TimeKAN: KAN-based Frequency Decomposition Learning Architecture for Long-term Time Series Forecasting
Authors: Songtao Huang
Zhen Zhao
Can Li
LEI BAI
Venue: ICLR 2025 Poster (2025)
Keywords: Kolmogorov-Arnold Network; Time Series Forecasting
Primary area: learning on time series and dynamical systems

Abstract: 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.

Published as a conference paper at ICLR 2025
TIMEKAN: KAN-BASED FREQUENCY DECOMPOSI- TION LEARNING ARCHITECTURE FOR LONG-TERM
TIME SERIES FORECASTING
Songtao Huang--, Zhen Zhao', Can Li,, Lei Bai
1 Shanghai Artificial Intelligence Laboratory, Shanghai, China
2 School of Information Science and Engineering, Lanzhou University, Lanzhou, China Thhe Key Laboratory of Road and Traffic Engincering the Ministry of Education, Tongji University, Shanghai, China
huangscngtao@pjlee org. en, zhen. www.zcattooooooooooo com, Lchelen1005egmail.. com, baisanshi gmaal..c.ooooooooo
ABSTRACT
Real-world time series often have multiple frequency components that are inter- twined with each other, making accurate time series forecasting challenging. De- composing the mixed frequency components into multiple single frequency com- ponents is a natural choice. However, the information density of patterns varies across different frequencies, and employing uniform modeling approach for dif- ferent frequency components can lead to inaccurate characterization. To address this challenges, inspired by the flexibility of the recent Kolmogorov-Amold Net- work (KAN), we propose a KAN-based Frequency Decomposition Learning ar- chitecture (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 bottom-up 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 recombine the fre- queney bands into the original format. Extensive experimental results across mul- tiple real-world time series datasets demonstrate that TimeKAN achieves state-of- the-art performance as an extremely lightweight architecture. Code is available at
INTRODUCTION
Time series forecasting (TSF) has garnered significant interest due to its wide range of applications,
including finance (Huang et al.) 2024), energy management (Yin et al.) 2023), traffic flow planning
(Jiang Luo (0022). and weather forecasting (Lam al.) 2023). Recently, deep learning has led
to substantial advancements in TSF, with the most state-ofthh-ap performances achieved by CNN-
based methods (Wang al. 2023 donghao & wang xue
2024), Transformer-based methods
et al.
2023
Liu et al. 2024b) and MLP-based methods (Zeng et al. 2023 Wang et al. 2024a).
Due the complex nature of the real world, observed multivariate time series are often non- stationary and exhibit diverse patterns. These intertwined patterns complicate the internal rela- tionships within the time series, making challenging to capture and establish connections between historical observations and future targets. To address the complex temporal patterns in time series, an increasing number of studies focus on leveraging prior knowledge to decompose time series into simpler components that provide a basis for forecasting, For instance, Autoformer (Wu et al.) 2021) decomposes time series into seasonal and trend components. This idea is also adopted by DLinear (Zeng et al.) 2023) and FEDFormer (Zhou et al.] (022b). Building this foundation, TimeMixer WWang et al. 2024a) further introduces multi-scale seasonal -renn decomposition and highlights the importance of interactions between different scales. Recent models like TimesNet (Wu al.

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Published as a conference paper at ICLR 2025
2023). PDF (Dai et al. 2024), and SparseTSF (Lin et al. 2024 emphasize the inherent periodicity in time series and decompose long sequences into multiple shorter ones based on the period length, thereby enabling the separate modeling of inter- inter-period and intra-period dependencies within tem- poral patterns. In summary, these different decomposition methods share a common goal: utilizing the simplified subsequences provide critical information for future predictions, thereby achieving accurate forecasting.
is worth noting that time series are often composed of multiple frequency components, where the low-frequency components represent long-term periodic variations and the high-frequency compo- nents capture certain abrupt events. The mixture of different frequency components makes accurate forecasting particularly challenging. The aforementioned decomposition approaches motivate us to design frequency decomposition framework that decouples different frequeney components in time series and independently learns the temporal patterns associated with each frequency. However, this introduces another challenge: the information density of patterns varies across different frequen- cies, and employing uniform modeling approach for different frequency components can lead to inaccurate characterizations, resulting in sub-optimal results. Fortunately, a new neural network ar- chitecture, known Kolmogorov-Amool Networks (KAN) (Liu et al. 2024c), has recently gained significant attention in the deep learning community due to its outstanding data- fittt--ittingg capabilities and flexibility, showing potential optional kernels and allows for the adjustment of kernel order to control its fitting capacity. This consideration leads to explore the use Multi-order KANs represent temporal patterns across different frequencies, thereby providing more accurate information for forecasting.
substitute for traditional MLP. Compared MLP, KAN offers
Motivated by these observations, we propose a KAN-based Frequency Decomposition Learning ar- chitecture (TimeKAN) to address the complex prediction challenges caused by multiple frequency mixtures. Specifically, TimeKAN first employs moving average progressively remove relatively high-frequency components from the sequence. Subsequently, Cascaded Frequency Decomposition (CFD) blocks adopt bottom-up eascading approach to obtain sequence representations for each frequency band. Multi-order KAN Representation Learning (M-KAN) blocks leverage the high flexibility of KAN to learn and represent specific temporal patterns within each frequency band. Finally, Frequency Mixing blocks recombine the frequency bands into the original format, ensuring Leppootttioo-Laannngggg -Mixing process is repeatable, thereby modeling different tempo-
that this ral patterns various frequencies more accurately. The final high-level sequence is then mapped to the desired forecasting output via a simple linear mapping. With our meticulously designed ar- chitecture, TiMeKAN achieves state-of-the-art performance across multiple long-term time series forecasting tasks, while also being a lightweight architecture that outperforms complex TSF models with fewer computational resources.
Our contributions are summarized as follows:
We revisit time series forecasting from the perspective of frequency decoupling, effectively disentangling time series characteristics through frequency Aecomposition-Leariiie Mixing architecture address challenges caused by complex information coupling in time series.
We introduce TimeKAN as a lightweight yet effective forecasting model and design a novel M-KAN blocks to effectively modeling and representing patterns at different frequencies by maximizing the flexibility of KAN.
TimeKAN demonstrates superior performance across multiple TSF prediction tasks, while having a parameter count significantly lower than that of state-of-the-art TSF models.
2 RELATED WORK
2.1
KOLMOGOROV-AROOLD NETWORK
Kolmogoroy-Amon representation theorem states that any multivariate continuous function can be expressed combination of univariate functions and addition operations. Kolmogorov-Amoll Network (KAN) (Liu et al. (024c) leverages this theorem to propose an innovative alternative to traditional ML.. Unlike MLP. which use fixed activation functions the nodes, KAN introduces
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learnalle activation functions along the edges. Due to the flexibility and adaptability, KAN is con- sidered as a promising alternative to MLP.
The original KAN was parameterized using spline functions. However, due to the inherent com- plexity of spline functions, the speed and scalability of the original KAN were not satisfactory. Consequently, subsequent research explored the of simpler basis functions to replace splines, thereby achieving higher efficiency. ChebyshevKAN (SS) 2024) incorporates Chebyshev polyno-
mials parametrize the learnable functions. FastKAN (Li 2024 uses faster Gaussian radial basis
functions to approximate third-order B-spline functions.
Moreover, KAN has been applied alternatives MLP in various domains. Convolutional KAN [Bodner et al. 2024) replaces the linear weight matrices in traditional convolutional networks with learnable spline function matrices. U-KAN (Li et al., 2024) integrates KAN layers into the U-Net architecture, demonstrating impressive accuracy and effciency in several medical image segmen- tation tasks. KAN has also been used to bridge the gap between Al and science. Works such as PIKAN (Shukla et al. 2024) and PINN (Wang et al. (024b) utilize KAN to build physics-informed machine learning models. This paper aims to introduce KAN into TSF and demonstrate the strong potential of KAN in representing time series data.
2.2 TIME SERIES FORECASTING
Traditional time series forecasting (TSF) methods, such as ARIMA (Zhang 2003), can provide suf- ficient interpretability for the forecasting results but often fail to achieve satisfactory accuracy. In recent years, deep learning methods have dominated the field of TSF, mainly including CNN-based, Transformer-basee and MLP-based approaches. CNN-based models primarily apply convolution operations along the temporal dimension to extract temporal patterns. For example. MICN (Wang et al. 2023) and TimesNet (Wu et al., (023) enhance the precision of sequence modeling by adjust- ing the receptive field to capture both short-term and long-term views within the sequences. Mod- em (donghao & wang xue 2024) advocates using large convolution kernels along the temporal dimension and capture both cross-time and cross-variable dependencies. Compared to CNN-based methods, which have limited receptive field, Transformer-basee methods offer global modeling ca- pabilities, making them more suitable for handling long and complex sequence data. They have become the cornerstone of modern time series forecasting. Informer (Zhou et al. (021) is one of the early implementations of Transformer models in TSF, making efficient forecasting possible by care- fully modifying the internal Transformer architecture. PatchTST (Nie et al.) 2023) divides the quence into multiple patches along the temporal dimension, which are then fed into the Transformer, establishing as an important benchmark in the time series domain. In contrast, iTransformer (Liu) et al. 2024b) treats each variable as an independent token to capture cross-variable dependencies multivariate time series. However, Transformer- based methods face challenges due to the large number of parameters and high memory consumption. Recent research on MLP-based methods has shown that with appropriately designed architectures leveraging prior knowledge, simple MLPs can outperform complex Transformer-bases methods. DLinear (Zeng al. 2023). for instance, prepro- cesses sequences using a season decomposition strategy. FITS [u et al. (024b) performs linear transformations in the frequency domain. while TimeMixer (Wang et al. 2024a) uses MLP to facilitate information interaction at different scales. These MI.--ased methods have demonstrated strong performance regarding both forecasting accuracy and efficiency. Unlike the aforementioned methods, this paper introduces the novel KAN to TSF to represent time series data more accurately. It also proposes a well-designed Decomposition-Leammii Leariion--Lammin--Miiii Mixxng architecture to fully unlock the potential of KAN for time series forecasting.
2.3 TIME SERTES DECOMPOSITION
Real-world time series often consist of various underlying patterns. To leverage the characteristics of different patterns, recent approaches tend to decompose the series into multiple subcomponents, including trend-seasonal decomposition, multi-scale decomposition, and multi-period decomposi-
tion. DLinear (Zeng et al. (023) employs moving averages to decouple the seasonal and trend components. SCINet (Liu et al.) 2022) uses hierarchical downsampling tree to iteratively tract and exchange information at multiple temporal resolutions. TimeMixer (Wang et al.] (024a) follows fine-to-coare principle to decompose the sequence into multiple scales across different
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Input Series
(e]Cascaded Frequency Decomposition
(c) Frequency Mixing
XL
Representation Learning
High Frequency
High-order KAN
Linear
Output
Embeddiin
Desthwise Conv
Frequancy Upaampling
Frequency
Moving Average
Upsampling
Middle Frequency
Embodiing
Deptheine Corv
Frequency Upsampling
Moving
Frequency Upsampling
Average
Low-arder KAN
Low Frequency
Emibeddin
Depthwise
Figure/ I:
The architecture of TimeKAN, which mainly consists of Cascaded Frequency Decompo- sition block, Multi-order KAN Representation Learning block, and Frequency Mixing block. Here, we divide the frequency range of the time series into three frequency bands as an example.
time spans and further splits each scale into seasonal and periodic components. TimesNet (Wu er al. 2023) and PDF (Dai et al.] (024) utilize Fourier periodic analysis to decouple sequence into multiple sub-period sequences based on the calculated period. Inspired by these works, this paper proposes novel Decomposition- Learning-Miist architecture, which examines time series from multi-fryyuency perspective to accurately model the complex patterns within time series.
3
TIMEKAN
3.1
OVERALL ARCHITECTURE
Given a historical multivariate time series input X E R Nx the aim of time series forecasting is to predict the future output series X, RNF, where T, F is the look-back window length and the future window length, and N represents the number of variates. In this paper, we propose TimeKAN to tackle the challenges arising from the complex mixture of multi-frequency components in time series. The overall architecture of TimeKAN is shown in Figure [ We adopt variate-independent manner (Nic al. 2023) predict each univariate series independently. Each univariate input time series is denoted as X E IRT and we consider univariate time series as the instance in the following calculation. our TimeKAN, the first step is to progressively remove the relatively high high-freeuency components using moving averages and generate multi-level sequences followed by projecting each sequence into a high-dimensional space. Next, adhering to the eecomppsitionn-Leaminn-- architecture design principle, we first design Cascaded Frequency Decomposition (CFD) blocks to obtain sequence representations for each frequency band, adopting a bottom-up cascading ap- proach. Then, we propose Multi-order KAN Representation Learning (M-KAN) blocks to learn and represent specific temporal patterns within each frequency band. Finally, Frequency Mixing blocks recombine the frequency bands into the original format, ensuring that the Decomposition-Learee Learsing----Lning--- Mixing process is repeatable. More details about our TimeKAN are described as follow.
3.2 HIERARCHICAL SEQUENCE PREPROCESSING
Assume that we divide the frequency range of raw time series X into predefined & frequency bands. We first use moving average to progressively remove the relatively high-frequency components and generate multi-level sequences { ,I where 2; E b E {1...- ,,}), Iı is equal to the input series and denotes the length of moving average window. The process of producing lulti-level sequences is as follows:
ArgPool(Paddiggrr..)))
After obtaining the lulti-level sequences, each sequence is independently embedded into higher dimension through a Linear layer:
2t Linear(x;)
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where Fi RT-TXD and D is embedding dimension. We define 41 as the highest level sequence and F& as the lowest level sequence. Notably, each lower-level sequence is derived from the se- quence one level higher by removing portion of the sigh-freueenyy information. The above process is a preprocessing process and only occurs once in TimeKAN.
3.3 CASCADED FREQUENCY DECOMPOSITION
Real-world time series are often composed of multiple frequency components, with the low- frequency component representing long-term changes in the time series and the high-frequency component representing short-term fluctuations or unexpected events. These different frequency components complement each other and provide a comprehensive perspective for accurately mod- eling time series. Therefore, we design the Cascaded Frequency Decomposition (CFD) block to accurately decompose each frequency component in a cascade way, thus laying the foundation for accurately modeling different frequency components.
The aim CFD block is to obtain the representation each frequency component. Here, we take obtaining the representation of the i-th frequency band example. To achieve it, first employ the Fast Fourier Transform (FFT) to obtain the representation of 24+1 in the frequency domain. Then, Zero-Padding is used to extend the length of the frequency domain sequence, that it can have the same length as the upper sequence I; after transforming back to the time domain. Next, use Inverse Fast Fourier Transform (IFFT) transform it back into the time domain. We refer to this upsampling process as Frequency Upsampling, which ensures that the frequency information remains unchanged before and after the upsampling. The process of Frequency Upsampling can be described as:
2 = (FFT((Padding(FF((:::)))
(3)
Here, it, and x, have the same sequence length. Notably, compared to z4, å, lacks the i-th frequency component. The reason is that 2241 is originally formed by removing i-th frequency component from 2; in the hierarchical sequence preprocessing and Zi+1 is now transformed into è; through a lossless frequency conversion process, thereby aligning length with in the time domain. There- fore, to get the series representation of the i-th frequency component Si in time domain, we only need get the residuals between x; and f::
ft .4
3.4 MULTI-ORDER KAN REPRESENTATION LEARNING
Given the multi-level frequency component representation fu) generated by the CFD block, we propose Multi-order KAN Representation Learning (M-KAN) blocks to learn specific representations and temporal dependencies at each frequency. M-KAN adopts allel architecture to separately model temporal representation learning and temporal dependency learning in frequency-specific way, using Multi-order KANs learn the representation of each frequency component and employing Depthwise Convolution to capture the temporal dependency. The details of Depthwise Convolution and Multi-order KAN will be given as follows.
par-
Depthwise Convolution To separate the modeling of temporal dependency from learning quence representation, we adopt a specific type of group convolution known as Depthwise Con- volution, in which the number of groups matches the embedding dimension. Depthwise Convolu- tion employs / groups of convolution kernels to perform independent convolution operations on the series of each channel. This allows the model focus capturing temporal patterns without interference from inter channel relationships. The process of Depthwise Convolution is:
fi, = CooWD-D(fi group = D)
(5)
Multi-order KANs Compared with traditional MLP, KAN replaces linear weights with learnable univariate functions, allowing complex nonlinear relationships to be modeled with fewer parame- ters and greater interpretability. (Xu et al. 2024a). Assume that KAN is composed of L + layer neurons and the number of neurons in layer is ny. The transmission relationship between the j-th neuron in layer [+1 1 and all neurons in layer can be expressed as 24 >'n (L,,,(21,)). where 21 is the j-th neuron at layer I + and 21,is the i-th neuron layer 7. We simply understand
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that each neuron is connected to other neurons in the previous layer through learnable univariate function 6. The vanilla KAN (Liu et al.) 2024c) employs spline function as the learnable univariate basic functions o, but suffering from the complex recursive computation process, which hinders the efficiency of KAN. Here, we adopt ChebyshexKAN (SS, 2024) to learn the representation of each frequency component, i.e., channel learning. Chebyshev KAN is constructed from linear combina- tions of Chebyshev polynomial. That is, using the linear combination of Chebyshev polynomial with different order to generate learnable univariate function p. The Chebyshev polynomial is defined by:
Tn (ar) = cos(n arccos(
where n is the highest order of Chebyshev polynomials and the complexity of Chebyshev polyno- mials is increasing with increasing order. A I-layer ThebyshevKAN applied to channel dimension can be expressed as:
2o(x)
Σ>
e o. (i(tnh((z))
j=l 1=0
di(x)
KAN(+)=
(8)
ΦD(2)
where is the index of output neuron and θ R RD are the learnable coefficients used to linearly combine the Chebyshev polynomials. It is worth noting that the frequency compo- nents within the time series exhibit increasingly complex temporal dynamics as the frequency in- creases, necessitating network with stronger representation capabilities to learn these characteris- tics. ChebyshevKAN allows for the adjustment of the highest order of Chebyshev polynomials n to enhance its representation ability. Therefore, from the low-frequency to high-frreuenny compo- nents, we adopt an increasing order of Chebyshev polynomials to align the frequency components with the complexity of the KAN, thereby accurately learning the representations of different fre- quency components. We refer to this group of KANs with varying highest Chebyshev polynomials orders Multi-order KANs. We set an lower bound order b, and the representation learning process for n, can be expressed as:
f12 = KAN(f,, order = b k --i)
The final output of the M-KAN block is the sum of the outputs from the Multi-order KANs and the Depthwise Convolution.
f= fú, + fa.
(10)
3.5 FREQUENCY MIXING
After specifically learning the representation of each frequency component, we need to re-transform the frequency representations into the form of multi-level sequences before entering next CFD block, ensuring that the Decomposition-Leanee Leatoinn---aannnn---ia Mixxin process is repeatable. Therefore, we designed Frequency Mixing blocks to convert the frequency component at i-th level f; into multi-level se- quences Tú, enabling it serve as input for the next CFD block. To transform the frequency com- ponent at i-th level fü into multi- sequences I; we simply need to to supplement the frequency information from levels i + 1 to & back into the i-th level. Thus, we employ Frequency Upsampling again incrementally reintegrate the information into the higher frequency components:
F3 = (FFF((Paddin(FFF((e...
+ fi
(11)
For the last Frequency Mixing block, we extract the highest-level sequence æ and use a simple linear layer to produce the forecasting results Xo.
Xo Linear(x1)
(12)
Due to the use of a aariate-indeeendet strategy, we also need to stack the predicted results of all variables together obtain the final multivariate prediction Xo-

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Published as a conference paper at ICLR 2025
Table I: Full results of the multivariate long-term forecasting result comparison. The input sequence length is set to 96 for all baselines and the prediction lengths F E {96, 192, 336, 720). Avg means the average results from all four prediction lengths.
Models
TumeANN
Dass
Tumsnfurerr 20244
TIMeFEM FEM FOAe
PaSSTST 2023]
Tinna 2022
MICN 2033
Dlissar A02
Fr
2024
FILM K1laan
FEDHerser 20213
Aussfoemer N0O
Mernc
MSB
MAE
MSE
MAE
MSS
MAE
MSE
MAE
MSS
MAE
MSE
MAE
MSB
MAE
MSE
MAE
MSS
MAB
MSE
MAB
MSE
MAE
MSE
MAE
0n
0.000:
0.930
18:40
0.447
0.38c
0.402
0.436
0.466
0.397
0412
0.995
0.407
0.443
0.513
0.47
0.436
14454
0.464
0446
0,200
0.546
0.4%6
0638
1.403
0.887
0443
0.50
0:531
11506
0.176
0.915
0.500
(1514
Avy
0.450
[1475
0.480
0:457
[0.516
0n
0.155
0:340
[1575
0:434
0.340
192
0.393
0.415
0.402
11414
11492
0.492
0.482
336
0.421
0.436
0.452
01452
11507
0.355
0.591
0.54
0.006
0.473
0.435
0.886
01436
0.450
01442
0.468
11524
0.655
01874
0.405
0,457
01,65
Any
[0.583
0.404
0.390
0.409
[0:333
0.407
00.302
0.406
[0.50]
0.411
10:414
0.427
0.504
0.531
[0.563
0.509
[0.43
0.446
[0.402
0.430
0.437
0.649
0.450
0.459
96
0.302
0.361
0.317
0.356
11.334
0.368
0.336
0.359
10352
0.374
0.338
11575
[1365
0.387
0.346
0.374
11.337
0.374
[03553
0.370
0.379
0.415
0.500
0,475
152
1.357
0.363
0.367
0:388
01337
0,361
0.378
0.359
1.390
0,355
0.37%
11:87
[1403
0.404
0.352
0.391
0.341
0.308
0.399
0.357
0.436
[1641
0.553
0.4%6
380
1:003
9:001
11:00
0/131
01111
0.470
[1431
0.11
0410
11413
11:36
0:431
0415
11:20
0.423
0.421
0.40%
01445
11:59
01621
0.557
220
9:428
[0000
0.439
0.441
0.461
0.449
0.478
11450
[U489
0.462
04433
0:451
0.400
0.471
0.481
0.441
0.545
0.000
0.671
0.561
0.315
0.363
0:397
00:00
0.410
0.399
0.902
00:406
0.407
00.400
1.405
[14423
0.422
0.404
0.406
00:407
0.417
014422
0.400
00:446
0.452
0.588
0.517
96
1:174
0:365
0.133
0.180
0266
0.257
10.83
0.270
0.187
11267
1.197
0.296
0.199
0.233
10.86
0.275
00183
0.256
0.208
0.381
0.255
0,339
193
1:239
8:30M
11:00
11350
02305
1226
0.30%
01255
0.314
0.249
11309
11284
0:361
0.300
01259
0.323
0.248
0.306
0.259
0.723
(.200
0.340
320
01311
0.347
0.300
0:347
0:31
1135)
11281
02229
0.369
0.289
0.30
0.345
0.335
0339
0:772
220
0:505
6.39
0.39
0-112
0,407
0.400
0.404
0.411
0.404
0.408
0.403
01549
0.523
0.558
0.525
0.559
0.511
04410
0.400
0.421
0.415
0.483
6.422
102277
6:37
0:310
0.333
00.200
0.354
0.291
0.333
11351
0.400
0.354
00:00
01:39
0.374
10.288
0.328
11:27
01:29
11174
0:11
01220
16118
0:227
0.12
0.220
1.194
0.361
0.195
01:53
[.177
0.27
10:155
0.226
(0217
0.296
133
0330
01:07
0.366
0.21
11:60
11:39
0.200
0:277
0211
0.390
730
0.344
0334
0.388
0.345
0.381
0.77%
11:55
10.70
0.244
11360
Eleetterr
192
0.174
0.183
0.274
1.29
0.351
0.199
0.265
0.190
0.190
0.304
0.2%6
0.168
0.184
0.302
01272
0.190
0.199
0.302
0.285
0.210
0.210
0.305
0.302
0.171
0.200
0.268
[0:98
0.278
0.201
0.315
[1201
0.222
0.354
336
0.197
0.286
0.215
0.178
0.369
0212
0.296
0.217
0.319
0.198
0.300
0.198
0.312
0.223
0.309
0.100
0.264
0.217
0.300
0.214
0,329
0.231
0.443
720
0.236
0.330
0.234
0,312
0.225
0.317
0.253
0,330
0.255
0.352
0.230
0.320
0.217
0.350
0.258
0.350
0.226
0.316
0.278
0,356
0.246
0,355
0.254
0.361
Ang
[0.197
0.206
0.182
01272
11.178
0.370
(0270
0.248
10226
0,318
0.199
11304
11196
0.31M
0.225
U.319
0.282
(0.223
0,300
(0.214
0.327
(0.27 0,338
Duzri
17
22
9
0
0
1
0
0
9
6
0
4 EXPERIMENTS
Datasets We conduct extensive experiments on six real-world time series datasets, including Weather, ETThI, ETTh2, ETTml, ETTm2 and Electricity for long-term forecasting. Following previous work (Wu et al. 202T), we split the ETT series dataset into training, validation, and test sets in a ratio of 6:2.2. For the remaining datasets, we adopt a split ratio of 7:1:2.
Baseline We carefully select eleven well-acknowledged methods in the field of long-term time se- ries forecasting as our baselines, including (1) Transformer-based methods: Autoformer (2021),
FEDformer (20226). PatchTST (2023), iTransformer (2024b); (2) MI.P-based methods: DL.in-
ear (023) and TimeMixer (2024a) (3) CNN-based method: MICN (2023). TimesNet (2023); (4)
Frequency-basee methods: FreTS (024)) and FiLM (202aa). And a time series foundation model
Time-FFM (0024)).
Experimental Settings To ensure fair comparisons, we adopt the same look-back window length T = 96 and the same prediction length F = (96, 192, 336, 7203. We utilize the 1.2 loss for model training and use Mean Square Error (MSE) and Mean Absolute Error (MAE) metrics to evaluate the performance of each method.
4.1 MAIN RESULTS
The comprehensive forecasting results are presented in Table I. where the best results are high- lighted in bold red and the second-best are underlined in blue. A lower MSE/MAE indicates a more accurate prediction result. We observe that TimeKAN demonstrates superior predictive per- formance across all datasets, except for the Electricity dataset, where iTransformer achieves the best result. This is due to iTransformer'' use of channel-wise self-attention mechanisms to model inter- variable dependencies, which is particularly effective for high- high-dimensional datasets like Electricity. Additionally, both TimeKAN and TimeMixer perform consistently well in long-term forecasting tasks, showcasing the generalizability of well-designed time-series decomposition architectures for accurate predictions. Compared with other state-of-the-art methods, TN introduces novel
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Table 2: Ablation study of the Frequeney Upsampling. The best results are in bold.
ETThi
ETTh2
ETTml
ETTm2
Weather
Electricity
Datasets
Metric
MSE
MAE
MSE
MAE
MSE
MAE]
MSE
MAE
MSE
MAE
MSE
MAE
Linear Mapping
0.401 0.413
0.312 0.362 0.328 0.365
0.180 0.263
0.164 0.211
0.184
0.275
Linear Interpolation
0.383 0.398
0.296 0.347
0.336 0.370
0.181 0.263
0.165 0.210
0.196 0.277
Transposed Convolution
0.377
0.407 0.290 0.344
0.326 0.366
0.178
0.261
0.163
0.211
0.188
0.274
Frequency Upsamping 0.367 0.395
0.290 0.340
0.322 0.361
0.174 0.255
0.162
0.208
0.174 0.266
Table 3: Ablation study of the Multi-order KANs. The best results are in bold.
ETTh1
ETTh2
ETTml
ETTm2
Weather
Datasets
Metric
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAB
MSE
MAE
MLPs
0.376 0.397
0.298
0.348
0.319 0.361
0.178 0.264 0.162 0.211
Fixed Low-order KANs
0.376 0.398
0.292 0.341
0.327 0.366
0.175 0.257
0.164 0.211
Fixed High-order KANs
0,380 0.407
0.310
0.363
0.327
0.269
0.176
0.257
0.164 0.212
Multi-order KANs
0.367 0.395
0.290 0.340
0.322
0.361
0.174
0.255
0.162 0.208
Lewpiiiion-Leannnnn----aa framework, closely integrating the characteristics of Multi-order KANs with this hierarchical architecture, enabling superior performance in wide range of long- term forecasting tasks.
4.2 ABLATION STUDY
In this section, we investigate several key components of TimeKAN, including Frequency Upsam- pling, Depthwise Convolution and Multi-order KANs.
Frequency Upsampling To investigate the effectiveness of Frequency Upsampling, we compared it with three alternative upsampling methods that may not preserve frequency information before and after transformation: (1) Linear Mapping; (2) Linear Interpolation; and (3) Transposed Con- volution. As shown in Table 2 replacing Frequency Upsampling with any of these three methods resulted in a decline in performance. This indicates that these upsampling techniques fail to maintain the integrity of frequency information after transforming, leading to the cecomposition-Learnni Mixing framework ineffective. This strongly demonstrates that the chosen Frequency Upsampling, as a non- -arammetttri method, is an irreplaceable component of the TimeKAN framework.
We designed the following modules to investigate the effectiveness of Multi- order KANs: (1) MLPs, which means using MLP to replace each KAN; (2) Fixed Low-order KANs, which means using a KAN of order 2 at each frequency level; and (3) Fixed High-order KANs, which means using KAN of order 5 at each frequency level. The comparison results are shown in Table B. Overall, Multi-order KANs achieved the best performance. Compared to MLPs, Multi- order KANS perform significantly better, demonstrating that well-designed KANs possess stronger representation capabilities than MLPs and are a compelling alternative. Both Low-order KANs and High-order KANs performed worse than Multi-order KANs, indicating the validity of our de- sign choice to incrementally increase the order of KANs to adapt to the representation of different frequency components. Thus, the learnable functions of KANS are indeed double-edged sword; achieving satisfactory results requires selecting the appropriate level of function complexity for spe- cific tasks.
Multi-order KANS
Depthwise Convolution To assess the effectiveness of Depthwise Convolution, we replace it with the following choice: (1) w/o Depthwise Convolution; (2) Standard Convolution; (3) Multi-head Self-Attention. The results are shown in Table A Overall, Depthwise Convolution is the best choice. We clearly observe that removing Depthwise Convolution replacing it with Multi-head Self-Attention leads to a significant drop in performance, highlighting the effectiveness of using con- volution to learn temporal dependencies. When Depthwise Convolution is replaced with Standard
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Table 4: Ablation study of the Depthwise Convolution. The best results are in bold.
ETThl
ETTh2
ETTml
ETTm2
Weather
Datasets
Metric
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
w/o Depthwise Conv
0.379 0.397
0.296 0.343 0.337 0.373
0.180
0.263
0.168 0.211
Standard Conv
0.364
0.393
0.295 0.345
0.323
0.364
0.180
0.264
0.162 0.210
Self-Attention
0.377
0.406
0.293
0.342
0.329
0.365
0.184
0.272
0.174 0.225
Depthwise Conv
0.367
0.395
0.290
0.340
0.322
0.361
0.174
0.255
0.162 0.208
ETTm2
Weather
0.21
TimaKAN iThansooror
0.190
0.20
mranstomer PatchTST
PatchTsr
MSE
0.180
MSF
0.18
0.170
0.16
0.160
0.14
192
512
720
96
192
336
512
720
T
Figure 2: Comparison forecasting performance between TimeKAN and other three models with varying look-back windows ETTm2 and Weather datasets. The look-back windows are selected to be E {48,90 96, 192, 336, 512, 720}, and the prediction length fixed to -6.
Convolution, there declines in metrics, which implies that focusing on extracting tempo- ral dependencies individually with Depthwise Convolution, without interference from inter-channel relationships, reasonable design.
Varing Look-bace Window In principle, extending the look-back window can provide more in- formation for predicting future, leading to a potential improvement in forecasting performance. A fective long-term TSF method equipped with strong temporal relation extraction capability should be able to improve forecasting performance when look-back window length increasing (Zeng et al. 2023 As model based frequency decomposition learning, TimeKAN should achieve better predictive performance as the look-back window lengthens, since more incremental frequency in- formation is available for prediction. To demonstrate that TimeKAN benefits from a larger look-back window, we select look-back window lengths from T {48, 96, 192, 336, 512, 720} while keeping the prediction length fixed 96. As demonstrated in Figure 2 our TimeKAN consistently reduces the MSE scores as the look-bace back window increases, indicating that TimeKAN can effectively learn from long time series.
4.3 MODEL EFFICIENCY
We compare TimeKAN with MLP-based method TimeMier and Transformer-based methods i'Trans- former and PSt in terms of model parameters and Multiply-Accumuaat Operations (MACs), validate that TimeKAN we fix the prediction length F = 96 and input length T = 96, and set the input batch size to 32. The comparison results are summarized in Table is clear that our TimeKAN demonstrates significant advantages in both model parameter size and MACs, particularly when compared to Transformer- based models. For instance, on the Electricity dataset, the parameter count of PatchTST is nearly 295 times that of TAMeKAN, and its MACs to the relatively lightweight MLP-based method TimeMixer, TimeKAN shows superior efficiency. On the Weather dataset, Time KAN requires only 20.05% of the parameters needed by TimeMixer and only 36.14% of the MACs. This remarkable efficiency advantage is primarily attributed to the lightweight architectural design. The main computations of the VimeKAN model are concentrated
lightweight and efficient architecture. To ensure fair comparison,
almost 118 times greater. Even when compared

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Published as a conference paper at ICLR 2025
Table 5: A comparison of model parameters (Params) and multiply-accumulat operations (MACs) for TimeKAN and three other models. To ensure fair comparison, we fix the prediction length F 96 and the input length T 96, and set the input batch size 32. The lowest computational cost is highlighted in bold.
FITHI
ETTH2
ElTml
FTTm2
Weather
Electricity
Datasets
Metric
Params
MACs
Params
MACs
Params
MACs
Params
MACs
Params
MACs
Params
MACs
TimeMixer
75.50K
20.37M
75.50K
20.37M
75.50K
20,37M
77.77K
24.18M
104.43K
82.62M
106.83K
1.26G
Transformer
841.57K
77.46M
224.22K
19.86M
224.22K
19.86M
224.22K
19.86M
4.83M
1.16G
4.83M
16.29G
PStchTST
3.75M
5.90G
10.06M
17.66G
3.75M
5.90G
10.06M
17.66G
6.90M
35.30G
6.90M
539.38G
TImeKAN
12.84K 7.63M
15.00K
8.02M
14.38K
7.63M
38.12K
16.66M
20.94K
29.86M
23.34K
456.50M
in the M-KAN block, and the Depthwise Convolution we employed significantly reduces the num- ber of parameters through grouped operations. Additionally, the powerful representation capabilities afforded by Multi-order KANs allow us to represent time series with very few neurons. Therefore, cannot overlook that TimeKAN achieves outstanding forecasting performance while requiring minimal computational resources.
CONCLUSION
We proposed an efficient KAN-based Frequency Decomposition Learning architecture (TimeKAN) for long-term time series forecasting. Based on caanmpooittion-Laannnn -Mining architecture, TimeKAN obtains series representations for each frequency band using a Cascaded Frequency Decomposition blocks. Additionally. Multi-order KAN Representation Learning blocks further leverage the high flexibility of KAN to learn and represent specific temporal patterns within each frequency band. Finally, Frequency Mixing blocks recombine the frequency bands into the original format. Extensive experiments on real-world datasets demonstrate that TimeKAN achieves the state of the art forecasting performance and extremely lightweight computational consumption.
ACKNOWLEDGEMENTS
This work is supported by Shanghai Artificial Intelligence Laboratory. This work done during Songtao Huang's 's internship at Shanghai Artificial Intelligence Laboratory.
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A ADDITIONAL MODEL ANALYSIS
Table 6: Full comparison results of model parameters (Params) and multiply-accumular operations (MACs) for TimeKAN and other models. To ensure a fair comparison, we fix the prediction length F = 96 and the input length T = 96, and set the input batch size to 32. The lowest computational
cost is highlighted in bold.
ETTHI
ETTH2
ETTml
ETTm2
Weather
Electricity
Datasets
Metric
Purams
MACs
Purams
MACs
Parums
MACs
Params
MACs
Params
MACs
Params
MACs
TimeMixer
75.50K
20.37M
75.50K
20.37M
75.50K
20.37M
77.77K
24.18M
104.43K
82.62M
106.83K
1.26G
Transformer
841.57K
77.46M
224.22K
19.86M
224.22K
19.86M
224.22K
19.86M
4.83M
1.16G
4.83M
16.29G
PTtchTST
3.75M
5,90G
10.06M
17,66G
3.75M
5,90G
10.06M
17.66G
6.90M
35.30G
6.90M
539,38G
TimesNet
605.48K
18.13G
1.19M
36.28G
4.71M
144G
1.19M
36.28G
1.19M
36.28G
150.30M
4.61T
MICN
25.20M
71.95G
25.20M
71,95G
25.20M
71.95(i
25.20M
71.950
111.3KK
295.07M
6.64M
19.50
Dlinear
18.62K
0.6M
18.62K
0.6M
18.62K
0.6M
18.62K
0.6M
18.62K
0.6M
18.62K
0.6M
FreTS
3.24M
101.46M
3.24M
101.46M
3.24M
101.46M
3.24M
101.46M
3.24M
101.46M
3.24M
101.46M
12.58M
2.82G
12.58M
2.82G
12.58M
2.82G
12.58M
2.82G
12.58M
8.46Ci
12.58M
8.46Gi
FEDFormer
23.38M
24.96G
23.38M
24.96G
23.38M
24.96G
23.38M
24.96G
23.45M
25.23G
24.99M
30.89G
AutoFormer
10.54M
22.82G
10.54M
22.82G
10.54M
22.82G
10,54M
22.82G
10.61M
23,08G
12.14M
28.75G
TimeKAN
12.84K
7.63M
15.00K
8.02M
14.38K
7.63M
38K
16.66M
20.94K
29.86M
23.34K
456.50M
A.1 COMPUTATIONAL COMPLEXITY ANALYSIS
In our TimeKAN, the main computational complexity lies in Fast Fourier Transform (FFT), Depth- wise Convolution block and Multi-order KAN block. Consider a time series with length L and the hidden state of each time point is D. For FFT, the computation complexity is OL logL). For Depthwise Convolution block, if we set the convolutional kernel to M and stride to 1, the com- plexity is OLDMM). Finally, assuming that the highest order of Chebyshev polynomials is K, the complexity of Multi-order KAN block is (LLD*)). Since M, D, K are constants that are indepen- dent of the input length L, the computational complexity of both the Depthwise Convolution block and the Multi-order KAN block can be reduced to (L). which is linear about the sequence length. In summary, the overall computational complexity is max(O) (L logL), (L) = O logL). When the input is a multivariate sequence with M variables, the computational complexity will expand to ((M. L logL) due to our inddpeee-n strategy.
A.2 MODEL EFFICIENCY
Here, we provide the complete results of model efficiency in terms of parameters and MACs in Table b As can be seen, except for DLinear, our TimeKAN consistently demonstrates significant advantage in both parameter count and MACs compared to any other model. DLinear is model consisting of only single linear layer, which makes it the most lightweight in terms of parameters and MACs. However, the performance of DLinear already shows significant gap when compared to state-of-the-art methods. Therefore, our TimeKAN actually achieves superior performance in both forecasting accuracy and efficiency.
A.3 ERROR BARS
To evaluate the robustness of TimeKAN, we repeated the experiments on three randomly selected seeds and compared it with the second-best model (TimeMixer). We report the mean and standard deviation of the results across the three experiments, as well as the confidence level of TimeKAN's superiority over TimeMixer. The results are averaged over four prediction horizons (96, 192, 336, and 720). As shown in the Table [ in most cases, we have over 90% confidence that TimeKAN outperforms the second-best model and demonstrates good robustne of TimeKAN.
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Metric Published as a conference paper at ICLR 2025
Table 7: Standard deviation and statistical tests for our TimeKAN method and second-best method (TimeMixer) on five datasets.
MSE
MAE
Dataset
TimeKAN
TimeMixer
Confidence
TimeKAN
TimeMixer Confidence
ETThi
0,422+0.00 0.462+0.006
99%
0.430=#0.002
0,448=0.004
99%
ETTh2
0.387=0.003 0.392+0.000
99%
0.408=0.003
0.412=0.004
90%
ETTml
0.378+0.002 0.386+0.003
99%
0.396=0.001
0.399#0.001
99%
ETTm2
0.278:0.001 0.278+0.001
0.324#0.001 0.325+0.001
90%
Weather
0.243+0.001
0.245+0.001
99%
0.273+0.001
0.276#0.001
99%
Table 8: Comparison the Electricity dataset when the look back window is expanded to 512.
96
192
336
720
Models
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MOMENT
0.136 0.233
00.152
0.247
0.167
0.264
0.205
0.295
TimeMixer
0.135
0,231
0.119
0.245
0.172
0.268
0.203
0.295
TimeKAN
0.133
0.230
0.149
0.247
0.165
0.261
0.203 0.294
A.4 FREQUENCY LEARNING WITH LONGER WINDOW
In Table 1. TimeKAN performs relatively poorly on the Electricity dataset. We infer that its poor performance on the electricity dataset is due to the overly short look-back window (T 96), which cannot provide sufficient frequency information. To verify this, we compare the average number of effective frequency components under specific look-back window. Specifically, we randomly select sequence of length T from the electricity dataset and transform it into the frequency do- main using FFT. We define effective frequencies those with amplitudes greater than 0.1 times the maximum amplitude. Then, we take the average number of effective frequencies obtained across all variables to reflect the amount of effective frequency information provided by the sequence. When = 96 (the setting in this paper). the average number of effective frequencies is 10.69. When we extend the sequence length to 512, the average number of effective frequencies becomes 19.74. Therefore, the effective frequency information provided by 512 time steps is nearly twice that of 96 time steps. This indicates that T 96 loses substantial amount of effective information.
To validate whether using T = 512 allows us to leverage more frequency information, we extend the look back window of TimeKAN to 512 on the electricity dataset and compare it with the state-of- the-art methods TimeMixer and time series foundatiom model MOMENT (Goswami et al. 2024). The results are shown in T Although TimeKAN performs significantly worse than Mixer when 7 96, it achieves the best performance on the electricity dataset when the look-back window extended to 512. This also demonstrates that TimeKAN can benefit significantly from richer frequency information.
A.5 IMPACT OF NUMBER OF FREQUENCY BANDS
To explore the impact of the number of frequency bands on performance, we set the number of
frequency bands to 2, 3, 4, and 5. The effects of different frequency band divisions on performance are shown in the Table 9 As we can see, in most cases, dividing the frequency bands into 3 or 4 layers yields the best performance. This aligns with our prior intuition: dividing into two bands results in excessive frequency overlap, while dividing into five bands leads to too little information within each band, making it difficult to accurately model the information within that frequency range.
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Published as a conference paper at ICLR 2025
Table 9: Impact of number of frequency bands performance under the 96-to-96 prediction setting.
ETTh2
Weather
Electricity
Number of Frequency
MSE
MAE
MSE
MAE
MSE
MAE
2
0.292 0.340
0.164
0.2019
0.183
0.270
3
0.290 0.339
0.163
0.209
0.177
0.268
4
0.290
0.340
0.162
0.208
0.174
0.266
5
0.295
0.346
0.164
0.211
0.177
0.273
B MATHEMATICAL. DETAILS
B.1 KOLMOGOROV-ARNON NETWORK
Kolmogorov-Armole representation theorem states that any multivariate continuous function can be expressed combination of univariate functions and addition operations. More specifically, multivariate continuous function g [0. 1 + R can be defined as:
2n++1
( g ...)
Σ + Σ eon(T)))
(13)
where diy and D; are univariate functions. Following the pattern of MLP, Kolmogorov-Aroold Network (KAN) (Liu al. 2024c) extends the Kolmogoro--Amolthherre to deep representations, ie., stacked multilayer Kolmogorov- Amold representations. Assume that KAN is composed of L+ layer neurons and the number of neurons in layer | is ny. The transmission relationship between the j-th neuron in layer + and all neurons in layer can be expressed as:
J4 Σ (11,)
(14)
We can simply understand that each neuron is connected to other neurons in the previous layer through a univariate function Φ. Similar to MLP, the computation of all neurons at layer can be reorganized function matrix multiplication - Therefore, given input vector R the final output of KAN network is:
KAN(z) = ( 11 Lo).
(15)
In vanilla KAN (Liu et al. which is a class of smooth curves constructed via segmented polynomial basis functions. To ensure stability and enhance the representational capacity, KAN overlays the spline function on fixed basis function b, which is typically the SiLU function:
2024c), the univariate function Φt.j. is parametrized using B-splines,
<(x) = w)b(x) + w,spline(x)
(16)
spline(x) = Σ C.).(x)
(17)
where wy and w, learnable weights and spline(x) is the spline function constructed from the linear combination B-spline basis functions B, However, the complex recursive computation process of high-order B-spline functions hinders the efficiency of KAN. Therefore, in this work, adopt the simpler Chebyshev polynomial as the univariate function to replace the B-spline function (SS 2024). The univariate function defined by the Chebyshev polynomial given follows:
(z) cos(k arccos(z))
(18)
Here, k represents the order of the polynomial. Then, we consider the univariate function & as linear combination of Chebyshev polynomials with different orders:
K
THL1 = Σ (L..)
Σ
Σ Tu (Lanh((())) k=
(19)
Where θi, is the coefficients of k-th order Chebyshev polynomials acting on the Ii, and tanh the tanh activation function used normalize the inputs between -1 and 1. By adjusting the highest order of the Chebyshev polynomial K, we can control the fitting capability of KAN. This also inspires tour design of the Multi-order KAN dynamically represent different frequencies.
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Published as a conference paper at ICLR 2025
B.2 FOURIER TRANSFORM
Time series are often composed of multiple frequency components superimposed on each other, and it is difficult observe these individual frequency components directly in the time domain. There- fore, transforming time series from the time domain to the frequency domain for analysis is often necessary. The Discrete Fourier Transform (DFT) is a commonly used domain transformation algo- rithm that converts discrete-ting signal from the time domain to the complex frequency domain. Mathematically, given sequence of real numbers x(n] in time domain, where n 0, ,,...,N............. the DFT process can be described as:
[
Σ ==0
N-1
2/n| -
Σ x|n] n=0
2nT kn N
sin
2xT km )) k 0, 0...,....,AA N-1 N
(20)
where X[k] is the k-th frequency component of frequency domain signal and i is the imaginary unit. Similarly, we can use Inverse DFT (iDFT) to convert a frequency domain signal back to the time domain.
N-1 Σ K=
N-1
x
Inn|
A km
Σ k=0
[
cos
2T N
km
+ isn
2TT N
kn
(21)
N
N
The computational complexity of the DFT typically ((N2) Zhou et al. 2022b). In practice,
use the Fast Fourier Transform (FFT) to efficiently compute the Discrete Fourier Transform (DFT) of complex sequences, which reduces the computational complexity to ((N log.N). Additionally, by employing the Real FFT (rFFT), we can compress an input sequence of N real numbers into signal sequence in the complex frequency domain containing N/2 frequency components.
16


Reviews:
[{"review_id": "KjFQ0Guitu", "reviewer": "Reviewer_XpEF", "summary": "The paper presents TimeKAN, a Kolmogorov-Arnold Network (KAN)-based model for long-term time series forecasting, designed to handle complex multi-frequency patterns in real-world data. Traditional models struggle with mixed frequencies, but TimeKAN addresses this with a three-part architecture: Cascaded Frequency Decomposition to separate frequency bands, Multi-order KAN Representation Learning to model each band’s specific patterns using adaptable polynomial orders, and Frequency Mixing to recombine frequencies effectively. Experiments show that TimeKAN achieves superior accuracy and efficiency compared to existing models, making it a robust, lightweight solution for complex TSF tasks.", "questions": "**To my knowledge, KAN itself has not been formally accepted, meaning it has not undergone rigorous peer-reviewed validation. If KAN’s theoretical foundation is later found to be flawed, would this impact the validity of this paper?** – If the underlying theory or structure of KAN is later shown to have limitations or inaccuracies, would the overall reliability of TimeKAN be compromised? Has the author considered this risk, and are there alternative solutions in place?\n\n**In the experimental section, this paper does not compare TimeKAN with current state-of-the-art models (such as large language models or foundation models), making it difficult to assess its actual performance** – Without direct comparisons with these advanced models, can TimeKAN demonstrate a significant advantage? If the authors believe TimeKAN holds particular value in computational efficiency or predictive accuracy, could more data be provided to quantify this advantage?\n\n**A major trend in time series research is developing foundation models, inspired by large language models, to generalize across domains and tasks. However, it is unclear if TimeKAN’s current design can support such robust representation learning** – Can TimeKAN truly compete with established frameworks like Transformers in terms of generalization and adaptability? If not, have the authors considered alternative approaches to enhance TimeKAN’s structural robustness and flexibility for broader applications?", "limitations": "", "rating": 3, "confidence": 5, "soundness": 1, "presentation": 2, "contribution": 1, "strengths": "**Exploratory Application of KAN to Time Series Forecasting**: The paper attempts to introduce Kolmogorov-Arnold Networks (KAN) into time series forecasting, using multi-order polynomial representations to handle the complexities of different frequency components. While KAN has not been widely applied in this area, this effort demonstrates its potential flexibility in data fitting and offers an alternative approach to traditional MLPs.\n\n **Comprehensive Experimental Design**: The paper includes experiments across various time series datasets, such as weather, electricity, and energy data, covering diverse scenarios. Additionally, it conducts ablation studies to examine the effects of each module. These experiments help to assess TimeKAN’s performance and may provide a reference point for further research.", "weaknesses": "**Lack of Innovation**: The primary contribution of this paper is the integration of the Kolmogorov-Arnold Network (KAN) into time series forecasting, yet the work does not introduce novel methods or substantial breakthroughs in methodology. While the inclusion of KAN is somewhat new, other components, such as frequency decomposition and mixing, are mature techniques, and the paper does not propose innovative applications or enhancements to these. Overall, this work appears more like a combination of existing technologies rather than a genuinely innovative study.\n\n**Absence of Comparison with Cutting-Edge Models**: The experiments lack direct comparisons with state-of-the-art models, especially those in high demand for time series forecasting, such as large language models (LLMs) and foundation models. Given current research trends, these models have become widely adopted benchmarks. Without such comparisons, the effectiveness of the proposed method remains unclear, especially as the improvements presented are relatively limited compared to advancements in mainstream approaches.\n\n **Reliance on KAN, a Model with Limited Validation**: The foundation of TimeKAN is the KAN model, which has not yet been widely validated or accepted. Its theoretical correctness and practical effectiveness remain uncertain, which casts doubt on the reliability and generalizability of TimeKAN as a whole. If there are inherent issues with KAN, the predictive performance and stability of TimeKAN could be compromised, making the paper's conclusions less convincing.\n\n**Insufficient Analysis of Computational Efficiency**: While the paper claims that TimeKAN is more lightweight than existing methods, it lacks an in-depth analysis of its actual computational efficiency, especially compared to more mainstream and optimized time series models. Additionally, there is no quantification of the computational cost associated with KAN’s multi-order polynomial calculations when handling long-sequence data. Given that many time series tasks require efficient real-time computations, focusing solely on parameter reduction does not adequately demonstrate TimeKAN’s advantage in computational efficiency; the absence of data on inference speed and computational cost undermines its practical applicability.\n\n\n**Focus on Single-Task Performance Rather Than Generalized Representation**: A dominant trend in time series modeling now follows the approach of large language models (LLMs) to develop foundation models and leverage self-supervised representation learning. This approach enables generalization across various tasks and domains, ultimately aiming for a “one-fit-all” solution. However, this paper remains focused on improving single-task performance in time series forecasting (TSF), which may be of limited value in light of the broader goals of the field. Furthermore, the improvements reported in the experimental results are relatively modest, and without statistical significance testing, it remains unclear if these gains are truly meaningful or could simply be attributed to random variation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "bNRGCWmTRz", "reviewer": "Reviewer_gWX6", "summary": "In this paper, the authors propose a time series forecasting method based on KAN. TimeKAN uses frequency decomposition and KAN to effectively capture temporal correlations of the data. Experiments show the effectiveness of TimeKAN based on real-world datasets.", "questions": "1. See weaknesses.\n2. Could you provide a short theoretical analysis about why in some cases in time series forecasting, KAN is better than MLP?\n3. For Table 2, why not include the electricity dataset?\n4. For KAN, you mentioned that the Kolmogorov-Arnold representation theorem states that any multivariate continuous function can be expressed as a combination of univariate functions and addition operations. Could you explain more about how it can capture multivariate correlations?", "limitations": "", "rating": 5, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. A vivid introduction and related work section to explain the background of time series forecasting and KAN.\n2. A clear figure to illustrate the overall framework of TimeKAN. In the methodology section, all components are detailed. \n3. Good ablation study to test the effectiveness of the different components of the model.", "weaknesses": "1. The author does not explain why TimeKAN does not perform well in the Electricity dataset. It would be helpful if the authors could provide potential reasons or hypotheses for why TimeKAN underperforms on the Electricity dataset specifically. \n2. From Table 4, I do not see a huge increase with KAN compared to MLP models. Generally, if these results are similar, mostly, KAN is much slower than MLP. It would be good to see runtime comparisons between KAN and NLP implementations. Additionally, if KAN is slower than MLP in practice, it would be beneficial for authors to discuss more reasons why we prefer KAN over MLP. \n3. For the look-back window, the authors do not compare TimeKAN with other models. For most models, when the prediction length is fixed, the prediction accuracy will increase as the look-back window increases. It is beneficial to provide a comparative analysis of TimeKAN's performance with varying look-back windows compared to other baseline models. This would provide a more comprehensive evaluation of TimeKAN's capabilities relative to existing methods.\n4. For baseline methods, it is better to choose more frequency-based (such as FreTS) methods since frequency decomposition is a key contribution.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "AWIOqw3Ijr", "reviewer": "Reviewer_8Zzz", "summary": "TimeKAN is a time series forecasting model that combines frequency decomposition, representation learning, and mixing. It first uses a moving average to separate high and low frequencies, creating multi-level sequences that are embedded into a high-dimensional space. Cascaded Frequency Decomposition (CFD) blocks progressively isolate each frequency band. The Multi-order KAN Representation Learning (M-KAN) blocks use Kolmogorov-Arnold Networks to capture temporal patterns within each frequency band independently. Finally, the Frequency Mixing blocks recombine these decomposed bands to restore the original sequence, which is then used for forecasting through a linear layer", "questions": "What if we split the frequency band to more layers (more than 3 for example ). Will it increase performance ?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "1\tClarity and Structure: The paper follows a logical flow from problem statement to conclusions, making complex ideas accessible.\n\t2.\tThorough Background: A strong review of related work provides valuable context, situating the contribution within the field.\n\t3.\tDetailed Experiments: Comprehensive experiments across multiple datasets support the model’s performance claims, with ablation studies highlighting component effectiveness.\n\t4.\tFocused Writing: The paper stays on topic, avoiding unnecessary details and maintaining focus on the core contribution", "weaknesses": "Depth in Explanation: The methodology section could offer more detail on complex components like Kolmogorov-Arnold Networks for greater accessibility.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lmVlqIuiPP", "reviewer": "Reviewer_BGvS", "summary": "This paper explores a method for decomposing mixed frequency components into distinct single-frequency components to improve time series forecasting accuracy. The proposed approach, called TimeKAN, is based on the Kolmogorov-Arnold Network (KAN). TimeKAN's process consists of three key components: (1) Cascaded Frequency Decomposition (CFD) blocks, which use a bottom-up cascading approach to obtain series representations for each frequency band; (2) Multi-order KAN Representation Learning (M-KAN) blocks, which capture and represent specific temporal patterns within each frequency band; and (3) Frequency Mixing blocks, which recombine the separated frequency bands back into the original series format.\n\nThe study demonstrates that TimeKAN outperforms several state-of-the-art forecasting methods, including Autoformer, FEDformer, and iTransformer, by achieving lower MSE and MAE across multiple time series datasets such as Weather, ETTh2, and ETTm2.", "questions": "N/A", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "1. Figure 1 is beautifully designed and provides an intuitive overview of each component in the new TimeKAN method, as well as how they connect.\n\n2. The study makes effective use of large-scale datasets and performs comparisons with a variety of other methods (including CNN-based and Transformer-based models), demonstrating the advantages of TimeKAN. The model is also tested across different prediction lengths, and for datasets where performance is less optimal, the paper offers thorough explanations and detailed insights.\n\n3. The analysis delves into several key components of TimeKAN, such as Upsampling, Depthwise Convolution, and Multi-order KANs. I especially appreciated this section, as it not only establishes that TimeKAN outperforms other deep learning methods but also shows that each individual component of TimeKAN is optimally designed.", "weaknesses": "1. Section 3.2 appears somewhat disorganized. While the overall logic is clear, the expression could be refined for clarity. Additionally, more mathematical details and background should be provided, which can be included in the appendix.\n\n2. If possible, please add more data to Table 5 in Section 4.3. Supplement it with the performance of other methods in Table 1 on parameters (params) and MAC across these six datasets.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)