Factorized axis convolutional gated recurrent unit with dynamic adaptive pooling for remaining useful life prediction of rolling bearings

Hanbyeol Park; Jungho Choo; Hyerim Bae

cs.AISep 25, 2026

Factorized axis convolutional gated recurrent unit with dynamic adaptive pooling for remaining useful life prediction of rolling bearings

Authors: Hanbyeol Park, Jungho Choo, Hyerim Bae

Organizations: Department of Industrial Engineering Pusan National University Pusan, Republic of Korea

Abstract

Convolutional neural networks (CNN) are widely used to predict the remaining useful life (RUL) of rolling bearings from time-frequency representations (TFRs) of vibration signals. However, during degradation, characteristic structures in TFRs align predominantly along the frequency or time axis, making it challenging for conventional CNN isotropic kernels to capture directional structure. Furthermore, global average pooling (GAP) averages across axes, potentially obscuring the locations and concentrations of salient activations. This study introduces a factorized-axis convolutional gated recurrent unit (GRU) that employs multiscale anisotropic convolution and dual-axis convolution block attention module to enhance directional features and highlight salient time-frequency regions. Dynamic adaptive pooling (DAP) adaptively aggregates the time-frequency-axis information from the extracted feature maps, whereas a GRU captures temporal dynamics in the latent representations and Monte Carlo dropout enables predictive uncertainty estimation. Experiments on two public bearing datasets demonstrate that the proposed model outperforms existing RUL prediction methods across operating conditions. Ablation experiments demonstrate that the factorized axis-wise design achieves lower mean errors than convolutional isotropic kernels. DAP yields clear improvements on one dataset while matching GAP on the other, highlighting the importance of anisotropic feature extraction and adaptive feature aggregation for TFR-based RUL prediction.

Figures & tables

Figure 1: Motivation for axis-wise kernels and input-dependent aggregation in TFR feature maps. (a) Isotropic convolutional kernels tend to progressively weaken strong, localized activations, whereas anisotropic kernels preserve them more effectively. (b) GAP assigns equal weights to all time–frequency locations and thus does not distinguish between degradation-relevant and irrelevant frequency regions.

Figure 3: Architectures of the CA and SA modules.

Notation	Description
$n$	Bearing index.
$t$	Observation index.
$T_{n}$	Failure-time index of bearing $n$ .
$P$	Number of samples in a single raw vibration record.
$\mathbf{x}_{n}(t)$	Raw vibration record of bearing $n$ at observation $t$ , $\mathbf{x}_{n}(t)\in\mathbb{R}^{P}$ .
$C$	Number of ordered contiguous segments within each vibration record; $C=5$ .

Table 1: Notation utilized in the proposed method.

Figure 4: Segment-wise input construction. Each raw record is divided into five ordered contiguous segments, and each segment is independently transformed into a single-channel magnitude scalogram. The segment-wise scalograms from the latest $L$ observations are stacked to construct $\mathbf{X}_{n}(t)$ .

Figure 5: Architecture of the FAAC feature extractor, comprising multi-scale anisotropic convolution, dual-axis convolutional block attention module, and dynamic adaptive pooling.

Figure 6: Proposed FAAC-GRU architecture.

Figure 7: Bearing test rigs for the two public datasets. (a) FEMTO-ST bearing dataset, released for the 2012 PHM challenge, (b) XJTU-SY bearing dataset, provided by xi’an Jiaotong University.

Data set	Condition 1	Condition 2	Condition 3
Train set	Bearing 1-1 (FB1-1)	Bearing 2-1 (FB2-1)	Bearing 3-1 (FB3-1)
	Bearing 1-2 (FB1-2)	Bearing 2-2 (FB2-2)	Bearing 3-2 (FB3-2)
Test set	Bearing 1-3 (FB1-3)	Bearing 2-3 (FB2-3)	Bearing 3-3 (FB3-3)
	Bearing 1-4 (FB1-4)	Bearing 2-4 (FB2-4)
	Bearing 1-5 (FB1-5)	Bearing 2-5 (FB2-5)
	Bearing 1-6 (FB1-6)	Bearing 2-6 (FB2-6)

Table 2: Training and test splits for the FB dataset, following the PHM 2012 challenge protocol [ 35 ] .

Data set	Condition 1	Condition 2
Leave-one-out cross validation	Bearing 1-1 (XB1-1)	Bearing 2-2 (XB2-2)
	Bearing 1-2 (XB1-2)	Bearing 2-4 (XB2-4)
	Bearing 1-3 (XB1-3)	Bearing 2-5 (XB2-5)
	Bearing 1-5 (XB1-5)

Table 3: Evaluation scheme for the XB dataset, limited to bearings with outer-race faults under Condition 1 and 2 [ 36 ] .

Model	Description
	Input	Network architecture
SAL-CNN [ 26 ]	STFT spectrogram.	CNN–CBAM feature extractor followed by an LSTM-based temporal encoder and a fully connected regression head
CDCT [ 33 ]	Raw vibration signal, Time-, and Frequency-domain feature.	Dual-branch causal CNN encoders followed by transformer-based feature integration and a fully connected regression head
HA-CLSTM [ 37 ]	Multichannel wavelet packet coefficient.	ConvLSTM backbone with channel and spatial attention, followed by global pooling and a fully connected regression head
TT-CLSTM [ 10 ]	Time-domain feature and CWT scalogram.	Parallel transformer and ConvLSTM branches with late feature fusion, followed by a fully connected regression head and Kalman filtering
MLP-DT [ 9 ]	CWT scalogram.	MLP-based feature compression followed by an integrated DeepAR–transformer and a fully connected regression head

Table 4: Inputs and network architectures of the comparison models.

Test set	SAL-CNN		CDCT		HA-CLSTM		TT-CLSTM		MLP-DT		FAAC-GRU
	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE
FB1-3	0.068	0.073	0.057	0.068	0.091	0.113	0.066	0.104	0.043	0.066	0.035	0.037
FB1-4	0.257	0.291	0.104	0.138	0.102	0.144	0.397	0.486	0.241	0.262	0.037	0.054
FB1-5	0.154	0.184	0.206	0.261	0.146	0.170	0.281	0.343	0.142	0.184	0.079	0.116
FB1-6	0.262	0.298	0.155	0.201	0.128	0.162	0.384	0.445	0.176	0.266	0.120	0.150
FB1-7	0.141	0.219	0.082	0.112	0.112	0.194	0.253	0.296	0.178	0.188	0.111	0.151

Table 5: MAE and RMSE of the RUL prediction models for FB Condition 1.

Test set	SAL-CNN		CDCT		HA-CLSTM		TT-CLSTM		MLP-DT		FAAC-GRU
	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE
FB2-3	0.137	0.165	0.281	0.356	0.420	0.516	0.159	0.205	0.188	0.298	0.123	0.154
FB2-4	0.169	0.224	0.356	0.416	0.454	0.538	0.281	0.340	0.220	0.251	0.279	0.305
FB2-5	0.198	0.235	0.315	0.388	0.466	0.535	0.328	0.403	0.212	0.241	0.156	0.202
FB2-6	0.135	0.149	0.287	0.350	0.397	0.473	0.317	0.356	0.118	0.164	0.147	0.188
FB2-7	0.325	0.336	0.402	0.472	0.402	0.493	0.216	0.249	0.278	0.310	0.285	0.333

Table 6: MAE and RMSE of the RUL prediction models for FB Conditions 2 and 3.

Test set	SAL-CNN		CDCT		HA-CLSTM		TT-CLSTM		MLP-DT		FAAC-GRU
	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE
XB1-1	0.137	0.164	0.130	0.182	0.165	0.193	0.177	0.215	0.126	0.143	0.129	0.161
XB1-2	0.228	0.256	0.205	0.265	0.166	0.197	0.174	0.227	0.295	0.322	0.163	0.201
XB1-3	0.213	0.241	0.136	0.157	0.233	0.263	0.267	0.304	0.148	0.186	0.167	0.197
XB1-5	0.224	0.272	0.198	0.230	0.192	0.243	0.218	0.301	0.277	0.309	0.145	0.201
$\mu$	0.201	0.233	0.167	0.209	0.189	0.224	0.209	0.261	0.211	0.240	0.151	0.190

Table 7: MAE and RMSE of the RUL prediction models for XB Condition 1.

Test set	SAL-CNN		CDCT		HA-CLSTM		TT-CLSTM		MLP-DT		FAAC-GRU
	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE
XB2-2	0.180	0.226	0.087	0.097	0.182	0.206	0.120	0.161	0.232	0.267	0.081	0.101
XB2-4	0.178	0.205	0.243	0.282	0.248	0.291	0.201	0.234	0.238	0.265	0.168	0.190
XB2-5	0.240	0.285	0.185	0.219	0.119	0.162	0.161	0.211	0.430	0.509	0.160	0.199
$\mu$	0.199	0.239	0.172	0.199	0.183	0.220	0.161	0.202	0.300	0.347	0.136	0.163

Table 8: MAE and RMSE of the RUL prediction models for XB Condition 2.

Metric	Test sets	SAL-CNN	CDCT	HA-CLSTM	TT-CLSTM	MLP-DT	FAAC-GRU
RMSE	FBs	0.234	0.281	0.338	0.330	0.235	0.181
RMSE	XBs	0.235	0.204	0.222	0.236	0.285	0.178
Score	FBs	23.064	30.083	36.282	39.519	19.003	16.246
Score	XBs	2.643	2.024	2.196	2.357	3.501	1.915

Table 9: Mean RMSE and score for the FB and XB test bearings.

Figure 8: RUL predictions over the RtF sequences of selected FB and XB test bearings.

Figure 9: RUL prediction trajectories of the evaluated models for the (a) FB and (b) XB test bearings.

Model	Computational complexity
	Parameters (M)	FLOPs (G)	Inference time (ms)
SAL-CNN	0.111	0.149	0.078
CDCT	1.419	0.018	0.069
HA-CLSTM	0.053	2.119	0.340
TT-CLSTM	0.638	1.804	0.195
MLP-DT	0.772	0.451	0.109

Table 10: Model parameter counts and computational cost per input sample

Test set	CBAM		DCBAM		Test set	CBAM		DCBAM
Condition 1	MAE	RMSE	MAE	RMSE	Condition 2&3	MAE	RMSE	MAE	RMSE
FB1-3	0.039	0.048	0.035	0.037	FB2-3	0.139	0.167	0.123	0.154
FB1-4	0.058	0.066	0.037	0.054	FB2-4	0.344	0.411	0.279	0.305
FB1-5	0.076	0.111	0.079	0.116	FB2-5	0.175	0.209	0.156	0.202
FB1-6	0.099	0.137	0.120	0.150	FB2-6	0.088	0.125	0.147	0.188
FB1-7	0.127	0.163	0.111	0.151	FB2-7	0.337	0.397	0.285	0.333

Table 11: RUL prediction performance of FAAC-GRU with CBAM or DCBAM on FB test bearings under three operating conditions.

Test set	CBAM		DCBAM		Test set	CBAM		DCBAM
Condition 1	MAE	RMSE	MAE	RMSE	Condition 2	MAE	RMSE	MAE	RMSE
XB1-1	0.112	0.155	0.129	0.161	XB2-2	0.084	0.105	0.081	0.101
XB1-2	0.186	0.216	0.163	0.201	XB2-4	0.160	0.183	0.168	0.190
XB1-3	0.209	0.224	0.167	0.197	XB2-5	0.176	0.207	0.160	0.199
XB1-5	0.150	0.206	0.145	0.201
$\mu$	0.164	0.200	0.151	0.190	$\mu$	0.140	0.165	0.136	0.163

Table 12: RUL prediction performance of FAAC-GRU with CBAM or DCBAM on XB test bearings under two operating conditions.

Model	Computational complexity
	No. of parameters (M)	No. of FLOPs (G)	Inference time (ms)
FLAT	5.083	10.018	1.329
GAP	0.743	9.811	1.301
DAP	0.939	9.814	1.306

Table 13: Parameter counts and computational cost per sample for three feature map aggregation variants of FAAC-GRU.

Condition	Test set	FLAT		GAP		DAP
		MAE	RMSE	MAE	RMSE	MAE	RMSE
FB Condition 1	FB1-3	0.039	0.041	0.046	0.055	0.035	0.037
	FB1-4	0.209	0.256	0.109	0.140	0.037	0.054
	FB1-5	0.109	0.163	0.053	0.109	0.079	0.116
	FB1-6	0.135	0.185	0.161	0.195	0.120	0.150
	FB1-7	0.148	0.187	0.155	0.198	0.111	0.151

Table 14: RUL prediction performance of three FAAC-GRU aggregation variants on FB test bearings under three operating conditions.

Condition	Test set	FLAT		GAP		DAP
		MAE	RMSE	MAE	RMSE	MAE	RMSE
XB Condition 1	XB1-1	0.147	0.183	0.124	0.160	0.129	0.161
	XB1-2	0.204	0.248	0.178	0.206	0.163	0.201
	XB1-3	0.199	0.227	0.189	0.209	0.167	0.197
	XB1-5	0.148	0.171	0.142	0.200	0.145	0.201
	$\mu$	0.174	0.207	0.158	0.194	0.151	0.190

Table 15: RUL prediction performance of three FAAC-GRU aggregation variants on XB test bearings under two operating conditions.

Figure 10: DAP attention weights for FB1-3 and FB2-3. Input samples were selected at 25% of each RtF sequence; the first feature channel is shown.

Figure 11: DAP attention weights for XB1-3 and XB2-2. Input samples were selected at 25% of each RtF sequence; the first feature channel is shown.

Test set	ISO		ANISO		Test set	ISO		ANISO
Condition 1	MAE	RMSE	MAE	RMSE	Condition 2&3	MAE	RMSE	MAE	RMSE
FB1-3	0.046	0.052	0.035	0.037	FB2-3	0.165	0.204	0.123	0.154
FB1-4	0.147	0.168	0.037	0.054	FB2-4	0.295	0.338	0.279	0.305
FB1-5	0.085	0.127	0.079	0.116	FB2-5	0.187	0.212	0.156	0.202
FB1-6	0.123	0.165	0.120	0.150	FB2-6	0.118	0.133	0.147	0.188
FB1-7	0.149	0.179	0.111	0.151	FB2-7	0.319	0.374	0.285	0.333

Table 16: RUL prediction performance of ISO and ANISO on FB test bearings under three operating conditions.

Test set	ISO		ANISO		Test set	ISO		ANISO
Condition 1	MAE	RMSE	MAE	RMSE	Condition 2	MAE	RMSE	MAE	RMSE
XB1-1	0.146	0.189	0.129	0.161	XB2-2	0.109	0.112	0.081	0.101
XB1-2	0.228	0.255	0.163	0.201	XB2-4	0.157	0.170	0.168	0.190
XB1-3	0.197	0.221	0.167	0.197	XB2-5	0.181	0.216	0.160	0.199
XB1-5	0.161	0.214	0.145	0.201
$\mu$	0.183	0.220	0.151	0.190	$\mu$	0.149	0.166	0.136	0.163

Table 17: RUL prediction performance of ISO and ANISO on XB test bearings under two operating conditions.

Figure 12: Input CWT scalograms (5 segments) and (a) ISO (b) ANISO Grad-RAM maps for FB1-4 at the end of its RtF sequence.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Figure A1: Axis preference of local structures.

Figure A2: Distribution of dominant orientation.

Figure A3: Difference in coherency between TFRs and pixel-permuted maps.

Figure B1: Raw vibration signals over the RtF for FB under three operating conditions.

Figure B2: Raw vibration signals over the RtF for XB under two operating conditions.

Section	Layer / Operation	Output shape
Input	Segment-wise TFR sequence	(B, L, C, 1, 64, 64)
Segment batching	Merge B, L, C for shared extraction	(BLC, 1, 64, 64)
Shared FAAC feature extractor	MSAC block 1	(BLC, 32, 64, 64)
	(2, 2) max pooling	(BLC, 32, 32, 32)
	MSAC block 2	(BLC, 64, 32, 32)
	(2, 2) max pooling	(BLC, 64, 16, 16)

Appendix

Table C1: FAAC-GRU layer configuration and output structures.

RUL Estimation

Remaining useful life estimation predicts how long machines, batteries, and other assets can operate before failure or end of life.

See all 57 papers

Multi-Term Fourier Graph Neural Network with Sample Relationship Learning for Enhanced Remaining Useful Life Prediction
Sep 21, 2026Ya Song, Laurens Bliek, Yaoxin Wu +1Graph Neural Networks RUL Estimation
Information Systems, Department of Industrial Engineering & Innovation Sciences, Eindhoven University of Technology, Eindhoven, The Netherlands
A Novel Preprocessing-Driven Approach to Remaining Useful Life (RUL) Prediction Using Temporal Convolutional Networks (TCN)
May 4, 2026Florent Imbert, Tosin Adewumi, Hui HanRUL Estimation Predictive Maintenance
Machine Learning Group · Lulea University of Technology · Lulea, Sweden
Time-Series Foundation Model Embeddings for Remaining Useful Life Estimation
Jun 10, 2026Amir El-Ghoussani, Michele De Vita, Ronald Naumann +1RUL Estimation Predictive Maintenance
Friedrich-Alexander-Universität Erlangen–Nürnberg (FAU), Germany · Nokia Solutions and Networks GmbH & Co. KG, Germany

Sep 21, 2026J/K move · Enter open · S save

Multi-Term Fourier Graph Neural Network with Sample Relationship Learning for Enhanced Remaining Useful Life Prediction

Ya Song, Laurens Bliek, Yaoxin Wu, Yingqian Zhang

Information Systems, Department of Industrial Engineering & Innovation Sciences, Eindhoven University of Technology, Eindhoven, The Netherlands

Graph Neural Networks RUL Estimation Multiscale Time-Series Modeling Spatiotemporal GNNs

Open paper AskPDF arXiv

Predicting the remaining useful life (RUL) is essential for effective predictive maintenance. Spatio-Temporal Graph Neural Networks (ST-GNNs), which can model both temporal and spatial relationships by representing time series data as a sequence of graphs, have shown exceptional performance in RUL prediction. However, current ST-GNNs face several drawbacks. First, they require domain expertise or significant computational power to establish graph structures prior to deploying GNNs. Second, the models are restricted to capture temporal dependencies within a predefined fixed-size lookback window. This restriction ignores the common issue of varying time series lengths, leading the prediction model to miss short-term or long-term dependencies. Finally, conventional models often fail to capture the inherent relationships between samples generated from adjacent time windows, which are crucial for improving both the accuracy and robustness of predictions. To address the aforementioned issues, we introduce a novel framework called Multi-Term Fourier Graph Neural Network with Sample Relationship Learning (MTFGN-SRL). Rather than treating the sample as a sequence of graphs, we consider it as a single complete graph and utilize a Fourier Graph Neural Network (FGN) to capture the spatio-temporal information in the frequency domain. We propose a multi-term learning module that utilizes multiple lookback windows to generate samples with varying terms, which are then fed into the FGN to enhance the extraction of useful information from the data. Finally, we develop a sample relationship learning module by training a heterogeneous GNN to identify inter-sample relationships, resulting in enhanced accuracy and robustness in predictions. Evaluations on the CMAPSS dataset demonstrate MTFGN-SRL's superior performance over state-of-the-art methods in RUL prediction.

Abstract

RUL Estimation

Explore similar work

Multi-Term Fourier Graph Neural Network with Sample Relationship Learning for Enhanced Remaining Useful Life Prediction

A Novel Preprocessing-Driven Approach to Remaining Useful Life (RUL) Prediction Using Temporal Convolutional Networks (TCN)

Time-Series Foundation Model Embeddings for Remaining Useful Life Estimation