Kairos: Toward Adaptive and Parameter-Efficient Time Series Foundation Models

Kun Feng; Shaocheng Lan; Yuchen Fang; Wenchao He; Sihan Lu; Shuqi Gu; Xingyu Lu; Lintao Ma

cs.LGSep 30, 2025

Kairos: Toward Adaptive and Parameter-Efficient Time Series Foundation Models

Authors: Kun Feng, Shaocheng Lan, Yuchen Fang, Wenchao He, Sihan Lu, Shuqi Gu, Xingyu Lu, Lintao Ma, +1 more

Organizations: School of Information Science and Technology, ShanghaiTech University, Shanghai, China · Ant Group, Shanghai, China

Abstract

Inherent temporal heterogeneity, such as varying sampling densities and periodic structures, has posed substantial challenges in zero-shot generalization for Time Series Foundation Models (TSFMs). Existing TSFMs predominantly rely on massive parameterization to absorb such heterogeneity, as their static tokenization and positional encoding schemes entangle diverse temporal patterns into a fixed representation space, encouraging memorization rather than adaptation. To address this limitation, we propose Kairos, a flexible and parameter-efficient TSFM dedicated to forecasting tasks, which decouples temporal heterogeneity from model capacity through a novel tokenization perspective. Kairos introduces a dynamic patching tokenizer and a mixture-of-size encoding that adapt observational granularity to local information density, enabling fine-grained temporal abstraction without increasing model width or depth. In addition, we design a multi-granularity positional embedding based on dynamic rotary encodings, which conditions on instance-level spectral features and temporal structure induced by dynamic patching tokenization, allowing robust modeling of diverse temporal dependencies. Trained on a novel Predictability-Stratified Time-Series (PreSTS) corpus, Kairos achieves superior zero-shot performance with substantially fewer parameters on two mainstream benchmarks, GIFT-Eval and Time-Series-Library. The project page is at https://foundation-model-research.github.io/Kairos .

Figures & tables

Figure 1 : (a) Comparison on the GIFT-Eval benchmark [ 3 ] demonstrates that our Kairos achieves superior zero-shot forecasting performance (lower normalized MASE) while requiring significantly fewer parameters than existing TSFMs. (b) (c) Significant variation exists in information density (i.e., signal complexity) across and within different time series datasets. (d) Existing TSFMs primarily use tokenization methods like point-wise or fixed-size patching, while our Kairos uses Mixture-of-Size Tokenization to dynamically adapt to information density, enabling the learning of generalizable rules rather than memorization.

Figure 2 : The architecture of Kairos , which includes (i) The Mixture-of-Size Encoder (right) adaptively tokenizes the input series by routing segments to optimal granularity experts and fusing their embeddings into a unified sequence $\boldsymbol{H}$ . Embedding subscripts identify the expert and patch within a segment (Eq. 2 ). (ii) The Heterogeneity-Aware Transformer (middle) processes these tokens using Dynamic Rotary Position Embedding (DRoPE) (left), which modulates frequencies based on instance-level spectral features and calibrates positions to account for varying patch sizes. (iii) The Multi-Patch Decoder (top-middle) employs learnable forecast tokens to predict multiple future patches in parallel via cross-attention.

Type	Statistical	DL (Full-Shot)		TSFMs (TestData Leakage)				TSFMs (Zero-Shot)
Method	Seasonal Naïve	DLinear	PTST.	TTM	Chronos	Chronos Bolt	TimesFM	Moirai	VisionTS	Ying.	Toto	Sundial	$\textsc{Kairos}_{s}$ (ours)	$\textsc{Kairos}_{b}$ (ours)
#Params	-	-	-	5M	709M	205M	500M	311M	112M	300M	151M	128M	23M	53M
MASE $\downarrow$	1.000	1.061	0.849	1.020	0.870	0.808	0.758	0.875	0.863	0.798	0.750	0.750	0.748	0.738
CRPS $\downarrow$	1.000	0.846	0.587	0.873	0.574	0.574	0.550	0.599	0.755	0.548	0.517	0.559	0.554	0.548

Table 1 : Performance evaluation on the GIFT-Eval. We evaluate using normalized MASE and CRPS (Section 4.1 ). Baselines encompass statistical, Deep Learning (DL), and TSFM methods, with TSFMs categorized into TestData Leakage and Zero-Shot following official protocols. TestData Leakage signifies that the model’s training set included test data, whereas Zero-Shot indicates that the model was trained without any data from the test set or its corresponding training split. Baseline results are officially reported by GIFT-Eval.

Figure 3 : Zero-shot performance on TSLib averaged over prediction lengths {96, 192, 336, 720}. The subscripts

l

b

s

m

, and

a

represent model sizes of large, base, small, mini and advanced, respectively. The complete experimental results are presented in Appendix J .

Model Variants	Short	Medium	Long	AVG
Impact of Encoder Design
w/ Fixed Patch Size (32)	0.724	0.802	0.820	0.761
w/ Mixture-of-Size (w/o Null Experts)	0.720	0.770	0.800	0.748
Impact of Positional Embedding
w/ Standard RoPE (Fixed $\theta$ )	0.729	0.807	0.835	0.767
w/ Only Instance-Level Frequency Modulation	0.719	0.767	0.797	0.746

Table 2 : Ablation study comparing encoder design, positional embedding and decoder strategy. Models were evaluated using the normalized MASE (detailed in Section 4.1 ) across prediction horizons and as an aggregate across all tasks.

Figure 4 : Adaptive tokenization visualization. Tokens are demarcated by dotted lines and varying colors, where deeper red curves indicate finer granularity. Kairos adapts granularity to information density, applying finer processing to volatile regions.

Method	DRoPE	Intra-Dataset	Standard RoPE	Inter-Dataset
	(Ours)	Shuffle	(Fixed $\theta$ )	Shuffle
MASE $\downarrow$	0.738	0.751	0.767	0.947
Degradation	–	1.76%	3.93%	28.32%

Table 3 : Causal analysis of adaptive modulation by DRoPE. We evaluate the impact of disrupting or removing the instance-wise

\theta

adjustment. Degradation indicates relative performance drop compared to DRoPE.

Appendix figures & tables26 assets

Supplementary material from the paper’s appendix.

Appendix

Model	Accuracy $\uparrow$	Rank Mean $\downarrow$	Rank Std $\downarrow$
MOMENT (Frozen)	0.794	1.90	0.80
GPT4TS (Fine-tuned)	0.567	3.27	0.81
TimesNet (Fine-tuned)	0.573	3.32	0.74
Kairos -Base (Ours, Frozen)	0.818	1.51	0.71

Appendix

Table 4 : Representation transfer results on 91 UCR classification datasets. Baseline results follow MOMENT [ 16 ] .

Figure 5 : Performance analysis of multi-patch prediction on the GIFT-Eval benchmark across short, medium, and long horizons. We vary the number of forecast patches and find that a forecast patch number of 2 achieves the optimal trade-off, resulting in the best overall performance (lowest normalized MASE).

Forecast tokens $J$	Points per round $H_{\mathrm{step}}$	Samples with $T_{\mathrm{sample}}\geq 2H_{\mathrm{step}}$
1	64	85.73%
2	128	82.69%
6	384	75.54%
12	768	71.53%

Appendix

Table 5 : PreSTS training coverage as the number of forecast tokens increases, with fixed patch length

H_{s}=64

Setting	Short	Medium	Long	AVG
$J=1,H_{s}=128$	0.721	0.807	0.837	0.763
$J=2,H_{s}=64$	0.709	0.761	0.794	0.738

Appendix

Table 6 : Decoder comparison at a fixed direct forecast length of 128. Results are normalized MASE on GIFT-Eval.

Round	Forecast range	AR MAE	Oracle MAE	MAE penalty
1	1–128	0.381	0.381	0.00%
2	129–256	0.437	0.395	+10.61%
3	257–384	0.422	0.368	+14.85%
4	385–512	0.419	0.382	+9.79%
5	513–640	0.462	0.407	+13.62%

Appendix

Table 7 : AR decoding versus a teacher-forced oracle on ETTh1 using Kairos -Base and a context length of 2048. Forecast ranges are relative to the initial forecast origin.

Setting	Short	Medium	Long	AVG
Kairos (Full Model)	0.709	0.761	0.794	0.738
Structural Replacement
w/ Sequence-Level Routing (Pathformer)	0.730	0.782	0.819	0.759
Causal Interventions at Inference
w/ Uniform Granularity Weights	0.856	0.780	0.819	0.831
w/ Shuffled Routing Decisions	1.154	1.282	1.271	1.205

Appendix

Table 8 : Empirical validation of segment-level adaptation on GIFT-Eval. Results report the normalized MASE across prediction horizons and as an aggregate across all tasks.

Patch sizes	Experts $G$	Top- $K$	Normalized MASE $\downarrow$
$\{64,128,256\}$	3	3	0.747
$\{32,64,128,256\}$	4	4	0.738
$\{16,32,64,128,256\}$	5	5	0.737

Appendix

Table 9 : Sensitivity of Kairos -Base to the granularity configuration on GIFT-Eval. The routing budget follows

K=G

, and all configurations use two null experts.

Model	AVG	Short	Medium	Long
Kairos	0.750	0.791	0.699	0.703
Kairos (fixed patch)	0.763	0.804	0.712	0.714
Sundial	0.790	0.843	0.720	0.733

Appendix

Table 10 : Normalized high-pass MAE on GIFT-Eval using first differences. Scores are normalized by Seasonal Naïve per task and aggregated with a geometric mean. Lower values indicate better performance.

Model	Inference Time (s) $\downarrow$	MASE $\downarrow$
TTM-Advanced*	0.009	1.020
Timer-XL	0.014	0.795
Kairos -Mini	0.030	0.753
ChronosBolt-Base	0.055	0.808
Kairos -Base	0.061	0.738
Moirai-Large	0.070	0.875

Appendix

Table 11 : Comparison of single-batch inference speeds and zero-shot performance on GIFT-Eval. All models were tested with an input length of 2048 and an output length of 96, except for TTM-Advanced (*), which used its maximum input length of 1536. Inference times are averaged per batch on a single GPU.

Variant	MASE $\downarrow$
Data Curation Ablations
w/o tier-stratified sampling	0.748
w/o tier-stratified sampling & synthetic data	0.755
Architectural Ablations
w/o Multi-Patch Decoder	0.753
w/o Mixture-of-Size Encoder	0.761

Appendix

Table 12 : Ablation study decoupling data curation and architectural components. Models were evaluated using the normalized MASE on GIFT-Eval.

Model	Params	Training Data	MASE $\downarrow$
Kairos	53M	PreSTS	0.738
Kairos	53M	Chronos corpus	0.761
ChronosBolt	205M	PreSTS	0.781
ChronosBolt	205M	Chronos corpus	0.808
Chronos	709M	Chronos corpus	0.870

Appendix

Table 13 : Matched-data comparison on the GIFT-Eval benchmark separating architecture and training data contributions.

Model	Params	Normalized MASE $\downarrow$
Parameter-matched Transformer	53M	0.797
Kairos -Base	53M	0.738

Appendix

Table 14 : Parameter-matched comparison on GIFT-Eval. Both models use the same pre-training data, training steps, and optimization settings.

Horizon	Kairos -Mini	PatchTST	DLinear	TimesNet
96	0.152 / 0.202	0.149 / 0.205	0.174 / 0.238	0.174 / 0.222
192	0.197 / 0.246	0.195 / 0.248	0.214 / 0.272	0.229 / 0.265
336	0.246 / 0.286	0.254 / 0.293	0.256 / 0.304	0.282 / 0.304
720	0.316 / 0.335	0.346 / 0.354	0.309 / 0.346	0.360 / 0.352
AVG	0.228 / 0.267	0.236 / 0.275	0.238 / 0.290	0.261 / 0.286

Appendix

Table 15 : From-scratch forecasting on Weather. Each entry reports MSE / MAE, and AVG is the arithmetic mean over the four forecast horizons. The best value for each metric at each horizon is highlighted in bold .

	Layers	Heads	$D_{h}$	$d_{\text{ff}}$	$d_{\text{expert}}$	$\{P_{i}\}_{i=1}^{G}$	$K$	$G$	$Z$	$\tau$	$\eta_{b}$	$\omega$	$\theta$	Params
$\textsc{Kairos}_{\text{mini}}$	4	4	256	1024	1408	$\{32,64,128\}$	3	3	2	$\{0.55,0.1,0.05,0.15,0.15\}$	0.01	128	$10000^{-2d/64}$	10M
$\textsc{Kairos}_{\text{small}}$	4	8	384	1536	1408	$\{32,64,128\}$	3	3	2	$\{0.55,0.1,0.05,0.15,0.15\}$	0.01	128	$10000^{-2d/64}$	23M
$\textsc{Kairos}_{\text{base}}$	6	8	512	2048	1408	$\{32,64,128,256\}$	4	4	2	$\{0.45,0.1,0.1,0.05,0.15,0.15\}$	0.01	128	$10000^{-2d/64}$	53M

Appendix

Table 16 : Details of Kairos model configurations.

Figure 6 : Comparison of real and synthetic datasets across (a) ARCH-LM statistic, (b) spectral entropy, and (c) dominant period distributions, illustrating complementary volatility, complexity, and periodicity characteristics.

Dataset	Domain	Frequency	# Time Series	# Time points
Wind Power	Energy	4S	1	7,397,147
Residential Load Power	Energy	T	813	437,983,677
Residential PV Power	Energy	T	699	376,016,850
Los-Loop	Transport	5T	207	7,094,304
PEMS03	Transport	5T	358	9,382,464
PEMS04	Transport	5T	921	15,649,632

Appendix

Table 17 : Detailed descriptions of second-level, minute-level, and hourly datasets.

Dataset	Domain	Frequency	# Time Series	# Time points
Bitcoin	Econ/Fin	D	18	81,918
Covid Mobility	Transport	D	362	148,602
Extended Web Traffic	Web	D	145,063	370,926,091
Favorita Sales	Sales	D	111,840	139,179,538
Favorita Transactions	Sales	D	54	84,408
Subseasonal	Climate	D	3,448	56,788,560

Appendix

Table 18 : Detailed descriptions of daily datasets.

Dataset	Domain	Frequency	# Time Series	# Time points
CDC Fluview ILINet	Healthcare	W	375	319,515
CDC Fluview WHO NREVSS	Healthcare	W	296	167,040
Kaggle Web Traffic Weekly	Web	W	145,063	16,537,182
Project Tycho	Healthcare	W	1,258	1,377,707
Traffic Weekly	Transport	W	862	82,752
NN5 Weekly	Econ/Fin	W	111	12,543

Appendix

Table 19 : Detailed descriptions of weekly datasets.

Dataset	Domain	Frequency	# Time Series	# Time points
GoDaddy	Econ/Fin	M	6,270	257,070
CIF 2016	Econ/Fin	M	72	7,108
FRED MD	Econ/Fin	M	107	77,896
M1 Monthly	Econ/Fin	M	617	55,998
M3 Monthly	Econ/Fin	M	1,428	167,562
Tourism Monthly	Econ/Fin	M	366	109,280

Appendix

Table 20 : Detailed descriptions of monthly, quarterly, and yearly datasets.

Dataset	Domain	Frequency	# Time Series	# Target	# Time points
ETTh1	Energy	H	1	7	17,420
ETTh2	Energy	H	1	7	17,420
ETTm1	Energy	15T	1	7	69,680
ETTm2	Energy	15T	1	7	69,680
Weather	Nature	10T	1	21	52,696
Saugeen (D)	Nature	D	1	1	23,741

Appendix

Table 21 : Detailed descriptions of evaluation datasets.

Method

DLinear

iTrans.

TimesNet

PatchTST

Path.

Chronos

Moirai

TimesFM-2.0

Timer-XL

\text{TTM}_{a}

ChronosBolt

Kairos

Context length

{96, 2048}

{336, 512, 2048}

{96, 2048}

512

2048

1536

2048

Appendix

Table 22 : Context Lengths for Models on the TSLib Benchmark.

Comparison	Wins/Losses	Sign test $p$	Sign test Holm $p$	HL reduction [95% CI]	Wilcoxon $p$	Wilcoxon Holm $p$
Kairos -Base vs. Sundial	20/8	0.0357	0.0714	3.45% [0.14%, 7.98%]	0.0402	0.0402
Full vs. fixed patch	19/9	0.0872	0.0872	1.06% [0.10%, 2.51%]	0.0179	0.0358
Full vs. no null experts	23/5	0.00091	0.00365	0.91% [0.32%, 2.17%]	0.0011	0.0032
Full vs. vanilla RoPE	23/5	0.00091	0.00365	2.62% [1.42%, 4.80%]	0.00022	0.00087

Appendix

Table 24 : Dataset-level statistical comparisons on GIFT-Eval using two-sided exact tests. Wins and losses are counted across 28 source datasets. Holm correction covers the four comparisons within each test family separately. HL denotes the Hodges–Lehmann reduction estimate, reported with its 95% confidence interval.

Aggregation	Wins	Losses	Reduction among wins	Worsening among losses
Task	64	33	8.1%	12.3%
Source dataset	20	8	9.36%	8.02%

Appendix

Table 25 : Magnitudes of gains and losses for Kairos -Base relative to Sundial. Reduction and worsening are geometric-average changes within the winning and losing groups, respectively.

	Kairos -Base			Full-shot baselines
Dataset	Zero-shot	1% few-shot	10% few-shot	PatchTST	DLinear
ETTh1	0.437 / 0.415	0.405 / 0.413	0.398 / 0.412	0.427 / 0.437	0.430 / 0.436
ETTh2	0.340 / 0.377	0.328 / 0.367	0.319 / 0.363	0.361 / 0.402	0.495 / 0.480
ETTm1	0.351 / 0.365	0.321 / 0.352	0.312 / 0.351	0.346 / 0.376	0.354 / 0.385
ETTm2	0.255 / 0.304	0.233 / 0.293	0.230 / 0.290	0.256 / 0.312	0.271 / 0.342
Weather	0.230 / 0.251	0.210 / 0.245	0.208 / 0.239	0.236 / 0.275	0.238 / 0.290

Appendix

Table 26 : Few-shot forecasting results for Kairos -Base. Each entry reports MSE / MAE averaged over prediction horizons

\{96,192,336,720\}

. The percentages indicate the fraction of randomly sampled training windows used for full-parameter fine-tuning. PatchTST and DLinear use the full training split.

Figure 7 : Several distinct case types are generated by the synthetic dataset algorithm. Specifically, the Custom designation refers to a composite signal type, which is systematically constructed by combining seasonal, trend, and noise components. Concurrently, the Perfect periodic designation denotes synthesized, idealized industrial signals.

Panel of Figure 8 — Figure 8 : Example of forecasts from $\textsc{Kairos}_{b}$ on the test datasets used in experiments.

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Inspired by recent breakthroughs in large language models for natural language processing, foundation models have emerged as a promising paradigm for zero-shot time series forecasting, enabling accurate predictions on datasets never seen during pre-training. Ranging from tens to hundreds of millions of parameters, these models are pre-trained on vast and diverse collections of time series, learning generalizable representations that support both point and probabilistic forecasting. This approach alleviates the need for dataset-specific model design and manual tuning, offering a unified solution across forecasting problems. In this work, we review the main architectures, pre-training strategies, and optimization methods underpinning these models. We further investigate post-pre-training fine-tuning of selected foundation models to enhance their performance on specific datasets. Our empirical results demonstrate that this step consistently improves forecasting accuracy over the zero-shot baseline.

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