CTRL: Control-Based Time Series Forecasting with LLM-Guided Residual Learning

Minkyoung Kim; Daeun Ji; Yohan Lee; Beomsoo Kim; Beakcheol Jang

cs.LGSep 20, 2026

CTRL: Control-Based Time Series Forecasting with LLM-Guided Residual Learning

Authors: Minkyoung Kim, Daeun Ji, Yohan Lee, Beomsoo Kim, Beakcheol Jang

Organizations: Graduate School of Information, Yonsei University, Seoul, Republic of Korea

Abstract

Time series forecasting underpins critical decision-making across diverse domains. While large language models (LLMs) offer promising reasoning capabilities, existing LLM-based time series forecasting approaches either reduce them to numerical predictors that bypass their strengths, or allow direct forecast generation that destabilizes predictions in non-stationary settings. We introduce CTRL, a framework that decouples semantic reasoning from quantitative prediction. A frozen backbone generates base forecasts, while specialized LLM agents function as controllers that analyze backbone prediction errors through decomposed trend, seasonal, and irregular components, grounding reasoning in interpretable temporal structure. Each agent outputs compact control signals that a lightweight residual decoder translates into forecast corrections. CTRL incorporates label-free test-time adaptation that detects distribution shift from input statistics alone and readapts control signals with only 3-24 LLM calls via caching. CTRL is explicitly designed to improve robustness under non-stationary temporal dynamics and distribution shift, while remaining competitive on highly stationary time series where adaptive correction provides limited additional benefit.

Figures & tables

Figure 1: LLM integration paradigms. (a) Prior methods use LLMs as direct numerical predictors. (b) CTRL uses LLMs as controllers generating compact control signals for a residual decoder; a frozen backbone provides base forecasts. Final prediction: $\tilde{\mathbf{y}}=\hat{\mathbf{y}}+\delta$ .

Figure 2: CTRL Framework Overview. A frozen backbone generates initial predictions $\hat{y}$ . In Stage 1, three specialized LLM agents analyze STL-decomposed held-out training statistics via in-context reasoning: Trend and Seasonal agents output 4D control signals [scale, bias, gate, confidence], while the Irregular agent produces text encoded via frozen GPT-2. In Stage 2, concatenated control signals guide component-specific MLPs in the residual decoder, trained on held-out training data with ground truth. In Stage 3, test-time adaptation detects distribution shift from input statistics alone and adjusts control signals without accessing ground-truth labels. Final prediction: $\tilde{y}=\hat{y}+\delta$ .

Data	H	Ours(DL)		Ours(PT)		DLinear		PatchTST		TimeLLM †		CALF		TEMPO		GPT4TS
		MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE
ETTh1	48	.378	.340	.381	.342	.382	.346	.386	.346	.398	.369	.400	.372	.419	.395	.425	.412
	96	.398	.371	.399	.372	.399	.372	.405	.380	.429	.414	.410	.393	.469	.471	.464	.473
	192	.429	.410	.439	.429	.430	.412	.439	.431	.432	.425	.436	.430	.494	.514	.527	.598
	336	.457	.442	.430	.415	.459	.444	.431	.416	.452	.450	.476	.490	.477	.485	.599	.738
	720	.510	.498	.458	.443	.511	.497	.459	.444	.477	.462	.522	.540	.502	.502	.600	.783

Table 1: Forecasting results on multivariate benchmarks. Bold : best. (DL) and (PT) indicate DLinear and PatchTST backbones, respectively. † Time-LLM Jin et al. (2024) uses GPT-2 backend with a single seed; all other results averaged over 3 seeds.

	Exch.	ETT				ECL	Weather
		h2	m2	h1	m1
ADF	-1.9	-4.1	-5.7	-5.9	-15.0	-8.5	-26.7
Shift	2.33	1.48	1.49	0.50	0.50	0.62	0.71

Table 2: Dataset stationarity. Lower ADF indicates stronger stationarity; Shift measures train-test distribution divergence via normalized mean difference.

Figure 3: Backbone prediction vs CTRL-refined prediction on ETTm2

	ETTm2		Weather
	MSE	MAE	MSE	MAE
LLMTime	.245	.151	.237	.150
CTRL	.197	.070	.132	.039
Improv.	19.6%	53.2%	44.3%	74.2%

Table 3: Comparison with LLMTime. CTRL uses DLinear backbone.

Baseline
Configuration	ETTh2	ETTm2
Backbone only	+9.45	+7.20
Learned control signal	+3.13	+4.42
Random control signal	+5.84	+4.54
Zero control signal	+5.70	+5.18
Test-time Adaptation

Table 4: Design ablation on ETTh2 and ETTm2 (384

\rightarrow

96).

\Delta

indicates

\Delta

MSE(%), performance degradation vs. CTRL (full). Results are averaged over 3 seeds.

	Ours	LLMTim	GPT4TS	CALF	TEMPO
Params	400K	0	4.4M	18.2M	12.4M
Train	1.8m	0	2.1m	62m	106m
LLM	3–24	$N$	0	0	0

Table 5: Model complexity on ETTh2 (384

\rightarrow

96).

Dataset	Test Samples	Checks	Max Calls
ETTh1/h2	2,402	2	9
ETTm1/m2	11,042	7	24
Weather	10,060	7	24
ECL	4,781	3	12
Exchange	1,038	1	6

Table 6: LLM calls per dataset (calibration interval

K{=}50

H{=}96

). Total calls

=3+3\times\text{Checks}

in the worst case.

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Total	Train	Val	Test	Freq.
ETTh1/h2	17,420	12,194	1,742	3,484	Hourly
ETTm1/m2	69,680	48,776	6,968	11,042	15-min
Weather	52,696	36,887	5,269	10,540	10-min
ECL	26,304	18,412	2,631	5,261	Hourly
Exchange	7,588	5,311	759	1,518	Daily

Appendix

Table 7: Dataset statistics and split configuration.

Dataset	LR	Epochs	STL (Seas., Trend)
ETTh1	1e-4	40	24, 48
ETTh2	1e-4	40	24, 48
ETTm1	1e-4	40	24, 48
ETTm2	1e-4	40	24, 48
ECL	5e-6	40	12, 24
Exchange	1e-4	40	7, 24

Appendix

Table 8: Dataset-specific hyperparameters. †

LLM Sampling (Llama 3.3 70B)
Configuration	$\Delta$ MSE (%)
temp=0.2, top-p=0.8	+0.22
temp=0.3, top-p=0.8	–
temp=0.5, top-p=0.8	+0.26
Adaptation Threshold ( $\tau$ )
$\tau$ =OFF (no adaptation)	+1.03

Appendix

Table 9: Hyperparameter sensitivity on ETTh2 (384

\to

96). Bold indicates selected configuration.

\Delta

shows degradation relative to selected setting.

Component	Params
DLinear backbone (frozen)	74K
GPT-2 encoder (frozen)	124M
MLP projector (trained)	100K
Residual decoder wo proj. (trained)	215K
Total trainable	$\sim$ 400K

Appendix

Table 10: CTRL parameter breakdown (ETTh2, DLinear backbone).

Figure 4: Three-way ablation on ETTh2. Scatter plots (top): zero control signal clusters around the diagonal with near-random win rate (49%), indicating the decoder alone learns minimal correction. Adding LLM control signals shifts points below the diagonal, but gains concentrate in low-to-mid MSE regions. The full system extends improvements to harder samples where distribution shift invalidates cached control signals. Histograms (bottom): test-time adaptation compresses negative-skewed tails by detecting when cached control signals fail and triggering re-reasoning.

	Zero	w/o TTA	Full
Mean Imp.	+4.3%	+12.6%	+13.3%
Win Rate	49%	76%	82%

Appendix

Table 11: Three-way ablation on ETTh2.

Comparison	$\Delta$ Improv.
w/o Adaptation vs Zero	+8.3%
CTRL (Full) vs w/o Adaptation	+0.7%
CTRL (Full) vs Zero	+9.0%

Appendix

Table 12: Relative contribution of each component.

Before Adaptation (Stage 1)
	Scale	Bias	Gate	Conf.
Trend	0.90	$+$ 0.10	0.40	0.60
Seasonal	0.95	$+$ 0.10	0.40	0.60
Irregular	0.95	$+$ 0.10	0.50	0.60
After Adaptation (Stage 3)
Trend	0.50	$-$ 0.50	0.69	0.21

Appendix

Table 13: Control signal values before and after adaptation (Case 1: Trend Reversal).

Before Adaptation (Stage 1)
	Scale	Bias	Gate	Conf.
Trend	0.90	$+$ 0.10	0.40	0.60
Seasonal	0.95	$+$ 0.10	0.40	0.60
Irregular	0.95	$+$ 0.10	0.50	0.60
After Adaptation (Stage 3)
Trend	0.58	$-$ 0.29	0.59	0.28

Appendix

Table 14: Control signal values before and after adaptation (Case 2: Seasonal Amplification).

Component	$\boldsymbol{\Delta}$ (ETTh2)
Irregular text emb.	0.032
Trend bias	0.009
Trend confidence	0.006
Trend scale	0.005
Trend gate	0.005
Seasonal confidence	0.001

Appendix

Table 15: Control signal sensitivity on ETTh2 (384

\to

96, DLinear). The irregular text embedding is 3.4

\times

more influential than the most responsive numeric parameter (trend bias).

Text Variant	$\boldsymbol{\Delta}$
No text (zeroed embedding)	0.0323
Random unrelated words	0.0074
Technical jargon	0.0045
Crisis/volatility description	0.0010
Random characters	0.0005
Calm/stable description	0.0004

Appendix

Table 16: Decoder output change when replacing LLM-generated irregular text with alternative inputs on ETTh2.

Trend Specialist Agent
Role	First specialist handling TREND correction
Focus	Low-frequency patterns: direction, level shifts, drift, slope accuracy
Key Questions	(1) Slope sign match? (2) Slope value match? (3) Systematic over/under-prediction?
Output: 4D control signal
scale	Trend amplitude adjustment ( $>$ 1 if underestimate, $<$ 1 if overestimate)
bias	Level correction ( $+$ if too low, $-$ if too high)

Appendix

Table 17: Trend Agent prompt structure.

Seasonal Specialist Agent
Role	Second specialist handling SEASONAL correction
Focus	High-frequency periodic patterns: daily/weekly cycles, amplitude, phase
Key Questions	(1) Correct periodicity? (2) Amplitude correct? (3) Phase aligned?
Output: 4D control signal
scale	Amplitude adjustment ( $>$ 1 if dampened, $<$ 1 if exaggerated)
bias	Baseline shift (usually $\approx 0$ )

Appendix

Table 18: Seasonal Agent prompt structure.

Irregular Pattern Agent
Role	Third specialist handling noise/anomalies not explained by trend or seasonality
Task	Generate concise text description converted to numerical embedding
Analysis priorities (ordered)
1. Noise	Variance level: high/low/changing
2. Anomalies	Outliers, spikes present?
3. Autocorr.	Correlated or white noise?

Appendix

Table 19: Irregular Agent prompt structure.

Trend/Seasonal Adaptation
Input	Component shift analysis: validation slope/dominance vs test, shift ratio (%), direction change
Output (JSON)
shift_diagnosis	Description of detected shift
impact_assessment	Expected effect on predictions
scale_multiplier	Multiplicative adjustment to scale
bias_shift	Additive adjustment to bias

Appendix

Table 20: Trend/Seasonal adaptation prompt structure.

Irregular Adaptation
Input	Original analysis text, statistics comparison (std, mean, spike ratio, autocorr) for validation vs test
Output (JSON)
needs_adjustment	Boolean flag
updated_text	Revised irregular analysis
adapted_params	{scale, bias, gate, confidence} adjustments
reasoning	Explanation

Appendix

Table 21: Irregular adaptation prompt structure.

Unified Distribution Shift Adaptation
Input: STL decomposition comparison
Trend shift	Validation vs test slope, shift ratio, significance
Seasonal shift	Validation vs test dominance, shift ratio, significance
Raw statistics	Mean, std comparison, overall shift score
Output (JSON)
Base params	shift_diagnosis, impact_assessment, scale_multiplier, bias_shift, gate_multiplier, confidence_penalty

Appendix

Table 22: Unified adaptation prompt structure.

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Zero-Shot Time Series Forecasting Time Series Forecasting LLM-Based Time Series Forecasting Mixture-of-Experts Models Autoregressive Modeling

Open paper AskPDF arXiv

Recent advances in Large Language Models (LLMs) have spurred cross-modal solutions for time-series forecasting. However, existing methods rely heavily on textual prompts for modality alignment-introducing nontrivial computational overhead and failing to leverage the rich spectral dynamics inherent in time-series data. To enable prompt-free, frequency-aware adaptation of frozen LLMs, we propose FM-LLM (Frequency-Enhanced Mixture-of-Experts for adapting LLMs to Time Series Forecasting), an autoregressive framework grounded in constrained asymmetric coupling. A Fourier Analysis Network (FAN)-based spectral token aligner injects structured harmonic representations directly into the frozen LLM with numerical compatibility. An asymmetric Mixture-of-Experts (MoE) decoder enforces role separation: shared experts with lightweight FAN layers reconstruct the global periodic backbone, while routed experts-restricted to standard FFNs-specialize in modeling non-periodic residual dynamics. A time-frequency hybrid loss function jointly optimizes temporal accuracy and spectral consistency, mitigating error accumulation during long-horizon autoregressive rollouts. Evaluated across eleven public benchmarks, FM-LLM achieves state-of-the-art performance on 59 out of 78 evaluation metrics. Compared to the strongest autoregressive LLM-based baseline, it delivers average improvements of 5.3% in MSE and 5.6% in MAE, with maximum gains reaching 8.0% for MSE and 8.4% for MAE. FM-LLM also demonstrates robust transferability, maintaining superior performance in 10% few-shot and zero-shot forecasting scenarios.

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