Linearly constrained multi-representation domain generalization for unseen-bearing RUL prediction

1 Introduction

In modern industrial systems, Prognostics and Health Management (PHM) for rotating machinery such as bearings, gears, and motors are crucial for improving equipment reliability, reducing maintenance costs, and avoiding catastrophic failures (Kim et al., 2017; Zio, 2022; Meng and Li, 2019; Gharib and Kovács, 2023). Among them, Remaining Useful Life (RUL) prediction, as one of the core tasks of PHM, aims to estimate the remaining available time before equipment failure based on operational state data (Wang et al., 2020; Ren et al., 2018; Huang et al., 2021). Bearings, as one of the most vulnerable critical components in rotating machinery, often lead to overall equipment failure due to performance degradation (He et al., 2020). Therefore, accurate bearing RUL prediction can help formulate reasonable maintenance plans, prevent economic losses caused by unexpected downtime, and avoid potential casualties resulting from sudden bearing failures (Heng et al., 2009; Li et al., 2021; Yan et al., 2016; Qian et al., 2014).

Existing RUL prediction methods can generally be divided into two categories: Physics model-based methods and data-driven methods (Lei et al., 2018; Tsui et al., 2015). Traditional Physics model-based RUL prediction methods primarily rely on in-depth understanding of bearing failure mechanisms and mathematical modeling (Jin et al., 2019). For example, Based on the Paris law for crack propagation, (Bechhoefer et al. 2008) establishes a mathematical mapping between the Health Index (HI) and equivalent crack length proposing a deterministic degradation model for RUL prediction of transmission components. (Orsagh et al. 2003) integrated the Yu-Harris stress theory with the Kotzalas-Harris expansion model and used Monte Carlo methods to quantify uncertainties in load and lubrication conditions, generating probabilistic RUL distributions. (Qiu et al. 2002) proposed a stiffness degradation-driven RUL prediction method for bearings, correlating system stiffness with natural frequency/acceleration response through a single-degree-of-freedom vibration model and constructing a nonlinear life prediction equation using the Damage Curve Approach (DCA) and Dual-Linear Damage Rule (DLDR). However, physics-based methods have three main limitations: First, the accuracy of physics-based RUL prediction heavily relies on prior knowledge, and prediction performance significantly declines when bearing operating conditions exceed modeling assumptions (Lu et al., 2024). Second, complete modeling is often challenging under complex operating conditions (Deng et al., 2023). Finally, Physics model-based methods typically require destructive testing to obtain parameters, which is costly (Cubillo et al., 2016; Sikorska et al., 2011).

With the widespread deployment of Internet of Things (IoT) devices and advancements in intelligent technologies, data-driven RUL prediction methods have gradually become a research hotspot. (Yao et al. 2023) proposed an RUL prediction method based on a Convolutional Denoising Autoencoder (CDAE) and Bidirectional Long Short-Term Memory (Bi-LSTM) network. This method directly uses vibration signals as input, reduces noise impact through CDAE, and employs Bi-LSTM to extract high-dimensional degradation features of bearings, ultimately achieving RUL estimation. (Cheng et al. 2020) proposed an RUL prediction framework combining Hilbert-Huang Transform (HHT) and Convolutional Neural Networks (CNN). This method uses HHT to construct nonlinear degradation energy indicators as training labels and employs CNN to learn the implicit relationship between degradation features and raw vibration signals, thereby predicting the RUL of test bearings through Support Vector Regression (SVR). (Zhu et al. 2019) proposed a deep feature learning method based on Time-Frequency Representation (TFR) and Multi-Scale Convolutional Neural Network (MSCNN). This method uses wavelet transform to obtain the TFR of bearing degradation signals and inputs them into MSCNN after dimensionality reduction via bilinear interpolation. The advantage of this approach is its ability to simultaneously preserve global and local features, leading to significant improvements in prediction accuracy. Although data-driven methods overcome many limitations of physics-based models and achieve higher prediction accuracy, they still face two key challenges: (1) Most existing models rely on a single representation of vibration signals, making it difficult to jointly characterize raw temporal fluctuations and time-frequency energy redistribution during degradation; (2) most RUL models require target-bearing samples for training or calibration, and this dependence is particularly evident in transfer-learning and domain-adaptation methods, which often still require target-bearing or target-domain samples during model training or adaptation (Ding et al., 2023; Xu et al., 2024; Lei et al., 2025). In real industrial scenarios, however, target-bearing data may be scarce or entirely unavailable before deployment.

These challenges result in significant performance degradation due to domain shift when target bearing data is missing, especially considering the complex failure modes of bearings. To address these issues, this paper develops a multi-representation domain generalization framework for unseen-bearing RUL prediction. The main contributions include: (1) A dual-channel multi-representation input design that aligns raw time-domain vibration and CWT time-frequency representations, enabling the model to more fully exploit time-domain fluctuations and frequency-domain degradation information during the degradation process. (2) A domain generalization strategy that learns transferable degradation representations from source bearings and improves RUL prediction for unseen target bearings. (3) A lightweight linearly constrained temporal prior based on the irreversibility of bearing damage, used to encourage physically plausible monotonic degradation behavior.

The remainder of this paper is organized as follows. Section 2 details the proposed methodology. Section 3 covers the experimental setup, presents the results, and provides a comprehensive analysis. Section 4 offers a detailed discussion of key model components, and Section 5 concludes with future research directions.

2 Methodology

2.1 Overview

The proposed multi-representation domain generalization framework for unseen-bearing RUL prediction comprises two key components: (1) a bidirectional multiscale ConvLSTM architecture for spatiotemporal feature extraction, and (2) a linearly constrained temporal prior that encourages monotonic degradation trends. These components work jointly to learn degradation patterns from source bearings. Figure 1 illustrates the overall architecture.

2.2 Input representation

For the acquired raw time-series data X_1D = {x₁, x₂, …, x_*T*}, we construct a dual-channel multi-representation tensor. In the implementation, each time window is stored as two aligned image-like channels: the CWT scalogram and the reshaped raw-vibration matrix. Consecutive windows are then stacked into a temporal input. For a mini-batch, the input can be written generically as X∈ℝB×L×C×H×W, where B is the batch size, L is the number of temporal windows, C = 2 denotes the two vibration-derived representations, and H and W denote the height and width of each aligned representation map. The concrete values used in the experiments are reported in the training-parameter table. The two channels comprise:

• Time-domain representation: The raw vibration signal X_1D = {x₁, x₂, …, x_*T*} is segmented by the same temporal window as the CWT representation and reshaped into an H×W matrix. This operation preserves the original amplitude sequence in a form spatially aligned with the time-frequency map.
• Time-frequency representation: Continuous Wavelet Transform (CWT) using the Morlet wavelet: where a is the scale parameter, b is the translation parameter, and ψ^* denotes the complex conjugate of the Morlet mother wavelet function defined as: with ω₀ = 6 being the central frequency parameter. The normalization factor 1/a ensures energy preservation across scales. The CWT scalogram is resized to the same H×W resolution as the time-domain representation so that the two channels are aligned at the input level.

2.3 Multiscale ConvLSTM cell

This study uses a multi-scale ConvLSTM cell that augments the representational capacity of conventional ConvLSTM through a multi-scale spatial feature extraction mechanism. Following the standard ConvLSTM gating idea, the current input X_*t* and the previous hidden state H_*t*−1 jointly determine the gate updates. In implementation, they are concatenated channel-wise and then processed by convolutional gates, which is a common engineering form equivalent in spirit to using input-to-state and state-to-state convolutions jointly in the gate computation. Three parallel convolutional layers with kernel sizes (3 × 3, 5 × 5, 7 × 7) extract features at different receptive-field sizes, each generating an output with 4 × hidden_dim channels. These features are fused by channel-wise concatenation followed by a 1 × 1 convolution, and the fused tensor is split into the input gate, forget gate, output gate, and candidate cell state. The detailed procedure is presented in Algorithm 1.

Algorithm 1 Multi-scale ConvLSTM cell. Pseudocode diagram describing an advanced neural network cell update mechanism that includes input concatenation, multi-scale feature extraction with three convolutional kernels, feature fusion, gate formation and activation using sigmoid and tanh, and cell and hidden state updates, ending with the return of updated states.

2.4 Bidirectional processing

To comprehensively capture both past and future contextual information within sequential spatiotemporal data, we employ a bidirectional processing architecture based on parallel Multi-Scale Convolutional Long Short-Term Memory (MS-ConvLSTM) networks.

Let X={X1,X2,…,XT} represent the input sequence where each element Xt∈ℝC×H×W denotes the spatiotemporal data at time step t, with C channels, height H, and width W. The forward hidden state sequence H⃗ is computed temporally from t = 1 to t = T, while the backward sequence H⃖ is processed in reverse from t = T to t = 1, governed by the recurrence relations: Here H⃗t∈ℝK×H′×W′ and H⃖t∈ℝK×H′×W′ represent the forward and backward hidden states at time t, respectively, with K feature channels and reduced spatial dimensions H′ ≤ H, W′ ≤ W due to convolutional operations. The MS-ConvLSTM transformation function Φ_MS-ConvLSTM operates identically in both directions, with boundary conditions initialized as zero tensors: H⃗0=0 for the forward pass and H⃖T+1=0 for the backward pass.

H⃗t=ΦMS-ConvLSTMfwd(Xt,H⃗t-1)H⃖t=ΦMS-ConvLSTMbwd(Xt,H⃖t+1)(3)

The final spatiotemporal feature representation at each time step is obtained through channel-wise concatenation: yielding fused features Ht∈ℝ2K×H′×W′ where the concatenation operator ∥ combines features along the channel dimension. This architecture enables each H_*t* to integrate both causal (t′ ≤ t) and non-causal (t′≥t) dependencies from the entire sequence X, preserving complementary spatiotemporal patterns while maintaining spatial coherence.

Ht=C(H⃗t,H⃖t)≡[H⃗t∥H⃖t](4)

2.5 Linearly constrained temporal prior

Based on the damage-irreversibility assumption in bearing degradation, we introduce a lightweight trainable linear temporal prior into the RUL prediction process. Monotonicity and physical-consistency constraints have already been studied in RUL prediction. For example, (Lu et al. 2024) proposed a physics-guided LSTM network in which time-frequency degradation indicators are selected using trend-related measures such as monotonicity, and a regularization term is constructed according to the consistent variation between these indicators and RUL. In their method, the variation of the selected feature indicators is further used as a dynamic weight, so the physical prior mainly enters the model through the training objective and loss regulation. The motivation of their method and ours is related in that both use the general irreversibility of degradation as physical guidance, but the implementation is different. Lu et al. use feature-indicator consistency and dynamic weighting to guide LSTM training, whereas our method uses a prediction-layer linear temporal prior after the multi-representation ConvLSTM has extracted spatiotemporal features. The proposed prior is therefore not designed as a full physics-informed degradation model, but as a lightweight linear inductive bias that encourages the normalized RUL output to follow a globally decreasing tendency. This design keeps the additional complexity low and is suitable for the leave-one-bearing-out setting considered in this study, where target-bearing samples are unavailable for adapting a more complex physical model. It can reduce physically implausible upward drift in the predicted RUL trajectory while still allowing the data-driven term to capture local fluctuations and bearing-specific degradation patterns.

The model prediction combines a base regression module f_θ(·) with a temporal decay term: where ŷ_*t* is the predicted normalized RUL at time step t, f_θ(X_1:*t*) is the neural regression output extracted from the input sequence, τ_*t*∈[0, 1] is the normalized operational time index within the prediction stage, and α is a trainable scalar initialized with a negative value and constrained to remain non-positive during optimization. The term ατ_*t* encourages the predicted RUL to decrease as operating time increases, while the flexible neural term f_θ(·) allows data-driven local adjustments.

ŷt=fθ(X1:t)+α·τt(5)

2.6 Dataset preprocessing and training strategy

To validate the proposed method for bearing RUL prediction, we conduct experiments on two widely recognized public datasets: CWRU (Wang et al., 2020) and PRONOSTIA (Nectoux et al., 2012). These datasets provide complete run-to-failure vibration signals collected under various operating conditions, including variable loads and rotational speeds. Their use enables evaluation of whether degradation representations learned from source bearings can generalize to a held-out target bearing.

Before model training, all raw vibration signals were preprocessed in a unified manner. For each bearing, the prediction stage starts from the first prediction time point (FPT). Let T_*i* denote the failure time of bearing i and F_*i* denote its FPT. For a sample collected at time t within the prediction stage, the normalized RUL label is defined as Thus, y_*i*(t) = 1 denotes the beginning of the prediction stage and y_*i*(t) = 0 denotes failure. This normalization removes the direct effect of different FPT-to-failure durations and allows bearings with different lifetime lengths to be modeled on a common 0–1 degradation scale. In this sense, the normalized output should be interpreted primarily as a degradation-progress or relative health-state estimate. It is not, by itself, an absolute remaining-time prediction on the real time scale. This formulation relies on complete run-to-failure records and is therefore well suited to public benchmark evaluation, where the failure time is known after data collection. The aim of the present domain-generalization experiment is to compare how well different models perceive degradation progression when trained on source bearings and tested on a held-out target bearing under similar benchmark settings.

yi(t)=Ti-tTi-Fi, Fi≤t≤Ti.(6)

To avoid target-bearing leakage, we adopt a leave-one-bearing-out partitioning strategy. Under each operating condition, one bearing is selected as the unseen target bearing, while the remaining bearings are used as source-domain data for model training. The target bearing is completely excluded from training, validation, normalization-parameter estimation, hyperparameter adjustment, and model selection, and is used only for final testing. Table 1 details the training environment configuration, while Table 2 specifies the training parameters and model-architecture settings.

Table 1. Training environment configuration.

Table 2. Training parameters and model architecture of the proposed model.

2.7 Evaluation metrics

The evaluation of our proposed bearing RUL prediction method employs two complementary statistical metrics - Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) - which collectively provide rigorous quantitative assessment of predictive performance on the normalized RUL scale. RMSE, calculated as the square root of the average squared differences between actual and predicted normalized RUL values, inherently emphasizes larger prediction errors through its quadratic formulation, making it sensitive to large deviations in degradation-stage estimation. In parallel, MAE measures the arithmetic mean of absolute prediction errors, offering robust interpretation of overall model accuracy that remains less affected by extreme values. This dual-metric approach is selected because of their complementary characteristics in error evaluation, with RMSE highlighting critical large deviations while MAE reflecting consistent prediction capability. Their widespread adoption in prognostic literature ensures meaningful benchmarking against state-of-the-art methods.

3 Experiment and analysis

3.1 Data description

The experimental setup to acquire the vibration data of the XJTU-SY bearing is shown in Figure 2, which included an AC motor, a motor speed controller, a support shaft, two support bearings, two accelerometers, and a test bearing. This testing platform is capable of conducting accelerated degradation testing, providing experimental data that closely reflect real-world conditions, and comprehensively capturing the entire lifecycle of bearings during the degradation process. The vibration signals were collected via accelerometers at a sampling frequency of 25.6 kHz, with a sampling interval of 1 min and a duration of 1.28 s for each sampling event.

Table 3 provides detailed specifications of our data usage. However, bearing1_4 does not include first prediction point (FPT) data due to its sudden fault characteristics, and bearing3_2 also lacks FPT data because of severe vibration signal fluctuations caused by multi-component faults. Therefore, we conducted experiments using the remaining 13 sets of bearing data.

Table 3. Detailed information of the XJTU-SY bearing dataset.

PRONOSTIA is a publicly available dataset developed by the FEMTO-ST Institute in France, specifically designed for research in bearing fault PHM. The dataset aims to provide standardized experimental data to support the analysis of bearing degradation processes and the development of RUL prediction algorithms. The test platform, as shown in Figure 3, consists of three parts: a rotating section, a degradation generation section applying radial force to the tested bearing, and a measurement section.

In all these experiments, three operating conditions were applied: 1,800 rpm and 4 kN, 1,650 rpm and 4.2 kN, and 1,500 rpm and 5 kN. Both vibration and temperature signals were collected. The vibration signals were sampled at 25.6 kHz, with 0.1 seconds of data sampled every 10 seconds, resulting in 2,560 samples recorded. The temperature signals were sampled at 10 Hz, with 600 samples recorded per minute. We employed vibration signal data under operating condition 1, with detailed data usage specifications presented in Table 4.

Table 4. Detailed information of the PRONOSTIA bearing dataset.

3.2 Ablation experiments

To rigorously assess the effectiveness of each component in the proposed framework, we conducted ablation studies on the usable bearings of the XJTU-SY dataset. Due to manuscript length limitations, the main text reports the results under operating condition 1, while the results under operating conditions 2 and 3 are provided in the (Appendix Figures 1, 2). As illustrated in Figure 4, removing the multiscale (MS), bidirectional (BI), or temporal constraint (TC) module generally increases the prediction error or weakens trajectory stability on the displayed condition-1 bearings. These results provide empirical support for retaining the three components in the complete model, while the appendix reports the corresponding results under the other two operating conditions.

3.3 Comparison experiment

To ensure a comprehensive and objective evaluation, the proposed model was compared with two benchmark methods [BiLSTM (Schuster and Paliwal, 1997) and MSCNN (Cai et al., 2016)] and three state-of-the-art approaches [TT-ConvLSTM (Niazi et al., 2024), DTAMM (Jin et al., 2025), and DCNN-Bootstrap (Huang et al., 2021)], representing diverse paradigms in bearing degradation modeling and RUL prediction. This selection covers both conventional deep learning architectures and advanced hybrid frameworks, providing a balanced basis for comparison. All experiments were conducted on two widely used datasets, XJTU-SY and PRONOSTIA, to verify the model's effectiveness and generalization capability.

Tables 5, 6 present the RMSE and MAE comparison results on the XJTU-SY and PRONOSTIA datasets, respectively. On the XJTU-SY dataset, the proposed method achieves the lowest mean RMSE and MAE values, with particularly clear advantages on bearings with complex degradation characteristics such as 1-5 and 2-1. Although methods such as DTAMM and DCNN-Bootstrap perform well in several individual cases, their results fluctuate more across bearings. On the PRONOSTIA dataset, the proposed method also obtains the lowest mean RMSE and MAE values and remains competitive across the tested bearings. Compared with single-representation approaches, the dual-channel architecture provides complementary degradation information from raw vibration and CWT time-frequency representations, while the bidirectional multiscale temporal modeling further improves the stability of the predicted degradation trend.

Table 5. Performance comparison on XJTU-SY dataset.

Bold values indicate the lowest (best) RMSE or MAE in each row; tied minima are also shown in bold.

Table 6. Performance comparison on PRONOSTIA dataset.

Bold values indicate the lowest (best) RMSE or MAE in each row; tied minima are also shown in bold.

To further explain these RMSE and MAE comparison results, Student's t-tests were conducted by comparing cases where the proposed method outperformed the competing methods with cases where it underperformed. As shown in Figure 5, when RMSE and MAE are evaluated jointly, statistically significant differences are observed in the superior-performing cases, whereas no significant differences are observed in the underperforming cases. The separate analyses of RMSE and MAE in Figure 6 show a consistent pattern: significant differences appear in the superior-performing cases, while the inferior cases remain statistically non-significant. These results indicate that the performance gains of the proposed method are not accidental, but are statistically supported across different metrics and datasets.

Notably, although the proposed method achieves the lowest average error metrics, it exhibits relatively higher variance. This variance can be interpreted together with the statistical analysis: significant improvements are consistently observed in the superior-performing cases, while the few inferior cases show no significant differences compared with the competing models. Thus, the increased variance mainly reflects the coexistence of large improvements in many cases and statistically neutral outcomes in a small number of cases. On PRONOSTIA, the moderate result on bearing 1-5 is related to its short degradation cycle, whose failure pattern differs from the degradation trajectories captured in the source data.

Finally, because aggregate RMSE and MAE values cannot fully describe the temporal behavior of the predicted RUL trajectories, representative RUL prediction curves are provided for six XJTU-SY bearings: 1-5, 2-1, 2-2, 2-4, 3-1, and 3-4. These cases cover the three operating conditions, while the full quantitative comparison is reported in Table 5. As shown in Figure 7, the proposed model more closely follows the ground-truth normalized RUL trajectory and produces fewer large temporal deviations than BiLSTM and MSCNN in the displayed cases. This visualization complements the quantitative tables by showing the temporal evolution of the predicted degradation trend.

4 Discussion

To enhance the persuasiveness of the results, further validate the effectiveness of the proposed model, and improve its interpretability, this section focuses on discussing the key design elements of the model and the limitations of the methodology.

4.1 MS module interpretability validation

To enhance the interpretability of our proposed model, we performed a visualization analysis of the learned weight parameters of the multiscale convolutional kernels within the Bidirectional Multiscale ConvLSTM module. Specifically, we investigated the characteristics of different-scale convolutional kernels (3 × 3, 5 × 5, and 7 × 7) across early, mid, and late stages of bearing degradation. We used Bearings 1-5 as test subjects, with their results during the early, mid, and late degradation stages shown in Figure 8.

The visualization in Figure 8 is used as qualitative interpretability evidence for the multi-scale convolution design. The red boxes mark regions that are strongly emphasized by convolutional kernels of different scales in the early, middle, and late degradation stages. Viewed horizontally within the same degradation stage, the highlighted regions of the 3 × 3, 5 × 5, and 7 × 7 kernels are not identical; instead, they are distributed over complementary areas and together provide broader coverage of the degradation-related response pattern. This indicates that the three kernel scales do not simply duplicate the same feature map. Rather, they can focus on different local-to-global regions and jointly describe the evolving signal pattern more completely across the bearing lifecycle. This observation supports the interpretability of the multi-scale design.

4.2 Multi-representation input validation

Our designed experiments systematically validate the impact of multi-representation fusion on the predictive performance of our proposed model. The results are shown in Figure 9. This is achieved by comparing three input configurations:

• Single-channel time-frequency representation (using only grayscale time-frequency maps generated via continuous wavelet transform).
• Dual-channel (time-frequency maps and raw data) combining time-frequency maps with raw vibration data.
• Dual-channel (time-frequency maps and random noise) augmenting time-frequency maps with identically-sized random noise.

In the experimental results, the dual-channel time-frequency plots and the input of the original signal outperformed the other two methods, achieving the best results on all four bearings. This highlights their advantages. First, compared to a single representation, the multi-representation input that combines time-frequency image features with the original vibration representation can provide more comprehensive degradation information, significantly improving prediction accuracy. Second, by introducing a noise channel as a control experiment, the model's anti-interference capability was rigorously tested when the second channel contained invalid information, further demonstrating the effectiveness of multi-representation fusion—that is, the original data indeed provides valuable supplementary information, rather than merely increasing the model's parameter capacity.

4.3 Theoretical upper-bound reference and deployable SRC-FPT analysis

The original Proposed method is retained throughout this manuscript. For the temporal-coordinate analysis in this subsection only, the originally reported Proposed configuration is denoted as the Target-duration reference. This designation does not indicate a different network architecture, input representation, training procedure, or learned parameter set. For target bearing i, the temporal coordinate used by the original Proposed configuration is where F_*i* and T_*i* denote the FPT and failure time of the target bearing, respectively. Because T_*i*−F_*i* is available only after the complete run-to-failure record has been observed, the Target-duration reference may be viewed as a theoretical upper-bound reference for the temporal-coordinate construction. It preserves the originally reported Proposed results while making explicit the information available in that retrospective setting. This use of the completed target record is distinct from model development: the target bearing remains excluded from training, validation, normalization-statistics estimation, hyperparameter tuning, and model selection.

τi,ttgt=t-FiTi-Fi,(7)

To obtain a deployable temporal coordinate, the target-duration denominator is replaced with a source-only reference fixed before testing the held-out bearing: where Si contains only the source bearings in the corresponding leave-one-bearing-out split. The minimum source duration is used as a conservative reference rather than as an optimal lifetime estimator. The raw output is then aligned at the known or causally detected FPT: The complete mathematical formulation is presented in Equations 1–9.

DiSRC=minj∈Si(Tj-Fj), τi,tSRC=t-FiDiSRC,(8)

ŷi,tSRC-FPT=ŷi,traw+(1-ŷi,Firaw).(9)

This operation neither updates model parameters nor uses T_*i*; it preserves the pairwise temporal differences, slopes, and local shape of the prediction trajectory. The resulting source-referenced, FPT-calibrated configuration is denoted SRC-FPT.

Tables 7, 8 compare the original Proposed configuration (denoted as the Target-duration reference in this subsection) with SRC-FPT in the original peer-method context. Tables 9, 10 provide the direct comparison between SRC-FPT and the peer methods.

Table 7. Performance comparison on the XJTU-SY dataset with the Target-duration reference and the deployable SRC-FPT configuration.

Bold values indicate the lowest (best) RMSE or MAE in each row; tied minima are also shown in bold.

Table 8. Performance comparison on the PRONOSTIA dataset with the Target-duration reference and the deployable SRC-FPT configuration.

Bold values indicate the lowest (best) RMSE or MAE in each row; tied minima are also shown in bold.

Table 9. Direct comparison of the deployable SRC-FPT configuration with peer methods on the XJTU-SY dataset.

Bold values indicate the lowest (best) RMSE or MAE in each row; tied minima are also shown in bold.

Table 10. Direct comparison of the deployable SRC-FPT configuration with peer methods on the PRONOSTIA dataset.

Bold values indicate the lowest (best) RMSE or MAE in each row; tied minima are also shown in bold.

With the same FPT boundary calibration, SRC-FPT obtains mean RMSE/MAE values of 0.166/0.139 on XJTU-SY and 0.179/0.146 on PRONOSTIA, compared with 0.165/0.139 and 0.182/0.147 for the Target-duration reference. The Target-duration reference is marginally lower in RMSE on XJTU-SY, whereas SRC-FPT is lower in both metrics on PRONOSTIA. Because the peer-method outputs were not subjected to the same FPT boundary calibration, the cross-method comparisons are provided for context; the primary controlled evidence is the within-model comparison between the Target-duration reference and SRC-FPT.

These results support a limited conclusion: the competitive aggregate performance of SRC-FPT is not dependent on knowing the exact FPT-to-failure duration of the target bearing.

4.4 Comparative performance analysis

To quantitatively assess the computational efficiency of our proposed model, we compared it against the baseline ConvLSTM model using four key metrics: parameter count, FLOPs, inference time, and training time. The results, summarized in Table 11, indicate that our multi-representation domain generalization approach entails higher computational demands. Specifically, the dual-channel input processing and the bidirectional multi-scale ConvLSTM architecture contribute to increased parameter count and FLOPs. For instance, the use of BiLSTM doubles the parameters compared to a unidirectional LSTM, while multi-scale convolutional kernels expand the model's capacity to capture global and local features.

Table 11. Model performance comparison.

Although these design choices lead to longer training and inference times, they are essential for enabling unseen-bearing RUL prediction without target-bearing training data. The computational overhead is a direct consequence of our model's ability to learn shared degradation patterns across different bearings—a core requirement for domain generalization. In practical scenarios where target data is unavailable, the proposed method eliminates the need for costly data collection and model retraining, thereby offering significant operational savings despite higher on-line computational costs.

Moreover, the performance gains demonstrated in Sections 3.2, 3.3 justify the computational trade-offs. Our method achieves more stable and accurate predictions under complex degradation conditions, which is critical for real-world industrial applications. Future work will focus on optimizing computational efficiency through model compression and lightweight architectures without compromising multi-representation integrity.

5 Conclusion

This paper proposed a linearly constrained multi-representation domain generalization framework for bearing RUL prediction when target-bearing samples are unavailable during model development. The method aligns raw time-domain vibration and CWT time-frequency representations as dual-channel inputs, uses a bidirectional multiscale ConvLSTM network to extract spatiotemporal degradation features, and introduces a lightweight linear temporal prior to encourage physically plausible RUL evolution.

Experiments on the XJTU-SY and PRONOSTIA datasets show that the proposed method achieves lower average RMSE and MAE than the comparison methods. The ablation results indicate that the multiscale, bidirectional, and temporal-constraint components each contribute to the complete model under the reported settings, and representative RUL curves further show improved tracking of the normalized degradation trajectory. These results suggest that combining complementary vibration representations with temporal degradation modeling is beneficial for cross-bearing RUL prediction under target-data scarcity.

The present study predicts normalized RUL, which reflects the relative degradation stage rather than absolute remaining time. Future work will focus on mapping normalized RUL to real time-scale RUL using industrial historical data, online degradation-rate estimation, lifetime statistics, probabilistic degradation models, or Bayesian updating mechanisms. Additional work will also consider broader operating-condition variations, stronger physical degradation modeling, and lightweight architectures for more efficient deployment.

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Appendix: Additional XJTU-SY ablation results

This appendix reports additional ablation comparisons for the XJTU-SY dataset under operating conditions 2 and 3.

Footnotes

Edited by: Kaleem Siddiqi, McGill University, Canada

Reviewed by: Peter Karsmakers, KU Leuven Campus Geel, Belgium

Yuchen Lu, Harbin Engineering University, China