BEAT: Balanced Frequency Adaptive Tuning for Long-Term Time-Series Forecasting

Zhixuan Li; Naipeng Chen; Seonghwa Choi; Sanghoon Lee; Weisi Lin

cs.LGJan 31, 2025

BEAT: Balanced Frequency Adaptive Tuning for Long-Term Time-Series Forecasting

Authors: Zhixuan Li, Naipeng Chen, Seonghwa Choi, Sanghoon Lee, Weisi Lin

Organizations: College of Computing and Data Science, Nanyang Technological University, Singapore · Department of Electrical and Electronic Engineering, Yonsei University, Korea

Abstract

Long-term time-series forecasting supports a wide range of applications, including weather prediction and electricity demand planning. Frequency-domain methods address this task by decomposing observations into components that describe temporal variations at different scales. However, separate representations do not by themselves provide an explicit mechanism for adjusting the training emphasis across components. Under a shared forecasting objective, the frequency-specific networks can retain different levels of coefficient prediction error, motivating an error-dependent adjustment to their gradients. To this end, we propose BEAT (Balanced frEquency Adaptive Tuning), a framework that combines frequency-specific error monitoring with adaptive gradient modulation. We design a Frequency-Specific Monitor that compares predicted and target wavelet coefficients in a common normalized space and expresses each discrepancy relative to a reference error computed from the detail components. We further introduce a Dynamical Gradient Balancer that converts these ratios into positive, bounded coefficients. Components with higher relative errors receive larger gradient weights, whereas those with lower relative errors receive smaller weights. A shared modulation-strength parameter controls the departure from unmodulated training, and the monitoring and balancing operations are used only during training. Experiments on seven real-world datasets show that BEAT achieves competitive performance against state-of-the-art forecasting methods.

Figures & tables

Fig. 1: An illustrative case of error-dependent gradient modulation. The two panels show identical target and predicted coefficient series in the same normalized space, rather than experimental results before and after training. (a) The naive approach applies no additional gradient scaling. (b) BEAT attenuates the gradient of the illustrated lower-frequency branch, whose fitting error is below the detail-based reference, and amplifies that of the illustrated higher-frequency branch, whose error is above the reference. The two displayed branches are examples from a multilevel decomposition. Their error ordering is case-specific, not a fixed preference for low or high frequencies. Arrow sizes indicate scaling relative to each branch’s own unmodulated gradient, not equal gradient magnitudes across branches.

Symbol	Definition
$T$ , $K$ , $N$	Historical window length, forecasting horizon, and number of variables, respectively.
$X$	Historical observations in $\mathbb{R}^{T\times N}$ , from $x_{1-T}$ to $x_{0}$ .
$Y$ , $\hat{Y}$	Future target and prediction in $\mathbb{R}^{K\times N}$ , with target observations $y_{1},\dots,y_{K}$ .
$\mathcal{N}_{X}$	RevIN transform using statistics from $X$ and the same affine parameters for input and target.
$X_{norm}$ , $Y_{norm}$	Historical input and future target transformed by $\mathcal{N}_{X}$ , respectively.
$\hat{Y}_{norm}$	Reconstructed forecast before inverse RevIN.

TABLE I: Main notation used in BEAT.

Fig. 2: Overall architecture of BEAT. The forecasting path decomposes historical observations into approximation and detail coefficient series, predicts their future values with frequency-specific networks, and reconstructs the forecast. (a) Wavelet reconstruction using the inverse discrete wavelet transform. (b) Multi-level wavelet decomposition. During training, (c) FSM compares predicted and target coefficients in the same normalized space to obtain relative discrepancy ratios, and (d) DGB converts these ratios into bounded coefficients that modulate the corresponding branch gradients. For clarity, panel (c) omits the target-normalization step $Y_{norm}=\mathcal{N}_{X}(Y)$ before wavelet decomposition, where $\mathcal{N}_{X}$ is the same history-conditioned RevIN transform used for the input. FSM and DGB are omitted at inference. Best viewed in color and zoomed in.

Dataset	# VAR	Size	FRQ
Weather	21	(36792, 5271, 10540)	10 min
ETTh1/h2	7	(8545, 2881, 2881)	1 hour
ETTm1/m2	7	(34465, 11521, 11521)	15 min
Traffic	862	(12185, 1757, 3509)	1 hour
ECL	321	(18317, 2633, 5261)	1 hour

TABLE II: Datasets used for evaluation.

Index	FSM	DGB	Weather		ETTh1
Index	FSM	DGB	MSE	MAE	MSE	MAE
(1)			0.243	0.269	0.422	0.424
(2)	✓	✓	0.239	0.263	0.415	0.419

TABLE III: Ablation study of designed components in BEAT. FSM and DGB represent the proposed Frequency-Specific Monitor and Dynamical Gradient Balancer. The average performance over four prediction lengths, including

\{96,192,336,720\}

, is reported for each dataset’s test set.

Methods		BEAT		UP-TimeMixer		Delayformer		WPMixer		TimeMixer		iTransformer		TSMixer		PatchTST		Crossformer
Methods		Ours		2026		2026		2025		2024		2024		2023		2023		2023
Metric		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
	96	0.160	0.198	0.179	0.232	0.160	0.210	0.162	0.204	0.163	0.209	0.174	0.214	0.175	0.247	0.186	0.227	0.195	0.271
	192	0.207	0.241	0.221	0.264	0.208	0.252	0.209	0.246	0.208	0.250	0.221	0.254	0.224	0.294	0.234	0.265	0.209	0.277
	336	0.258	0.279	0.265	0.304	0.265	0.293	0.263	0.287	0.251	0.287	0.278	0.296	0.262	0.326	0.284	0.301	0.273	0.332
	720	0.329	0.332	0.349	0.358	0.346	0.346	0.339	0.338	0.339	0.341	0.358	0.347	0.349	0.348	0.356	0.349	0.379	0.401

TABLE IV: Long-term forecasting results on seven datasets with prediction lengths

\{96,192,336,720\}

. BEAT uses an input length of 96. Avg is calculated from the four displayed horizon-specific results and rounded to three decimal places. First-place counts include all 35 rows and count ties for each method. The best and second-best results are highlighted in bold red and underlined blue, respectively.

Index	Wavelet Type	Weather		ETTh1
Index	Wavelet Type	MSE	MAE	MSE	MAE
(1)	Coiflets	0.242	0.268	0.415	0.419
(2)	Daubechies	0.239	0.263	0.418	0.420
(3)	Biorthogonal	0.240	0.265	0.422	0.419
(4)	Symlets	0.244	0.269	0.419	0.421

TABLE V: Ablation study of the wavelet type’s design choice.

Index	Levels of Decomposition $f$	Weather		ETTh1
Index	Levels of Decomposition $f$	MSE	MAE	MSE	MAE
(1)	1	0.239	0.264	0.416	0.419
(2)	2	0.239	0.263	0.417	0.420
(3)	3	0.240	0.266	0.415	0.419
(4)	4	0.243	0.265	0.416	0.421
(5)	5	0.244	0.266	0.418	0.422

TABLE VI: Ablation study of levels of decomposition in the wavelet transform.

Index	Discrepancy Calculation	Weather		ETTh1
Index	Discrepancy Calculation	MSE	MAE	MSE	MAE
(1)	RMSE	0.244	0.268	0.421	0.425
(2)	$1-R^{2}$	0.241	0.265	0.418	0.423
(3)	MSE	0.239	0.263	0.415	0.419
(4)	MAE	0.240	0.264	0.416	0.419

TABLE VII: Ablation study of the design for discrepancy calculation.

Index	Modulation	Weather		ETTh1
Index	Modulation	MSE	MAE	MSE	MAE
(1)	Loss	0.242	0.268	0.420	0.421
(2)	Gradient	0.239	0.263	0.415	0.419

TABLE VIII: Ablation study of the design for learning process modulation.

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