores into stress signal S(t) L EiEr() & f
// Step 5: Simulate fold-bifurcation ODE
simulate Eq. (fold-ode) with input S(t) to get z(t;;
// Step 6: Record maximum risk of this sequence Misin Musin U (max, 2(t)):
// Step 7: Define anomaly threshold from high percenteeet of normal maxima
Zun Ouantile,,(Muain
B EXPERIMENTAL ENVIRONMENTS
B. DATASETS
We evaluate FOLD 40 widely used public benchmarks. Table summarizes their key statis- ties: the value parentheses denotes the number of sequences provided by each repository, and all reported scores averaged over sequences within dataset.
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Algorithm 2: Point-wise Anomaly Detection(FOLDD)
Input: X, fa. [P:] 6, X, (a, 3,7), h, L, Zitr
Output: z(t), ((t) 1[z(t) > Ziml
Y x (x));
// Forecast H H--tep outputs from the input sequence X
for each patch P: in X do
V. - (x X P:):
// Perturbed prediction by masking P. // Stress signal combining
a 8 D(Y P) + + -
sensitivity and uncertainty
S() 1 Eiez &, fort= // Aggregate patch- stress into a time --vaaiiinn signal
[I(R]
simulate Eq. with input (() J to obtain z(t):
// Risk tra jratory evolves under fold-bifureation dynamics return z(t) and ÿ(t) 1(z(() > Zuil:
// Anomalies are flagged once z(t) leaves the stable basin Zehre
realizingFon. anomaly ly detection without labels
Table 6: Overview of dataset characteristics for the 40 benchmarks in TSB-AD. The symbol " in the second column indicates that the dataset transformed from multivariate source. The 'Category' column specifies whether the dataset features point anomalies (P) sequence anomalies (Seq).
Name
Collected
# TS
IS Curated
Avg 13mm
Avg TS
Avg f Anomaly
Arg Anomaly Len
Anomuly Ratio
Category
UCR
Dau et al.
2019
228
67818.7
1
198.9
0.6%
P&Sec
NAB Almon d
2017
58
28
5099.7
1.6
370.1
10.6%
Seq
1OPS
T0PS.3
17
72792.3
25.6
48.7
1.3%
Sog
MGAB
TTill al. 2020
10
9
97777.8
9.7
20.0
0.2%
Sci
WSD
Zhhan
210
111
17444.5
5.1
25.4
0.6%
Seg
Sog
TODS
Lare
N074
13
5000.0
97.3
18.7
6.3%
S&
NEK Si St 2024
9
1073.0
2.9
51.1
8.0%
P&
1246.9
P&Seq
TSR-AD-U
Power
Ke0goreear 2007
35040.0
750
8.5%
Seg
Daphine
2019
(U)
38774.0
6
384.3
5.9%
Seq
SWaT
Malnur 2 Ibyeenheer
2016
(U)
1
419919.0
27.0
1876.0
12.19
Seg
LTDB
Gokibberger et ai 2000
9
9920010
127.5
144.5
18.695
Seq
(ao
all
32
3.1
Soq
MITDB
Gooder Qual 2000
(U)
8
631250.0
68.7
451.9
4,2%
Soi
MSL HHlminan
9
3492.0
1.3
130.0
5,8%
Seg
2013
19
Soq
SMD
38
24207.7
2.4
173.7
2.0%
Soi
SVDB
1990
20
171380.0
36.4
292.5
3.6%
Seq
Roggea
Soy
GHI.
Filonos et al. 2016
25
19
199001.1 .0
2.2
1038.2
1.1%
Seq
Daphin
38774.0
6.0
384.3
5.9%
Sot
Soy
Genesis
Woth Biryeich xV Niggemann
2018
18
16220.0
3.0
16.7
0.34%
Seg
OPP
Rod
2010
24
8
248
17426 75
1.4
394.3
4,1%
Set
SMD
22
25466
8.9
3,8%
SWaT
Motiia Tippenhuerr
2016
59
207457.5
16.5
1093.6
12.7%
Seq
ISB-AD-M
PSM
(Arulnis de 2011
I
25
217624.0
72.0
338.6
11.2%
P&55c
27
25
7855.9
1.3
196.3
2,9%
Seq
MSL
Hundparne P
16
55
3119.
1.3
111.7
5.1%
Seq
Creditt and (Staralalinn et
2018
29
281807.0 D
165.0
1.1
0.2%
P&5cq
GECCO
Monize
I
138521.0
$1.0
33,8
1.2%
Soy
MITDB
Seg
SVDB
(Treanaand 1991
31
207122.6
69.3
268.2
4.8%
Seq
LTDB
Gooller
2000
100000.0
105
134.4
15.59
Sol
Seg
TAO (20
13
10000.0 3
788.2
1.1
8.7%
P&Seq
B. BENCHMARK PROTOCOL: TSB-AD
We adhere to the standard evaluation protocol of the TSB-AD benchmark (Liu & Paparrizos 2024).
which features comprehensive collection of 1,070 high-quality time series curated from 40 diverse datasets. The benchmark structured into two primary tracks: Univariate (TSB-AD-U) and Mul- tivariate (TSB-AD-M).
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Consistent with the official leaderboard guidelines, the data is partitioned into three distinct subsets to ensure rigorous and fair evaluation:
Evaluation Data:
This set comprises 350 univariate and 180 multivariate time series re- served strictly for testing. The average anomaly ratios 4.5% and 5.0%, respectively. We report the final performance metrics (VUS-PR, F1) on this split.
Training Data: A short, anomaly-free historical segment is provided for each evaluation series. In framework, we utilize this segment to train the forecasting backbone (fo) and to calibrate the threshold Zekr using the distribution of normal risk scores.
time series (48 for univariate, 20 for multivariate), distinet from the evaluation set, is provided for hyperparameter optimization (HPO). We utilize this split to select optimal hyperparameters (e.g., patch size P, margin p) without leaking information from the test set.
Tuning Data: separate
B.2 SETTINGS
GPU: NVIDIA A6000
2. OS: Ubuntu 20.04
3. Framework: PyTorch 2.1.1+cu121
4. CUDA: nvec 112.1.66 (build cuda 12.1. 1co.pp/://oommiee222225222 0)
B.3 HYPERPARAMETERS
Hyperparameter search ranges.
We conducted a grid search over the following ranges:
Forecasting models:
Sequence length {96, 48).
Output length E {12, 24).
Learning rate E {0.001, 0.005, 0.0001}
Stress signal calculation:
D(:,)) E (cosine similarity, MSE, MAE),
å E {0.5, 1.0, 1.5, 2.0}.
E {0.5, 1.0, 1.5, 2.0}
TMC E (30, 50}
Fold-bifurcation ODE:
(0.5, 1.0, 2.0, 3 E (0.1, 0.3, 0.5, 0.7,
7 E (0.001, 0.01, 0.110000,01,
Number of patches E {6,8)
Best hy peprraarameters. The same best configuration was applied consistently across all datasets:
Forecasting models:
Sequence length 48,
Output length 12,
Learning rate 0.005
Stress signal caleulation:
D(.,)) cosine similarity.
5 = 1.0,
1.0
Tare E (50)
Fold-bifurcation ODE:
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1.0,
0.5,
0.001,
Number of patches = 6
B.4 POINT-WISE EVALUATION
We reuse TranAD Tuli al. 2022) splits/prerrocessing and re-score all methods under a unified point-wise protocol. Labels are per timestep: decisions are j(t) IK (sscore(t) > threshold), and Precision/Recal//F1 are computed over timesteps (no anomaly-range dilation, no delay tolerance, no smoothing).
Baselines. For each baseline, we follow the thresholding strategy originally used in the respective paper (e.g., reconstruction- or prediction-error based detectors use their default criteria.. We did not re-tune thresholds, ensuring that FOLD is compared against baselines under their standard settings.
FOLD. score(() = =2((): the threshold Zhr is calibrated from normal training windows as in Sec. 3.4
prediction (() 16 (x(1)) > }-
Point-wise micro F1 score. We evaluate at the timestep level with no temporal tolerance (k=0). For each time t, let (t) E 1) be the ground-true -trtthhhhhhhhh label and j(t) E f0,11 1} be the binarized prediction (obtained by thresholding the anomaly score).
Dataset-level counts are aggregated over all 8, t:
TP
Σ ((o(t)=1
(11)
FP Σ (0000=1A D(t
(12)
FN Σ 17(00-N w()=1).
(13)
Micro-precision and recall are
TP+FP and R TP+FN and the point- wise micro F1 is
2PR
F1
(14)
P+
No dilation/collar or range merging applied; prediction is counted as correct only when matches the exact timestep ()))), Unless noted otherwise, all Fl scores reported in this
paper use this definition.
Multivariate setting and interactions. Let X E R4XD denote window of D variables. The forecaster fo is multivariate (PatchTST, TimeMixer with cross-feature mixing), hence its prediction Ý = (X) E IR # D already encodes cross- variabva dependencies. We compute patch scores {e+} by locally masking X with patch m, and measuring the change of a joint multivariate loss, e.g..
E1 Y fo(X @ mi)ll I, timex feat
A (VarlL) Var[Jo(X m2)))),
sensitivity
uncertainty
where the loss aggregates over all D features. Because is multivariate, masking feature j at time I can change predictions of many other features, and this cross-effect is reflected in e;, Patch scores aligned to time and feature axes form feature-wise stresses (c) and their aggregation S(I):
(t())
Σ e, (((,k) E span(m:)),
S(I)
Σ Wh Sa(1) (wx 2 0, Σ wk 1).
We then evolve feature-wise risk dynamics
(t( = a, (t) B, 24 (t)² (t)
yielding z(t) = (2, 2D(()) E R For readability, figures report gystem-levell risk obtained by summation, e.g. (t) = Σe (t). Anomalies can be declared either when (xy(() crosses a calibrated threshold.
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B.5 IMPLEMENTATION DETAILS ABOUT FOLD(CHRONOS)
To demonstrate the model-agnostic nature of FOLD, we replaced the trainable forecasting backbone (/a) with Chronos (Ansari et al. 2024), a pre-trained probabilistic time-series foundation model, in zero-shot setting. Specifically, we utilized the amazon/ htt///hhoott-eeeeee
variant (approx. 46M parameters) to evaluate the framework's capability even with a lightweight foundation model. Since Chronos natively outputs predictive distribution rather than a point estimate, we adapt the stress signal calculation (Eq. as follows:
Sensitivity Term. Instead of retraining or fine-tuning, use the pre-trained Chronos model to generate the median forecast Ymedian. The sensitivity is computed as the deviation of this median forecast under local masking:
D(Y V)
11 Meddian -yi.medionl
(15)
Uncertainty Term. Unlike standard deep learning models that require MC Dropout for uncer- lainty estimation, Chronos provides distribution of sample trajectories. We directly utilize the
variance of these generated samples as the uncertainty measure:
Uncertainty = IV (Var(y (Rr,) - Var(V))
(16)
where Var(-) denotes the variance across the sampled forecasts from Chronos.
This adaptation allows FOLD to leverage the superior zero-shot generalization of foundation mod- els without any additional training or architectural modifications. The quantitative results this integration are presented in Table
C ADDITIONAL EXPERIMENTAL RESULTS
We follow TranAD (Tuli et al. 2022 for splits and preprocessing (native dimensionality: NAB/UCR univariate: others multivariate) and re-evaluate all methods under a unified point-wise protocol: for each timestep f, i() = I (z(t) > Zur) } and Precision/Recall/F are computed over timesteps (no dilation, no delay tolerance, no smoothing). This stricter protocol replaces window-level scoring common in prior work. See Appendix B for dataset, hyperparameter, and evaluation details.
C.1 EXPERIMENTAL RESULTS
Table reports the performance of FOLD compared with 10 statc-of-the-art baselines across benchmarks. Under the point-wise anomaly detection setting, which demands anomalies to be iden- tified at the exact timestep, many prior methods show degraded performance, suggesting that earlier window-based results may have overstated fine-grained accuracy.
Despite this stricter setting, FOLD achieves consistently strong F1-scores across domains. On datasets dominated by sudden deviations (e.g.. NAB, UCR), FOLD matches surpasses prediction- based approaches. On benchmarks characterized by gradual stress accumulation (c.g.. SMAP., MSL), FOLD further outperforms prediction- and distance-based methods. Notably, on NAB and SMAP, FOLD improves over the strongest basclines by 216.4% % and 170.1% respectively, highlighting its robustness across both low-resource and performero settings.
Another key finding is that FOLD maintains competitive performance regardless of the forecast- ing backbone. Substituting PatchTST with simpler models such as DLinear or TimeMixer yields comparable results, indicating that FOLD does not hinge on forecaster sophistication. Instead, the strength of FOLDD lies in transforming sensitivity and uncertainty signals into a dynamical stress process via signals but their principled accumulation and state-transition modeling that drive accurate anomaly detection.
modeling. This demonstrates that it is not merely the presence of error
We also compare with NRDetector (Wang et al. 2025 a recent method explicitly designed for point-wise anomaly detection. While NRDetector narrows the evaluation gap by aligning its design with stricter metrics, FOLD consistently outperforms it across multiple benchmarks. This advantage arises from our dynamical-systene formulation, which unifies sudden and gradual stresses under
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Table 7: Performance comparison of FOLD with baseline methods across all datasets. P: Precision, R: Recall, F1: Fl-score. The best F1-scores are shown in bold and the second-bes in italic All results are averaged over 3 independent runs standard deviations are reported in Appendix D
NAB
UCR
MBA
Method
R
F1
P
R
F1
P
R
F1
LSTM-AD
0.0501
0.2917
0.0855
0.2083
0.0113
0.0214
0.5822
0.2234
0.3229
OmniAnomaly
0.0757
0.3073
0.1215
0.0161
0.3125
0.0306
0.5045
0.1712
0.2556
MSCRED
0.0616
0.2187
0.0961
0.0053
0.0045
0.0049
0.7414
0.2261
0.3465
MAD-GAN
0.0807
0.3958
0.1341
0.1609
0.1863
0.1727
0.7018
0.2299
0.3463
USAD
0.0843
0.6719
0.1498
0.0412
0.0824
0.0549
0.6565
0.2234
0.3333
MTAD-GAT
0.0571
0.1875
0.0875
0.0625
0.0039
0.0073
0.6313
0,1876
0,2892
GDN
0.0617
0.2188
0.0963
0.1250
0.1250
0.1250
0.4607
0.1577
0.2349
Anomaly Transformer
0.0591
0,5000
0.1057
0,0303
0.1666
0.0513
0.2487
0,0285
0.0511
TranAD
0.0746
0.3437
0.1226
0.1333
0.0180
0.0317
0.7478
0.1334
0.2264
NRdetector
0,0502
0.0581
0.0539
0.6361
0.0542
0.0999
0,5000
0,3920
0.4394
DOLD(DLinnea
0.3519
0.6875
0.4655
0.4267
0.4058
0,4160
0.7664
0.3807
0,5087
POLD(PatchTST)
0.3147
0.9600
0.4740
0.1361
0.5981
0.3218
0,4184
FOLDD (TmeMixer)
0.2993
0.9607
0.4564
0.2725
0.6726
0,3879
0.4578
0.3139
0.3724
SMAP
MSL
SWaT
Method
P
R
F1
LSTM-AD
0.1197
0.2969
0.1706
0.0392
0.3351
0.0702
0.7778
0.0108
0.0213
OmniAnomaly
0.0580
0.1762
0.0873
0.0424
0.3189
0.0748
0.9742
0.6475
0.7779
MSCRED
0.0928
0.0085
0.0156
0.0538
0.2665
0.0895
1.0000
0.3897
0.5608
MAD-GAN
0.0682
0.1910
0.1005
0.0217
0.0197
0.0207
0.9243
0.3602
0.5183
USAD
0.0743
0.2831
0.1177
0.0393
0.3209
0.0700
0.9870
0.4721
0.6387
MTAD-GAT
0.0170
0.7429
0.0332
0.1453
0.3556
0.2063
0.9731
0.6195
0.7570
GDN
0,4056
0.0495
0.0882
0.0602
0.3298
0.1018
0.9673
0.6428
0.7723
Anomaly Transformer
0.3195
0.0255
0.0472
0.1557
0.0159
0.0289
0.4663
0.1514
0.2285
TranAD
0.1185
0.2197
0,1540
0,2326
0.0323
0.0567
0.9964
0.4363
0.6068
NRdetector
0.6372
0.1608
0,2568
0.4884
0,1511
0,2308
0.7336
0.5772
0.6460
FOLD(DLinear)
0.6820
0.7059
0.6937
0,3514
0,5821
0,4382
0.6824
1,0000
0.8112
FOLD(PatchTS
0.7238
0.5214
0.6062
0,5429
0.6727
0.6009
0.8212
0.7882
0. 8043
0.5103
0.6928
0.5877
0.5956
0.5117
0.5505
0.7804
0.8046
0.7923
WADI
SMD
MSDS
Method
R
F1
P
R
F1
P
R
F1
LSTM-AD
0.0472
0.0012
0.0023
0.0389
0.003
0.0057
0.0075
0.2692
0.0145
OmniAnomaly
0.1249
0.0275
0.0451
0.4779
0.0543
0.0975
0.8414
0.7997
0.8200
MSCRED
0.0519
0.2069
0.0830
0.9783
0.0921
0.1684
0.4324
0.0075
0.0147
MAD-GAN
0.0069
0.0852
0.0128
0.9999
0.0064
0.0127
1.0000
0.1542
0.2672
USAD
0.0969
0.3584
0.1526
0.6320
0.14
0.1821
0.7263
0.9999
0.8414
MTAD-GAT
0.0846
0.2894
0.1309
0.4803
0.0879
0.1486
0.0282
0.5962
0.0538
GDN
0.0820
0.2256
0.03
0.4658
0.1277
0.2004
0.0004
0.9999
0.0008
Anomaly Transformer
0.1428
0.6571
0,2346
0.1698
0.0362
0.0597
0.0374
0.5103
0.0696
TranAD
0.2157
0.3433
0,2649
0.6536
0.0660
0.1199
0.7264
0,9999
0.8414
NRdetoctor
0.4849
0.2712
0.3479
0.2232
0.0722
0.1091
0.4943
1.0000
0.6615
FOLD(DD inner))
0.5299
0.7037
0.6046
0.3095
0.4258
0.3585
0.8531
0.8822
0.8674
FOLD(PathHTTT)
0.4144
0.6731
0,5130
0.4151
0.3025
0.3498
0.8812
0.8781
0.879
(Oiee(Timeexer)
0.4065
0.6900
0.5116
0.4224
0.3095
0.3572
0.8023
0.8692
0.8344
single framework. Thus, FOLD not only competes strongly with state-of -ot--------tr-r-- baselines but also advances pose anomaly detection beyond current specialized methods.
D EXPERIMENTS RESULTS WITH STANDARD DEVIATION
The reported mean and standard deviation are computed by repeating the forecasting model training three times with different random seeds. Since the anomaly detection pipeline builds the trained forecaster, the reported deviations reflect both variability in the forecasting stage and its propagation into stress-signal based detection.
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Table 8: FOLD anomaly detection results with mean and standard deviation three independent runs of the forecasting model.
NAB
UCR
MBA
Method
F1
F1
F1
FOLD(DLinear)
0.4655 0.000
0.4160
0.008
0.5087
0.001
FOLD(PathhTST)
0.4740 +: 0.003
0.2396 + 0.007
0.3018
0.007
FOLD(TimeMinn
0.4564 + 0.001
0.3879 + 0.001
0.2072 0.005
SMAP
MSL
SWaT
Method
FOLD(DLinear)
0.6937 + 0.002
0.4382 0.001
0.8112
0.005
FOLD(PatchTST)
0.6062 + 0.010
0.6009 + 0.000
0.8043 + 0.002
FOLD
0.5877 + 0.012
0.5505 + 0.002
0.7923
0.005
WADI
SMD
MSDS
Method
F1
FOLD(DLinear))
0.6046 + 0.001
0.3585 + 0.004
0.8674 + 0.001
FOLD(PatchTST)
0.5130 + 0.001
0.3498 0.005
0.8796 0.010
FOI.D(TimeMixer)
0.5116 + 0.002
0.3572 0.002
0.8344 0.012
E ADDITIONAL ABLATION
E. CHOICE OF DISTANCE METRIC FOR SENSITIVITY TERM
To evaluate the impact of different distance metrics in the sensitivity term, we conducted an ablation study comparing MSE, MAE and cosine similarity as the function D(. )). All other components of FOLD (DLinear) were kept fixed. As shown Table cosine similarity consistently improvements. These results confirm that FOLD is robust to the choice of distance metric, with cosine similarity selected as the default due to its stable performance across benchmarks.
Table 9: Ablation study on distance metric for sensitivity term.
NAB
MSL
SMD
Method
R
F1
P
R
F1
P
R
F1
FOLD (MSE)
0.2984
0.9351
0.4524
0,3409
0.5901
0.4322
0.2942
0,3087
0,3013
FOLD (MAE)
0,2812
0.8426
0.4216
0.3381
0.5206
0,4099
0.3112
0.2724
0.2905
FOLD (Cosine similarity)
0.2993
0,9607
0.4564
0.3514
0.5821
0.4382
0.3095
0.4258
0.3585
E.2 SENSITIVITY TO ODE PARAMETERS
We analyze the sensitivity of FOLD to the coefficients a, 3, in the fold-bifurcation inspired ODE (Eq. 77 Each parameter is varied individually while the others fixed, and the resulting F1- scores are reported across representative benchmarks. As shown in Figure FOLD maintains stable performance
wide range of values. indicating robustness to hyperparameter choices. Only extreme values lead to noticeable degradation, underscoring the importance of avoiding pathological settings rather than requiring precise tuning.
E.3 SENSITIVITY TO THE NUMBER PATCHES
We further analyze the effect of the patch number N, which controls the temporal granularity of stress signal estimation. As shown in Figure FOLD maintains consistently superior performance compared to baselines across wide range of N. While moderate patch sizes (e.g., N 6-10) yield slightly better results, the overall performance does not degrade significantly even at extreme values. This robustness indicates that FOLD does not rely finely tuned patch size: the forecasting
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FOLD
Baseline
0.7
UCR
MBA
SMAP
MSL
SWAT
0.
WADI
MSDS
01
0.5 0,0011
00031
0.01
0.05
03
18 - 0.5,
0.001 fixed)
B to 1.0, y 0.001 flsnd)
(10=1.0, 1.0,8 0.5 fixed)
Figure 7: Sensitivity of FOLD to ODE parameters 3,y, Performance is stable across broad ranges.
backbone provides sufficiently stable sensitivity and uncertainty signals, and the fold-bifurcation dynamics remain effective regardless of the precise partitioning. Therefore, FOLD's advantage stems from its principled stress accumulation modeling rather than from sensitive hyperparameter choices.
0.6
0.40
0.35
0.5
FOLD D
Sootm
0.4
FOLD
0.4
FOLD
0.35
Nitletacctr
0.20
0.3
02-
0.15
0.2
0.10
Number Phitches
Paitches
Number
(a) NBA
(b) SMAP
(c) MSL
Figure 8: Sensitivity study on the number of patches N. We report FI FI-scores of FOLD compared with representative baselines on NAB, SMAP, and MSL. Across datasets, FOLD consistently outperforms the baselines, and its performance remains stable across a broad range of N, demon- strating robustness patch partitioning choices.
E.4 THRESHOLD ROBUSTNESS ANALYSIS
To complement the theoretical discussion in Section 3.4 empirically assess the robustness of the threshold Z
Bootstrap confidence intervals. For each dataset, we resample the calibration set Mrai (B 1000 replicates) and recompute Zeh Quantile, (Mrrii) (Mnai))),(MMmaan) with p 0.99. Table 10 reports the mean and 95% confidence interval, showing that the variation of Z is below 5% across all datasets.
Table 10: Bootstrap confidence intervals of 7ih selected datasets.
Datasel
Zar (mean)
95%
UCR
2.518
[2.502, 2.534]
SMAP
[0.590, 0.623
SMD
0.713
[0.699.0.75 0.731]
MSL
0.382
10.369, 0.401
Calibration robustness. We probe the stability of the calibrated threshold Z against E- contamination, fraction of normal calibration windows are replaced by anomalous ones. For each E (0,3,5 5, 10) we recompute Zin and report the relative shift AZ (Z Zin / Z and the resulting wise F1 on the test set (no re-tuning of p,
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Table 11: Effect of calibration contamination threshold stability and detection performance
SMAP (S-1).
(86)
AZ (%)
AFI (pp)
0.5982
3
0.6105
+2.05
0.7918
-2.19
5 0.7075
+18.3
0,5366
-33.7
10
0.7204
+20.4
0.3832
-5..6
Even at 10%. FOLD retains an Fl 0.3832 on SMAP (S-1), which,despite the drop from clean calibration, remains substantially higher than strong baselines on the same machine (e.g.. NRDetec- tor 0.1032, TranAD Amall contaminations (c 3%) shift Zue by only few percent and lead to modest Fl changes, consistent with the high-quantile stability we exploit.
Madin MW
0.44
Ancenal
Trun Annmaly
Prodicted Anomaly
Predicted Anomaly
1200
2006
3090
5000
5006
$300
7100
1,00
2006
3703
2000
5003
5000
T006
(a) 0% Contamination
(b) Contamination
6.7
Tran Rick stote
Train Risk state 2E0
Thresbald
Thresbole
1000
1500
320
6.7
6.7
Med MMM ma
0.5
04
044
Thet
state
Thresheld
Threshold
True Anomaly
Anomaly
Anumaly
0.1
6:11
Tizze Strpe
Tima Stepe
(c) 5% Contamination
(d) 10% Contamination
Figure 9: Robustness of threshold on SMAP.
F EFFICIENCY AND COMPLEXITY ANALYSIS
We analyze the computational efficiency of FOLD from both theoretical complexity and empirical resource usage perspectives.
Theoretical Complexity. Unlike many SOTA anomaly detectors built Transformer backbones with quadratic complexity O(L?) (where the sequence length), FOLD utilizes lightweight back- bones (e.g., DLinear) with linear complexity O(L). During inference, FOLD computes the stress signal by evaluating P patches. Even with MC dropout samples for uncertainty estimation, the total scoring complexity is P...). Since K and P are small constants independent of L., FOLD
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Table 12: Computational efficiency comparison the MSI. dataset.
GPU Memory Usage (MB)
Additional Memory at Inference (MB)
System Memory Usage (MB)
Model
LSTM-AD
26.73
8.180
1094.76
OmniAnomaly
19.73
1.320
1343.11
MSCRED
42.87
12.54
80409.57
MAD-GAN
29.87
1.300
1036.21
USAD
29.92
1.200
1409.09
MTAD-GAN
36.87
9.872
3024.28
GDN
38.41
10.18
4886.08
Anomaly Transformer
28.32
351.4
3687.15
TranAD
40.42
1.240
1016.51
NRdetector
31.22
214.1
2965.31
FOLD (DLincar)
18.04
1.170
893.49
maintains an asymptotic linear complexity O(L), offering a fundamental efficiency advantage over O(L²) methods. Furthermore, the patch evaluations independent and massively parallelizable, allowing them to be processed in single batch on GPUs.
Empirical Efficiency. Table 12 compares peak GPU memory, additional memory inference, and overall system memory on the MSL dataset. FOLD achieves the lowest memory footprint across all metries. Specifically, requires only 18.04 MB GPU memory and minimal additional inference memory (1.17 MB), significantly outperforming complex deep learning baselines like Anomaly Transformer (28.32 MB 351.4 MB) and MSCRED (42.87 MB). This confirms that FOLD's linear complexity translates directly to practical, real-time efficiency suitable for industrial deployment.
Practical Implications (Training-Free). Crucially, the anomaly detector FOLD is training- free extension of the forecasting model. Unlike other methods that require training separate detector or fine-tuning on anomaly scores, FOLD is derived directly from the pre-trained forecaster without additional parameter optimization. Combined with the massively parallelizable inference computations (as discussed in Theoretical Complexity), this training- free nature minimizes deploy- ment costs and makes FOLD highly amenable to real-time industrial applications.
G ADDITIONAL VISUALIZATION
H THEORETICAL ANALYSIS OF HOLD-BIFURCATUUS DYNAMICS
In this section, we provide a formal derivation demonstrating that the unified detection capability of FOLD is not an ad-hoe design choice but an intrinsic mathematical property of the governing ODE. We analyze the behavior of Eq. (7) under different time-scales of the input stress S()).
H.1 UNIFIED MECHANISM VIA TIME- SMME-SCALE SEPARATION
The system state =() evolves according to the single governing equation:
dz
a.S(t) (32² + +72)
(17)
dt
Forcing
Restoring
We analyze the dominant terms in this equation based the time-scale of the input stress relative to the system's intrinsic relaxation time.
Regime 1:
Quasi-Static Limit (Gradual Accumulation). When the stress S(t) varies slowly 0), the time derivative da becomes negligible compared to the algebraic terms. In this regime,
ds
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Toa Srum
(a) SMAP(GT)
(b) NAB(GT)
(c) UCR(GT)
Tiget CAN
(d) SMAP(FOLD)
(c) NAB(FOLD)
(f) UCR(FOLD)
(g) SMAP(NRdetector)
(h) NAB(NRdetector)
UCR(NRdetector)
Chanddl
Tam
ii SMAP(TranAD)
(k) NAB(TranAD)
(I) UCR(TranAD)
Figure 10: Qualitative comparison on three benchmark datasets representing distinct anomaly types: Gradual (SMAP), Abrupt (NAB), and Periodic (UCR). We compare FOLD against state-of-the- art point-wise (NRDetector) and window-based (TranAD) baselines. FOLD demonstrates robust detection across diverse scenarios, accurately capturing gradual drifts SMAP and abrupt spikes in NAB, whereas baselines exhibit missed detections or false positives (e.g., NRDetector on NAB, TranAD on SMAP).
the system operates slow manifold, tracking the instantaneous moving equilibrium where forces balance:
as((I) 3z(1)² xz(()
(18)
Consequently, the risk state =({) algebraically tracks the magnitude 5(1). Anomaly detection triggered via Saddle-Node Bifurcation when the stress exceeds the system' 's capacity to maintain this stable equilibrium. This mathematically explains the detection gradual drifts (c.g., in SMAP).
Impulsive Limit (Abrupt Spike). When the stress S(t) acts as large-magnitude impulse over short duration At (i.e., S(t) > 8z" + y2), the forcing term dominates the restoring dynamics. The ODE asymptotically simplifies to a pure integrator:
Regime 2:
d= R aS(t)
At
A
aS(T)dT
(19)
di
Here, 2(t) undergoes instantancous state jump proportional to the total accumulated energy of the spike. This explains why the model reacts immediately to abrupt anomalies (e.g., in NAB) without delay.
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Toa Srum
(a) SMAP(GT)
(h) NAB(GT)
(c) UCR(GT)
(d) SMAP(FOLD)
(c) NAB(FOLD)
(f) UCR(FOLD)
Figure 11: Visualization of uncertainty assessments to demonstrate reliability and confidence. The blue line represents the input data, and the shaded region indicates the uncertainty band (derived from MC dropout). Note that the uncertainty band remains narrow during normal states (indicating high confidence) but significantly widens the onset of anomalies, triggering the risk accumula- tion mechanism. This visually confirms that FOLD's detections are driven by model-intrinsic risk assessment rather than random fluctuations.
10
(a) OPPORTUNITY
(b) SMAP
(c) SMD
Figure 12: Visualization of Risk state (Zeys (t) on Multivariate Datasets. Although FOLD computes risk feature-wise, the final decision based on the aggregated system-level risk (blue line). This
plot demonstrates how the aggregated Zays (t) successfully captures anomalies in high-dimensional datasets (OPPORTUNITY, SMAP, SMD) by integrating stress from multiple features.
H.2 THEORETICAL GUARANTEE OF ROBUSTNESS
The robustness of FOLD is guaranteed by the Linear Stability Analysis of the restoration term. In the absence of strong forcing (i.e., S(t) - 0), the dynamics simplify to da (B2² + 72). Near
the stable equilibrium (2 z 0). the linearized dynamics are governed by the eigenvalue À ~ -y. Since 7 > O, we have À < which guarantees exponential decay of perturbations:
=tt
(20)
This proves that the system is mathematically guaranteed return to the stable basin after tran- sient shock, ensuring that the model naturally heals itself and prevents false positives from error propagation.
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MM ww Ma Mowa Mant
(a) NAB
(b) UCR
(c) MBA
mm
Walh Wy Yom Jani Mhyn
Way Ao
(d) SMAP
(e) MSL
SWaT
Jar
(g) WADI
(h) SMD
MSDS
Figure 13: Visualization of anomaly detection results
27


Reviews:
[{"review_id": "nTOdbS2RUO", "reviewer": "Reviewer_nbGe", "summary": "The paper proposed a method for point anomaly detection in time series, which is an important task in time series domain. The proposed method employs forecasting model and provide masked signal to produce a sensitivity score, which is then utilized with the fold bifurcation theory to get anomaly scores. The method seems to outperform some baselines across 9 datasets in F1 score.", "questions": "See weaknesses.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper has the following strong points:\n1. The proposed method is novel, and well motivated. Particularly, the paper does a good job in motivating the need for the fold bifurcation theory and it's usage for anomaly detection.\n2. The proposed method is theoretically motivated and well formulated.\n3. Some experimental evidence has been provided.\n4. The paper is well written.", "weaknesses": "However, the paper has the following weaknesses.\n1. Does the proposed method work only for point anomalies? The author(s) say \"However, most approaches are evaluated under coarse window-level settings, which can mask their limitations in the stricter point-wise anomaly detection scenario.\" This is not entirely true. There are recent papers with multiple types of anomaly detection capabilities. Example include: [1], [2], and many more. \n2. The evaluation metric F1 score has been establised as a biased metric in recent literature [1]. Why does the author not use mathematically proven good metrics like VUS-PR defined in [1]?\n3. The evaluation should be done on well establised anomaly detection benchamarks such as in [1], and should be compared with recent SOTA methods on more datasets.\n4. Since the proposed method does patching, masking etc., what is the runtime complexity of the method? In terms for both theoretical and empirical evidence. Since anomaly detection is needed in realtime in most usecases, runtime is a very important factor.\n5. The authors say \"many real-world failures arise not from isolated spikes but from the gradual accumulation of stress that drives a system toward a critical transition.\" This is not true in several situation when external factors create anomalies, or even periodic anomalies. \n6. How does the method perform on multivariate time series? \n\n[1] Liu, Qinghua, and John Paparrizos. \"The elephant in the room: Towards a reliable time-series anomaly detection benchmark.\" Advances in Neural Information Processing Systems 37 (2024): 108231-108261.\n[2] Ekambaram, V., Kumar, S., Jati, A., Mukherjee, S., Sakai, T., Dayama, P., ... & Kalagnanam, J. (2025). TSPulse: Dual Space Tiny Pre-Trained Models for Rapid Time-Series Analysis. arXiv preprint arXiv:2505.13033.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "3MgeOJrlcb", "reviewer": "Reviewer_DxMr", "summary": "This paper introduces a point-wise anomaly detection method grounded in fold-bifurcation dynamics. The approach is fully unsupervised, requiring neither anomaly labels nor detector training.", "questions": "- Can you please discuss control parameter r in Eq. 1 and Eq. 2 and why is it appropriate to capture it as a time-varying stress signal S(t) in Eq. 3. \n\n- In what sense is FREE better than early warning systems that monitor Eigenvalues trends?  Why cannot we use early warning systems for anomaly detection?\n\n- Perhaps it is obvious to those working with these ODEs, it isn't immediately obvious to me how will one determine that risk trajectory z(t) has left its stable basin?\n\n- In Sec 3.2, the forecasting model f_{\\theta} predicts the next H elements?\n\n- Purturbed sequence contains X except those that belong to patch i?  Is this is what we mean by purturbed sequence?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper develops an intersting view of anomaly detection where by anomaly is seen as the cause of accumulating stress within the system.  Stress signals are extracted from time series data and integrated using a fold-bifurcation ODE.  These signals accumulate over time till they reach a tipping point and an anomalous condition is raised.\n\nThe proposed method is reported to address a key shortcoming of prior approaches that show strong results on window-based metrics but often fail to generalize under point-wise evaluations.\n\nThe proposed scheme is evaluated on 9 benchmarks and compared against 10 baselines, and the results suggest that the proposed scheme achieves good performance under strict point-wise evaluation.\n\nThe premise of this work appears sound given the work in (Scheffer et al. 2009) that shows that variance increases as a system approaches a critical transition.", "weaknesses": "Please state the paper’s novelty relative to the early-warning systems you cite---what does this scheme enable that prior EWS methods do not?\n\nPerhaps I missed it, but it seems that z(t) is computed independently for each feature?  Is that so?  If it is true then how does this method captures subtle interactions between multiple variables present in the time series?  I couldn't following Figure 3, for example, since I was confused to see a single z(t) curve.  Shouldn't there be one for each variable?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "HtiBgzGq3f", "reviewer": "Reviewer_u6SV", "summary": "The paper presents FREE (Free point-wise Anomaly Detection via fold-bifurcation), a framework for time-series anomaly detection that targets strict point-wise evaluation rather than coarse window-level metrics. FREE computes stress signals from a forecasting model and feeds them into a fold-bifurcation–inspired ODE to track a system’s risk state; anomalies are flagged when this state crosses a threshold calibrated on normal data.", "questions": "My primary concerns center on actionable guidance from theory to practice, the method’s novelty relative to comparable baselines, a thorough efficiency analysis, and a clear discussion of threshold selection and robustness.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "FREE, grounded in fold-bifurcation dynamics, recasts anomaly detection as a point-wise decision process driven by the progression from stress to critical transition, providing strong theoretical footing and inherent interpretability. By integrating forecast-derived stress signals into a bifurcation equation, it captures the buildup of gradual pressures while remaining sensitive to sudden tipping events. Extensive multi-benchmark evaluation demonstrates strong performance—particularly under strict point-wise settings—achieving a balanced mix of accuracy and efficiency and underscoring its practical value.", "weaknesses": "1.The central assumption—that many real-world failures stem from gradually accumulating stress leading the system toward a critical transition rather than isolated spikes—seems reasonable, but the paper would benefit from additional concrete scenarios to broaden and clarify its practical applicability.\n\n2.For the fold-bifurcation–inspired ODE modeling, the linkage between theory and specific application domains is underdeveloped; the discussion remains largely theory-driven without fully mapping the formulation to domain variables, constraints, and operational workflows.\n\n3.On visualization, it’s hard to see—at a glance—where the method outperforms alternatives. Clear comparative plots would help, ideally accompanied by visualized uncertainty assessments to show reliability and confidence.\n\n4.Regarding thresholding, the confidence-interval analysis feels insufficient. The large differences in preset thresholds—are they driven by dataset standards? Could thresholds be informed by priors or domain knowledge to improve calibration stability?\n\n5.The efficiency results are compelling, but a deeper analysis is needed to understand the sources of improvement relative to other methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)