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 2: Architecture of CBAM.
Figure 3: Architectures of the CA and SA modules.
Notation
Description
n
Bearing index.
t
Observation index.
Tn
Failure-time index of bearing n .
P
Number of samples in a single raw vibration record.
xn(t)
Raw vibration record of bearing n at observation t , xn(t)∈RP .
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 Xn(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
μ
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
μ
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
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
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
μ
0.164
0.200
0.151
0.190
μ
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
μ
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
μ
0.183
0.220
0.151
0.190
μ
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.
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.
Ya Song, Laurens Bliek, Yaoxin Wu +1
Information Systems, Department of Industrial Engineering & Innovation Sciences, Eindhoven University of Technology, Eindhoven, The Netherlands
Accurate prediction of Remaining Useful Life (RUL) in aero-engines is vital for predictive maintenance, improved operational reliability, and reduced lifecycle costs. While deep learning approaches have demonstrated strong potential in this area, most existing methods focus primarily on model architecture design and treat input features uniformly, often neglecting the influence of data preprocessing. In this work, we propose a novel preprocessing pipeline that enhances RUL prediction by improving data quality and temporal representation before model training. Our approach leverages complete temporal sequences and generates RUL estimates at each timestep, enabling the model to capture fine-grained degradation dynamics and deliver continuous prognostic insights throughout the engine's operational life. To validate the effectiveness of the proposed pipeline, we conduct experiments on the NASA C-MAPSS dataset. Comparative evaluations against a suite of state-of-the-art neural models including CNN, RNN, LSTM, DCNN, TCN, BiGRU-TSAM, AGCNN, and ATCN, demonstrate that our approach consistently achieves superior accuracy and robustness in aero-engine RUL prediction. These results highlight the critical role of preprocessing in maximizing the effectiveness of neural prognostic models.
Florent Imbert, Tosin Adewumi, Hui Han
Machine Learning Group · Lulea University of Technology · Lulea, Sweden
Remaining Useful Life (RUL) prediction is essential for industrial predictive maintenance, yet many learning-based approaches rely on extensive feature engineering or large labeled datasets to train task-specific sequence models. In this work, we introduce a lightweight learning approach, in which we leverage a frozen pretrained time-series foundation model (TSFM) and combine it with a small regression head for RUL estimation from multivariate sensor streams. More specifically, we use Chronos-2 as a frozen backbone to extract context window features and train a lightweight regression neural network for RUL prediction. Experiments on real-world industrial sensor data from two device types show that Chronos-2 features consistently improve over recurrent, convolutional, Transformer-based, and gradient-boosting baselines under the same preprocessing and evaluation protocol. We further analyze the impact of context length and find that performance improves significantly with longer histories, indicating that TSFM representation offer a practical and data-efficient alternative for RUL estimation in industrial settings.
Amir El-Ghoussani, Michele De Vita, Ronald Naumann +1
Friedrich-Alexander-Universität Erlangen–Nürnberg (FAU), Germany · Nokia Solutions and Networks GmbH & Co. KG, Germany