PaperlayerView exact revision on arXiv

Source-linked AI summary

TimeKAN: KAN-based Frequency Decomposition Learning Architecture for Long-term Time Series Forecasting

Songtao Huang, Zhen Zhao, Can Li, Lei Bai

arXiv:2502.06910v2cs.LGcs.AI

TL;DR

Intertwined frequency components and differing pattern densities make accurate time-series forecasting difficult. TimeKAN addresses this with cascaded frequency decomposition, KAN-based frequency-specific representation learning, and frequency mixing, achieving state-of-the-art forecasting performance with lightweight computation.

  • Problem

    Intertwined frequency components and different pattern information densities make accurate long-term time-series forecasting challenging.

  • Method

    TimeKAN decomposes sequences into frequency bands, learns band-specific patterns with CFD and M-KAN blocks, and recombines them through Frequency Mixing.

  • Results

    TimeKAN achieves state-of-the-art forecasting performance across multiple real-world datasets while remaining extremely lightweight.

  • Takeaways & Limitations

    Frequency Decomposition-Learning-Mixing provides a lightweight framework for modeling temporal patterns at different frequencies.

Abstract

from arXiv · show

Real-world time series often have multiple frequency components that are intertwined with each other, making accurate time series forecasting challenging. Decomposing the mixed frequency components into multiple single frequency components is a natural choice. However, the information density of patterns varies across different frequencies, and employing a uniform modeling approach for different frequency components can lead to inaccurate characterization. To address this challenges, inspired by the flexibility of the recent Kolmogorov-Arnold Network (KAN), we propose a KAN-based Frequency Decomposition Learning architecture (TimeKAN) to address the complex forecasting challenges caused by multiple frequency mixtures. Specifically, TimeKAN mainly consists of three components: Cascaded Frequency Decomposition (CFD) blocks, Multi-order KAN Representation Learning (M-KAN) blocks and Frequency Mixing blocks. CFD blocks adopt a bottom-up cascading approach to obtain series representations for each frequency band. Benefiting from the high flexibility of KAN, we design a novel M-KAN block to learn and represent specific temporal patterns within each frequency band. Finally, Frequency Mixing blocks is used to recombine the frequency bands into the original format. Extensive experimental results across multiple real-world time series datasets demonstrate that TimeKAN achieves state-of-the-art performance as an extremely lightweight architecture. Code is available at https://github.com/huangst21/TimeKAN.

1 INTRODUCTION

TimeKAN addresses the difficulty of forecasting intertwined frequency components by decomposing, learning, and recombining frequency-specific patterns with flexible KAN-based modeling. The architecture is presented as lightweight and effective across multiple time-series forecasting tasks.

  • Motivation: Prior decomposition methods simplify time series into seasonal, trend, periodic, or shorter subsequence components to support forecasting.These approaches model different temporal structures separately, but the introduction motivates frequency-specific decomposition as a further direction.
  • Motivation: Intertwined frequency components complicate forecasting because low frequencies represent long-term variations while high frequencies capture abrupt events.Uniform modeling across frequencies can inaccurately characterize patterns because their information density differs.
  • Proposed architecture: TimeKAN uses a Decomposition-Learning-Mixing architecture with Cascaded Frequency Decomposition, Multi-order KAN Representation Learning, and Frequency Mixing blocks.The process separates frequency bands, learns their temporal patterns, and recombines them into the original format.
  • Proposed architecture: The M-KAN blocks model patterns at different frequencies by exploiting KAN flexibility, while the overall model is designed to remain lightweight.This design targets frequency-specific characterization rather than applying one uniform model to every component.
  • Results: TimeKAN demonstrates superior performance across multiple time-series forecasting tasks with significantly fewer parameters than state-of-the-art models.The contribution statement characterizes the model as lightweight yet effective.

2 RELATED WORK

Related work develops time-series models around KAN-based alternatives to MLPs and decomposition strategies for separating diverse temporal patterns. TimeKAN extends these directions through multi-frequency Decomposition-Learning-Mixing.

  • KAN-based models: KAN replaces fixed node activations in MLPs with learnable activation functions along edges, providing a flexible alternative to traditional MLPs.The original KAN uses spline functions, while later variants explore simpler basis functions for improved efficiency.
  • KAN-based models: KAN variants and applications include ChebyshevKAN, FastKAN, Convolutional KAN, and U-KAN across efficiency, convolutional modeling, and medical image segmentation.These studies position KAN as a reusable alternative to MLP components in different architectures and domains.
  • Time-series forecasting: Deep-learning time-series forecasting methods include CNN-, Transformer-, and MLP-based approaches that extract or model temporal patterns in different ways.The related-work discussion contrasts these families with traditional methods such as ARIMA, which offer interpretability but often lower accuracy.
  • Time-series decomposition: Recent forecasting models decompose series into trend-seasonal, multi-scale, or multi-period components to exploit distinct underlying patterns.Examples include moving-average decomposition, hierarchical downsampling, scale-based seasonal and periodic splitting, and Fourier-based sub-period decomposition.
  • Time-series decomposition: TimeKAN examines time series from a multi-frequency perspective through a Decomposition-Learning-Mixing architecture for modeling complex patterns.This approach is presented as a frequency-oriented extension of prior decomposition work.

3 TIMEKAN

TimeKAN decomposes multivariate time series into frequency-specific representations, models each band with tailored KAN capacity, and repeatedly mixes the bands for forecasting. Its pipeline combines hierarchical preprocessing, cascaded frequency decomposition, Multi-order KAN learning, depthwise convolution, and frequency mixing.

  • Overall Architecture: TimeKAN independently predicts each univariate series after progressively removing high-frequency components with moving averages and projecting multi-level sequences into a higher-dimensional space.The preprocessing generates hierarchical sequences, with each lower-level sequence formed by removing part of the high-frequency information from the preceding level.
  • Cascaded Frequency Decomposition: CFD blocks recover each frequency-band representation by frequency-upsampling adjacent sequences and computing residuals between aligned sequences.FFT, zero-padding, and IFFT align sequence lengths while preserving frequency information before residual extraction.
  • Multi-order KAN Representation Learning: M-KAN blocks use parallel Multi-order KANs and depthwise convolution to learn frequency-specific representations and temporal dependencies.Depthwise convolution independently processes each channel, while KAN replaces linear weights with learnable univariate functions.
  • Multi-order KAN Representation Learning: Chebyshev polynomial orders increase from low- to high-frequency components, providing stronger representation capacity for increasingly complex temporal dynamics.The paper names this collection of KANs with varying highest polynomial orders Multi-order KANs.
  • Frequency Mixing: The M-KAN output sums Multi-order KAN and depthwise-convolution outputs, after which Frequency Mixing converts frequency representations back into multi-level sequences for subsequent CFD blocks.The final Frequency Mixing block uses the highest-level sequence and a linear layer to produce forecasts, then stacks variable predictions into the multivariate output.

4 EXPERIMENTS

Experiments evaluate TimeKAN on six real-world datasets against established forecasting baselines, using standard long-term forecasting settings and ablations. TimeKAN generally delivers superior forecasting with substantially lower computational cost, while its frequency upsampling, Multi-order KANs, and Depthwise Convolution contribute materially to performance.

  • Experimental Settings: Experiments use six real-world datasets, eleven forecasting baselines, fixed input length T = 96, prediction lengths F = {96, 192, 336, 720}, and MSE/MAE evaluation.ETT datasets use a 6:2:2 split, while the remaining datasets use 7:1:2.
  • Main Results: TimeKAN achieves superior predictive performance across all datasets except Electricity, where iTransformer performs best.The passage attributes iTransformer’s Electricity advantage to channel-wise self-attention for modeling inter-variable dependencies in high-dimensional data.
  • Ablation Study: Replacing Frequency Upsampling with Linear Mapping, Linear Interpolation, or Transposed Convolution decreases performance.The ablation indicates that these alternatives do not preserve frequency information adequately, weakening the Decomposition-Learning-Mixing framework.
  • Ablation Study: Multi-order KANs outperform MLPs and fixed low- or high-order KANs across the ablation comparisons.The comparison supports using different KAN orders across frequency levels rather than one fixed order.
  • Ablation Study: Depthwise Convolution is the strongest tested choice for temporal dependency learning, outperforming its removal, Standard Convolution, and Multi-head Self-Attention.Replacing it with Standard Convolution causes declines in most metrics, consistent with separately extracting temporal dependencies without inter-channel interference.
  • Model Efficiency: On Weather, TimeKAN uses 20.05% of TimeMixer’s parameters and 36.14% of its MACs; on Electricity, PatchTST has nearly 295 times more parameters and almost 118 times more MACs.The efficiency comparison fixes F = 96, T = 96, and batch size 32; grouped Depthwise Convolution and compact Multi-order KAN representations support the low cost.

5 CONCLUSION

TimeKAN is an efficient KAN-based architecture for long-term forecasting that decomposes, learns, and recombines frequency bands. Experiments on real-world datasets report state-of-the-art forecasting with lightweight computational consumption.

  • TimeKAN uses a Decomposition-Learning-Mixing architecture for long-term time series forecasting.It obtains representations for each frequency band, learns patterns within them, and recombines the bands into the original format.
  • Cascaded Frequency Decomposition blocks obtain series representations for each frequency band.
  • Multi-order KAN Representation Learning blocks model specific temporal patterns within each frequency band.
  • Extensive real-world dataset experiments report state-of-the-art forecasting performance with extremely lightweight computational consumption.

A ADDITIONAL MODEL ANALYSIS

Table 6 compares TimeKAN and other models using parameter counts and multiply-accumulate operations under fixed forecasting and batch settings. The table highlights the lowest computational cost.

  • Table 6 compares model parameters (Params) and multiply-accumulate operations (MACs) for TimeKAN and other models.
  • The comparison fixes prediction length F = 96, input length T = 96, and input batch size 32.
  • The lowest computational cost is highlighted in bold.

A.1 COMPUTATIONAL COMPLEXITY ANALYSIS

TimeKAN’s main computational costs come from FFT, depthwise convolution, and the Multi-order KAN block, with complexity determined by sequence length, hidden dimension, kernel size, and polynomial order.

  • FFT contributes computational complexity O(L logL) for a sequence of length L.
  • A depthwise convolution block with kernel size M and stride 1 has complexity O(LDM).
  • The Multi-order KAN block’s complexity depends on sequence length, hidden-state dimension, and the highest Chebyshev polynomial order K.

A.2 MODEL EFFICIENCY

TimeKAN is evaluated for model efficiency using parameter count and MACs. The reported comparison indicates that it combines strong forecasting accuracy with low computational cost relative to other models except DLinear.

  • TimeKAN consistently has significantly fewer parameters and MACs than other models except DLinear.DLinear remains the lightest because it consists of a single linear layer.
  • DLinear’s lightweight design is accompanied by a significant performance gap compared with state-of-the-art methods.
  • TimeKAN achieves superior performance in both forecasting accuracy and efficiency.

A.3 ERROR BARS

TimeKAN’s robustness was assessed across three random seeds against TimeMixer over four prediction horizons. In most cases, the experiments provided over 90% confidence that TimeKAN outperformed the second-best model.

  • Robustness evaluation: Three random seeds were used to compare TimeKAN with TimeMixer, reporting means, standard deviations, and confidence levels.Results were averaged over prediction horizons 96, 192, 336, and 720.
  • Robustness evaluation: Over four prediction horizons, TimeKAN outperformed TimeMixer with over 90% confidence in most cases.
  • Reported comparisons: Table 7 reports standard deviations and statistical tests for TimeKAN and TimeMixer across five datasets.

A.4 FREQUENCY LEARNING WITH LONGER WINDOW

TimeKAN performs poorly on Electricity with a 96-step look-back because that window provides limited effective frequency information. Extending the window to 512 steps nearly doubles the available effective frequencies and restores the model’s best performance on the dataset.

  • Frequency information: 10.69 effective frequencies are available at T = 96, compared with 19.74 at T = 512 on Electricity.Effective frequencies have amplitudes greater than 0.1 times the maximum amplitude.
  • Frequency information: The 96-step look-back loses substantial frequency information because it provides nearly half as many effective frequencies as 512 steps.
  • Forecasting performance: With T = 512, TimeKAN achieves the best Electricity performance, despite performing significantly worse than TimeMixer at T = 96.

A.5 IMPACT OF NUMBER OF FREQUENCY BANDS

Frequency-band division affects performance because bands must balance separation against retaining sufficient information. The reported experiments favor three or four bands, while frequency-domain tools support analyzing these components efficiently.

  • Frequency-band selection: Dividing the spectrum into 3 or 4 frequency bands yields the best performance in most cases.
  • Frequency-band selection: Two bands create excessive frequency overlap, whereas five bands leave too little information within each band.
  • Frequency-specific representation: The highest Chebyshev polynomial order K controls KAN’s fitting capability and motivates dynamically representing different frequencies.
  • Frequency-domain analysis: The DFT converts time-domain signals into complex frequency components, enabling analysis of superimposed frequencies.
  • Frequency-domain analysis: FFT reduces DFT complexity from O(N^2) to O(N logN), while rFFT represents N real inputs with N/2 + 1 frequency components.
Loading 2502.06910v2…