Towards Robust Time Series Learning via Capacity-Centric Modulation

Siru Zhong; Senzhang Wang; James T. Kwok; Yuxuan Liang

cs.LGSep 30, 2026

Towards Robust Time Series Learning via Capacity-Centric Modulation

Authors: Siru Zhong, Senzhang Wang, James T. Kwok, Yuxuan Liang

Organizations: Data Science & Analytics Thrust, The Hong Kong University of Science and Technology (GZ), Guangzhou, China · School of Computer Science & Engineering, Central South University, Changsha, China · Dept. of Computer Science & Engineering, The Hong Kong University of Science and Technology, Hong Kong

Abstract

Sample-level reliability heterogeneity is common in deep time series learning. Standard training pipelines apply a uniform regularization setting to all samples, which can under-regularize corrupted samples and over-restrict clean samples. Common robustness approaches filter observations in data space or impose priors on latent representations. We propose Capacity-Centric Modulation (CCM) as a complementary, sample-adaptive regularization principle. Under this principle, we introduce SACM (Sample-Adaptive Capacity Modulation), a task-agnostic framework that exploits spectral sparsity to assign sample-wise dropout probabilities along internal activation paths. SACM integrates into existing backbones without architectural redesign and preserves the deterministic inference pipeline. Across 301 real-world dataset-backbone pairs covering 9 forecasting, 32 classification, and 4 anomaly-detection datasets, SACM reduces forecasting MSE by 6.7% on average and improves classification accuracy and point-adjusted F1 by 3.04% and 17.05%, respectively, relative to unmodified backbones, with zero test-time overhead.

Figures & tables

Fig. 1: Overview of robustness paradigms and intervention levels. a) Data-Centric Selection removes or reweights observations at the data level; b) Prior-Centric Modeling constrains latent representations; c) Capacity-Centric Modulation modulates effective capacity along internal activation paths for predictive (forecasting), discriminative (classification), and reconstructive (anomaly detection) tasks.

Paradigm	Intervention Space	Core Philosophy	Representative Methods	Key Characteristics & Limitations
Data-Centric Selection	Data Space ( $\mathcal{X}$ )	“Filter Noise”	RobustTSF [ 7 ]	$\bullet$ Discrete keep-or-drop screening
(e.g., Hard Selection)		Exclude unreliable observations	Selective Learning [ 12 ]	$\bullet$ Risks discarding valid rare events; task-coupled
Prior-Centric Modeling	Latent Space ( $\mathcal{Z}$ )	“Disentangle Noise”	BayesTSF [ 34 ]	$\bullet$ Imposes explicit priors (e.g., Gaussian VAEs/BNNs)
(e.g., Complex Priors)		Constrain latent distributions	RSTIB [ 6 ]	$\bullet$ Sensitive to non-stationarity; heavy inference cost
Capacity-Centric Modulation	Hypothesis Space ( $\mathcal{H}$ )	“Coexist with Noise”	SACM (Ours)	$\bullet$ Sample-adaptive active capacity allocation
(Sample-Adaptive Principle)		Match capacity to reliability		$\bullet$ Task-agnostic, zero inference overhead, complementary

TABLE I: Comparison of robustness paradigms in time series modeling. SACM instantiates Capacity-Centric Modulation on the active hypothesis space (

\mathcal{H}

), operating complementarily to conventional data- and prior-centric approaches.

Fig. 2: Schematic illustration of representative synthetic temporal components and corruption profiles. Top: Structured dynamics

\mathbf{x}^{\star}

from stationary periodicity to non-stationarity in mean (Trend), frequency (Chirp), and amplitude (AM). Bottom: Residual variations

\mathbf{r}

representing Gaussian noise, heavy-tailed spikes, and randomly scattered missing observations.

Fig. 3: Illustration of Spectral Sparsity. Composite (Periodic + Trend + Chirp + AM) under corruption. Left: Corrupted inputs. Middle: Frequency spectra: structured dynamics concentrate in discrete peaks while corruptions often spread energy across more frequencies. Right: Reconstruction from dominant modes faithfully recovers the underlying ground-truth signal.

Fig. 4: Spectral proxy analysis across real-world benchmarks. (a) Proxy SNR, defined as the ratio of spectral energy in the top 10% highest-amplitude Fourier coefficients to that in the remaining coefficients; (b) strong negative correlation between SFM and proxy SNR (

r=-0.846

p<0.05

); (c) dataset rankings illustrating heterogeneous spectral concentration.

Fig. 5: Unified cross-task architecture of SACM across forecasting, classification, and anomaly detection . The spectral scorer extracts spectral residual scores via detrending, log-scale normalization, and SFM-anchored soft filtering; the rate mapper maps scores to bounded dropout rates and differentiable activation masks. As a task-agnostic framework , SACM applies sample-adaptive regularization during training while leaving the inference pipeline unchanged.

Type	Category	Mathematical Formulation	Physical Interpretation	Example
Signal	Stationary (Periodic)	$x_{t}=\sum_{k}A_{k}\sin(2\pi f_{k}t+\phi_{k})$	Stable equilibrium; constant freq. & amp.	Power grid voltage
	Non-stat. (Mean)	$x_{t}=\alpha t+\beta+\sum_{k}A_{k}\sin(2\pi f_{k}t)$	Trend & Seasonality; drifts with fluctuations	Macroeconomic growth (GDP)
	Non-stat. (Freq.)	$x_{t}=A\sin(2\pi(f_{0}t+\frac{1}{2}kt^{2}))$	Spectral Drift; time-varying frequency	Doppler effects (Radar/Sonar)
	Non-stat. (Var.)	$x_{t}=(1+\mu\sin(2\pi f_{m}t))\sin(2\pi f_{c}t)$	Time-varying amplitude envelope	Vibration or communication signals
Noise	Gaussian Noise	$\tilde{x}_{t}=x_{t}+\epsilon_{t},\kern 4.62497pt\epsilon_{t}\sim\mathcal{N}(0,\sigma^{2})$	Random measurement noise	Sensor thermal noise
	Heavy-tail (Student-t)	$\tilde{x}_{t}=x_{t}+\epsilon_{t},\kern 4.62497pt\epsilon_{t}\sim t_{\nu}\,(\nu=2.5)$	Heavy-tailed random corruption	Impulsive sensor disturbances

TABLE II: Definitions of clean and corrupted signals. Overview of the four clean regimes and three corruption profiles (

\tilde{x}_{t}

\sigma

Metric

Informer ( 2021 )

Crossformer ( 2023 )

PatchTST ( 2023 )

TimesNet ( 2023 )

iTransformer ( 2024 )

TimeMixer ( 2024 )

WPMixer ( 2025 )

TimeFilter ( 2025 )

MultiPatchFormer ( 2025 )

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

0.1

MSE

0.966

0.514 (

\uparrow

46.790\,890\,269\,151\,14

0.450

0.386 (

\uparrow

14.222\,222\,222\,222\,22

0.542

0.529 (

\uparrow

2.398\,523\,985\,239\,852

0.902

0.848 (

\uparrow

5.986\,696\,230\,598\,67

0.523

0.519 (

\uparrow

0.764\,818\,355\,640\,535\,4

0.545

0.540 (

\uparrow

0.917\,431\,192\,660\,550\,5

0.539

0.519 (

\uparrow

3.710\,575\,139\,146\,568

0.543

0.530 (

\uparrow

2.394\,106\,813\,996\,317

0.547

0.523 (

\uparrow

4.387\,568\,555\,758\,684

MAE

0.775

0.581 (

\uparrow

25.032\,258\,064\,516\,13

0.502

0.471 (

\uparrow

6.175\,298\,804\,780\,876

0.561

0.554 (

\uparrow

1.247\,771\,836\,007\,13

0.763

0.739 (

\uparrow

3.145\,478\,374\,836\,173

0.544

0.541 (

\uparrow

0.551\,470\,588\,235\,294\,1

0.555

0.554 (

\uparrow

0.180\,180\,180\,180\,180\,2

0.552

0.539 (

\uparrow

2.355\,072\,463\,768\,116

0.558

0.550 (

\uparrow

1.433\,691\,756\,272\,401

0.558

0.546 (

\uparrow

2.150\,537\,634\,408\,602

0.3

MSE

0.978

0.507 (

\uparrow

48.159\,509\,202\,453\,99

0.439

0.378 (

\uparrow

13.895\,216\,400\,911\,16

0.571

0.543 (

\uparrow

4.903\,677\,758\,318\,739

0.847

0.821 (

\uparrow

3.069\,657\,615\,112\,161

0.554

0.548 (

\uparrow

1.083\,032\,490\,974\,729

0.553

0.551 (

\uparrow

0.361\,663\,652\,802\,893\,3

0.556

0.543 (

\uparrow

2.338\,129\,496\,402\,878

0.577

0.572 (

\uparrow

0.866\,551\,126\,516\,464\,5

0.559

0.550 (

\uparrow

1.610\,017\,889\,087\,657

MAE

0.779

0.577 (

\uparrow

25.930\,680\,359\,435\,17

0.495

0.464 (

\uparrow

6.262\,626\,262\,626\,263

0.580

0.564 (

\uparrow

2.758\,620\,689\,655\,172

0.743

0.730 (

\uparrow

1.749\,663\,526\,244\,953

0.566

0.560 (

\uparrow

1.060\,070\,671\,378\,092

0.565

0.561 (

\uparrow

0.707\,964\,601\,769\,911\,5

0.563

0.554 (

\uparrow

1.598\,579\,040\,852\,575

0.583

0.581 (

\uparrow

0.343\,053\,173\,241\,852\,5

0.568

0.562 (

\uparrow

1.056\,338\,028\,169\,014

0.5

MSE

0.936

0.494 (

\uparrow

47.222\,222\,222\,222\,22

0.431

0.391 (

\uparrow

9.280\,742\,459\,396\,752

0.573

0.556 (

\uparrow

2.966\,841\,186\,736\,475

0.816

0.794 (

\uparrow

2.696\,078\,431\,372\,549

0.561

0.554 (

\uparrow

1.247\,771\,836\,007\,13

0.558

0.552 (

\uparrow

1.075\,268\,817\,204\,301

0.556

0.550 (

\uparrow

1.079\,136\,690\,647\,482

0.585

0.578 (

\uparrow

1.196\,581\,196\,581\,197

0.581

0.550 (

\uparrow

5.335\,628\,227\,194\,492

TABLE III: Forecasting results on Synth-12 (

L=96

, averaged across

H\in\{96,192,336,720\}

). + SACM denotes backbones augmented with SACM ; bold marks the lower error per pair. Avg. Gain macro-averages relative improvements across noise levels

\sigma\in[0.1,0.9]

$\boldsymbol{\sigma}$	SNR (dB)	SFM	$\boldsymbol{\Delta}$ SFM	MSE	Stress Regime
0.1	23.77	0.008	0.005	0.004	Mild
0.3	16.57	0.023	0.020	0.022	Moderate
0.5	12.39	0.046	0.043	0.058	High
0.7	9.54	0.076	0.073	0.111	Severe
0.9	7.39	0.109	0.106	0.182	Extreme

TABLE IV: Synth-12 corruption gradient statistics across noise levels

\sigma\in\{0.1,\dots,0.9\}

(

N=33,600

, clean SFM

=0.003

Dataset

Metric

Informer ( 2021 )

Crossformer ( 2023 )

PatchTST ( 2023 )

TimesNet ( 2023 )

iTransformer ( 2024 )

TimeMixer ( 2024 )

WPMixer ( 2025 )

TimeFilter ( 2025 )

MultiPatchFormer ( 2025 )

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

ETTh1

MSE

1.337

1.077 (

\uparrow

19.446\,522\,064\,323\,11

0.449

0.441 (

\uparrow

1.781\,737\,193\,763\,92

0.459

0.446 (

\uparrow

2.832\,244\,008\,714\,597

0.534

0.520 (

\uparrow

2.621\,722\,846\,441\,948

0.448

0.445 (

\uparrow

0.669\,642\,857\,142\,857\,1

0.458

0.453 (

\uparrow

1.091\,703\,056\,768\,559

0.431

0.431 (

0

0.428

0.427 (

\uparrow

0.233\,644\,859\,813\,084\,1

0.429

0.429 (

0

MAE

0.823

0.743 (

\uparrow

9.720\,534\,629\,404\,617

0.445

0.439 (

\uparrow

1.348\,314\,606\,741\,573

0.432

0.429 (

\uparrow

0.694\,444\,444\,444\,444\,4

0.492

0.483 (

\uparrow

1.829\,268\,292\,682\,927

0.431

0.431 (

0

0.429

0.427 (

\uparrow

0.466\,200\,466\,200\,466\,2

0.455

0.454 (

\uparrow

0.219\,780\,219\,780\,219\,8

0.452

0.449 (

\uparrow

0.663\,716\,814\,159\,292

0.441

0.439 (

\uparrow

0.453\,514\,739\,229\,024\,9

ETTh2

MSE

2.657

1.392 (

\uparrow

47.610\,086\,563\,793\,75

0.795

0.626 (

\uparrow

21.257\,861\,635\,220\,13

0.384

0.378 (

\uparrow

1.5625

0.480

0.456 (

\uparrow

5

0.384

0.381 (

\uparrow

0.781\,25

0.386

0.380 (

\uparrow

1.554\,404\,145\,077\,72

0.377

0.373 (

\uparrow

1.061\,007\,957\,559\,682

0.381

0.377 (

\uparrow

1.049\,868\,766\,404\,199

0.395

0.384 (

\uparrow

2.784\,810\,126\,582\,278

MAE

1.120

0.827 (

\uparrow

26.160\,714\,285\,714\,29

0.599

0.521 (

\uparrow

13.021\,702\,838\,063\,44

0.403

0.398 (

\uparrow

1.240\,694\,789\,081\,886

0.460

0.447 (

\uparrow

2.826\,086\,956\,521\,739

0.400

0.399 (

\uparrow

0.25

0.402

0.399 (

\uparrow

0.746\,268\,656\,716\,417\,9

0.398

0.396 (

\uparrow

0.502\,512\,562\,814\,070\,4

0.401

0.399 (

\uparrow

0.498\,753\,117\,206\,982\,5

0.411

0.405 (

\uparrow

1.459\,854\,014\,598\,54

ETTm1

MSE

1.480

1.194 (

\uparrow

19.324\,324\,324\,324\,32

0.413

0.394 (

\uparrow

4.600\,484\,261\,501\,211

0.396

0.393 (

\uparrow

0.757\,575\,757\,575\,757\,6

0.519

0.512 (

\uparrow

1.348\,747\,591\,522\,158

0.399

0.399 (

0

0.393

0.392 (

\uparrow

0.254\,452\,926\,208\,651\,4

0.391

0.388 (

\uparrow

0.767\,263\,427\,109\,974\,4

0.394

0.393 (

\uparrow

0.253\,807\,106\,598\,984\,8

0.402

0.399 (

\uparrow

0.746\,268\,656\,716\,417\,9

TABLE V: Real-world forecasting averaged across four prediction horizons per dataset (settings in Sec. VI-A ). + SACM denotes backbones augmented with SACM ; bold marks the lower error per pair. Avg. Gain macro-averages relative improvements across datasets.

Fig. 6: Error trajectories under increasing corruption. (a) Informer exhibits a mid-noise error peak under fixed capacity (red), which is absent with SACM (blue). (b) Crossformer shows decreasing baseline error across the plotted corruption levels; SACM remains lower and smoother.

Dataset	Mean $\pm$ Std.	Variance	Rel. Var.	Allocation
ETTm2	$0.329\pm 0.070$	0.0049	$1.0\times$	Lower variation
ILI	$0.267\pm 0.139$	0.0194	$4.0\times$	Higher variation

TABLE VI: Distribution of sample-adaptive dropout rates during final epochs. Statistics after batch-relative rate mapping.

Training-Data Fractions on ILI
Scenario	Raw		+ SACM
	MAE $\downarrow$	MSE $\downarrow$	MAE $\downarrow$	MSE $\downarrow$
10% Data	1.333	4.185	1.295 ( $\uparrow$ $2.850\,712\,678\,169\,542$ %)	4.074 ( $\uparrow$ $2.652\,329\,749\,103\,943$ %)
25% Data	1.156	3.201	1.144 ( $\uparrow$ $1.038\,062\,283\,737\,024$ %)	3.195 ( $\uparrow$ $0.187\,441\,424\,554\,826\,6$ %)
50% Data	1.116	3.169	1.106 ( $\uparrow$ $0.896\,057\,347\,670\,250\,9$ %)	3.143 ( $\uparrow$ $0.820\,448\,090\,880\,403\,9$ %)
75% Data	1.111	3.169	1.096 ( $\uparrow$ $1.350\,135\,013\,501\,35$ %)	3.128 ( $\uparrow$ $1.293\,783\,527\,926\,791$ %)

TABLE VII: PatchTST performance across training-data fractions (ILI, top) and zero-shot OOD transfer (ETT, bottom). Shaded cells denote SACM ; arrows indicate relative changes.

Dataset

iTransformer

PatchTST

NSFormer

TimesNet

InceptionTime

MiniROCKET

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

ArticularyWordRecognition

97.67

98.33 (

\uparrow

0.66

pp)

97.33

97.67 (

\uparrow

0.34

pp)

97.00

98.00 (

\uparrow

1

pp)

97.33

97.67 (

\uparrow

0.34

pp)

99.00

99.00 (

0

pp)

96.67

97.67 (

\uparrow

1

pp)

AtrialFibrillation

33.33

40.00 (

\uparrow

6.67

pp)

46.67

53.33 (

\uparrow

6.66

pp)

33.33

40.00 (

\uparrow

6.67

pp)

33.33

33.33 (

0

pp)

33.33

33.33 (

0

pp)

40.00

40.00 (

0

pp)

BasicMotions

92.50

95.00 (

\uparrow

2.5

pp)

67.50

70.00 (

\uparrow

2.5

pp)

85.00

87.50 (

\uparrow

2.5

pp)

90.00

90.00 (

0

pp)

25.00

25.00 (

0

pp)

85.00

92.50 (

\uparrow

7.5

pp)

CharacterTrajectories

98.40

98.61 (

\uparrow

0.21

pp)

97.84

97.98 (

\uparrow

0.14

pp)

98.33

98.33 (

0

pp)

98.33

98.54 (

\uparrow

0.21

pp)

99.79

99.79 (

0

pp)

96.45

97.49 (

\uparrow

1.04

pp)

Cricket

87.50

88.89 (

\uparrow

1.39

pp)

84.72

90.28 (

\uparrow

5.56

pp)

90.28

98.61 (

\uparrow

8.33

pp)

79.17

84.72 (

\uparrow

5.55

pp)

98.61

98.61 (

0

pp)

93.06

93.06 (

0

pp)

TABLE VIII: Classification accuracy (%) on 32 datasets and six backbones. Shaded + SACM cells denote SACM and show changes from Raw in percentage points (pp); bold marks the higher accuracy within each pair.

Dataset

Metric

PatchTST

iTransformer

NSFormer

AnomTrans

TranAD

DCdetector

TimesNet

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

Raw

+ SACM

SMD (%)

44.74

70.27 (

\uparrow

25.53

pp)

53.71

79.98 (

\uparrow

26.27

pp)

63.71

67.10 (

\uparrow

3.39

pp)

66.01

66.06 (

\uparrow

0.05

pp)

60.78

70.86 (

\uparrow

10.08

pp)

66.63

66.17 (

\downarrow

0.46

pp)

44.78

45.66 (

\uparrow

0.88

pp)

99.04

96.80 (

\downarrow

2.24

pp)

99.35

86.84 (

\downarrow

12.51

pp)

78.52

79.61 (

\uparrow

1.09

pp)

84.97

85.16 (

\uparrow

0.19

pp)

84.99

83.63 (

\downarrow

1.36

pp)

83.15

84.46 (

\uparrow

1.31

pp)

99.92

99.92 (

0

pp)

61.64

81.43 (

\uparrow

19.79

pp)

69.73

83.27 (

\uparrow

13.54

pp)

70.34

72.82 (

\uparrow

2.48

pp)

74.30

74.40 (

\uparrow

0.1

pp)

70.87

76.72 (

\uparrow

5.85

pp)

73.98

74.20 (

\uparrow

0.22

pp)

61.85

62.68 (

\uparrow

0.83

pp)

MSL (%)

72.93

88.19 (

\uparrow

15.26

pp)

83.01

88.22 (

\uparrow

5.21

pp)

55.96

70.44 (

\uparrow

14.48

pp)

45.14

59.26 (

\uparrow

14.12

pp)

66.08

67.18 (

\uparrow

1.1

pp)

74.01

74.69 (

\uparrow

0.68

pp)

86.48

88.72 (

\uparrow

2.24

pp)

91.09

77.61 (

\downarrow

13.48

pp)

38.93

76.18 (

\uparrow

37.25

pp)

45.72

86.64 (

\uparrow

40.92

pp)

4.27

96.77 (

\uparrow

92.5

pp)

34.02

72.33 (

\uparrow

38.31

pp)

82.65

92.64 (

\uparrow

9.99

pp)

53.70

94.94 (

\uparrow

41.24

pp)

TABLE IX: Point-adjusted anomaly-detection precision (P), recall (R), and F1 scores across four datasets (%). Shaded + SACM columns denote SACM ; bold marks higher values per pair. Avg.

\Delta

F1 reports macro-averaged F1 gains in percentage points (pp).

	ILI ( $H{=}48$ )					Exchange Rate ( $H{=}96$ )
Backbone	$\boldsymbol{p^{\star}}$	Raw	Best Fixed	Learned Global	SACM	$\boldsymbol{p^{\star}}$	Raw	Best Fixed	Learned Global	SACM
TimeFilter	0.100	2.466/0.974	2.619/1.012	2.469/0.980	2.461 / 0.968	0.000	0.108/0.232	0.108/0.232	0.106/0.230	0.105 / 0.229
PatchTST	0.000	2.939/1.099	2.956/1.111	2.879/1.095	2.874 / 1.093	0.100	0.107 /0.230	0.108/0.231	0.109/0.230	0.107 / 0.229
TimeMixer	0.000	3.055 /1.130	3.227/1.207	3.217/1.205	3.064/ 1.129	0.000	0.102/ 0.224	0.102/ 0.224	0.101 /0.225	0.101 / 0.224
Informer	0.450	7.173/1.920	6.569/1.807	6.650/1.936	6.348 / 1.784	0.050	4.411/1.676	2.754/1.169	2.777/1.143	2.078 / 1.050

TABLE X: Dropout strategy comparison on forecasting benchmarks (

H=96

for Exchange Rate,

H=48

for ILI). Entries report test MSE/MAE.

p^{\star}

is validation-selected; bold marks lowest MSE.

Method	Components			MSE $\downarrow$		MAE $\downarrow$
Method	Detrend	Norm	log-SFM	Avg.	Gain (%)	Avg.	Gain (%)
Baseline	-	-	-	1.159	-	0.836	-
Minimal Model	None	✗	✗	1.514	$\downarrow$ 30.6	0.960	$\downarrow$ 14.8
w/o Detrend+Norm	None	✗	✓	1.468	$\downarrow$ 26.7	0.942	$\downarrow$ 12.7
w/o Detrend	None	✓	✓	1.642	$\downarrow$ 41.7	1.014	$\downarrow$ 21.3
Simple Detrend	Simple	✓	✓	1.574	$\downarrow$ 35.8	0.987	$\downarrow$ 18.1

TABLE XI: Component ablation on Synth-12 (

\sigma\in[0.1,0.9]

). Gain denotes relative change vs. baseline (

p=0.1

Fig. 7: Hyperparameter sensitivity analysis. (a) Effect of sensitivity parameter

\gamma

across corruption scales

\sigma\in[0.1,0.9]

; (b) MSE grouped by sensitivity setting, with separate markers for the five corruption scales.

	Overhead (Cost)		Training Cost		Net Gain
Model	Params	s/Epoch	Epochs	Total (s)	Saved	Speedup
TimeMixer	+16	102.3 $\to$ 113.7	20 $\to$ 16	2045 $\to$ 1819	11.0%	1.12 $\times$
Informer	+16	0 95.8 $\to$ 123.9	30 $\to$ 16	2873 $\to$ 1982	31.0%	1.45 $\times$

TABLE XII: Training cost for the paired runs (Raw

\to

+ SACM ). Epochs denotes epochs to stopping; total time includes SACM overhead.

Compatibility with SL
Method Strategy	Mechanism	Error Metric $\downarrow$		Rel.
Method Strategy	Mechanism	MSE	MAE	$\boldsymbol{\Delta_{\mathrm{rel}}}$ (%)
Baseline (Raw)	-	7.140	1.916	-
SACM	Capacity Modulation	6.429	1.846	$\uparrow$ 10.0
SL	Data Selection	6.461	1.902	$\uparrow$ 9.5
SL + SACM (Ours)	Combined	6.336	1.774	$\uparrow$ 11.3

TABLE XIII: Compatibility with Selective Learning (SL) and computational cost on ILI (

L=24,H=60

, Informer). Rel. Gain denotes relative MSE reduction from Raw.

Fig. 8: Qualitative comparison of trajectories. Input context (gray), clean ground truth (green), Raw WPMixer (dashed red), and SACM (blue).

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Dim.	Size	Split	Frequency	Prediction Length	Domain
ETTh1	7	14,400	6:2:2	1 hour	{96, 192, 336, 720}	Temperature
ETTh2	7	14,400	6:2:2	1 hour	{96, 192, 336, 720}	Temperature
ETTm1	7	57,600	6:2:2	15 min	{96, 192, 336, 720}	Temperature
ETTm2	7	57,600	6:2:2	15 min	{96, 192, 336, 720}	Temperature
Weather	21	52,696	7:1:2	10 min	{96, 192, 336, 720}	Meteorology
Electricity	321	26,304	7:1:2	1 hour	{96, 192, 336, 720}	Electricity

Appendix

TABLE XV: Statistics of the nine real-world forecasting benchmarks. Columns report dimensionality, time steps, chronological train/val/test split, sampling frequency, prediction horizons, and domain; ILI and ECG use task-specific shorter horizons.

Dataset	Dim.	Train	Test	Anomaly Rate	Domain
SMD	38	708,405	708,420	4.16%	Server machine
MSL	55	58,317	73,729	10.72%	Spacecraft
SMAP	25	135,183	427,617	13.13%	Spacecraft
PSM	25	132,481	87,841	27.76%	Server machine

Appendix

TABLE XVI: Statistics of the four anomaly-detection benchmarks. Columns report dimensionality, training and test time steps, anomaly ratio in the test split, and domain. Evaluation follows the point-adjusted protocol defined in Appendix A-C .

Dataset	Sequence/target	Dataset	Sequence/target
ArticularyWordRecognition	Articulatory trajectories; spoken words	AtrialFibrillation	ECG; cardiac rhythm
BasicMotions	Wearable sensors; basic motions	CharacterTrajectories	Pen-tip trajectories; characters
Cricket	Motion sensors; umpire gestures	DuckDuckGeese	Audio features; bird calls
EigenWorms	Posture sequences; locomotion	Epilepsy	Wrist acceleration; seizure-like motions
EthanolConcentration	Spectral sequences; concentration	ERing	Ring-sensor trajectories; gestures
FaceDetection	Brain signals; face detection	FingerMovements	Brain signals; finger movements

Appendix

Table 28

$\sigma$	$H$	Informer				Crossformer				PatchTST				TimesNet				iTransformer				TimeMixer				WPMixer				TimeFilter				MultiPatchFormer
		[ 56 ]				[ 54 ]				[ 33 ]				[ 43 ]				[ 28 ]				[ 42 ]				[ 31 ]				[ 16 ]				[ 32 ]
		Raw		+ SACM		Raw		+ SACM		Raw		+ SACM		Raw		+ SACM		Raw		+ SACM		Raw		+ SACM		Raw		+ SACM		Raw		+ SACM		Raw		+ SACM
		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
0.1	96	1.214	0.854	0.695	0.683	0.260	0.377	0.227	0.352	0.262	0.381	0.262	0.377	0.476	0.556	0.472	0.550	0.260	0.375	0.255	0.371	0.261	0.378	0.260	0.377	0.258	0.374	0.253	0.367	0.267	0.380	0.264	0.379	0.267	0.383	0.255	0.375
	192	0.811	0.722	0.479	0.561	0.475	0.511	0.405	0.478	0.603	0.595	0.586	0.589	0.957	0.797	0.949	0.791	0.567	0.564	0.564	0.563	0.608	0.590	0.601	0.587	0.593	0.583	0.567	0.563	0.594	0.591	0.569	0.569	0.612	0.580	0.565	0.568
	336	0.723	0.688	0.436	0.535	0.516	0.537	0.403	0.493	0.661	0.635	0.632	0.620	1.052	0.845	1.015	0.827	0.623	0.609	0.618	0.606	0.651	0.620	0.647	0.620	0.654	0.628	0.621	0.605	0.646	0.621	0.646	0.619	0.640	0.620	0.639	0.620

Appendix

TABLE XVII: Complete per-horizon forecasting results on Synth-12 for nine backbones. Each horizon reports Raw and SACM (+ SACM ) MSE/MAE; lower errors are bold and higher errors are underlined. Up and down arrows report relative improvements and regressions, respectively, averaged over horizons.

Dataset	$H$	Informer				Crossformer				PatchTST				TimesNet				iTransformer				TimeMixer				WPMixer				TimeFilter				MultiPatchFormer
		[ 56 ]				[ 54 ]				[ 33 ]				[ 43 ]				[ 28 ]				[ 42 ]				[ 31 ]				[ 16 ]				[ 32 ]
		Raw		+ SACM		Raw		+ SACM		Raw		+ SACM		Raw		+ SACM		Raw		+ SACM		Raw		+ SACM		Raw		+ SACM		Raw		+ SACM		Raw		+ SACM
		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	96	1.569	0.902	1.305	0.833	0.394	0.404	0.384	0.397	0.394	0.392	0.383	0.390	0.488	0.475	0.456	0.454	0.384	0.391	0.382	0.390	0.401	0.395	0.388	0.387	0.388	0.386	0.386	0.385	0.390	0.390	0.387	0.388	0.391	0.383	0.395	0.385
	192	1.410	0.856	1.063	0.732	0.436	0.431	0.427	0.421	0.447	0.423	0.443	0.422	0.513	0.478	0.509	0.475	0.438	0.422	0.437	0.422	0.443	0.420	0.437	0.419	0.420	0.441	0.419	0.442	0.420	0.443	0.421	0.440	0.421	0.429	0.421	0.429
	336	1.221	0.762	0.981	0.673	0.471	0.453	0.463	0.446	0.490	0.444	0.480	0.440	0.581	0.505	0.565	0.496	0.487	0.446	0.487	0.446	0.492	0.441	0.492	0.441	0.448	0.492	0.447	0.490	0.441	0.488	0.438	0.484	0.441	0.474	0.441	0.471

Appendix

TABLE XVIII: Complete per-horizon real-world forecasting results for nine backbones. Each horizon reports Raw and SACM (+ SACM ) MSE/MAE; lower errors are bold and higher errors are underlined. Up and down arrows report relative improvements and regressions, respectively, averaged over horizons.

ETTh1 (

L{=}96,H{=}96

)

ILI (

L{=}24,H{=}24

)

Exchange Rate (

L{=}96,H{=}96

)

Backbone

Raw MSE

+ SACM MSE

Gain (%)

W/T/L

p

Raw MSE

+ SACM MSE

Gain (%)

W/T/L

p

Raw MSE

+ SACM MSE

Gain (%)

W/T/L

p

Informer

1.2658\pm 0.1879

\mathbf{1.0658}\pm 0.0453

\uparrow

15.8

4/0/1

0.076

6.2605\pm 0.7293

\mathbf{5.8442}\pm 0.2124

\uparrow

6.7

3/0/2

0.301

4.0671\pm 0.8449

\mathbf{1.2747}\pm 0.3771

\uparrow

68.7

5/0/0

0.003

PatchTST

0.3943\pm 0.0047

\mathbf{0.3897}\pm 0.0039

\uparrow

1.2

4/0/1

0.056

3.3997\pm 0.3373

\mathbf{2.8384}\pm 0.1953

\uparrow

16.5

5/0/0

0.004

0.1040\pm 0.0018

\mathbf{0.1022}\pm 0.0014

\uparrow

1.7

5/0/0

0.005

TimeMixer

0.3946\pm 0.0025

\mathbf{0.3920}\pm 0.0020

\uparrow

0.7

5/0/0

0.007

3.3904\pm 0.1844

\mathbf{3.3582}\pm 0.3473

\uparrow

1.0

2/0/3

0.764

0.1024\pm 0.0005

\mathbf{0.1017}\pm 0.0007

\uparrow

0.7

4/0/1

0.166

TimeFilter

0.3895\pm 0.0010

\mathbf{0.3891}\pm 0.0009

\uparrow

0.1

4/0/1

0.104

1.8652\pm 0.1738

\mathbf{1.8227}\pm 0.1565

\uparrow

2.3

4/0/1

0.550

0.1044\pm 0.0014

\mathbf{0.1033}\pm 0.0014

\uparrow

1.0

5/0/0

0.124

Appendix

TABLE XIX: Paired-seed stability across five random seeds (2022–2026). MSE entries show mean

\pm

std. W/T/L denotes wins/ties/losses.

Dataset

Backbone

\boldsymbol{p^{\star}}

Fixed dropout rate

p

(test MSE/MAE)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

ETTh1

Informer

0.50

1.478/0.872

1.508/0.879

1.367/0.835

1.425/0.860

1.360/0.836

1.432/0.858

1.362/0.844

1.506/0.884

1.314/0.824

1.284/0.815

0.975/0.739

Exchange Rate

Informer

0.05

4.024/1.589

2.754/1.169

4.107/1.601

4.227/1.626

4.261/1.625

4.192/1.617

4.214/1.619

4.277/1.625

4.213/1.612

4.170/1.601

4.274/1.621

ILI

Informer

0.45

7.662/2.060

9.094/2.213

9.310/2.254

8.394/2.102

8.853/2.181

7.877/2.023

8.567/2.122

8.519/2.106

8.573/2.119

6.569/1.807

8.792/2.146

ETTh1

PatchTST

0.05

0.392/0.392

0.386/0.390

0.383/0.390

0.391/0.390

0.390/0.390

0.388/0.390

0.387/0.390

0.387/0.391

0.388/0.391

0.391/0.393

0.399/0.396

Exchange Rate

PatchTST

0.10

0.107/0.230

0.102/0.225

0.108/0.231

0.101/0.223

0.101/0.225

0.103/0.226

0.103/0.227

0.104/0.228

0.105/0.229

0.106/0.230

Appendix

TABLE XX: Fixed-dropout sweep at

H=96

for ETTh1 and Exchange Rate and

H=48

for ILI. Cells report test MSE/MAE;

p^{\star}

is selected solely by validation MSE and marked in bold. Underlined cells mark the second-lowest test MSE in each row.

Method	Configuration			$\sigma=0.1$		$\sigma=0.3$		$\sigma=0.5$		$\sigma=0.7$		$\sigma=0.9$		Average		$\Delta\%$ (vs. Base)
Method	Detrend	LogNorm	log-SFM	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
Baseline	-	-	-	1.228	0.862	1.189	0.846	1.180	0.843	1.140	0.831	1.060	0.798	1.159	0.836	-	-
Minimal Model	None	✗	✗	1.698	1.034	1.907	1.092	1.668	1.010	0.633	0.650	1.665	1.015	1.514	0.960	30% $\downarrow$	14% $\downarrow$
w/o Detrend+Norm	None	✗	✓	2.041	1.124	1.159	0.843	1.114	0.827	1.954	1.111	1.072	0.806	1.468	0.942	26% $\downarrow$	12% $\downarrow$
w/o Detrend	None	✓	✓	1.672	1.017	1.863	1.093	1.721	1.051	1.442	0.950	1.513	0.960	1.642	1.014	41% $\downarrow$	21% $\downarrow$
Simple Detrend	Simple	✓	✓	1.416	0.943	1.757	1.042	1.674	1.018	1.486	0.958	1.535	0.974	1.574	0.987	35% $\downarrow$	18% $\downarrow$

Appendix

TABLE XXI: Complete component ablation on Synth-12 across five corruption scales. Rows report MSE/MAE at each

\sigma

and on average;

\Delta

is relative to fixed dropout. OLS denotes global ordinary least-squares detrending. The full SACM configuration is labeled “Ours,” and bold marks the lowest error in each result column.

Backbone / dataset	Metric	Batch size $B$					MSE variation
Backbone / dataset	Metric	8	16	32	64	128	Range (%)	CV (%)
PatchTST ETTh1	MSE	0.390715	0.391424	0.383258	0.392519	0.384708	2.42	0.97
PatchTST ETTh1	MAE	0.392903	0.392274	0.390247	0.391016	0.388892	2.42	0.97
WPMixer Synth-12 ( $\sigma{=}0.5$ )	MSE	0.280752	0.285856	0.286771	0.278124	0.279373	3.11	1.24
WPMixer Synth-12 ( $\sigma{=}0.5$ )	MAE	0.390588	0.397123	0.394220	0.387984	0.388969	3.11	1.24

Appendix

TABLE XXII: Batch-size sensitivity at

H=96

for two representative backbones. Because the rate mapper normalizes residual scores within each mini-batch, we vary

B\in\{8,16,32,64,128\}

and report test MSE/MAE; bold marks the lowest MSE in each setting. All runs within a setting share a fixed control configuration, so the comparison isolates the effect of

B

. The last two columns summarize the MSE sweep: relative range is

(\max_{B}\mathrm{MSE}-\min_{B}\mathrm{MSE})/\min_{B}\mathrm{MSE}

, and CV is the population standard deviation across the five batch sizes divided by their mean.

Fig. 9: Stationary periodic signals under three isolated corruptions (Gaussian, heavy-tail, point-wise missing values). Columns: corrupted input, amplitude spectrum, and hard-threshold reconstruction. Narrow peaks remain separable from broadband residual energy, illustrating spectral separation in these examples.

Fig. 10: Trend-seasonal signals under three isolated corruptions. Columns: corrupted input, spectrum after global OLS detrending, and reconstruction with trend restoration. Detrending reduces DC leakage so the residual better reflects corruption rather than deterministic drift.

Fig. 11: Linear chirps sweeping from 0.5 to 3.0 Hz under three isolated corruptions. Columns: corrupted input, amplitude spectrum, and hard-threshold reconstruction. Energy forms a structured band rather than isolated peaks; the retained band preserves the sweep while corruptions raise residual energy.

Fig. 12: Amplitude-modulated signals (2.0 Hz carrier, 0.2 Hz envelope) under three corruptions. Columns: corrupted input, spectrum with carrier and sidebands, and hard-threshold reconstruction. The reconstruction retains the amplitude variation, while the residual can contain both corruption and modulation detail.

Fig. 13: Top-10% Fourier reconstruction on nine forecasting datasets. Each pair shows the original and reconstructed series with Pearson correlation, followed by the amplitude spectrum and retention threshold. Seven correlations exceed 0.95; Electricity and ECG yield 0.868 and 0.769. “Illness” denotes ILI. The hard mask is used for this diagnostic; SACM uses a learned soft mask to compute a training-time regularization cue.

Fig. 14: Comparison of no detrending, endpoint interpolation, and global OLS under four boundary stress tests, scored by reconstruction Edge-MAE near the window edges. Each row compares the three detrending strategies; panels show the observed input, clean reference, estimated trend, reconstruction, and Edge-MAE. Row 1 (start-point outlier): endpoint interpolation inherits the spike as the full slope, while global OLS dilutes its leverage. Row 2 (seasonal phase mismatch):

x_{1}\neq x_{L}

induces a join discontinuity; both detrending variants reduce leakage relative to no detrending. Row 3 (quadratic trend with endpoint noise): no detrending achieves the lowest Edge-MAE, showing that a linear trend model is not uniformly preferable under model mismatch. Row 4 (regime shift then final-step sensor failure): endpoint interpolation flattens the high-state evidence, whereas global OLS recovers a more suitable linear fit. Global OLS wins rows 1, 2, and 4. The plot label “Robust OLS refers to ordinary global OLS, not a robust-regression estimator.”

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Irregularly sampled multivariate time series (ISMTS) are prevalent in real-world applications, where both observation times and available measurements can vary substantially across domains. While recent models increasingly exploit such sampling information for prediction, its robustness under sampling pattern shifts remains underexplored. We introduce HAR-C, to the best of our knowledge the first controlled benchmark for sampling pattern shifts in ISMTS, and show that sampling shifts alone can substantially degrade performance, induce sampling-specific shortcuts, and remain challenging for existing domain generalization (DG) methods. Motivated by these findings, we propose PRISM, a DG framework that first learns complementary feature-centric and sampling-centric representations without task labels, and subsequently performs robust supervised training across diverse sampling variations to discourage brittle shortcut reliance. Extensive experiments on controlled and real-world ISMTS benchmarks demonstrate that PRISM consistently improves robustness to unseen sampling shifts over existing methods. Our code is available at https://anonymous.4open.science/r/PRISM.

Abstract

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