GRAM: Correcting Frozen Time-Series Foundation Models via Graph-Retrieved Amplitude Memory

Xiaoyun Yu; Xiangfei Qiu; Yonggui Huang; Shixiang Tang; Nanqing Dong; Wanli Ouyang; Geguang Pu; Honggang Qi

cs.LGOct 4, 2026

GRAM: Correcting Frozen Time-Series Foundation Models via Graph-Retrieved Amplitude Memory

Authors: Xiaoyun Yu, Xiangfei Qiu, Yonggui Huang, Shixiang Tang, Nanqing Dong, Wanli Ouyang, Geguang Pu, Honggang Qi,

Organizations: East China Normal University · Peking University · The Chinese University of Hong Kong · Shanghai Innovation Institute ·

Abstract

Time-series foundation models (TSFMs) enable zero-shot forecasting through large-scale cross-domain pretraining, while retrieval augmentation further improves their performance by leveraging historical information. However, existing methods typically correct TSFM forecasts using the ground-truth futures of similar historical windows, which contain both predictive components already captured by the foundation model and sample-specific random fluctuation that is difficult to transfer. In contrast, recurring systematic model bias within prediction errors more directly characterizes the failure modes of a frozen TSFM and therefore provides more valuable correction signals. Effectively exploiting such model bias, however, poses two challenges: prediction errors at different numerical levels are difficult to compare due to scale differences, and the recurring bias must be extracted from prediction errors contaminated by random fluctuation. To address these challenges, we propose GRAM, a general retrieval-augmented framework for frozen TSFMs. GRAM first introduces an Amplitude Memory Module (AMM) that scales prediction errors by amplitude and aggregates them into retrievable prototypes. It then employs a Prototype Graph Module (PGM) to model relations among prototypes to aggregate consistent bias information while suppressing random fluctuation. During online forecasting, GRAM retrieves and expands prototypes relevant to the current query and generates per-horizon corrections to refine the original TSFM forecast. Experiments across multiple datasets and foundation models demonstrate consistent forecasting improvements.

Figures & tables

Figure 1: Overview of Gram . Phase 1 (Offline Memory Construction): clustering (A) groups window embeddings into prototypes, the amplitude memory (B) stores each prototype’s centroid, amplitude, spread, coherence and support, and the prototype graph (C) links them by bias and shape edges. Phase 2 (Online Forecasting): retrieval (D) grows the top- $K$ prototypes into a subgraph and attaches the anchor node, and graph fusion (E), the only trainable module, outputs the error component added to the frozen forecast.

Models	Sundial				Chronos				TimesFM				Moirai
Models	Ori		+ Gram (Ours)		Ori		+ Gram (Ours)		Ori		+ Gram (Ours)		Ori		+ Gram (Ours)
Metric	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	0.397	0.412	0.389	0.408	0.448	0.418	0.436	0.414	0.421	0.409	0.417	0.407	0.620	0.504	0.530	0.473
ETTh2	0.343	0.383	0.339	0.379	0.367	0.380	0.362	0.377	0.372	0.388	0.369	0.385	0.451	0.438	0.405	0.415
ETTm1	0.366	0.370	0.360	0.367	0.421	0.383	0.411	0.379	0.381	0.372	0.378	0.370	0.740	0.523	0.617	0.485
ETTm2	0.275	0.321	0.264	0.313	0.291	0.317	0.287	0.315	0.291	0.318	0.285	0.315	0.404	0.388	0.344	0.360

Table 1: Zero-shot forecasting comparison between frozen foundation models and the same models corrected by Gram across seven datasets. For each TSFM, Ori is the bare frozen model and + Gram the corrected forecast; MSE is compared with MSE and MAE with MAE, and red marks the better of the two. Results are averaged over horizons

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

Models	Sundial				Chronos				TimesFM				Moirai
Models	FT		+ Gram (Ours)		FT		+ Gram (Ours)		FT		+ Gram (Ours)		FT		+ Gram (Ours)
Metric	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	0.403	0.415	0.411	0.417	0.461	0.421	0.452	0.422	0.489	0.438	0.489	0.438	0.594	0.493	0.505	0.462
ETTh2	0.338	0.376	0.335	0.374	0.375	0.384	0.370	0.383	0.366	0.388	0.365	0.387	0.432	0.422	0.394	0.404
ETTm1	0.351	0.362	0.343	0.361	0.349	0.354	0.341	0.352	0.304	0.355	0.303	0.354	0.566	0.451	0.477	0.422
ETTm2	0.274	0.319	0.262	0.311	0.275	0.308	0.274	0.307	0.269	0.305	0.268	0.305	0.318	0.342	0.293	0.329

Table 2: Forecasting with LoRA fine-tuned TSFMs. For each TSFM, FT is the LoRA fine-tuned model and + Gram the same fine-tuned model corrected by Gram ; MSE and MAE are averaged over

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

, and red marks the better of the two within each TSFM–metric pair. Lower is better.

Variants	ETTh1		ETTh2		ETTm1		ETTm2		Weather		Exchange		ECL
Variants	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae
w/o shape relation	0.412	0.421	0.318	0.362	0.378	0.371	0.196	0.268	0.192	0.223	0.101	0.227	0.149	0.243
w/o bias relation	0.427	0.411	0.331	0.353	0.362	0.389	0.205	0.276	0.185	0.212	0.098	0.234	0.142	0.252
w/o anchor	0.403	0.432	0.324	0.341	0.387	0.365	0.191	0.258	0.201	0.225	0.104	0.218	0.153	0.237
amplitude $\rightarrow$ mean residuals	0.438	0.431	0.327	0.378	0.396	0.382	0.212	0.285	0.194	0.235	0.113	0.245	0.147	0.241
recent level $\rightarrow$ naive ratio	0.409	0.413	0.342	0.392	0.371	0.360	0.214	0.295	0.189	0.221	0.125	0.258	0.151	0.253

Table 3: Ablation at

H=96

: absolute MSE / MAE, averaged over the four TSFMs and the seven datasets. The first three rows remove one module, the last two replace the stored quantity or the recent level; red marks the best average.

Figure 2: Diagnostics of the pattern-conditioned bias over all 112 settings. Panels (a)–(b) characterise the bias the memory stores, (c)–(e) the evidence retrieval returns and its link to the gain, and (f) the coverage across datasets and TSFMs.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Domain	Frequency	Channels	Split (train/val/test)
ETTh1	Electricity transformer	Hourly	7	6:2:2
ETTh2	Electricity transformer	Hourly	7	6:2:2
ETTm1	Electricity transformer	15 minutes	7	6:2:2
ETTm2	Electricity transformer	15 minutes	7	6:2:2
Electricity	Household consumption	Hourly	321	7:1:2
Weather	Meteorology	10 minutes	21	7:1:2

Appendix

Table 4: Benchmark characteristics and splitting protocol. The proportions are train/validation/test, are the published ones and are chronological. The memory-construction and calibration portions are cut from the training portion, and the reported numbers are computed on the test portion.

Models		Sundial				Chronos				TimesFM				Moirai
Models		Ori		+ Gram (Ours)		Ori		+ Gram (Ours)		Ori		+ Gram (Ours)		Ori		+ Gram (Ours)
Metric		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	96	0.356	0.383	0.352	0.381	0.385	0.380	0.376	0.378	0.384	0.382	0.380	0.381	0.426	0.411	0.416	0.408
	192	0.390	0.406	0.384	0.403	0.442	0.412	0.427	0.409	0.428	0.408	0.421	0.406	0.467	0.433	0.454	0.429
	336	0.403	0.416	0.400	0.415	0.481	0.433	0.467	0.430	0.446	0.419	0.441	0.418	0.478	0.446	0.463	0.443
	720	0.438	0.444	0.420	0.433	0.483	0.449	0.475	0.439	0.428	0.427	0.428	0.423	1.112	0.725	0.787	0.612

Appendix

Table 5: Complete zero-shot results behind Table 1 , every dataset and horizon. For each TSFM, Ori is the bare frozen model and + Gram the corrected forecast; MSE is compared with MSE and MAE with MAE, and red marks the better of the two.

Figure 3: Four frozen models (one per row, named on the left: TimesFM, Moirai, Chronos, Sundial) on three datasets (one per column) at

H=96

. Each panel is a single test window — the same window for the rows of a column — with the last

48

steps of the series before the dotted line and the

96

-step forecast after it; the two forecasts are drawn only inside the forecast window, and all values are standardised. The windows are chosen so that every model is clearly off before the correction (frozen RMSE between

18\%

and

55\%

of the panel’s own vertical range) and close after it (residual at most

42\%

of that range,

+25\%

+63\%

of the squared error); among the windows that qualify, the one drawn is the window whose worst-case residual is the smallest. The Moirai row is the jitteriest of the four, which is the model’s own behaviour on these series rather than the correction’s.

Models		Sundial				Chronos				TimesFM				Moirai
Models		FT		+ Gram (Ours)		FT		+ Gram (Ours)		FT		+ Gram (Ours)		FT		+ Gram (Ours)
Metric		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	96	0.368	0.387	0.371	0.388	0.411	0.387	0.400	0.387	0.414	0.411	0.414	0.411	0.410	0.403	0.402	0.401
	192	0.394	0.408	0.393	0.407	0.461	0.416	0.441	0.413	0.518	0.451	0.518	0.451	0.437	0.420	0.424	0.416
	336	0.407	0.418	0.410	0.418	0.488	0.434	0.473	0.432	0.602	0.479	0.602	0.479	0.454	0.433	0.442	0.431
	720	0.444	0.449	0.470	0.455	0.483	0.448	0.492	0.455	0.421	0.409	0.421	0.409	1.077	0.715	0.753	0.599

Appendix

Table 6: Complete results behind Table 2 , every dataset and horizon. For each TSFM, FT is the LoRA fine-tuned model and + Gram the same fine-tuned model corrected by Gram ; MSE is compared with MSE and MAE with MAE, and red marks the better of the two.

		ETTh1	ETTh2	ETTm1	ETTm2	Weather	Exchange	ECL
TS-RAG	MSE	0.434	0.348	0.395	0.282	0.258	0.332	0.157
TS-RAG	MAE	0.413	0.375	0.374	0.314	0.274	0.392	0.244
Gram (Ours)	MSE	0.394	0.341	0.360	0.264	0.236	0.439	0.147
Gram (Ours)	MAE	0.411	0.380	0.369	0.313	0.269	0.440	0.242

Appendix

Table 7: Comparison with the retrieval-augmented method that reuses the ground-truth futures of similar historical windows, TS-RAG Ning et al. (2025) , the state of the art of that family. Every entry is the average over the four horizons of the dataset; the better of the two methods is red in each column.

	$H=96$	$H=192$	$H=336$	$H=720$
read-out parameters (M)	0.26	0.32	0.40	0.63
prototype records (KB)	98	194	338	722
read-out cost ( $\mu$ s / window)	7.7	8.7	9.4	12.2

Appendix

Table 8: Efficiency of the correction, measured on an H200 with batches of

4096

windows. The read-out is the only trained part; the memory is what has to be stored next to the frozen backbone. Retrieval is a comparison against

C=128

centroids and stays below

0.01\,\mu

s per window, so the read-out dominates the measured cost.

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Retrieval-augmented forecasting promises to adapt frozen Time Series Foundation Models (TSFMs) to new domains without fine-tuning, but recent methods typically rely on learned fusion modules, i.e., trained adapters that merge retrieved examples into the backbone's forecast, based on the assumption that frozen backbones cannot dynamically incorporate retrieved context on their own. We show this assumption is unnecessary. We introduce Align-RAG, a training-free method that applies a closed-form per-pair amplitude rescaling and integer-lag phase shift to retrieved past-future windows before they enter a frozen backbone's context. With no learned parameters, Align-RAG outperforms the state-of-the-art trained retrieval adapter on a frozen Chronos-Bolt on all seven datasets of the standard benchmark (avg -3.75% MSE), showing that the gains previously attributed to learned fusion are recoverable without any training. Align-RAG further improves zero-shot MSE on four additional frozen TSFMs with various architectures by 2.5% to 13.7% per backbone with no per-backbone tuning. To probe why alignment helps, we compare the frozen backbone's prediction shift under aligned demonstrations to the closed-form ridge prediction shift on the same pairs. We find that aligned demonstrations induce prediction shifts that track a closed-form ridge predictor on the same pairs, with a future-shuffle control ruling out a futures-averaging account. Together, these results indicate that frozen TSFMs already support dynamic in-context use of retrievals, and that closed-form alignment should be the default baseline for retrieval-augmented forecasting before any fusion module is trained. Code available at: https://github.com/masadi-99/align-rag

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