A Unified Scaling Law for Time Series Foundation Models

Xilin Dai; Yiding Liu; Zewei Dong; Jiang-Ming Yang; Qiang Xu

cs.LGOct 4, 2026

A Unified Scaling Law for Time Series Foundation Models

Authors: Xilin Dai, Yiding Liu, Zewei Dong, Jiang-Ming Yang, Qiang Xu

Organizations: Ant International · The Chinese University of Hong Kong

Abstract

We develop a Unified Scaling Law and a Unified Theory of Time Series Learning to understand how model capacity and historical information support forecasting. Across different lookback lengths and forecast horizons, we analyze 18,768 experimental cells from 21 checkpoints on 23 dataset-frequency tasks spanning six domains. Our empirical methodology integrates local resource relations into a parsimonious, fitted five-parameter law: capacity gains increase with history, context gains diminish toward saturation, and horizon effects enter as a common shift. Fitted without Toto 2.0, the law predicts its horizon-averaged capacity-scaling curves with mean absolute percentage errors of 1.09% and 1.50% at input lengths 2048 and 4096. To understand how history supports prediction, our learning theory uses Gaussian regression to analyze rule identification and predictive capability. We hypothesize that full-shot models learn by accumulating information in weights, while frozen time series foundation models (TSFMs) use history by extracting information through activations. Matched-history comparisons establish the predictive value of additional history. Controlled parameter exchanges and activation interventions provide evidence that history-derived rule information can be retained, reused across queries, and used to recover a contribution to long-context prediction. Together, these findings inform capacity scaling, context allocation, and the development of models that retain and apply historical rules. Code and main results are available at https://github.com/Fifthky/UniScale.

Figures & tables

Source	TSFM Checkpoints	Dataset-Frequency Task	Experimental Cells	Points
Controlled GIFT-Horizon	21	23	3,128	390
Controlled Grid-Horizon	21	23	15,640	680
Combined Controlled	21	23	18,768	1,070
GIFT Public	37	23	2,257	199

Table 1: Grid-first selection retains dataset-level results (cells); points aggregate responses for fitting and prediction. Controlled GIFT-Horizon contains only cells absent from Controlled Grid-Horizon.

Figure 1: The Unified Scaling Law and observed resource profiles. (a) Predicted MASE over

N,L

H=48,192,720

(blue, green, orange). (b)–(d) Observed means (markers) and mean predictions (lines) over identical capacity, input-length, and horizon groups, using all 1,070 Controlled points. Resource axes are logarithmic;

R

is linear in (a) and logarithmic in (b)–(d). 1k denotes 1,024 steps.

Figure 2: Observed resource relations and reduced-law restrictions: (a) capacity response, (b) changing context returns, (c) history-dependent capacity gains, and (d) context elasticity for the representative 22M group. Unified curves use fitted coefficients; orange dashed curves impose restrictions without refitting. Faint points show individual Controlled contrasts; emphasized markers show medians, with interquartile-range (IQR) bars where shown. Appendix C.4 specifies each comparison.

Figure 3: Controlled/Public prediction and horizon-averaged Toto 2.0 capacity curves.

Figure 4: Forecasting with matched history budgets on 23 tasks. Full-shot curves average three seeds; TSFM curves show the mean and median over a fixed set of available checkpoints covering the grid. Seed error bars are too small to distinguish at this scale: mean pointwise relative standard deviations are

\pm 0.208\%

for DLinear and

\pm 0.502\%

for PatchTST. The line marks Seasonal Naive.

Process	Query only	Transfer	Natural	Erased	Recovered
AR(1)	0.26289	0.24181	0.17697	0.23986	0.21229
Lag-8	0.26972	0.25106	0.19702	0.25074	0.23755
Threshold	0.30168	0.27873	0.24983	0.28404	0.26195

Table 2: Shared-site transfer and recovery in TimesFM 2.5. Conditional-mean MASE, averaged over three rule magnitudes and two donor repetitions. Natural, erased, and recovered predictions share the 512-step recipient history.

Figure 5: Context gains and fitted saturation scales. Left: mean and median MASE improvements between adjacent inputs for matched checkpoint–horizon pairs. Right: fitted

\log R

H=96

for three capacities;

L_{\star}

markers locate approximate saturation regions within the evaluated range.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Sampling intervals	Domain
Bitbrains fast storage	5 min	Cloud/web operations
Bitbrains random	5 min	Cloud/web operations
BizITObs L2C	5 min, 1 h	Cloud/web operations
Electricity	15 min, 1 h	Energy
ETT1	15 min, 1 h	Energy
ETT2	15 min, 1 h	Energy

Appendix

Table 3: The fixed evaluation panel and its application domains. Sampling intervals span five, ten, and fifteen minutes, one hour, one day, and one week. Nineteen pairs have short, medium, and long GIFT configurations; the four daily/weekly pairs contribute four short configurations, for 61 in total.

Checkpoint	Active $N$	Total $N$	Checkpoint	Active $N$	Total $N$
Toto 2.0 4M	4.14	4.14	Moirai 1.1 large	310.97	310.97
Toto 2.0 22M	21.92	21.92	Moirai 2.0 small	11.39	11.39
Toto 2.0 313M	312.68	312.68	TimesFM 1.0	200.00	200.00
Toto 2.0 1B	1041.03	1041.03	TimesFM 2.0	500.00	500.00
Toto 2.0 2.5B	2454.28	2454.28	TimesFM 2.5	200.00	231.29
Chronos 2 small	27.93	27.93	TiRex 1.1 GIFT	35.00	35.00

Appendix

Table 4: Controlled checkpoints and active and total parameter counts, in millions.

Figure 6: Resource coverage across active capacity

N

, input length

L

, and scored horizon

H

(colors and markers). Public coordinates are grid-binned; Controlled coordinates use data-supported inputs. The Controlled GIFT-Horizon Panel shows 662 protocol points and contributes 390 additional points after Grid-first selection (Table 1 ). Small horizontal offsets separate overlapping horizons.

Form	Controlled	Public	Toto
Constant	14.83	16.92	17.10
Capacity-independent	9.96	13.01	11.84
Unified	10.13	12.82	11.53

Appendix

Table 5: Reference prediction MAPEs (%) on Controlled, Public, and held-out Toto response points.

Figure 7: Complete Toto 2.0 evaluation. (a,b) Observed versus predicted $\log R$ at all 150 original NHL coordinates for the all-checkpoint and leave-Toto-out fits; color encodes active parameters and the diagonal denotes exact prediction. (c,d) Capacity response after pairing adjacent input lengths and averaging the five horizons. Solid circles are observed Toto means and dashed crosses are predictions from the corresponding fit. Each context curve contains all five Toto checkpoints.

Model	Input $C$	History $T=4096$	History $T=8192$
DLinear	96	1.04264	1.03958
DLinear	512	0.88869	0.88328
PatchTST	96	0.91400	0.91361
PatchTST	512	0.86638	0.84896
PatchTST-compact	96	0.99099	0.98986
PatchTST-compact	512	0.90357	0.89159

Appendix

Table 6: Validation-selected MASE in the crossed experiment, averaged over 23 tasks and three seeds.

T

is training history and

C

is immediate input.

Process	Frozen model	$L=96$	$L=2048$	Rule projection
AR(1)	TimesFM 2.5	0.32424	0.12650	0.812
AR(1)	Chronos 2	0.33295	0.12958	0.666
Lag-8	TimesFM 2.5	0.33272	0.16004	0.739
Lag-8	Chronos 2	0.34423	0.26440	0.133
Threshold	TimesFM 2.5	0.37300	0.22949	0.454
Threshold	Chronos 2	0.37281	0.25215	0.369

Appendix

Table 7: Conditional-mean MASE with fixed population statistics at

a=0.3

. Rule projection measures the predicted contrast along the known conditional-mean contrast; one corresponds to the oracle amplitude.

Process	Model	$L=1024$	$L=4096$	$L=8192$	Transfer projection
AR(1)	DLinear	0.27871	0.16235	0.12634	0.986
AR(1)	PatchTST	0.32585	0.14790	0.09567	1.028
Lag-8	DLinear	0.32463	0.17748	0.12204	0.985
Lag-8	PatchTST	0.34635	0.14133	0.10034	0.946
Threshold	DLinear	0.35156	0.25287	0.22841	0.584
Threshold	PatchTST	0.37115	0.20446	0.18716	0.755

Appendix

Table 8: Conditional-mean MASE of trained predictors, averaged over three seeds. Transfer projection measures the donor-rule response after parameter exchange and shared-history normalization calibration at

L=8192

Model	Objective	Baseline	Matched	Opposite	Random	Shuffled
TimesFM 2.5	Transfer	0.27810	0.25720	0.30316	0.31468	0.26804
TimesFM 2.5	Recovery	0.25821	0.23726	0.29228	0.29514	0.25137
Chronos 2	Transfer	0.28774	0.28776	0.29987	0.28710	0.29144
Chronos 2	Recovery	0.21064	0.19798	0.20436	0.20984	0.20454

Appendix

Table 9: Conditional-mean MASE for the primary TimesFM setting and independently confirmed Chronos setting. Baseline is query-only prediction for transfer and erased prediction for recovery. Matched and opposite refer to the donor rule.

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Retrieval Mechanisms Surpass Long-Context Scaling in Time Series Forecasting

Rishi Ahuja, Kumar Prateek, Simranjit Singh, Vijay Kumar

Department of Information Technology, Dr. B.R. Ambedkar National Institute of Technology Jalandhar, Punjab, 144008, India.

Time Series Foundation Models Retrieval-Augmented Forecasting Wireless Foundation Models Volatility

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Time Series Foundation Models (TSFMs) have borrowed the long context paradigm from natural language processing under the premise that feeding more history into the model improves forecast quality. But in stochastic domains, distant history is often just high-frequency noise, not signal. Hence, the proposed work tests whether this premise actually holds by running continuous context architectures (PatchTST included) through the ETTh1 benchmark. The obtained results contradict the premise: an inverse scaling law shows up clearly, with forecasting error rising as context gets longer. A 3,000-step window causes performance to drop by over 68%, evidence that attention mechanisms are poor at ignoring irrelevant historical volatility. Retrieval-Augmented Forecasting (RAFT) is evaluated as an alternative. RAFT achieves a mean squared error (MSE) of 0.379 with a fixed 720-step window and selective retrieval, outperforming both long-context configurations and zero-shot foundation models (Chronos, Moirai) despite requiring far less computation. In addition, the retrieval step injects only the most relevant historical segments as dynamic exogenous variables, which gives the model a context-informed inductive bias it cannot build on its own from raw sequences. Therefore, foundation models going forward need to shift architecturally toward selective retrieval.

Abstract

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