SpikF: Spiking Fourier Network for Efficient Long-term Prediction
Authors: Wenjie Wu
Dexuan Huo
Hong Chen
Venue: ICML 2025 poster (2025)
Keywords: Time-series Forecasting
Spiking Neural Networks
Fourier Transform
Primary area: deep_learning->sequential_models_time_series
TL;DR: Spiking Fourier Network enables efficient long-term prediction while significantly reducing energy consumption.

Abstract: Spiking Neural Networks (SNNs) have demonstrated remarkable potential across many domains, including computer vision and natural language processing, owing to their energy efficiency and biological plausibility. However, their application in long-term prediction tasks remains underexplored, which is primarily due to two critical challenges: (1) current SNN encoding methods are unable to effectively encode long temporal information, leading to increased computational complexity and energy consumption; (2) though Transformer-based models have achieved state-of-the-art accuracy in temporal prediction tasks, the absence of proper positional encoding for spiking self-attention restricts Spiking Transformer from effectively utilizing positional information, resulting in performance degradation. To address these challenges, we introduce an attention-free framework, **Spik**ing **F**ourier Network (**SpikF**), that encodes input sequences in patches and employs an innovative frequency domain selection mechanism to effectively utilize the sequential properties of time-series data. Extensive evaluations on eight well-established long-term prediction datasets demonstrate that SpikF achieves an averaged $1.9\\%$ reduction in Mean Absolute Error (MAE) compared to state-of-the-art models, while lowering total energy consumption by $3.16\times$. Our code is available at https://github.com/WWJ-creator/SpikF.

Spiking Neural Networks (SNNs), as the third generation of neural networks (Maass, 1997), emulate the brain's, 's informa- tion processing mechanisms through discrete spike events rather than continuous value computations (Roy et al., 2019). SpikF: Spiking Fourier Network for Efficient Long-term Prediction
Wenjie Wu
Dexuan Huo
Hong Chen
Abstract
This biologically-inspiee approach addresses the critical limitation of Artificial Neural Networks (ANNs) in energy efficiency. While ANNs have dominated machine learning applications, their substantial computational demands create challenges for resource-constrainee environments. By lever- aging event-driven processing where computations occur only during sparse spike events, SNNs achieve comparable performance to ANNs with significantly reduced energy consumption (Roy et al., 2019). This efficiency benefit makes SNNs particularly suitable for edge computing and real-time processing applications (Maass, 1996).
Spiking Neural Networks (SNNs) have demon- strated remarkable potential across many domains, including computer vision and natural language processing, owing to their energy efficiency and biological plausibility. However, their applica- tion in long-term prediction tasks remains under- explored, which is primarily due to two critical challenges: (1) current SNN encoding methods are unable to effectively encode long temporal information, leading to increased computational complexity and energy consumption; (2) though Transformer-basee models have achieved state-of- the-art accuracy in temporal prediction tasks, the absence of proper positional encoding for spik- ing self-attention restricts Spiking Transformer from effectively utilizing positional information. resulting in performance degradation. To address these challenges, we introduce an attention free framework, Spiking Fourier Network (SpikF), that encodes input sequences in patches and em- ploys an innovative frequency domain selection mechanism to effectively utilize the sequential properties of time-series data. Extensive evalua- tions on eight well-established long-term predic- tion datasets demonstrate that SpikF achieves an averaged 1.9% reduction in Mean Absolute Error (MAE) compared to state-of-the models, while lowering total energy consumption by 3.16x. Our code is available at https://gihub..om//WV creator/SpikF.
With these advantages, SNNs have made significant strides across various machine learning domains, particularly in computer vision. In image classification tasks, SNNs in- tegrated with Transformer architectures (Vaswani al., 2017; Zhou et al., 2023; 2024) have achieved performance par with conventional ANNs. Similarly, in object de- tection tasks, notable works such as Spiking- YOLO (Kim et al., 2020) and Spike (Luo et al., 2024) have fur- ther demonstrated the potential of SNNs. Beyond computer vision, SNNs have also shown promising results in natu- ral language processing, exemplified by advancements like SpikeGPT (Zhu et al., 2024), highlighting their versatility across diverse domains.
However, the application of SNNs to long-sequence predic- tion remains relatively unexplored. Though recent work (Lv et al., 2024b), has investigated SNN performance in short- sequence prediction tasks, comprehensive benchmarking of SNNs on popular long-sequence prediction datasets (Élec- tricity, Weather, ETT, Traffic and Exchange) has not been extensively established.
The limited exploration of SNNs in long-term prediction tasks is due to two critical challenges. Firstly, the cur- rent spike encoding mechanisms employed in temporal pre- diction tasks, such as delta encoding (Amir et al., 2017). convolution- based encoding (L.v al., 2024b), and linear- based encoding (Lv et al., 2024b), treat the input sequence as monolithic entity. These approaches increase the hidden dimension of the model, leading to a substantial computa- tional burden that undermines the inherent energy efficiency advantages of SNNs. Secondly, while ANN Transformer- based models have achieved remarkable accuracy in long- term prediction tasks, such as Autoformer (Wu et al., 2022).
1. Introduction
'Tsinghua University, Beijing, China. Correspondence to: Hong Chen <hongchen <hongchen&tsinghuu. @tsinghua.cdu.cn.>...
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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SpikF: Spiking Fourier Network for Efficient Long-term Prediction
FEDformer (Zhou al., 2022), PatchTST (Nie al., 2023) and i'Transformer (Liu et al., 2024), the traditional self- attention mechanism suffers from permuuation-invariance (Zeng et al., 2023), as detailed in Appendix 1. This limi- tation necessitates the incorporation of positional encoding (Vaswani et al., 2017) to effectively embed positional infor- mation into the input data. However, positional encoding in Spiking Transformers remains underexplored. Despite inno- vative approaches, such as central pattern generator-inspiret spiking positional encoding (Lv et al., 2024a), Spikformer utilizing this mechanism exhibit only marginal performance improvements and remain unable to deliver satisfactory results in temporal prediction tasks. In fact, Spikformer equipped with central pattern cenerator-inspieee positional encoding demonstrate suboptimal performance compared to simpler SNN architectures, such as SpikeTCN (Tavanaei & Maida, 2017) and SpikeRNN (Kim et al., 2019), incurring accuracy losses of 2.6% and 7. 7.7% in Root Relative Squared Error (RSE), respectively. This performance gap under- scores the current limitations of spiking positional encoding techniques in effectively embedding positional information, thereby hindering the potential of Spiking Transformers in long-term prediction tasks.
We conduct empirieal evaluations across eight real- world long-term time-series benchmark datasets, demonstrating that SpikF achieves superior accuracy over SOTA ANN models, with a 1.9% reduction in MAF. Our extensive experiments across Electricity, Weather, ETT, Traffic and Exchange datasets validate the robustness of our approach and establish a foun- dational SNN benchmark for the research community. To our knowledge, SpikF is the first SNN-based bench- mark providing full coverage across these datasets. It not only expands the scope of SNN research by estab- lishing a benchmark for future SNN-based temporal prediction models, but also provides an efficient and in- novative solution for the temporal prediction research community.
2. Related Works
2.1. Time-series Prediction
The time-series prediction methodologies have developed from traditional statistical approaches, such as ARIMA (Box et al., 1978) and exponential smoothing (Hyndman et al., 2008), to advanced deep learning architectures. These in- clude Temporal Convolutional Networks (TCNs) (Bai et al., 2018; donghao & wang xue, 2024), Recurrent Neural Net- works (RNNs) (Zhang et al., 2023; Jia et al., 2024), and Transformer-base based models. In recent years, attention-based models ( (Vaswani et al., 2017) have emerged as advanced solutions for time-series forecasting. Significant progress in this field has been made with models such as Informer (Zhou et al., 2021) and Autoformer (Wu et al., 2022), which tackle the quadratic time complexity of self-attention and enhance temporal dependency modeling. More recent inno- vations, such as PatchTST (Nie et al., 2023), enhance the local semantic information of time-series by dividing input sequences into patches, while iTransformer (Liu et al., 2024) enhances the modeling of dynamic correlations among dif- ferent variates through inverted token mechanisms. Despite their remarkable performance. these models often require substantial computational resources and energy, which re- stricts their applicability in resource-constrained environ- ments.
In order to address the challenges, this paper proposes Spiking Fourier Network (SpikF), an attention-free archi- tecture. Specifically, SpikF adapts Spiking Patch Encoding from Patch Embedding in PatchTST (Nie et al., 2023) and incorporates a Spiking Frequency Selection mechanism to model dependencies between patches, replacing the self- attention mechanism and naturally leveraging the sequential properties of time-series data. Our approach outperforms ex- isting SOTA models by 1.9% across eight well-established long-term prediction real-world datasets while reducing en- ergy consumption by 75.05% in a representative study on the ECL dataset. SpikF provides a computationally efficient framework for long-term term forecasting, demonstrating excep- tional suitability for energy-constrained scenarios and edge device deployment. Our key contributions include:
We propose a Spiking Patch Encoder which divides in- put sequences into patches and independently encodes each sub-series into binary spike trains. By reducing computational complexity, this design enables SNNs to effectively handle longer input sequences, thereby enhancing their ability to utilize historical information.
In contrast, SNNs have garnered limited but growing atten- tion for time-series prediction. Several exploratory studies have demonstrated their potential, including a two-phase SNN for electricity load prediction (Kulkarni et al., 2013), polychronous spiking network for financial data predic- tion (Reid et al., 2014), NeuCube-based (Kasabov, 2014) architecture for crop yield prediction (Bose et al., 2016), and a multi-modal SNN architecture for financial stock pre- diction (AbouHassan et al., 2023). However, none of these works addresses the challenge of long-term prediction. A recent study proposed an SNN framework tailored for short-
We propose a novel Spiking Frequency Selection mech- anism that identifies and selects critical components from input sequences to enhance prediction perfor- mance. This mechanism establishes long-range tempo- ral dependencies not only within the input sequence but also between the sequence and prediction targets, lever- aging the positional information inherently embedded in the Fourier Transform.
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SpikF: Spiking Fourier Network for Efficient Long-term Prediction
sequence prediction tasks (Lv et al., 2024b), representing significant advancement in this emerging field. Despite these progresses, research on SNNs for long-term forecast- ing remains relatively limited, and a comprehensive bench- mark for evaluating their performance on widely-used long- sequence datasets is still absent. This gap arises from the inherent challenges faced by SNNs, including the efficient encoding of long sequences and the inadequate utilization of sequential information.
The LIF neuron receives the resultant current, integrates it to accumulate membrane potential, and compares this potential with predefined threshold to determine whether to generate a spike. The membrane potential V(t] evolves according to:
U|t| = Vlt - 1]+ Tm
(4[1) Vlt - 1]+ + Vrest)
(1)
b|s H (U(4 VAA)
(2)
2.2. Frequeney-based Methods in Time-series Forecasting
Vit] = U|t] (1 (1--Stt Sit) + Sieet
(3)
where Tm is the membrane time constant, Vreet is the resting potential, H represents the Heaviside step function and [[4] is the input current which can be calculated by:
Time-series forecasting has seen significant advancements through the integration of frequency domain analysis, which allows for more efficient processing and enhanced forecast- ing performance.
[|t] = W Spre
(4)
FITS (Xu et al., 2024) employs a complex linear layer to interpolate in the frequency domain, discarding high- frequency noise through a cutoff frequency strategy, thus achieving superior performance compared to DLinear.
where Spre [t] denotes the spikes emitted by the pre-synaptic neurons, and W represents the synaptic weights between different layers.
FreTS (Yi et al., 2023) further expands the utilization of fre- quency domain features through frequency domain MLPs, which facilitate time-series forecasting by providing a global view and energy compaction.
SNN Training Method The indiffcrentiable nature of the Heaviside step function makes challenging to train deep SNNs. There are two popular approaches to train deep SNNs. One ANN2SNN conversion (Cao et al., 2015; Bu et al., 2021; Wang et al., 2022), which trains an ANN model and then converts the model into rate-coded SNN model. However, this approach usually needs multiple simulating time steps to achieve comparable accuracy with the original ANN model, which causes large latency (Han et al., 2020).
FEDformer (Zhou et al., 2022) employs frequency domain selection to generate sparse attention, thereby reducing com- putational complexity and capturing detailed structures of time-series data.
FilterNet (Yi et al., 2024) adapts filters from the signal processing field, developing two kinds of filters to weaken or strengthen specific frequency components, thus benefiting from the utilization of the full spectrum.
The other approach utilizes surrogate gradient function to ap- proximate the derivative of the Heaviside step function, and thus conducting spatio-temporal backpropagation (STBP) to update the network parameters. In our work, adopt the latter approach.
Most previous works have focused on applying FFT to the entire series, which facilitates the utilization of high- frequency components. In contrast, our approach employs patch and grouping mechanisms to enhance the utilization of low and frequecy components in the original se- ries, where local information is emphasized by the spiking patches, which are discussed in following sessions.
Fast Fourier Transform (FFT) in SNNs While SNNs excel at processing temporal sequences, owing to their in- trinsic membrane dynamics, their ability to capture global temporal features is inherently limited by two fundamental characteristics: the exponential decay factor of membrane potentials and the reset mechanism of neuronal activation, which is discussed in Appendix A.2. Recent advancements (Lopez-Randulle et al., 2022; Orchard et al., 2021) have offered a promising solution to this limitation through the integration of Fast Fourier Transform (FFT) with SNN ar- chitectures.
2.3. Spiking Neural Networks
SNNs have gained increasing attention due to their energy efficiency and biological plausibility (Roy et al., 2019). Un- like ANNs that process continuous activation values, SNNs operate using discrete spikes, closely mimicking the behav- ior of the human brain.
Study (Lopez- et al., 2022) has demonstrated that matrix multiplication can be represented by spiking linear layer with an appropriately defined weight matrix. Accord- ingly, they initially express the FFT as a series of matrix multiplications and subsequently employ an SNN with an equivalent number of layers to avoid the challenges associ- ated with meating-poin operations.
Spiking Neuron Model One of the most popular spiking neuron models is Leaky Integrate-and and-Fre (LIF) Neuron (Abbott, 1999), due to its simplicity and widely utilization in recent research (Zhou et al., 2023; 2024; Lv al., 2024b).
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SpikF: Spiking Fourier Network for Efficient Long-term Prediction
Meanwhile, study (Orchard et al., 2021) leverages the mem- brane dynamics of the Resonate-and-Fitt neuron, tension of the LIF model, to naturally perform the Fourier Transform. This approach also successfully gets rid of the need for additional floating-poin operations.
3.3. Spiking Patch Encoder
Our spike encoding pipeline comprises three key stages, First, the input time-series signal is segmented into patches, each of which is processed through shared learnable linear layer that maps the data into a higher-dimensione space, as being put forward by PatchTST (Nie et al., 2023) to enhance the better utilization of local semantic information. These transformed patches are then temporally upsampled by a factor of T, using a zero-order hold (ZOH) strategy, increas- ing their temporal resolution. The upsampled signals serve as input currents to the LIF neurons and are transformed into binary spike trains, converting the analog information into neuromorphic representations.
3. Methodology
3.1. Problem Setup
The multivariate long-term forecasting framework can be formulated as follows: given an input time-series z!:h E R4XD of length L with D dimensions, our object is to predict the subsequent H values, denoted as yl: E R
The 1: will be divided into patches:
3.2. Overview
The overall architecture of SpikF is illustrated in Figure 1. SpikF comprises three components: (1) Spiking Patch Encoder (SPE) for transforming continuous-valued input time-series into binary spike representations; (2) a feature extraction module with a Spiking Frequency Selector (SFS) to select important frequency components and convolutional layers enhance local semantic information; (3) an MLP Decoder to generate the final output.
p² x x (R- 11+1)+
(5)
where N is the number of patches and assume T. can be divided by N.
Then each patch will be processed by a spiking linear layer:
Spiking Patch Encoder Although convolutional and delta encoding mechanisms have been proposed by (Lv et al., 2024b), they exhibit limitations in encoding extended his- torical information. To solve the problem, we introduce a novel patch-basee encoding mechanism that processes input sequences in segmented patches, thereby facilitating com- putationally efficient representation of temporal patterns.
(T,(K-1)+LTR SN(BN(LN((*))
enc
(6)
where LN denotes the linear transformation layer, BN rep- resents batch normalization, and SN denotes the spiking neuron layer.
3.4. Spiking Frequency Selector
Spiking Frequency Selector As we know, self-attention mechanisms have gained widespread adoption across various machine learning domains, but their inherent coamutatiooninvariann renders them inadequate in captur- ing positional information in time-series data. FFT-based models (Zhou et al., 2022; Xu et al., 2024) are effective at capturing sequential properties, but their frequency modes are restricted to maintain computational efficiency. Specif- ically, FEDformer (Zhou et al., 2022) relies on fixed or randomly selected frequency modes, while FITS (Xu et al., 2024) adopts a cut-off frequency strategy to discard high- frequency components. Both of the two approaches employ artificial selection strategies, failing take the characteris- tics of input sequences into consideration. To address these problems, we propose a novel frequency domain selection mechanism that dynamically identifies relevant frequency components of spike trains by leveraging the binary nature of spikes. This approach not only enhances the utilization of sequential properties but also alleviates the burden on researchers to manually determine key frequencies for the input sequence.
The encoded spike trains are processed through two blocks in parallel enhance modeling of positional information and energy efficiency. Initially, the spiking patches are grouped at fixed intervals. In the first block, the grouped patches are transformed into frequency components using a Spiking Fast Fourier Transform (S-FFT) layer. In the second block, the grouped patches are converted into frequency- domain spiking selector via a spiking selector generator, which consists of a linear projection layer, batch normal- ization layer, and spiking neuron layer. To improve the transmission efficiency of the spiking selector, a spiking max max-pooling layer is utilized. The frequency spectrum is then selected by the spiking selector. The selected frequency components are reconstructed in the time domain using a grouped Spiking Inverse Fast Fourier Transform (S-iFFT) layer. The resulting real-valued time-series are the input to the LIF neurons, generating spike trains with reduced noise. The sparsity of spike trains contributes to computational ef- ficiency in both S-FFT and inverse operations. Additionally, the S-FFT enables the spiking selector to maintain a global receptive field across the entire time-series while leveraging positional information.
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SpikF: Spiking Fourier Network for Efficient Long-term Prediction
XL
SPE
Lineor
BW
Intra Conv
Inter Conv
BN
Flatten
Decader
MLP
imput Sequence
SFS
Spliing Nouren
TiMe
Elerens-wiee Product
Elemank-wiia Add
IIFFT
Phaso
Complex
Frequency
Patsh Splitting
Con
Intra-group Conv
Sebector Goverator
Aranan
Inersgroup Cory
Doman
Figure 1. Architecture overview of the proposed SpikF framework. (a) The Spiking Patch Encoder (SPE) converts continuous time-series data into discrete spike train representations through patch-bass processing. (b) Spiking Frequency Selector (SFS) integrates S-FFT operations across spike train patches, thereby facilitating the natural utilization of positional information. (c) The decoder reconstructs real-valued predictions the processed spike patterns by employing an MI.P architecture.
The spiking patches generated by the Spiking Patch Encoder are first grouped at fixed intervals:
where SF denotes the grouped S-iFFT operation and O denotes the Hadamard element-wise product.
Then H serves as the stimulation current to spiking neu- rons, generating spikes with reduced noise:
G { Sita Denei...
1+(
NT. -1)9
where G' denotes ith group of patches and g represents the number groups.
S = ==(NNBB(HH)))
(12)
Then each G transformed into the frequency domain through the S-FFT algorithm:
3.5. Multilayer Perceptron (MLP) Decoder
As well known, challenging to realize high temporal resolution of spike trains for the spike decoder, as directly applying an MLP with a full receptive field reconstruct continuous-valle outputs would increase both the number of learnable parameters and computational overhead. To overcome this challenge while maintaining the temporal integrity of the spiking patches, we introduce a temporal synchronization mechanism.
F' = SF,
(8)
where SF, denotes the grouped S-FFT operation. In the meantime, G' passes through spiking linear layer to generate frequency domain spikes:
M set = SN(BN((NN("))
(9)
We first synchronize the temporal resolution between spik- ing patches and prediction series through upsampling the prediction target. Then, both spikes and prediction targets are divided into T, groups, and the spike trains are processed through an MLP to reconstruct the continuous-vaale tion. During the training phase, the loss function is defined as:
Then, spiking pooling layer is used to emphasize the mutual key frequencies of different groups:
predic-
M = SMP(M (MM M set
M
set)
(10)
where SMP represents the Spiking Max-pooling layer.
L
MMP((*) Y||
(13)
T.
Finally, Maet selects the important frequency components through the Hadamard product, and these components are sent back to the time domain through S-iFFT:
where S represents the spike train in the kth group and Y denotes the target prediction series.
During the inference phase, the predicted sequence under- goes dimensionality reduction through average pooling, and
H3 SF ② F*)
(11)
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SpikF: Spiking Fourier Network for Efficient Long-term Prediction
the final prediction sequence is generated. This process is formally expressed
data, which contribute its superior accuracy over these henchmarks.
T, Y
T3 Σ
MLP(S*)
(14)
4.2. Model Analysis
Computational Efficiency Analysis To evaluate the com- putational efficiency of our approach, we first analyze the variation in Synaptic Operations (SOPs) for S-FFT and S- iFFT as the firing rate of the input sequence increases. Based on the theoretical analysis provided in Appendix A.3, the SOPs of S-FFT and S-iFFT operations for an input length of = m 2 are:
where Y represents the downsampled prediction sequence.
4. Experiments
We conduct experiments to verify SpikF time-series pre- diction tasks across multiple dimensions. Our evaluation framework mainly includes: (1) predictive performance on eight well-established long-sequence prediction benchmarks (2) computational efficiency through energy consumption analysis (3) individual component contributions via ablation studies of SPE, SFS and temporal synchronization mech- anisms, and (4) generalization capability of the model for short-term prediction tasks and extended input sequences. The experimental results demonstrate that SpikF achieves superior accuracy and energy efficiency compared SOTA models.
SOPs(S FFFT
Σ L (1- gm k=1
6.3mx22-11 (1-
-m gmx2** x
(2ma 2
(15)
SOPs(S-
(1- gmxx2* ')8 6.gmx2 (1 (1- Bm gmx22
')
km
4.1. Long-term Prediction
+ L 12maa - -2 233h) ) + 69 x ma
In this section, we present the predictive performance of SpikF on well-established long-sequence prediction datasets.
(16)
where / 311-1 a.
Figure 2 shows the formulas, indicating that the SOPs of S-FFT and S-iFFT are significantly lower than the Floating- Point Operations (FLOPs) of traditional FFT iFFT when processing sparse input sequences. Since the energy con- sumption per SOP is 5. times lower than that per FLOP (Yao et al., 2023), S-FFT in our model achieves superior energy efficiency compared to FFT in ANN models.
Datasets The experiments are conducted on eight widely- used long-term prediction datasets including ECL, Weather, ETT (ETThI, ETTh2, ETTml and ETTm2), Traffic and Exchange, with detailed characteristics presented in Ap- pendix B.I.
Benchmarks We compare SpikF with three categories of models in- cluding Autoformer (Wu et al., 2022), Crossformer (Zhang & Yan, 2023), PatchTST (Nie et al., 2023), and I'Taanformee (Liu et al., 2024); TCN-based models such as SCINet (Liu al.. 2022) and TimesNet (Wu al., 2023); and Linear- based models including DLinear and RLinear (Zeng al., 2023). Notably, TimesNet (Wu et al., 2023) incorporates FFT into its model architecture.
advanced baseline models: Transformer-base
2543
Oppratonss
1507
FFTAFFT
SOFt of S-FFT ETTh
SOPs of S-FFT for ETTM
As shown in Table 1, SpikF demonstrates superior overall performance compared to other ANN-based models, achieving a 1.9% reduction prediction error com- pared iTransformer in terms of MAE. Notably, on ETTml dataset, it achieves 6.1% lower MAE than iTransformer. While iTransformer focuses correlations among different variates and PStchTST considers longer history information, both of them struggle to utilize positional information. In contrast, SpikF integrates patch encoding and S-FFT into its architecture, enabling utilization of local and global features while leveraging the sequential properties of time-series
Main Results
2.0
07
34
30
Riing Rale
Figure 2. Theoretical computational complexity of the FFT and iFFT and their spiking version with respect to the variation of for input length of L = 48. SOPs of S-FFT and S-iFFT on ETThl dataset are illustrated in this figure.
To verify the efficiency of the SFS module, we compare it with common ANN-based temporal manipulation mecha- nisms in terms of the number of learnable parameters, oper- ations (FLOPs or SOPs), and energy consumption caused
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SpikF: Spiking Fourier Network for Efficient Long-term Prediction
Table 1. In multivariate forecasting analysis, we conduct extensive comparisons against multiple competitive models across varying prediction horizons, following the methodology established in iTransformer (Liu et al., 2024). All baseline models are configured with a uniform look-back window of I. 96 timesteps. Performance metrics are averaged over Jf = (96, 192, 336, 720) prediction lengths. Lower MSE or MAE values indicate better performance. The best performance metrics are highlighted in bold red, while the second-best results are marked underlined blue for clarity. This convention consistently applied across all subsequent tables.
Model
SpikF
l'Transformer|
RLinear PatchTTT Crossformerr
TimesNet DLinear
SCINet
|Autoformer
Metric
MSE
MSE
MAE
AASE MAE MSE
MSE MAE
MSE MAS MAE
MSE MAE
MSE MAE
ECL
0.183
0.275
0.178
0.270
00.219
0.298
00.005
0.290
0.244
0.334
00.192
0.295
0.212
0.300
00.268
0.365
00.277 0.338
Weather
0.245
0.265
0.258
0.278
|0.272
0.291
0.259
0.281
0.2.59
0.315
|0.259
0.287
0.265
0.317
0.2992
0.363
0.338
0.382
ETTh1
0.440
0.428
0.454
0.447
0.446
0.434
0.469
0.454
0.5599
0.522
|0.458
0.450
0.456
0.452
0.747
0.647
0.496 0.487
ETTh2
0.372
0.394
0.383
0.407
0.374
0.398
0.387
0.407
0.942
0.684
|0.414
0.427
0.559
0.515
0.954
0.723
0.450
0.459
ETTml
0.388
0.385
0.407
0.410
|0.414
0.407
00.887
0.400
00.513
0.496
|0.400
0.406
0.403
0.407
0.485
0.481
0.588
0.517
ETTm2
00.281
0.320
|0.288
0.332
|0.286
0.327
0.281
0.326
0.757
0.610
|0.291
0.333
0.350
0.401
|0557
0.537
0.327 0.371
Traffic
|0.497
0.296
0.428
0.282
|0.626
0.378
0.481
0.304
0.550
0.304
|0.620
0.336
0.625
0.383
0.804
0.509
0.628 0.379
Exchange|
0.360
0.402
00.360
0.403
00.378
0.417
00.367
0.404
|0.940
0.707
|0.416
0.443
0.354
0.414
0..500
0.626
(0.613 0.539
by operations per sample. Table 2 provides a comparative analysis of SFS with linear transform in the time domain (Zeng et al., 2023), linear transform in the frequency do- main (Xu et al., 2024) and inverted self-attention (Liu et al., 2024). The results are summarized in Figure 2. The detailed calculation basis is provided in Appendix C.I.
reduces overall operational energy consumption by 75.05% compared to iTransformer on the ECL dataset. Notably, operational energy consumption is reduced by 96.94% for the encoder and 93.30% for the feature extraction module.
As shown in Table 2, the SFS module, which leverages the grouped S-FFT paradigm and the event-driven nature of SNNs, reduces 42. 26. of the operational energy consump- tion compared to the time domain linear transform utilized by DLinear in terms of operational energy consumption.
SpikF
10043
mandfommr
Eneegy Conmargtoappa
6003
2003
Table 2. Comparison of the number of learnable parameters, op- erations, and operational energy consumption on the Electricity dataset with look-back window of 96 and prediction length of 720. The parameters listed in the table correspond to the key feature extraction module, which represents the Spiking Fre- quency Selector in SpikF, time-domain linear transform for DLin- ear, frequency-domain linear transform for FITS and inverted self- attention for I'Tranffomer.
4461
Enooder
Feslas
Figure 3. Systematic comparison of energy consumption between iTTrasformerr and SpikF on ECL dataset, with a look-back window of 96 and a prediction length of 720. The comparison encompasses all modules, including the encoder, feature extraction, and decoder.
As indicated by (Lemaire et al., 2022; Shen et al., 2024), energy consumption comprises operational energy, memory- access energy, and addressing energy. To provide a more comprehensive analysis of SpikF's energy use, we follow the methods of (Lemaire et al., 2022; Shen et al., 2024) and present additional statistics on SpikF's energy efficiency in Table 3.
Model
Paras
Opts
Energy/JJ
SpikF
1.2K
0.13G
117.66
DLinear
0.14M
45M
205.19
FITS
18K
0.10G
475.62
iTransformer
1.6M
0.72G
3289.29
where ACE refers to S-ACE for iTransformer and NS-ACE for SpikF as proposed by (Shen et al., 2024), Esem denotes the energy consumption for membrane accessing, Eopte refers to the energy consumption of operations, EAddr refers to the energy consumption for addressing, and Erotal rep- resents the total energy consumption.
To further underscore the energy efficiency of SpikF from a holistic perspective, we have compared the operational energy consumption of SpikF and ITrrasffrmer across all modules, including the encoder, feature extraction module, and decoder. As shown in Figure 3, SpikF with T, =
7

---

SpikF: Spiking Fourier Network for Efficient Long-term Prediction
Table 3. Energy consumption of SpikF and iTTranffommer
Model
ACE/M
Essem/ pJ
Eoyts/ /J
EAddr pp
ETotal/ pJ
SpikF 0.39 16.27x
1.14 x 10a
2.77 10a
4.34 10
1.37 x 10 13.16x
iTransformer
2.47
3.22 x 10a
1.11 x 10a
3.22 x 10-1
4.34 x 104
In terms of ACE and Erotal, the energy consumption of SpikFis is 6.27x and 3.16x lower than i'Transformer respec- tively.
4.3. Model Generalizability
Short-term Prediction To demonstrate the versatility of SpikF, we extend our evaluation the Solar-energy dataset, with detailed characteristics presented in Appendix B.I, specifically focusing on short-sequence prediction. This assessment aims to validate the model's applicability across diverse temporal scales and forecasting paradigms.
Ablation Study To evaluate the contributions of the three modules proposed in Section 3, we conduct an ablation study. For the SPE, we explore alternative encoding mecha- nisms, including convolution encoding and delta encoding. For the SFS, we replace the SFS module with inverted spik- ing self-attention (iSSA) proposed in (Lv et al., 2024b), and we also examine the impact of completely discarding the SFS module. Additionally, for the decoder, we investigate the effect of removing the temporal synchronization mech- anism. The implementation settings are provided in Ap- pendix C.2. As demonstrated Table 4, the SPE achieves 6.6% and 3.2% accuracy improvement over the Delta En- coder and Convolutional Encoder respectively, due to their limited utilization of long-term history information, while the SFS shows 1.6% performance enhancement compared to iSSA, for limitations of self-attention mechanisms as discussed in Appendix 1. Additionally, temporal synchro- nization contributes 0.5% performance boost.
We select iTransformer (Liu et al., 2024) and iSpik- former (Lv et al.. 2024b) as benchmarks, as they are SOTA ANN-based and SNN-based models, respectively, indi- cated by (Lv et al., 2024b).
As demonstrated in Table 5, SpikF achieves lower predic- tion errors than both iSpikformer and i'Transformer, with reductions of 2.3% and 2.8% respectively, showcasing the strengthened predictive capabilities of SpikF in handling both long-term and short-term prediction tasks.
Table S. Short-term prediction results on the Solar-energy dataset. Following the experimental setup of (Lv et al., 2024b), the look- back window is fixed 128, and the prediction lengths set as // = (6. 21, 18, 96). The Root Relative Squared Error (RSE) is used as the evaluation metric compare the predictive perfor- mance of the models, where a lower RSE value indicates better performance. The performance results for iSpikformer and iTrrans
Parameter Sensitivity Analysis We evaluate the sensi- tivity of SpikF to temporal resolution T, by varying this parameter across five values 1, 2, 4, 8, and 16 while main- taining fixed look-back window of 96 and a prediction horizon of 720 across all datasets. As illustrated in Figure 4, SpikF maintains robust performance across temporal reso- lution variations, with an average variation of only 3.7%. This highlights its ability to deliver consistent results even under reduced temporal resolution. Furthermore, these perimental results offer approach for balancing accuracy and inference time, as lower temporal resolution typically corresponds to lower latency.
former are sourced from (Lv et al., 2024b).
Solar-energy
Model
6
24
48
96
SpikF
0.189
0.346
0.444
0.514
iSpikformer
0.204
0.333
0.465
0.521
iTransformer
0.191
0.348
0.448
0.563
Extension of Input Sequence It is widely believed that model performance should improve with longer input se- quences, as incorporating more historical information en- hances the model's ability to capture temporal patterns. Therefore, the ability to utilize longer historical informa- tion serves as an indicator of a model's generalizability and predictive performance. To explore this, we modify the look-back window across five temporal spans: 48, 96, 192, 336, and 720 time steps, while simultaneously adjusting the prediction length from 96, 192, 336, to 720 time steps. As shown in Figure 5, the loss values generally decrease as the input sequence length increases across all prediction lengths. Specifically, when the length of input sequence ex-
ETTh2 Weather Exchange
ETTh2 Weither Exchange
16
Figure 4. Performance of SpikF under varying temporal resolu- tions. Results show minimal fluctuations, indicating robustness to T, selection.
8

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SpikF: Spiking Fourier Network for Efficient Long-term Prediction
Table Ablation study results of the aforementioned strategies. The full ablation study results available in Appendix C.2.
ETTh1
ETTml
Weather
Module
Design
MSE
MAE
MSE
MAE
MSE
MAE
SpikF
original
0.440
0.428
0.388
0.385
0.245
0.265
Delta
0.468
0.450
0.431
0.424
0.260
0.281
Spike Encoder
Conv
0.467
0.446
0,404
0.397
0.249
0.272
w/o 0.441
0.428
0.388
0.387
0.246
0.266
Feature Extraction
iSSA
0.452
0.440
0.389
0.389
0.247
0.268
Spike Decoder
w/o
0.444
0.430
0.391
0.387
0.246
0.266
pands from 48 to 720, an averaged prediction promotion is 9.0% in terms of MSE and 2.4% in terms of MAE, demon- strating the model's effective utilization of longer history information.
Layar1
Layer2
1.0
23
08
50
Dineniiion
322
100
332
150
222
200
222
250
MSE
MAT
322
ho0a Fr0QUSTIR00
30
15
Frecuency
27
00
300
Kdy Frequonclos
10
13
Frequencyy
20
ETT12
Figure 6. Visualization of the SFS module across different layers. The data is obtained from the ECL dataset with look-back win- dow of 96 timesteps and a prediction horizon of 720 timesteps. The left panel shows the selection probabilities for the first SFS module, while the right panel corresponds to the second module. The noise frequency components and key frequency components is highlighted in this figure.
726
339
96
192
336
120
96
140
Prodicion Langhh
Lock Lock-back Window
96
152
Predetion Longhh
Window
USE
MAE
4.4
FTTm2
Lock-tacc
26
772
392 Prifition wigh
356
Look back
199
728
336
Prodicion
Lergh
5. Conclusion
Figure 5. MSE and MAE values for the ET (top) and ETTm2 (bottom) datasets across varying input and output lengths. The re- sults highlight by the consistent decrease in MSE and MAE values with the in- crease of look-back window.
In this study, we introduce SpikF, novel architecture de- signed for long-term prediction tasks, bridging the critical gap of SNN benchmarks in this domain. The Spiking Patch Encoder enables efficient encoding of extended historical information, while the Spiking Frequency Selector module inherently resolves the challenge of utilizing positional in- formation in SNN architectures and facilitates global feature extraction. This work not only significantly broadens the potential applications of SNN architectures but also offers an energy-efficient solution for the time-series prediction community.
model's generalization capability, evidenced
4.4. Visualization
To validate the feature extraction capability of the SFS mod- ule, we plot the averaged selection matrix M for different SFS layers on the ECL dataset, as shown in Figure 6.
In Figure 6, deeper SFS module tends to discard more fre- quency components, while shallower module focuses on selecting useful frequencies. This is because that the first SFS module enhances the predictive utilization of input spike trains by selecting relevant frequency modes. As result, the input to the second module contains less noise, leading to a higher tendency for the second SES module to retain more frequency components. This analysis highlights the necessity and effectiveness of the design of SFS.
Acknowledgements
This work is supported by National Natural Science Foun- dation of China (No. 62334014), and the authors gratefully acknowledge the insightful comments and constructive sug- gestions provided by the editor and the anonymous review- ers, which have substantially enhanced the clarity, rigor, and overall quality of this manuscript.

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SpikF: Spiking Fourier Network for Efficient Long-term Prediction
Impact Statement
donghao, L. and wang xue. ModemnTCN: A modern pure convolution structure for general time series analysis. In The Twelfth International Conference Learning Representationss 2024. URL https::////// httpp::///ppeeeviee.e net/ forum? d=vooMMerrXH..
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Fang, W., Chen, Y., Ding. J., Yu, Z., Masquelier, T., Chen, D., Huang, L., Zhou, H., Li, G., and Tian, Y. Spik- ingjelly: An open-source machine learning infrastruc- ture platform for spike-based intelligence. Science Ad- vances, 9(40):add1448) 2023. doi: 10.1 10.s126/s adi1480. URL httpt: /ecps//eee///sssseeeeeeeeceeeeee science org/ /oi/ abs/ /0ss/10.u
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SpikF: Spiking Fourier Network for Efficient Long-term Prediction
A. Theoretical Analysis
A.1. Permutation-Invarian of Self-attention and Inverted Self-attention Mechanisms
Consider an input sequence X E LxD,, where Xies,, denotes X with its ith and jth rows exchanged, and X X,,e denotes X with its ith and jth columns exchanged. For the standard self-attention mechanism, the queries Q, keys K, and values V computed as follows:
Q
Q:Q K V=XWy
(17)
When the rows of are permuted, the corresponding transformations yield:
X We =
Queji
X WK- = Kitto Xits, Wv = Vieje
(18)
Furthermore, the attention scores and output permuted accordingly:
Qu+t,, (QK") )143,1043
(19)
and
Qitzs,. Bitsj.,Teess Klngj VD
QK? VD
SoftMax
Viess. = SootMax
V setj.
(20)
For the inverted atffnnttention mechanism, where X E RXL, the formulas
X Lett)(WQiiee (Wolees.. =Q X ses (WK)iesj., X ( Wy)ies,
(21)
This indicates that permuting the input X along the temporal dimension leads to a corresponding permutation of the output attention map. As a result, both the standard self-attention mechanism and its inverted variant are unable to effectively capture positional information. In the absence of subsequent position-sensitive operations. the model's prediction output would remain invariant under the assumption that the optimization process guaranteed avoid local minima. Similarly, the spiking versions of self-attention and inverted self-attention exhibit insensitivity to the relative position of the input sequence due to the same limitation.
In contrast, our proposed Spiking Frequency Selector (SFS) module is inherently sensitive to the relative positions of input spikes, owing to the inclusion of the S-FFT operation. This property enables the SES module to better model temporal dependencies.
To experimentally evaluate the utilization of positional information, we uniformly shuffled the input sequence and compared the prediction accuracy with that of the ordered input sequence. As shown in Figure 7, SpikF achieves 0.8% improvement in MAE when sequential properties are incorporated, while i'Transfrmer shows no performance improvement, highlighting its ineffective utilization of sequential properties.
A.2. The Role of Fourier Transform in SNN Architectures
As (Lvv et al., 2024b) suggests, the dynamics of membrane potential in SNNs provide unique method for capturing temporal data intricacies. However, this can result in separated receptive field, potentially missing global temporal information.
Proof:
Given the dynamics of LIF neurons:
Ult V - +] (IiT)
V[t 1] + Vrest)
(22)
Tm
b1s H (U[) V1a.
(23)
bI l (1 S((t) Vrest S(t]
(24)
13

---

SpikF: Spiking Fourier Network for Efficient Long-term Prediction
Unordered
Ordered
Cirss Trum
0.2
0.-
2.0
1rout Seqpence
0.0
-22
-0.
1imn
MAE
SpliF
Groune Trast
0.0
Frx.. Sa2ponu
0.0
-.2
-0.
TimA
Timm
Figure 7. Comparison of model performance with unordered versus ordered input sequences.
For two series of stimulation [1] 1, [2), [t*) and [2[1]. _2., [2[+"] where S(1] = 5 Sit" - 1] = 0 and
S[(*] = 1, these sequences are equivalent in terms of membrane potential when 2 E", as -
- Vreat.
assume that Sit") S(() -..
and S(t) 0 otherwise. Then the receptive field of the LIF neuron is 1, 1º] and [t 1,7.
limited the regions [1, 1,t), [t
This limitation hinders high-prediction SNNs in long-term prediction domains, which require modeling long-term dependen- cies (Zeng et al., 2023). Thus, relying solely on SNN internal dynamics is insufficient; external dynamics are necessary for modeling long-term dependencies.
While typical methods to expand the receptive field of SNN involve linear layers and self-attention mechanisms, these are less suitable for sequential tasks due their permutation-invariann (Zeng al., 2023), which has been proved in Appendix A.I.
In contrast, FFT transforms temporal series into the frequency domain, expanding the receptive field to the entire time-series. Modifications in the frequency domain influence the entire series, and sequential information is inherently embedded in frequency components via FFT's rotation factors:
T Σ [ml
FlK]
SIyje 3p
(25)
Thus, selecting frequency components allows for global influence, as F[k] is function of S(1), S[T), thus making FFT an ideal approach for external dynamics.
In summary, incorporating FFT into SNN architecture is essential in the time-series domain to expand the receptive field and improve long-term dependency modeling.
A.3. Computational Complexity of S-FFT and S-iFFT Algorithm
While numerous optimizations have been proposed to enhance the computational efficiency of Fast Fourier Transform (FFT), we give a theoretical analysis based on the classical FFT algorithm and its spiking version, as shown in Algorithm 1, due to its fundamental nature and universal applicability.
14

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SpikF:
Spiking Fourier Network for Efficient Long-term Prediction
Algorithm 1 Fast Fourier Transform
Require: Input sequence x={] of length I. = x 2", where is not divisible by 2
Ensure: FFT output X [A]]
1: [Define a recursive procedure for FFT)
2: Procedure RecursiveFFT(z,n) m,
3: n = () then
4: (Base case: Directly compute DFT for segments length m)
X <- (FT m)
6: return X
7:
else
8:
(Divide the sequence into even and odd parts}
9: L + m x 2
10: Teven - 2 - 2]]
11: Fodd f | [3).. ...., I]]
12:
(Recursively compute FFT for even and odd parts)
13:
Xeven R m, 1)
14:
Xods t RecursiveFFT m, 1)
15:
(Combine the results using the Cooley-Tukey butterfly)
16: for + 0 to L/2 1 do
17:
+ W x Xata(k)
18:
Xík + Xeven [k] +
19: X L/2] X Xeven [k] -t
20: end for
21: return X
22: end if
23: {Main call to the recursive procedure)
24: X +- RecursiveFFT(r, m, n.)
25: return X
where W denotes the twiddle factor.
For S-FFT, we assume that each position of the input sequence fires randomly with a rate of a. This hypothesis remains valid despite the temporal dynamics of the model, as is supported by the reordering in the butterfly algorithm. In the following analysis, a complex addition is considered as 2 SOPs (addition of real and imaginary parts), and a complex multiplication considered as SOPs (4 multiplications and additions). The calculation formulas of SOPs are provided in Appendix C.I.
In the first stage, when the algorithm reaches the recursive base case S-FFT, the spiking components are aggregated perform traditional S-DFT transformation, which calculates the transformed sequence within the group. The number of SOPs in this stage is given by:
m1u 2x2" m
1) Pk (2ma -2 2,
(26)
where Ps represents the probability of exactly & neurons within the group spiking, and 3 1 denotes the non-spiking ratio.
In the second stage, when the odd and even sequences are used to compute the final S-FFT results, the following rules apply for each component:
If the corresponding odd S-FFT component equals zero, no calculation is performed.
If the odd component non-zero and the corresponding even component is zero, only complex multiplication is performed.
Otherwise, one complex multiplication and one complex addition are performed.
15

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SpikF: Spiking Fourier Network for Efficient Long-term Prediction
Thus, the number of SOPs in the second stage is:
2 d
m x 2 8 (1 - ) + 6.3mx2--1
(1- (1-gm gm x
(27)
Σ
L 8 1 - 1- pmxx)+
+ 6.3m224 (1-
Therefore, the total SOPs for S-FFT are:
n1 SOPs(S-FFT) Σ L (m11- 6,gm22" (1 (1- gmx 24
')] (2ma 2
(28)
For S-iFFT, the input sequence is real-valued, though many elements are zero due to the selection mechanism. Thus, the first stage, additional calculations involving the input sequence multiplied by the rotation factor are required. By adding this part, the SOPs for S-iFFT are:
(OPP(S-iFFF)-> = (1- gmx2t-11) + 6 (1 (1-
+ (2ma - 2 23"h) +6L x ma (29)
1=1
It is important to note that S-FFT and S-iFFT essentially transform real-number multiplication into inter-synaptic spike transitions and real-number addition into the accumulation of stimulation from different pre-synaptic neurons. Therefore, the above analysis on the SOPs of S-iFFT can be converted into FL.OPs of FFT and iFFT by setting a 1. Thus, the FLOPSs for FFT and iFFT operations are:
FLOPs(FFT) = (1FF) = L (8m + 8n 2)
(30)
A.4. The Role of Average Pooling
Under the assumption that the prediction errors independently follow the Laplace distribution, which is the theoretical foundation of the MAE loss function, we establish the following mathematical framework to analyze the contribution of the average pooling layer in the MLP Decoder. Let the prediction errors before average pooling be denoted as e1, ..,-.., er, where each eg is independently and identically distributed (i.id.) according to the Laplace distribution:
1 -11
e1, 02,62, - cT, iidd er [(x) 2
(31)
The characteristic function of the Laplace distribution is given by:
(t)) 1 + t
(32)
Consider the error after applying average pooling, defined as e = expressed as:
1
Ek=1
Cb. The characteristic function of e can be
T.
(t(T))
(33)
1 2 T
Using the characteristic function inversion method, the probability density function of e is derived as:
1 / exp( 1l(-ile itx))
fx(x)T,)
dt
(34)
2T _oc (1
T7
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SpikF: Spiking Fourier Network for Efficient Long-term Prediction
The expectation of the absolute error Ee [[x]]((T)) is given by:
exp(-itx) itx)
dt dx 2
[T, + 2) VT T(T, +
E. [T][(TT))
(35)
T.
00
1+
T,7
where 1((2) = -00000000 x- dt.
When T3 is incremented to T, + 1, the expectation of the absolute error decreases by a factor of T++ expressed as:
This relationship is
T,-1
T + 1 T +
(36)
E. (T.
It is important to note that as Ts approaches infinity, the expectation of the absolute error converges to zero:
(TT)) =
(37)
Above analysis implies that when Ta increases, the expectation of MAE will decrease because of the average pooling layer.
B. Experimental Details
B.1. Depiction of the Datasets
In our experimental evaluation, we utilize several well-established real-world datasets that have been extensively validated in prior time-series forecasting research. The Electricity dataset (Wu et al., 2022) records hourly interval electricity consumption data from 321 clients. The Weather dataset (Wu et al., 2022) includes meteorological measurements collected every 10 minutes from the Weather Station of the Max Planck Biogcochemistry Institute in 2020, containing 21 variables. The ETT dataset (Zhou al., 2021) consists of four subsets: ETTh and ETTh2 with hourly interval data, and ETTm1 and ETTm2 with 15-minute interval data, each containing two years (2016-2018 of measurements from electrical transformers, including variables like oil temperature and load. The Traffic dataset (Wu al., 2022) provides hourly road occupancy rates from 862 sensors in the San Francisco Bay area between 2015 and 2016. The Exchange dataset (Wu et al., 2022) comprises daily exchange rate data from eight countries spanning from 1990 to 2016. Solar-Energy (Lai et al., 2018) 8)), provides a dataset documenting the solar power generation of 137 photovoltaic plants throughout the year 2006, with measurements recorded at 10-minute intervals. All datasets are preprocessed to handle missing values and normalized to ensure consistent scaling across variables, following the approach of (Liu et al., 2024). These datasets represent diverse domains with varying temporal patterns, seasonality, and trends, making them ideal benchmarks for evaluating the robustness and generalizability of time-series forecasting models.
We fix the look-back window for all datasets except Solar-energy to 96, consistent with iTransformer (Liu et al., 2024) for long-term prediction tasks. The look-back window of Solar-energy is set to 128, following (Lv et al., 2024b). The train-validation-testt split method follow the iTrrnsformer (Liu et al., 2024) and iSpikformer (Lv et al., 2024b) approach to ensure fairness. Detailed dataset information is provided in Table
B.2. Metrics
For long-term forecasting tasks, the Mean Absolute Error (MAF) and Mean Squared Error (MSE) are employed as evaluation metries, defined as follows:
MAE = Y| Y||
(38)
MSE Y Y
(39)
where Y and represent the ground truth and predicted values, respectively.
For short-term forecasting tasks, the Root Relative Squared Error (RSE) is utilized to assess model performance by (Lv al., 2024b). The metric is defined as:
17

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SpikF: Spiking Fourier Network for Efficient Long-term Prediction
Table 6. Dataset descriptions.
The dataset size is organized in (Train, Validation, Test).
Task
Dataset
Dim
Sampling Frequency
Dataset Size
Depiction
Electricity
321
Hour
(18317, 2633, 5261)
Electricity
Weather
21
10 Minutes
(36792, 5271, 10540)
Weather
Long
ETTh
| 7
Hour
(8545, 2881, 2881)
Electricity
ETTm
7
15 Minutes
(34465, 11521, 11521)
Electricity
Traffic
862
Hour
(12185, 1757, 3509)
Transportation
Exchange
1 Day
(5120, 665, 1422)
Economy
Short Solar-energy
137
10 Minutes
(31362, 10506, 10506)
Energy
Ex B
Y; - Y.
If Ex=1 ش j=1 Ek=1 (Yis )2,,))
RSE
(40)
Y Y!
Σ プラ Ek (Yr x.,
where B denotes the number of samples, H represents the prediction length, and 1) corresponds to the number of dimensions.
B.3. Implementation Details
All experiments are built on the frameworks developed on Py Torch (Paszke al., 2019) and SpikingJelly (Fang et al., 2023), the latter being an SNN repository built upon PyTorch. SpikingJelly has been rigorously validated in prior studies, including Spikformer (Zhou et al., 2023) and QKFormer (Zhou et al., 2024). The experiments are executed on a single NVIDIA 3090 GPU.
The surrogate function for apatio-temponn backpropagation (STBP) is selected as the Sigmoid function, defined as:
(x)
(41)
1+e
where the parameter a is set to 4.0, consistent with previous studies (Zhou et al., 2023; 2024) across all experiments.
Figure 8 provides a visual comparison between the Heaviside step function and the Sigmoid function, along with the derivative of the Sigmoid function.
1.0
Sigmoid Function
Derivative of Sigmaid Function
Heaviside Stop Function
0.3
0.4
Value
9.4
0.2
0.
Figure 8. Comparison of the Heaviside step function and the Sigmoid function (left), and visualization of the derivative the Sigmoid function (right).
The parameters for the spiking neurons are predominantly time constant Tm fixed at 2.0 and the threshold potential VAa. at 1.0. An exception is made for the generator LIF neurons, where Vea was adjusted to 0.1 to improve passing rate of the selector. The Adam optimizer is employed with its parameters configured as follows: 3 0.9 and 832 = 0.999.
their default values within the SpikingJelly library, with the
The temporal resolution T, is set to 16 for datasets exhibiting minimal variables and for those with greater variables. The feature extraction layers of the architecture are chosen from the set {1, 2, 4). and the patch dimension are selected from
18

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SpikF: Spiking Fourier Network for Efficient Long-term Prediction
{8, 32, 64). The hidden layer dimension within the encoder's MLP chosen from (180, 360, 540, 720). Batch sizes selected from {4,8, 8, 16, 32, 64), and initial learning rates set from (10-1, 5 10-1, 10-30- The number of training epochs is chosen from (5, 10, 15, 20). An early-stopping strategy is implemented to optimize the use of the validation set for model evaluation. All model parameters are fine-tuned based on validation set performance. A comprehensive tabulation of parameter selections is provided Table 7.
Table 7. Detailed hyperparameter selection of long term prediction datascts.
Dataset
Tg
Layers
dpatch
dhiddenn
Batch Size
Ir
Epoches
Electricity
4
2
16
720
8
10-4
15
Weather
16
2
32
360
32
10-3
5
ETTh
16
32
720
32
5 10
5
ETTm
16
32
720
32
5 x 10-4
10
Traffic
4
16
540
4
5 x 10-4
15
Exchange
16
32
720
32
10-4
10
Solar-Energy
4
16
360
16
10-3
5
C. Model Analysis
C.1. Model Efficiency
For SNN models, the SOPs of each layer are given by:
SOPs = a x 7 x FLOPs
(42)
where a represents the firing rate of input spike trains, T is the simulation time steps of the SNN, and FLOPs denotes the floating-poin operations of the layer.
For the exact power cost of our SFS module, as detailed Table 2, we assume the operations are implemented on 45mm hardware (Yao al., 2023; Zhou et al., 2023), with an energy consumption 4.6 pJ per FLOP and 0.9 pl per SOP.
C.2. Ablation Study
To evaluate the contribution of each module in SpikF's design to its overall performance, we conduct detailed ablation studies. These experiments involve systematically replacing or removing individual components while keeping the rest of the model unchanged.
For spike encoder, choose Delta Encoder which can be formulated as:
H (X)
(43)
Sene (BZ)
(44)
where Diff represents the differential operation, and Convolutional Encoder:
= Conv (X)
(45)
Sene (N(2OH(BN(HH))
(46)
where Conv denotes the convolutional layer.
For feature extraction module, we consider removing the SFS and replacing it with inverted spiking self-attention mechanism which is proposed by (Ly et al., 2024b) which can be expressed as:
Q SBNNSSS We)): K = SNN(BN(SS Wk)), (BN(Seee Wy))
SN
(47)
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---

SpikF: Spiking Fourier Network for Efficient Long-term Prediction
Attn QK"V scaler
(48)
where scaler is pre-defined constant. For spike decoder module, we consider removing the temporal synchronization mechanism.
Full results of our ablation study are shown at Table 8.
Table 8. Comprchensive results of the ablation study.
Prediction Length
ETTh1
ETTml
Weather
Module
Design
MSE
MAE
MSE
MAE
MSE
MAE
96
0.379
0.391
0.317
0.345
0.163
0.200
192
0.432
0.421
0.372
0.372
0.209
0.241
SpikF
original
336
0.473
0.441
0.401
0.394
0.266
0.283
720
0.474
0.459
0.461
0.430
0.344
0.334
Avg
0.440
0.428
0.388
0.385
0.245
0.265
96
0.419
0.422
0.372
0.393
0.179
0.219
192
0.463
0.445
0.421
0.417
0.225
0.259
Delta
336
0.502
0.462
0.438
0.429
0,279
0.297
720
0.487
0.470
0.493
0.456
0.357
0.347
Spike Encoder
Avg
0.468
0.450
0.431
0.424
0.260
0.289
0.406
0.411
0.343
0.369
0.166
0.210
192
0.457
0.437
0.382
0.383
0.214
0.250
Conv
336
0.502
0.459
0.409
0.399
0.270
0.290
720
0.502
0.475
0.480
0.436
0.346
0.338
Avg
0.467
0.446
0.404
0.397
0.249
0.272
96
0.380
0.390
0.317
0.350
0.164
0.201
192
0.434
0.421
0.369
0.372
0.210
0.243
w/o
336
0.472
0.441
0.399
0.395
0.266
0.284
720
0.476
0.462
0.468
0.430
0.345
0.336
Feature Extraction
Avg
0.441
0.428
0.388
0.387
0.246
0.266
96
0.389
0.402
0.324
0.350
0.166
0.203
192
0.436
0.428
0.366
0.374
0.213
0.245
iSSA
336
0.488
0.454
0.400
0.396
0.267
0.287
720
0.495
0.476
0.465
0.435
0.344
0.338
Avg
0.452
0.440
0.389
0.389
0.247
0.268
96
0.383
0.391
0.319
0.346
0.163
0.200
192
0.435
0.422
0.374
0.373
0.211
0.243
Spike
w/o
336
0.483
0.445
0.407
0.398
0.265
0.283
Decoder
720
0.478
0.461
0.466
0.433
0.346
0.336
Avg
0.444
0.430
0.391
0.387
0.246
0.266
C.3. Sensitivity to Patch Dimension
To evaluate SpikF's sensitivity to patch dimension selection, conduct experiments with varying patch dimension configurations (4, 8, 16, 32} across the ETTh1, ETTml, and ETTm2 datasets. Using a fixed look-back window of 96 time steps and prediction horizon of 720 time steps, our analysis shows minimal performance fluctuations across different patch dimension configurations, with an average variation of 1.2%. Notably, the ETTm2 dataset demonstrates exceptional stability, exhibiting only 0.5% variation. This consistent behavior demonstrates SpikF's robust performance relative to patch dimension selection in the encoding process.
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SpikF: Spiking Fourier Network for Efficient Long-term Prediction
5/0
6.0
0.2
0.4
ETTh1
ETTh2
MAE
ETTi2
ETTm1
ETTml
0.2
0.42
0,4
0.43
12
Patch Dimansion
Patch Dimension
Figure 9. Sensitivity to patch dimension selection.
C.4. Error Bars
To assess the robustness of our proposed model, we perform three independent experimental trials with distinet random seeds and calculate the corresponding 95% confidence intervals, as detailed in Table 9 and Table 10. is worth noting that SpikF optimized using the MAE loss function, in contrast to the widely adopted MSE loss function for other benchmark models. Therefore, to ensure a more fair comparison, we also include the 95% confidence intervals for the iTransformer model using the Mean Absolute Error (MAE) loss function. As evidenced in Table 9 and Table 10, SpikF not only outperforms iTraasformer in prediction accuracy but also demonstrates significantly improved robustness. Specifically, the ETTm2, Weather, and Exchange datasets, SpikF achieves an average confidence interval length which is 85.1% shorter than that of iTransformer.
Table 95% confidence intervals of SpikF iTrrnsffrmee on ETThl, ETTh2, and ETTml datasets with MAE loss function. Better performance is highlighted in bold.
Model
ETThl
ETTh2
ETTml
Metric
MSE
MAE
MSE
MAE
MSE
MAE
96
0.009
0.091
0.290 0.001
0.336+20.003
0.317=0.005
0.045
192
0.402:10.008
0.42110.002
0.367
0.385110.011
0.302-10.007
0.372:10.005
SpikF
336
0.000
0,441:10.002
0.014
0.400:10.007
0.402
0.394110.004
720
0.474
0.459
0.416=00.008
0.436=10.001
0.461
0.430+10.004
Avg
0.440-10.005
0.0001
0.372:L0.044
0.394
0.388
0.385
96
0.383-:0.005
0.396:10.003
0.2941:0.001
0.342-10.000
0.324=10.001
0,351
192
0.003
0.428-10.002
0.376:10.006
0.392-+0.004
0.374= 0.002
0.376:100002
iTransformer
336
0.001
0.450=
0.000
0.407
0.401
0.401
720
0.403
0.477
0.003
0.439=00.002
0.480=10.002
0.441
Avg
0.008
0.438-10.001
0.379:10.003
0.400:10.002
0.397
0.393:10.001
D. Visualization
D.I. Selection Matrix Visualization
In this section, we provide more visualizations of the SFS module to fully validate its ability of temporal feature extraction.
Different Prediction Length As illustrated in Figure 10, the SFS module effectively captures the key frequency com- ponents of the spike series. As illustrated in Figure 10, the frequency characteristics of the input sequence with different prediction length are consistent: the 0 frequency component is typically discarded after selection, while frequencies that are multiples of 4 play crucial role in predicting the final target. These observations align with the findings of FilterNet (Yi et al., 2024), an ANN-based frequency filter model, confirming that our spiking selection mechanism successfully extracts important frequencies from the Fourier spectrum. Notably, the selection rate gradually decreases as the prediction length increases from 96 to 720 timesteps, as longer prediction horizons exhibit weaker correlations with the look-back 96 timesteps,
21

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SpikF: Spiking Fourier Network for Efficient Long-term Prediction
Table 10. 95% confidence intervals of SpikF and iTransformer on FTTm2, Weather and Exchange datasets with MAE loss function.
Better performance highlighted in bold.
Model
ETTm2
Weather
Exchange
Metric
MSE
MAE
MSE
MAE
MSE
MAE
96
0.005
0.251=1.0001
0.163
0.200=1.0001
0.084:10.001
0.001
192
0.242-10.002
0.206
0.209
0.0002
0.180:10.001
0.30011.000
SpikF
336
0.002=00002
0.306=10.022
0.266=00.002
0.283
0.334=:0.020
0.417=00.013
720
0.405
0.397
0.344+0.002
0.334::0.002
0.841
0.690=0.008
Avg
0.201=00002
0.300
0.245
0.265
0.360:10.002
0.40
96
0.102
0.265:10.024
0.176:10.004
0.216:40.006
0.088:10.003
0.208
192
0.011
0.009
0.204
0.257
0.179=00000
0.303
iTransformer
336
0.013
0.348+00022
0.201
0.20
0.337:10.003
0.421
720
0.001
0.403
0.009
0.350
0.851
0.698
Avg
0.209
0.31110.018
0.260=10.002
0.280
0.364110.008
0.407
thereby diminishing the influence of the original spike trains.
335
720
100
0.
Dimensoo
0.8
0.6
0.7
02
02
10
70
10
Frequoncy
Froquency
Froquency
Frequonay
Figure Visualization of the Spiking Selector mechanism across varying prediction horizons. The heatmap illustrates selection probabilities. The analysis is conducted on the ETThl test set with fixed look back window of 96 timesteps and multiple prediction horizons (96, 192, 336, and 720 timesteps). The horizontal axis represents frequency components, while vertical axis denotes the multidimensional features of the dataset.
Shared and Individual Spiking Selector Generator As illustrated in Figure 11, the individual spiking generator extracts key frequency components for each dimension the ETTh1 dataset. In comparison, the common generator strategy successfully identifies and highlights the shared significant frequency components across all dimensions, as the components selected by the common generator tend to overlap with the highlighted components of each individual dimension. By balancing the contributions of these components, the shared generator strategy achieves superior performance compared the individual generator. This analysis is further supported by the superior performance through weight sharing strategy of previous works (Xu et al., 2024; Yi et al., 2024).
D.2. Visualization of Prediction Results of Different Models
To provide intuitional comparison among different models, present visualized prediction results on ETThl datasets in Figure 12. The evaluated models include i'Transformer (Liu et al., 2024), DI -ineerr (Zeng et al., 2023), and our proposed SpikF. The results clearly demonstrate that SpikF achieves significantly more accurate prediction compared to iTransformer and DLinear.
D.3. Visualization of Encoded Spikes
This section presents visualization of the encoded spikes across multiple datasets. As depicted in Figure 13, the spike trains generated by the SPE module exhibit a strong alignment with the input sequence in terms of density. This alignment underscores the interpretability of the spike trains and highlights their effectiveness in capturing the essential features the input sequence. Furthermore, the sparsity of the spike trains suggests that the energy consumption for subsequent processing
22

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SpikF: Spiking Fourier Network for Efficient Long-term Prediction
Our Generator
Indid.al Generator
08
0.
Dirensron
Freqqeency
Frequenzyy
Figure f. Comparison of shared and individual generator strategies. The frequency components selected by the shared generator framed by dashed lines, along with the corresponding MSE MAE provided the bottom of the figure. The look-hack window and prediction horizon are set as 96.
Spiks
iTranformer
DLinear
1000
yexr Seguence
Viuut Sngunnce
Ioor Segaancc
-18
Grann Tram
-18
Grandt
-18
-10
-10
-10
-12
-12
-12
-1.4
+3.4
+1.4
-1.8
-16
-1.6
-1.
1100
325
330
3.5
200
1000
1350
3.75
200
Thme
Figure 12. Visualization of prediction results on the ETTh] dataset, comparing SpikF with other benchmark models under look- back window 96 and a prediction horizon 96.
will be minimal, rendering the approach highly efficient.
For these datasets, the prediction horizon is set to 96. The input sequence has been preprocessed through instance normalization (Kim al., 2021). enhance clarity, averaged spike trains have been resized to match the mean and variance of the absolute input sequence. This preprocessing step ensures a more intuitional comparison of the trends between averaged spike trains and input sequences.
E. Full Results
E.1. Long-term Prediction
This section presents a comprehensive evaluation of long-term forecasting performance across eight benchmark datasets: ECL, Weather, ETT (ETTh1, ETTh2, ETTml, ETTm2), Traffic, and Exchange. The detailed performance metrics are organized Table 11, which offers comparative analysis of the predictive capabilities of various models. Notably, on ETThl dataset, SpikF achieves the lowest MSE and MAE across all prediction lengths.
The results of all benchmarks except SpikF are extracted from (Liu et al., 2024).
E.2. Extended Benchmark Analysis ETT Datasets
Given SpikF's remarkable performance on ETT datasets, we expand our evaluation by incorporating additional FFT-based ANN and SNN-based benchmarks to comprehensively assess SpikF's forecasting capabilities.
For FFT-based ANN models, we select FEDformer (Zhou et al., 2022), which employs frequency domain self-attention mechanism to enhance feature extraction, and FITS (Xu et al., 2024), which utilizes a frequency domain complex linear layer for input sequence upsampling. These models represent advanced approaches in frequency domain analysis. On the SNN side, we include Spiking Recurrent Neural Network (SpikeRNN) (Kim et al., 2019) and Inverted Spiking Transformer (iSpikformer) (Lyv al., 2024b) as benchmarks, as their ANN counterparts are widely recognized for their effectiveness in
23

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SpikF: Spiking Fourier Network for Efficient Long-term Prediction
long-term prediction tasks. SNN-based models equipped with SPE. Results of FITS are derived from (Yi al., 2024) and results of FEDffrmer are sourced from (Liu al., 2024).
The results, presented in Table 12, demonstrate that SpikF achieves precision improvement of 1.4% and 0.7% in terms of MSE and MAE respectively. compared to the second-best SNN-based model. Furthermore, MSE and MAE of SpikF are reduced by 3.4% and 3.6%, relative to the second-best FFT-based model.
24

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SpikF: Spiking Fourier Network for Efficient Long-term Prediction
ECL
Tolffc
Absolltee Input Sequence
Impur Sequence
Avrrraged Spice Trains
3.5
1,3
Volue
Abooluse input Sequence Input Seduenge
-1.
-13
Averaged hprker Thairs
IIII
BII
111
111
111
BI
1001
1100
1III
1111
Spiking
Weather
Exchange
Absolure Imput Sequence
Impui Sequente
Ave Averaged Spike Trains
-25
Input Sequence
-10
Averoged Spkc Tains
Figure 13. Visualization of encoded spike trains alongside the input sequence for the ECL, Traffic, Weather and Exchange datasets.
The sparse yet interpretable representation of the spike trains underscores the module's capability to extract meaningful features while maintaining low energy consumption.
25

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SpikF: Spiking Fourier Network for Efficient Long-term Prediction
Table 11. Full results of long-term -erettermmm prediction.
Model
SpikF
|
RLinear
PatchTST
|Crossformer] TimesNet
DLinear
SCINet
AAtoffomer
Metric
MSE MAE
MSE
MAE
MSE MSE MAE
MSE
MAE
MSEMAE) MSE MAE
MSE MSE MAE
96
0.156
0.252
0.148
0.240
0.201 0.281
0.181 0.270
0.219
0.314
0.168 0.272
0.197 0.282 0.247 0.345
0.201 0.317
192
0.169
0.262
0.162
0.253
0.201
0.283
0.188 0.274
0.231
0.322
0.184
0.289
0.196 0.285 0.257 0.355
0.222
0.334
336
0.188
0.281
0.178
0.269
0.215
0.298
0.204
0.293
0.246
0.337
0.198
0.300
0.209
0.301 0.269 0.369
0.231
0.338
720
0.219
0.306
0.225
0.317
0.257
0.331
0.246
0.324
0.280
0.363
0.220
0.320
0.245
0.333 0.299 0.390
0.254
0.361
Avv
0.183
0.275
0.178
0.270
002219
0.298
0..00
0.290
0.244
0.334
|o.192
0.295
|.2122
0,300 0.268 0.365
|0.277
0.338
96
0.163
0.200
0.174
0.214
0.192
0.232
0.177
0.218
0.158
0.230
0..12
0.220
0.196
0.255 0.221 0.306
0.266
0.336
Weather
192
0.209
0.241
0.221
0.254
0.240
0.271
0.225
0.259
0.206
0.277
0.219
0.261
0.237
0.296 0.261 0.340
0.307
0.367
336
0.266
0.283
0.278
0.296
0.292
0.307
0.278
0.297
0.272
0.335
0.280
0.306
0.283
0.335 0.309
0.378
0.359
0.395
720
0.344
0.334
0.358
0.347
0.364
0.353
0.354
0.348
0.398
0.418
0.365
0.359
0.345
0.381 0.377 0.427
0.419
0.428
|Av
0.245
0.265
0.258
0.278
00.272
0.291
0.255
0.281
0.259
0.315
|0.259
0.287
0.266
0.317
0.292 0.363
0.,38
0.382
96
0.379
0.391
0.386
0.405
0.386
0.395
0.414
0.419
0.423
0.448
0.384
0.402
0.386
0,400
0.654 0.599
0.449
0.459
192
0.432
0.421
0.441
0.436
0.437
0.424
0.460
0.445
0.471
0.474
0.436
0.429
0.437
0.432 0.719 0.631
0.500
0.482
336
0.473
0.441
0.487
0.458
0.479
0.446
0.501
0.466
0.570
0.546
0.491
0.469
0.481
0.459
0.778 0.659
0.521
0.496
720
0.474
0.459
0.503
0.491
0.481
0.470
0.500
0.488
0.653
0.621
0.521
0.500
0.519
0,516 0.836 0.699
0.514
0.512
Avg
0.440
0.428
0.454
0.447
0..446
0.434
0.449
0.454
0.529
0.522
|0.458
0.450
0.456
0.452
0.747
0.647
0.496
0.487
96
0.290
0.336
0.2997
0.349
0.288
0.338
0.302
0.348
0.745
0.584
|0.340
0.374
0.333
0.387
0.707
0.621
0.346
0.388
hh2
192
0.367
0.385
0.380
0.400
0.374
0.390
0.388
0.400
0.877
0.656
0.402
0.414
0.477
0.476
0.860
0.689
0.456
0.452
336
0.414
0.420
0.428
0.432
0.415
0.426
0.426
0.433
1.043
0.731
0.452
0.452
0.594
0.541
1.000
0.744
0.482
0.486
720
0.416
0.436
0.427
0.445
0.420
0.440
0.431
0.446
1.104
0.763
0.468 0.462
0.831
0.657 1.249
0.838
0.515
0.511
AAg
0.372
0.394
0.383
0.407
0.374
0.398
0.387
0.407
0..922
0.684
|o.414
0.427
0.559
0.515
0.954
0.723
|0.450
0.459
96
0.317
0.345
0.334
0.368
0.355
0.376
0.329
0.367
0.404
0.426
0.338
0.375
0.345
0.372 0.418
0.438
0.505
0.475
192
0.372
0.372
0.377
0.391
0.391
0.392
0.367
0.385
0.450
0.451
0.374
0.387
0.380
0.389 0.439 0.450
0.553
0.496
336
0.401
0.394
0.426
0.420
0.424
0.415
0.399
0.410
0.532
0.515
0.410
0.411
0.413
0.413
0.490 0.485
0.621
0.537
720
0.461
0.430
0.491
0.459
0.487
0.450
0.454
0.439
0.666
0.589
0.478
0.450
0.474
0.453 0,595 0.550
0.671
0.561
Avg
0.388
0.385
|0.407
0.410
(0.414
0.407
0.387
0.400
0.513
0.496
|0,400
0.406
|0.403
0.407
0.485
0.481
|0.588
0.517
96
0.175
0.251
0.0
0.264
0..882
0.265
0.175
0.259
0.287
0.366
0..877
0.267
0.193
0.292 0.286
0.274
0.255
0.339
192
0.242
0.296
0.250
0.309
0.246
0.304
0.241
0.302
0.414
0.492
0.249
0.309
0.284
0.362 0.399 0.445
0.281
0.340
336
0.302
0.336
0,311
0.348
0.307
0.342
0.305
0.343
0.597
0,542
0.321
0.351
0.369
0.427 0.637
0,591
0.339
0.372
720
0.405
0.397
0.412
0.407
0.407
0.398
0.402
0.400
1.730
1.042
0.408
0.403
0.554
0.522 0.960 0.735
0.433
0.432
Avg
0.281
0.320
0.288
0.332
|0.286
0.327
0.281
0.326
0.757
0.610
|0.291
0.333
0.350
0.401
0.571 0.537
0.327
0.371
96
0.477
0.286
0.395
0.268
00.649
0.389
0.462
0.295
0.522
0.290
00.593
0.321
0.650
0.396 0.788 0.499
0.613
0.388
Traffic
192
0.481
0.289
0.417
0.276
0.601
0.366
0.466
0.296
0.530
0.293
0.617
0.336
0.598
0.370 0.789 0.505
0.616
0.382
336
0.499
0.295
0.433
0.283
0.609
0.369
0.482
0.304
0.558
0.305
0.629
0.336
0.605
0.373 0.797 0.508
0.622
0.337
720
0.533
0.312
0.467
0.302
0.647
0.387
0.514
0.322
0.589
0.328
0.640
0.350
0.645
0.394 0.841 0.523
0.660
0.408
Avg
0.497
0.296
0..428
0.282
|0.626
0.378
0.481
0.304
0.550
0.304
|0.620
0.336
00.255
0.383
0.804
0.509
|0.628
0.379
96
0.084
0.201
0.086
0.206
0.093
0.217
0.088
0.205
0.256
0.367
0..07
0.234
0.088
0.218
0.267
0.396
0.197
0.323
Exchange
192
0.180
0.300
0.177
0.299
0.184
0.307
0.176
0.299
0.470
0.509
0.226
0.344
0.176
0,315
0.351
0.459
0.300
0.369
336
0.334
0.417
0.331
0.417
0.351
0.432
0.301
0.397
1.268
0.883
0.367
0.448
0.313
0.427
1.324
0.853
0.509
0.524
720
0.841
0.690
0.847
0.691
0.886
0.714
0.901
0.714
1.767
1.068
0.964
0.746
0.839
0.695
1.058
0.797
1.447
0.941
|Ag
0.360
0.402
0.360
0.403
|0.378
0.417
0.367
0.404
0.940
0.707
|0.416
0.443
0.354
0.414
0.750
0.626
00.633 0.539
26

---

SpikF: Spiking Fourier Network for Efficient Long-term Prediction
Table 12. More benchmarks on ETT dataset.
Model
SpikF
JSpikeRNNNiSpiiffomm
FITS
FEDformer
Metric
MSE MAE
MSE MAE MSE MAE
MSE MAE
MSE MAE
96
0.379
0.391
0.392
0.395
0.410
0.408
0.386
0.396
0.376 0.419
192
0.432
0.421
0.437
0.424
0.459
0.438
0.436
0.423
0.420
0.448
336
0.473
0.441
0.482
0.447
0.514
0.461
0.478
0.444
0.459
0.465
720
0.459 0.474
0.499
0.471
0.511
0.476
0.502
0.495
0.506
0.507
Avg
0.440
0.428
0.452
0.434
0.473
0.446
0.451
0.440
0.440
0.460
96
0.290
0.336
0.295
0.335
0.308
0.349
0.295
0.350
0.358
0.397
ETTh2
192
0.367
0.385
0.375
0.387
0.382
0.397
0.381
0.396
0.429
0.439
336
0.414
0.420
0.422
0.422
0.439
0.433
0.426
0.438
0.496
0.487
720
0.416
0.436
0.428
0.438
0.438
0.439
0.431
0.446
0.463
0.474
Avg
0.372
0.394
0.380
0.396
0.392
0.407
0.383
0.408
0.437
0.449
96
0.317
0.345
0.326
0.351
0.337
0.361
0.355
0.375
0.379
0.419
192
0.372
0.372
0.376
0.377
0.380
0.383
0.392
0.393
0.426
0.441
336
0.401
0.394
0.396
0.394
0.415
0.406
0.424
0.414
0.445
0.459
720
0.461
0.430
0.459
0.433
0.476
0.443
0.487
0.449
0.543
0.490
Avg
0.388
0.385
0.389
0.389
0.402
0.398
0.408 0.415
0.448
0.452
96
0.175
0.251
0.176
0.251
0.178
0.256
0.183
0.266
0.203
0.287
m2
192
0.242
0.296
0.243
0.296
0.243
0.298
0.247
0.305
0.269
0.328
336
0.302
0.336
0.302
0.335
0.304
0.338
0.307
0.342
0.325
0.366
720
0.397 0.405
0.409
0.396
0.398 0.406
0.407
0.399
0.421 0.415
Avg
0.281
0.320
0.283
0.319
0.283
0.322
0.286
0.328
0.305 0.349
27


Reviews:
[{"review_id": "cIiX2vAhIH", "reviewer": "Reviewer_Prww", "summary": "The authors propose **SpikF**, a novel SNN-based architecture designed for long-term prediction tasks. Technically, the **spiking patch encoder** is introduced to efficiently convert sub-series into spikes with low computational complexity. Additionally, a **spiking frequency selection mechanism** is implemented to identify and retain core components, thereby enhancing overall performance. Experimental results demonstrate that SpikF outperforms state-of-the-art (SOTA) methods across eight long-term prediction tasks and exhibits exceptional suitability for deployment on edge devices. This work highlights the potential of SNNs in handling complex temporal tasks while maintaining computational efficiency, making it a promising solution for real-world applications.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate your careful review and the insightful questions you have raised. We provide a detailed point-by-point response to each of your valuable comments to ensure clarity and address all aspects thoroughly.\n\n>**Q1:** Lack of Inference Time. ... a detailed analysis and discussion of inference time comparisons would greatly enhance the practical relevance of the work.\n\nWe totally agree that including inference time of SpikF will increase the practical relevance of our work. We utilize the data in [1] to estimate the inference time of SpikF and iTransformer, which are $2.47ms$ and $7.87ms$ respectively. This indicates that the inference time of SpikF is $3.19\\times$ less than iTransformer.\n\n[1] Sumit Bam Shrestha et al., \"Efficient Video and Audio Processing with Loihi 2,\" IEEE ICASSP 2024.\n\n>**Q2:** Improvements in Writing. ... Additionally, ensuring consistent font sizes and minimizing unnecessary blank spaces would enhance the overall quality of the paper.\n\nWe have rearranged the layout and adjusted the font sizes of the figures to ensure consistency. Furthermore, we have reviewed our writing to ensure clarity of expression. The improved figures can be found at [Anonymous Link](https://anonymous.4open.science/r/CE55/).\n\n>**Q3:** Why are SNNs more advantageous than ANNs in temporal prediction tasks?\n\nThe temporal dynamics of the membrane potential in SNNs enables the processing of complex time-series data while maintaining energy efficiency [2]. Specifically, SNNs have two advantages over ANNs in temporal prediction tasks [2-5]:\n\n1. Energy efficiency. For information is transformed through binary spikes in SNNs, SNNs empirically consume less energy than their ANN counterparts, making them suitable for edge applications in resource-constrained environments [2]. Additionally, the event-driven nature of SNNs is capable of utilizing the dynamic changes in time-series data in real-world scenarios [3] [4], which further promote energy efficiency of SNNs in real-world applications.\n\n2. The subtle dynamics of the membrane potential naturally incorporate the time dimension into the model architecture. According to equations (1) to (4), these dynamics are non-linear and can efficiently process complex time-series data, whereas such a mechanism is absent in traditional ANN architectures [5].\n\n[2] Roy, K. et al. Towards spike-based machine intelligence with neuromorphic computing. Nature.\n\n[3] Liu, S.-C. et al. Neuromorphic sensory systems. Current Opinion in Neurobiology.\n\n[4] Vanarse, A. et al. A review of current neuromorphic approaches for vision, auditory, and olfactory sensors. Frontiers in Neuroscience.\n\n[5] Lv, C. et al. Efficient and Effective Time-Series Forecasting with Spiking Neural Networks. ICML 2024.\n\n>**Q4:** While the power efficiency advantage is clear, could the authors provide some examples of future edge device applications?\n\nEdge devices have high requirements for low power consumption to extend standby time. As SpikF has the power efficiency advantage, it is suitable for the edge applications, according to [6-8]:\n\n1. Industrial sensors [6]. According to [6], SpikF can be deployed in industrial sensors to autonomously monitor safety accidents.\n\n2. Wearable healthcare devices [7]. According to [7], SpikF can be used in healthcare devices to predict diseases and accelerate the diagnostic procedure.\n\n3. Automation systems [8]. [8] shows the possibility of SpikF being adopted in automation systems. For example, SpikF can be used to process the sequential data collected by gas and pressure sensors, thus facilitating the control of robots.\n\n[6] Zhou, Y. et al. Computational event-driven vision sensors for in-sensor spiking neural networks. Nature Electronics.\n\n[7] Maji, P. et al. SNN Based Neuromorphic Computing Towards Healthcare Applications. IFIP Advances in Information and Communication Technology.\n\n[8] Jiang, X. et al. Fully Spiking Neural Network for Legged Robots. IEEE ICASSP 2025.\n\n>**Q5:** A more tailored energy analysis specific to SNNs, such as spike-based operations, would strengthen the paper's contributions.\n\nIn our manuscript, we use AC and MAC [9] to evaluate ANN-based models and synaptic operations [10] for SNN-based models. Both metrics have been widely acknowledged, ensuring a fair comparison between ANNs and SNNs. Fourthermore, for a more comprehensive analysis,, we adopt methods in [11] and [12], and find that SpikF is $6.27\\times$ and $3.16\\times$ more efficient than iTransformer, in terms of ACE and $E_{Total}$ respectively ([Table 1](https://anonymous.4open.science/r/D401/)).\n\n[9] Zhijian Xu et al. FITS: Modeling Time Series with 10k Parameters. ICLR 2024.\n\n[10] Zhou et al. Spikformer: When Spiking Neural Network Meets Transformer, ICLR 2023.\n\n[11] Shen G, et al. Are Conventional SNNs Really Efficient? A Perspective from Network Quantization. CVPR.\n\n[12] Lemaire E, et al. An analytical estimation of spiking neural networks energy efficiency. NeurIPS.", "claims_and_evidence": "Yes, the claims are very clear.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes, I have checked the correctness of all proofs.", "experimental_designs_or_analyses": "The effectiveness of the proposed method has been extensively validated through numerous experiments.", "relation_to_broader_scientific_literature": "No", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "Strengths:\n\n1. SNNs for long-term time-series prediction is very novel and interesting topic.\n\n2. The writing is very easy to understand and straightforward.\n\n3. Eight real-world long-term benchmark datasets are conducted to verify the effectiveness of the proposed SpikF.\n\nWeaknesses:\n\n1. Energy Efficiency Analysis. While the authors reference previous methods to analyze the energy efficiency of SNN algorithms, the use of AC (Accumulate) or MAC (Multiply-Accumulate) operations for power consumption calculations may not be entirely convincing for SNNs. Given the critical importance of this metric, the rationale behind this approach should be thoroughly justified. The authors are encouraged to provide their insights or at least discuss this limitation in detail, as it significantly impacts the credibility of the energy efficiency claims. A more tailored energy analysis specific to SNNs, such as spike-based operations, would strengthen the paper's contributions.\n\n2. Lack of Inference Time. Although the authors compare computational complexity across different methods in Fig. 2, they should also provide a comparison of inference times. Since both contributions of the paper (i.e., the spiking patch encoder and the spiking frequency selection mechanism) are directly related to algorithmic inference efficiency, a detailed analysis and discussion of inference time comparisons would greatly enhance the practical relevance of the work.\n\n3. Improvements in Writing. The writing and presentation of the paper could be further refined. For example, there is significant blank space around Equation (13), and the font sizes in the figures are inconsistent. It is recommended that the authors optimize the manuscript's layout and formatting in future revisions to improve readability and professionalism. Additionally, ensuring consistent font sizes and minimizing unnecessary blank spaces would enhance the overall quality of the paper.", "other_comments_or_suggestions": "No", "questions_for_authors": "Please see the weaknesses and respond to each comment. Besides, two questions are listed below:\n\nWhy are SNNs more advantageous than ANNs in temporal prediction tasks? While the power efficiency advantage is clear, could the authors provide some examples of future edge device applications?\n\nBased on the above comments, I am currently leaning toward a weak reject, or perhaps borderline. However, I will also consider the other reviewers' opinions and the authors' responses. If the authors provide satisfactory answers, I will likely raise my score.", "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": "JzkGiMaXNA", "reviewer": "Reviewer_PTJc", "summary": "This work addresses two key challenges in applying Spiking Neural Networks (SNNs) and Transformer architectures to long-term forecasting: (1) capturing long-range dependencies, which increases computational and energy costs, and (2) the lack of effective positional encoding for Spiking Transformers. To address these issues, the authors propose SpikF, a novel architecture that introduces a spiking version of Patch Embedding to divide inputs into patches and replaces self-attention with a Spiking Frequency Selection (SFS) mechanism to model dependencies.\nExperiments show that SpikF reduces error by 1.9% compared to state-of-the-art models. Additionally, the authors analyze energy efficiency by comparing Synaptic Operations (SOPs) in SpikF with the FLOPs of conventional models, demonstrating its superior energy efficiency. According to the authors, SpikF is the first SNN-based benchmark providing comprehensive evaluation across long-term forecasting datasets.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We deeply appreciate your thorough review and insightful feedback. We will address your questions one by one in the following sections.\n\n>**Q1:** Figure 2 lacks clarity regarding the meaning of the firing rate ($\\alpha$) ...\n\n$\\alpha$ denotes the firing rate of SNN and can be formulated by:\n\n$$\n\\alpha = \\frac{\\sum_{k=1}^{T_s}s_k}{T_sn}\n$$\n\nwhere $s_k$ represents the number of spikes released at time step $k$, and $n$ denotes the number of neurons.\n\n>**Q2:** The values of $\\alpha$ used in the experiments (Table 1) are unspecified, and it is unclear how performance depends on $\\alpha$.\n\nThe values of $\\alpha$ can be found in [Table 1](https://anonymous.4open.science/r/58FD/).\n\nWe vary the choice of $T_s$ to measure the impact of $\\alpha$ which is commonly used by previous works [1], and find that when $\\alpha\\approx0.2$ SpikF achieves the best performance ([Table 2](https://anonymous.4open.science/r/58FD/)).\n\n[1] Yao et al., “Attention Spiking Neural Networks,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 45, no. 8, pp. 9393-9410, 2023.\n\n>**Q3:** However, the role of frequency-based methods in time-series forecasting is not well-discussed.\n\nWe extend our discussion about frequency-based methods.\n\n**FITS**, **FreTS**, **FEDformer** and **FilterNet** employ a complex linear layer, MLPs, sparse attention and filters respectively to enhance the utilization of high-frequency information. In contrast, SpikF enhances high-frequency focus via patching and grouping while using FFT for low-frequency analysis, achieving comprehensive spectrum efficiency.\n\n>**Q4:** It is unclear whether Transformers with a sparsely active layer are a proper avenue for truly energy-efficient models.\n\nWe compare SpikF (1.08M) with FEDformer (20.68M), which is a sparse transformer with frequency domain analysis. SpikF outperforms FEDformer with a $19.15\\times$ smaller model size and $15.4\\\\%$ better accuracy, demonstrating SNN’s superiority in energy-efficient forecasting ([Table 3](https://anonymous.4open.science/r/58FD/)).\n\n>**Q5:** The impact of using patches is not discussed.\n\nThe use of patches enhances the utilization of local information and long history information. The experimental results show that patch mechanism reduces MSE by $2.6\\\\%$ ([Table 4](https://anonymous.4open.science/r/58FD/)).\n\n>**Q6:** The implementation details of the patch encoding layer are unclear.\n\nWe modify some of the equations and narratives to better describe the pipeline of the Spiking Pacth Encoder (SPE):\n\nThe input sequence $x^{1:L}$ is first divided into patches:\n\n$$p^k=x^{\\frac{L}{N}(k-1)+1:\\frac{L}{N}k}$$\n\nwhere $N$ is the number of patches.\n\nThen each patch is processed by a spiking linear layer:\n\n$$S_{enc}^{T_s(k-1)+1:T_sk}=\\mathcal{SN}(\\text{BN}(\\text{LN}(p^k)))$$\n\nThe SPE serves the role of utilizing local information and transforming continuous time-series into binary spikes.\n\n>**Q7:** Section 2.2: The variable $U$ is undefined, and it is unclear whether the membrane voltage has a reset mechanism.\n\nThe membrane potential will be reset to the resting potential once it reaches the threshold. And $U[t]$ represents the membrane potential according to the following formula:\n\n$$\nV[t]=\\begin{cases}\nU[t],\\ \\text{if}\\ U[t]<V_{th}\\\\\\\\\nV_{rest},\\ \\text{if}\\ U[t]\\geq V_{th}\n\\end{cases}\n$$\n\n>**Q8:** Section 3.4: The SFS module is not well described.\n\nThe spiking patches generated by the Spiking Patch Encoder are first grouped as $G^i$:\n\n$$\n\\mathbf{G}^i=\\{S_{enc}^i,S_{enc}^{i+g},\\dots,S_{enc}^{i+(\\frac{N}{g}-1)g}\\}\n$$\n\nwhere $g$ denotes the number of groups.\n\nA spiking max pooling layer is used to emphasize the mutual key frequencies of different groups:\n\n$$\\mathbf{M} _{sel} = \\text{SMP}(\\mathcal{M} _{sel}^1, \\mathcal{M} _{sel}^2, \\dots, \\mathcal{M} _{sel}^g)$$\n\nThe SFS module is responsible for selecting selecting key frequency components from the spiking patches.\n\n>**Q9:** Section 3.5: The motivation for the architecture choice is unclear, I would suggest to add more motivations.\n\nFollowing your suggestion, we refine our motivations in Section 3.5:\n\nGrouped feature utilization preserves time dynamics of SNN by avoiding information loss caused by averaging [2], reducing computational complexity of the spiking decoder.\n\n[2] Zhou et al. Spikformer: When Spiking Neural Network Meets Transformer, ICLR 2023.\n\n>**Q10:** Table 2 and Figure 3 likely underestimate energy consumption.\n\nOur method estimate energy consumption based on SOPs and FLOPs. To provide a more comprehensive analysis of energy consumption including ACE, $E_{Mem}$,$E_{Opts}$,$E_{Addr}$ and $E_{Total}$ according to [3-4] in [Table 5](https://anonymous.4open.science/r/58FD/).\n\n[3] Zhang et al. Sparse transformer with local and seasonal adaptation for multivariate time series forecasting. Sci Rep.\n\n[4] Tian et al. FEDformer: Frequency Enhanced Decomposed Transformer for Long-term Series Forecasting. ICML 2022.", "claims_and_evidence": "The authors make two main claims:\nPerformance Improvement: SpikF achieves a lower Mean Absolute Error (MAE) across multiple long-term time-series benchmark datasets. The claim is well-supported. Experimental results in Table 1 demonstrate clear improvements over baselines like PatchTST, iTransformer, and FEDformer.\nEnergy Efficiency: SpikF is significantly more energy-efficient (75.05% lower energy consumption) than existing ANN-based models. The claim is not well-supported.\n- Figure 2 lacks clarity regarding the meaning of the firing rate (α) in terms of the number of spikes per unit time.\n- The values of α used in the experiments (Table 1) are unspecified, and it is unclear how performance depends on α.\n- Table 2 and Figure 3 likely underestimate energy consumption. Since the LIF neuron dynamics are part of the computation, directly comparing SOPs and FLOPs may not be a fair measure of energy efficiency.", "methods_and_evaluation_criteria": "The performance evaluation is convincing, with comparisons to multiple baselines and ablation studies. However, as previously stated,  the impact of the firing rate (α) on performance is unclear. An additional analysis is necessary. The energy efficiency estimation (Appendix C.1) is inadequate. A more formal methodology is required. I suggest revising the energy analysis with a more rigorous approach.", "theoretical_claims": "No, but the claims seem correct.", "experimental_designs_or_analyses": "The experimental setup in Table 1 and Table 3 appears reasonable.", "relation_to_broader_scientific_literature": "The paper is well-positioned in spiking neural networks, time-series forecasting, and energy-efficient AI. Transformer-based forecasting models (Autoformer, FEDformer, PatchTST, iTransformer) are properly cited. However, the role of frequency-based methods in time-series forecasting is not well-discussed. More references to prior work in this area would strengthen the positioning.", "essential_references_not_discussed": "NA", "other_strengths_and_weaknesses": "- Strengths\nNovelty: The Spiking Fourier approach is an interesting and innovative alternative to traditional SNN architectures.\nImpact: Addresses real-world concerns in energy-efficient AI.\nClarity: The paper is generally well-written, with clear motivation and method descriptions.\n- Weaknesses\nEnergy efficiency is questionable: The paper does not convincingly justify how SpikF is more energy-efficient than ANN-based models. On a more general note, It is unclear whether Transformers with a sparsely active layer are a proper avenue for  truly energy-efficient models. \nUnclear limitations of the patch encoder: The impact of using patches is not discussed. The implementation details of the patch encoding layer are unclear. \nOther minor points:\n- Section 2.2: The variable U is undefined, and it is unclear whether the membrane voltage has a reset mechanism.\n- Section 3.4: The SFS module is not well described.\n- Section 3.5: The motivation for the architecture choice is unclear, I would suggest to add more motivations.", "other_comments_or_suggestions": "- Clearly define α and analyze its impact on performance.\n- Provide a more rigorous energy consumption analysis, ensuring a fair comparison between SOPs and FLOPs.", "questions_for_authors": "NA", "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": "xsnZTMyzQ5", "reviewer": "Reviewer_iCf2", "summary": "This paper introduces an attention-free framework, called Spiking Fourier Network (SpikF) for achieving long-term time series forecasting.\n\n## update after rebuttal\n\nThere is no external comment. And I think that it is a borderline paper.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate your thorough evaluation of our work and the expert feedback you have shared. In response to your constructive critique, we will provide clarifications addressing your question and incorporate these discussions into the revised manuscript to strengthen our theoretical framework.\n\n>**Q1:** However, I still care about the complexity and practicability of Fourier-related approaches in large-scale time series forecasting.\n\nWe would like to clarify the efficiency of SpikF when facing large-scale data:\n\n1. **Theoretical Analysis**: Theoretically, the computational complexity of Fast Fourier Transform is $O(NlogN)$, outperforming conventional approaches like linear transform ($O(N^2)$) and self-attention mechanism ($O(N^2)$). This fundamental advantage makes FFT-based solutions inherently scalable for long-horizon forecasting tasks.\n\n2. **Algorithmic Optimization**: Furthermore, FFT, as an algorithm with a long history, has been optimized by former researchers. For example, parallel distribution computation [1] and memory-sharing strategy [2] have further improved the efficiency of FFT when facing large-scale data. And these optimization can also be easily adapted into our architecture when facing real-world applications.\n\n3. **Experimental Validation**: As demonstrated in Figure 5, our method witnesses a precision promotion of $9.0\\\\%$ in terms of MSE when scaling look-back window from $48$ to $720$ timesteps, empirically indicating the effective utiliztion of larger-scale time-series. We provide the results of Figure 5 here, where LW represents look-back window, and PL denotes prediction length.\n\n   | LW \\ PL | 96 | 192 | 336 | 720 |\n   |------|------|------|------|------|\n   | 48 | 0.301 | 0.384 | 0.434 | 0.442 |\n   | 96 | 0.290 | 0.368 | 0.412 | 0.420 |\n   | 192 | 0.296 | 0.363 |0.386 | 0.415 |\n   | 336 | 0.287 | 0.351 | 0.371 | 0.395 |\n   | 720 | 0.290 | 0.351 | 0.368 | 0.410 |\n\n4. **Event-driven Paradigm**: When facing real-world large-scale time-series, SpikF can leverage temporal sparsity in online data streams [3], achieving computational efficiency through event-driven FFT triggering. This makes our approach particularly suitable for edge computing scenarios with limited resources when facing large-scale data.\n\nIn a word, Fourier-based approaches are more efficient than MLP-based or transformer-based algorithms for large-scale time series forecasting.\n\n[1] Yang, C. et al. A Parallel Fast Fourier Transform Algorithm for Large-Scale Signal Data Using Apache Spark in Cloud. ICA3PP 2018. vol 11336.\n\n[2] Eleftheriadis, Charalampos et al. Energy-Efficient Fast Fourier Transform for Real-Valued Applications. IEEE Transactions on Circuits and Systems II: Express Briefs. 69.\n\n[3] Jesus L. Lobo et al, Spiking Neural Networks and online learning: An overview and perspectives, Neural Networks, Volume 121, 2020, Pages 88-100.", "claims_and_evidence": "This paper aims at modifying SNNs and attention for long-term time series forecasting. The authors conduct experiments to verify SpikF in time-series prediction tasks across multiple dimensions. These empirical investigations are comprehensive, and the results look convincing.", "methods_and_evaluation_criteria": "The core of this paper is to replace attention by a Fourier-based methods. The workflow is introduced clearly by Section 3.4 and Figure 1. I care about whether and with what computational complexity the fast Fourier transform in Eq. (9) support large-scale data processing?", "theoretical_claims": "na", "experimental_designs_or_analyses": "These empirical investigations are comprehensive, and the results look convincing.", "relation_to_broader_scientific_literature": "na", "essential_references_not_discussed": "na", "other_strengths_and_weaknesses": "nothing.", "other_comments_or_suggestions": "Overall, I believe that this paper is a borderline paper. The authors provide comprehensive experiments, and the results look convincing. However, I still care about the complexity and practicability of Fourier-related approaches in large-scale time series forecasting. If this concerns are fixed in Rebuttal, I would consider raising my score.", "questions_for_authors": "nothing.", "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": "C28Czr47Hk", "reviewer": "Reviewer_oHDh", "summary": "This paper introduces the Spiking Fourier Network (SpikF), an attention-free framework designed to address key challenges in applying Spiking Neural Networks (SNNs) to long-term prediction tasks.\nThey encode input sequences in patches and employ a frequency-domain selection mechanism that better captures the sequential properties of time-series data.\nExtensive evaluations across eight long-term prediction datasets show that SpikF achieves a 1.9% reduction in Mean Absolute Error (MAE) compared to state-of-the-art models while reducing energy consumption by 75.05%.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are grateful for your comprehensive review and the valuable insights you have provided. In the subsequent sections, we will address your questions one by one. And we will integrate all relevant discussions into our article for the upcoming revision.\n\n>**Q1:** The proposed SpikF's hardware friendliness is ignored. I have checked the authors' code of the implementation of FFT operation in SpikF. I think that an operation like \"torch.fft\" is almost impossible to conduct on neuromorphic chips as it includes so much floating-point calculation.\n\nWe have discussed hardware friendliness in Section 2.2. For better clarity, we extend some explanation of our approach to hardware friendliness here.\n\nThe implementation of the Fast Fourier Transform (FFT) in neuromorphic hardware has been validated both theoretically and empirically by studies [1] and [2]. Your concerns regarding the floating-point calculations involved in FFT are addressed through their proposed methods.\n\nStudy [1] has demonstrated that matrix multiplication can be represented by a spiking linear layer with an appropriately defined weight matrix. Accordingly, they initially express the FFT as a series of matrix multiplications and subsequently employ an SNN with an equivalent number of layers to avoid the challenges associated with floating-point operations.\n\nMeanwhile, study [2] leverages the membrane dynamics of the Resonate-and-Fire neuron, an extension of the LIF model, to naturally perform the Fourier Transform. This approach also successfully gets rid of the need for additional floating-point operations.\n\n[1] Lopez-Randulfe et al., “Time-Coded Spiking Fourier Transform in Neuromorphic Hardware,” IEEE Trans. Comput., vol. 71, no. 11, pp. 2792–2802, 2022.\n\n[2] Orchard et al., “Efficient Neuromorphic Signal Processing with Loihi 2,” 2021 IEEE SiPS, pp. 254-259.\n\n>**Q2:** The novelty of this paper is questioned.\n\nYes, FFT is commonly used in sequential tasks. However, most previous works apply FFT to the entire time-series, which facilitates the utilization of high-frequency components. In contrast, our approach employs patch and grouping mechanisms to enhance the utilization of low and middle-frequency components from the original series, while local information is emphasized by the spiking patches. As a result, SpikF is able to use the **efficient utilization of the full spectrum** to improve accuracy, which has not been explored by former research.\n\n>**Q3:** The calculation of energy of SNNs is not accurate.\n\nAfter carefully reading the methods [3] [4] you referred to, we adopt the methods proposed by [3] and [4] to provide more metrics to comprehensive analyze the energy efficiency of SpikF and iTransformer ([Table 1](https://anonymous.4open.science/r/0D02/7)).\n\nIn terms of ACE and $E_{Total}$, the energy consumption of SpikF is $6.27\\times$ and $3.16\\times$ lower than iTransformer respectively.\n\n[3] Shen et al., “Are Conventional SNNs Really Efficient? A Perspective from Network Quantization,” IEEE/CVF CVPR, 2024, pp. 27538-27547.\n\n[4] Lemaire et al., “An analytical estimation of spiking neural networks energy efficiency,” Springer, 2022, pp. 574-587.\n\n[5] Yao et al., “Attention Spiking Neural Networks,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 45, no. 8, pp. 9393-9410, 2023.\n\n[6] Zhou et al., “Spikformer: When Spiking Neural Network Meets Transformer,” ICLR, 2023.\n\n>**Q4:** I think that a good study on SNNs should consider either hardware-friendliness or biological plausibility.\n\nWe totally agree with your perspective on SNN research. The hardware-friendliness has been discussed in **Q1**. Although it is less discussed in our work, biological plausibility has been considered in our design:\n\nThe basilar membrane in the inner ear exhibits a gradient of stiffness along its length. This mechanical property can process sound in different frequencies, and transform the series data into biological current signal, which is the inspiration of our SFS mechanism [7].\n\nIn addition, we believe that quality research in SNN domain should balance biological plausibility and precision. SpikF achieves $1.9\\\\%$ performance improvement across eight real-world datasets, demonstrating the practicality of our approach.\n\n[7] Moini, Piran, “Auditory System,” in Functional and Clinical Neuroanatomy, Academic Press, 2020, pp. 363-392.\n\n>**Q5:** Please report the standard deviation of SpikF in Table 1.\n\nWe report error bars of SpikF and iTransformer in [Table 2](https://anonymous.4open.science/r/0D02/), [Table 3](https://anonymous.4open.science/r/0D02/) and [Table 4](https://anonymous.4open.science/r/0D02/).", "claims_and_evidence": "Yes. Please refer to \"Questions For Authors\" and \"Weakness\".", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes. The equations in this paper is correct.", "experimental_designs_or_analyses": "Yes. Please refer to \"Questions For Authors\".", "relation_to_broader_scientific_literature": "The extension in how SNNs deal with long-term time series.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strengths:\n1. The method is simple and easy to understand.\n2. The proposed model achieves SOTA results and the ablation study is convincing.\n3. The design of patch-based splitting for long-term time series is remarkable. I like this point.\n\nWeaknesses:\n1. The proposed SpikF's hardware friendliness is ignored. I have checked the authors' code of the implementation of FFT operation in SpikF. I think that an operation like \"torch.fft\" is almost impossible to conduct on neuromorphic chips as it includes so much floating-point calculation.\n2. The novelty of this paper is questioned. The usage of FFT is quite common in sequential tasks, like sequential recommendation, time-series forecasting, and even natural language processing. It seems the authors just replaced the self-attention module with the FFT module.\n3. The calculation of energy of SNNs is not accurate. See comments.", "other_comments_or_suggestions": "I have to point out that: when designing SNN architectures, it is unwise for researchers to just take a common module from traditional ANNs to replace a certain module in SNNs and then report a SOTA performance.\nI think that a good study on SNNs should consider either hardware-friendliness or biological plausibility.\nFrom the perspective of pure deep learning, this paper is great.\nHowever, I think the authors ignore both the hardware-friendliness and the biological plausibility of SNNs.\nI do not think the FFT operation can be applied to neuromorphic chips.\n\nWhat's more, the energy calculation in Appendix A.2 is based on the discussion of [1][2]. Since both ANN and SNN process the same input data (with SNN using direct encoding in the first layer), the energy efficiency differences may only arise from variations in memory access or MAC/AC operations [3]. The paper only compares energy consumption during computation. Energy consumption is primarily determined by memory access rather than FLOPs or SOPs [4], but this impact is not included. **This omission should be addressed, as it is a common flaw in publications claiming energy savings.**\n\n[1] Yao M, Zhao G, Zhang H, et al. Attention spiking neural networks[J]. IEEE transactions on pattern analysis and machine intelligence, 2023, 45(8): 9393-9410.\n\n[2] Zhou Z, Zhu Y, He C, et al. Spikformer: When Spiking Neural Network Meets Transformer[C]//The Eleventh International Conference on Learning Representations, 2023.\n\n[3] Shen G, Zhao D, Li T, et al. Are Conventional SNNs Really Efficient? A Perspective from Network Quantization[C]//Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2024: 27538-27547.\n\n[4] Lemaire E, Cordone L, Castagnetti A, et al. An analytical estimation of spiking neural networks energy efficiency[C]//International Conference on Neural Information Processing. Cham: Springer International Publishing, 2022: 574-587.", "questions_for_authors": "1. Please report the standard deviation of SpikF in Table 1. I have to make sure the reported results are not just the best records among various random seeds.\n2. Please discuss the possibility of applying FFT operations to neuromorphic hardware.\n3. Please consider how memory access and other chip activities impact SNN energy consumption.", "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:

Meta-review: This paper proposes SpikF, a novel spiking neural network architecture for long-term time series forecasting that replaces attention mechanisms with a spiking Fourier selection module. While reviewers raised valid concerns about hardware compatibility of FFT operations and energy efficiency calculations, the work makes several strong contributions: (1) it demonstrates superior forecasting accuracy (1.9% MAE reduction) across eight benchmarks; (2) introduces an innovative patch-based spiking encoder for temporal data; (3) provides comprehensive ablation studies validating design choices. Two reviewers recommended acceptance, noting the method's simplicity and strong empirical results. Though hardware implementation requires further study (a common SNN challenge), the paper's novel fusion of frequency-domain processing with SNNs represents a meaningful advance for temporal modeling. Therefore, an acceptance is recommended.
Decision: Accept (poster)