Accurate remaining useful life prediction and uncertainty quantification for wind turbines using temporal convolutional variational deep Gaussian processes.

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Learn more: PMC Disclaimer | PMC Copyright Notice Sci Rep . 2026 Apr 10;16:16791. doi: 10.1038/s41598-026-47273-4 Accurate remaining useful life prediction and uncertainty quantification for wind turbines using temporal convolutional variational deep Gaussian processes Shuo Cui 1, 2, 3 , Xiuli Liu 1, 2, ✉ , Guoxin Wu 1, 2, 3 , Wan Qiao 1, 2, 3 , Jianyu Liu 1, 2, 3 , Xuanyu Chen 1, 2, 3 Author information Article notes Copyright and License information 1 Key Laboratory of Modern Measurement & Control Technology, Ministry of Education, Beijing Information Science and Technology University, Beijing, 100192 China 2 Beijing Key Laboratory of Mechanical and Electrical System Measurement and Control, Beijing Information Science and Technology University, Beijing, 100192 China 3 Mechanical Electrical Engineering School, Beijing Information Science and Technology University, Beijing, 100192 China ✉ Corresponding author. Received 2025 Jun 16; Accepted 2026 Mar 31; Collection date 2026. © The Author(s) 2026 Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/ . PMC Copyright notice PMCID: PMC13223215  PMID: 41963447 Abstract It remains highly challenging to extract features from monitoring signals of wind turbine components under varying operating conditions and complex structures, as well as to quantify the uncertainty of Remaining Useful Life (RUL) predictions. To address these issues, this paper proposes a method called Temporal Convolutional Variational Deep Gaussian Process (TCVDGP). This method introduces Deep Gaussian Processes (DGP) to model the nonlinear relationship between input features and RUL, and employs variational inference to efficiently estimate the posterior distribution for accurate uncertainty quantification. In addition, by integrating Temporal Convolutional Networks (TCN) and a Sequential Attention Mechanism (SAM), the model captures both short- and long-term dependencies in time series data, enhancing adaptability to complex and variable operating conditions. TCVDGP ultimately produces RUL predictions with confidence intervals. Experiments conducted on planetary gearbox and industrial wind turbine gearbox datasets, combined with correlation analysis between the model’s hidden layer outputs and internal gearbox parameters, reveal the reasons behind performance differences of TCVDGP across different degradation stages. The results show that TCVDGP not only effectively captures the health evolution of wind turbine gearboxes but also dynamically adjusts to key variables, significantly improving the reliability and interpretability of RUL predictions under complex operating conditions. Keywords: Remaining useful life prediction, Interpretability, Time convolutional variational deep Gaussian Process, Deep Gaussian Process, Wind turbine gearbox Subject terms: Mechanical engineering, Computer science Introduction With the continuous development of human society, reducing carbon emissions and achieving carbon neutrality have become global common goals, and the development of renewable energy is the key path to achieving these goals 1 , 2 . Wind energy, as an important renewable energy source, has been widely applied. However, with the ongoing construction and operation of large wind turbines, significant accidents caused by component failures occur frequently. Therefore, fault prediction and health management of wind turbines (Prognostics and Health Management, PHM) have attracted increasing attention 3 – 6 . Remaining Useful Life (RUL) prediction, as the core of the PHM system, not only provides critical decision support for equipment maintenance and management but also effectively helps prevent failures and extends the operational lifespan of equipment. Currently, the remaining useful life (RUL) prediction methods for wind turbine components can be mainly categorized into three types: physics-based life prediction methods, data-driven life prediction methods, and physics-data hybrid model-based life prediction methods 7 – 9 . Physics-based methods typically involve establishing performance degradation models through the equipment’s fault mechanisms. However, due to the complex structures of wind turbines in practical applications, building accurate physical models becomes extremely difficult. This limitation makes physics-based models less effective in describing the degradation process of wind turbine components 10 . Due to the limitations of physics-based models, the development of hybrid models has been hindered. In contrast, data-driven methods learn degradation patterns directly from sensor data, reducing dependence on physical knowledge, and are particularly suited for industrial scenarios in the era of big data. With the gradual development of data-driven technologies, these methods have become dominant in predicting the RUL of wind turbine components 11 . However, accurate RUL prediction still faces numerous challenges. These challenges include the poor adaptability of models under complex and variable operating conditions, as variations in rotor speed, load, and environmental factors during wind turbine operation can cause feature distribution drift and affect prediction stability; the highly nonlinear, stage-dependent, and non-stationary nature of component degradation processes, which makes it difficult for a single model to fully capture their evolution patterns; and the lack of effective uncertainty quantification in most existing methods, which limits their ability to provide confidence levels for prediction results to support maintenance decision-making. In recent years, deep learning techniques have been widely applied in the field of RUL prediction. To address these challenges, deep learning techniques have been widely applied in the field of RUL prediction in recent years, demonstrating strong capabilities in automatically extracting complex degradation features and handling high-dimensional data 12 , 13 . For example, the combination of Deep Belief Networks (DBN) and Self-Organizing Maps (SOM) can effectively extract degradation features of equipment, thereby improving the accuracy of RUL prediction 14 . The weighted time-embedded Transformer framework further optimizes prediction accuracy by integrating 3D convolution modules with moving window Transformer blocks 15 . Parallel Neural Networks (PNN), through parallel processing paths and ConvLSTM modules, efficiently capture degradation trends in complex time series data, excelling particularly in bearing RUL prediction 16 , 17 revealed bearing degradation by constructing health indicators (HI) that integrate multi-scale features with distribution similarity, optimizing HI characteristics using Wasserstein distance and multi-objective optimization, and enhancing prediction accuracy and generalization ability with gated recurrent unit networks. RUL prediction under varying operating conditions is a major challenge in industrial applications 18 , 19 , and researchers have proposed various methods to address this issue. Among them, the adaptive dynamic feedback framework significantly improves prediction robustness through feedback mechanisms and multi-model selection strategies, particularly in the prediction of wind turbine gearbox RUL under varying conditions 20 . Additionally, the adaptive generalized Cauchy model explains long-term dependencies by combining fractal dimension and Hurst index, and adapts to changes in degradation trends using dynamic drift coefficients, thereby effectively enhancing prediction accuracy 21 . In feature extraction, a novel dual-attention gated recurrent unit (GRU) neural network is proposed, which significantly improves model performance by enhancing adaptive feature extraction capabilities 22 . Meanwhile, the TCN-Transformer-based prediction framework enhances the stability of features under complex operating conditions through Kalman filtering for noise reduction and nonlinear smoothing techniques 23 . Finally, an RUL prediction method based on health state and trajectory similarity constructs a lifetime database through K-means clustering, and achieves higher prediction accuracy by combining similarity measurements and weighted matching strategies 24 . These methods provide diverse solutions for RUL prediction under varying operating conditions. At present, most studies on deep learning-based RUL prediction focus on achieving higher point estimation accuracy. For example, to overcome the limitations of manually designed auxiliary tasks 25 proposed a multi-task learning method based on a meta-auxiliary generative network; to balance accuracy and computational complexity 26 adopted a multi-objective evolutionary deep network 27 employed a GRU-LSTM fusion model to enhance feature extraction capability. These studies have significantly improved the prediction performance of RUL. However, there is significant uncertainty in the lifetime prediction of wind turbines. Taking the planetary gearbox as an example: First, the feature extraction process, which involves human participation, often relies on the prior experience of experts and has a certain degree of subjectivity. Secondly, under varying operating conditions, the fault characteristic frequencies and their amplitudes of the gears change with the speed and load. When the speed changes, the gear meshing frequency and fault characteristic frequency change accordingly. When the load changes, it causes speed fluctuations, which also result in changes to the gear meshing frequency and fault characteristic frequency. Fault features detected at a certain time may be caused by randomly changing operating conditions, resulting in pseudo-fault signals. Furthermore, the complex transmission structure causes the fault shock signals of components such as gears and bearings inside the gearbox to couple with vibration components from high-speed and low-speed shafts. These factors introduce significant uncertainty in the RUL prediction of wind turbine gearboxes. To improve prediction accuracy, an incremental contrastive hybrid model, combined with contrastive learning Transformer and enhanced generalized Wiener process, has been proposed for online prediction and uncertainty quantification 28 . Additionally, through the Gaussian Process Autoregressive-Variational Autoencoder framework, cognitive and stochastic uncertainties are decomposed, enhancing prediction reliability and optimizing prediction efficiency through active learning 29 . Moreover, the proposed multi-distribution fusion method adaptively integrates multiple candidate predictions, significantly improving adaptability and uncertainty quantification accuracy under complex operating conditions 30 . However, most existing deep learning methods can only perform point predictions and cannot provide uncertainty in prediction results, which greatly limits the practical application of deep learning in the RUL prediction of wind turbine components 31 , 32 . To address the challenges of extracting time-varying features from long sequences and the uncertainty in prediction results for wind turbine component lifetime prediction, this paper proposes a Time Convolutional Variational Deep Gaussian Process (TCVDGP) method. This method innovatively integrates the Temporal Convolutional Network (TCN) with the Sequential Attention Mechanism (SAM) into the internal structure of the Deep Gaussian Process (DGP). By improving the internal algorithm of the DGP, the method not only significantly enhances the model’s prediction performance but also combines correlation analysis between the model’s hidden layer outputs and internal gearbox parameters to deeply reveal the reasons for performance differences in RUL prediction across different stages (early, mid, and late), thereby significantly improving the model’s interpretability. Additionally, TCVDGP employs variational inference to efficiently estimate posterior distributions, providing accurate uncertainty quantification for the model, further enhancing the reliability and credibility of the prediction results. The main contributions of this paper are as follows: This paper introduces a novel attention mechanism—SAM (Sequential Attention Mechanism), which aims to address the issue of enhancing useful information and suppressing redundant information in one-dimensional convolution operations by learning more discriminative life-related features from the input one-dimensional signals. This paper proposes a TCVDGP method, which improves the internal structure of the Deep Gaussian Process (DGP) algorithm by deeply embedding SAM and Temporal Convolutional Network (TCN). SAM dynamically adjusts feature weights, compensating for the shortcomings of DGP in feature extraction; TCN captures both short-term and long-term temporal dependencies, optimizing DGP’s ability to model complex sequential data and varying operating conditions. On this basis, TCVDGP uses variational inference to efficiently estimate posterior distributions, providing precise uncertainty quantification. TCVDGP is able to effectively capture key features under varying operating conditions of wind turbine gearboxes and dynamically focus on important variables, significantly improving the model’s RUL prediction performance in dynamic environments. The TCVDGP method significantly improves the accuracy of RUL predictions while providing an interpretability analysis of the prediction results. Through correlation studies between the model’s hidden layer outputs and internal gearbox parameters, it deeply reveals the reasons for performance differences in RUL prediction at different stages (early, mid, and late). This model not only helps optimize wind turbine gearbox maintenance decisions, reduce downtime, and minimize resource waste, but also provides strong technical support for global carbon neutrality and carbon emission reduction goals. Model theory TCN In terms of causality, at time t, the value from the previous layer only depends on the value at time t and its previous time step at the next layer. Fig. 1 shows the receptive field size under different scaling factors. In the figure, represents the dilation factor of each convolution layer, and the value of is typically set as , where the filter size for each layer is 2. This structure achieves access to all historical inputs with a limited number of layers. The receptive field is calculated using the formula . As seen, by increasing the dilation factor , the output at the top layer can represent a larger range of inputs without increasing the filter size or network depth, thus effectively avoiding the issue of having too many layers in the network. Fig. 1. Open in a new tab Causal dilated convolution. To ensure the stability of the TCN network, residual blocks are used in place of convolution layers, as shown in Fig. 2 . In the residual block, the TCN network consists of two layers of dilated causal convolutions for convolution operations, two layers of ReLU for non-linear processing, two layers of weight normalization for normalization, and two layers of dropout for regularization. Additionally, to ensure that the input and output of the residual block have the same width, a convolution is introduced. Fig. 2. Open in a new tab Structure of the TCN residual block. DGP DGP captures more complex structures by stacking multiple GP layers, where the output of each layer serves as the input for the next layer. The schematic diagram is shown in Fig. 3 . The input–output relationship for each layer is as follows: , where, represents the raw input data, and is the Gaussian noise at the -th layer. Fig. 3. Open in a new tab Structure of DGP. Mathematically, the DGP model can be viewed as a composite multivariate function : 1 where ∘ denotes function composition. In the DGP framework, each intermediate node is associated with two GPs, i.e., output and input. Although the mapping between each layer is a GP, due to the marginalization of the latent variables being a non-Gaussian integral, the overall DGP prior is no longer a GP 33 , as shown in Eq. ( 2 ). 2 Training the DGP model is equivalent to maximizing the marginal likelihood of the output Y . For convenience, let represent the corresponding noiseless variables for , i.e., , and . Based on the conditional independence properties shown in Fig. 3 , the conditional probability density of the hidden layer , , can be simplified to and further decomposed as . Let represent the set of intermediate variables to be marginalized. Thus, the model likelihood can be represented as a vector integral, as shown in Eq. ( 2 ). The computation of Eq. ( 2 ) is mathematically infeasible. The main difficulty lies in the non-linear appearance of in the inverse of the kernel covariance matrix in . RUL prediction based on TCVDGP Temporal attention module In convolution operations, different feature maps have varying recognition capabilities for RUL prediction, and within each feature map, the importance of the lifetime features at different positions also differs. Therefore, it is essential to suppress redundant or irrelevant information during the computation process and enhance the lifetime-related information. To address this issue, this paper proposes a novel attention mechanism — the Temporal Attention Module (SAM). This module is designed to solve the problem of effectively enhancing useful information and suppressing redundant information in one-dimensional convolution operations by learning more discriminative lifetime features from the input one-dimensional signals 34 . SAM can specifically enhance the focus on lifetime-related information, improving the model’s sensitivity to key features, thereby effectively optimizing the performance of convolution operations. This is particularly significant when combined with TCN and DGP for RUL prediction. The basic framework of SAM is shown in Fig. 4 and can be divided into three parts: the channel path, the temporal path, and the residual path. Fig. 4. Open in a new tab Structure of the SAM algorithm. In the channel path, the input feature (where c is the number of channels and n is the length of the feature map) is first projected onto the temporal signal through a convolution layer. Next, the softmax function is used to compute the channel weight vector , as shown by the following formula: 3 Through this operation, the model assigns different weights to each feature map based on their contribution, thereby enhancing the feature maps related to RUL. In the temporal path, is first processed using impulse factor pooling (IFpool) to compress the global temporal information into a vector . Different lifetimes have different impulse factors, and IFpool computes the impulse factor for each feature map to reflect the variation of the impulses. The compression calculation for the -th feature map is as follows: 4 Next, a convolution layer is applied to encode the channel dependencies, and finally, the Sigmoid function is used to obtain the impulse weight vector , as shown in the following formula: 5 The channel weight vector and impulse weight vector are multiplied to obtain a weight matrix , with values in the range of [0, 1], and then point-multiplied with the input X . To prevent deep feature values from becoming too low and affecting the accuracy of lifetime prediction, residual connections are used to mitigate the “vanishing” problem of deep features. Finally, the output of SAM is obtained through the following formula: 6 Through the organic combination of the three paths, SAM can more effectively enhance the temporal features in the one-dimensional convolution operation, thus improving the model’s performance in lifetime prediction tasks. TCVDGP DGP, as a flexible non-parametric model, is capable of capturing complex nonlinear relationships and quantifying the uncertainty of prediction results. However, its application in high-dimensional time series data still faces two key challenges: first, the DGP method lacks a mechanism to handle the dynamic characteristics of time series data, making it difficult to fully explore the short-term and long-term dependencies in the data; second, DGP lacks dynamic adjustment capabilities, which makes it challenging to effectively focus on the key features in time series data. To address these issues, the TCVDGP method is proposed. By dynamically optimizing the covariance matrix and improving variational inference, TCVDGP enhances the model’s ability to capture temporal features and improves prediction accuracy. In the TCVDGP model, the TCN module first performs multi-scale extraction on the input feature generating time-dependent embedded features : 7 where represents the output features of the TCN, with channels, each of length . The TCN module, through one-dimensional convolution operations, effectively extracts both short-term and long-term dependency patterns from the data, providing a foundation for subsequent feature optimization. On this basis, to further dynamically adjust the importance of the features, the SAM attention mechanism is introduced. SAM dynamically optimizes the feature weight matrix by calculating the channel weights and impulse weights : 8 where represents the channel weights, which highlight the importance of each channel, and represents the impulse weights, which strengthen significant features from different time periods. The feature representation after attention optimization is: 9 In the GP model, the covariance matrix is central to describing the correlation between data points. The DGP model uses a fixed kernel function to compute the covariance matrix, making it difficult to dynamically adjust to accommodate the varying importance of different features. In the TCVDGP method, a dynamically optimized feature representation is introduced, and the covariance matrix is redefined as: 10 where “kernel” refers to a commonly used kernel function (such as the RBF or Matern kernel). This dynamic optimization mechanism allows the covariance matrix to adapt based on feature weights, enhancing the model’s ability to capture key features. On the basis of dynamically optimizing the covariance matrix, the model is efficiently trained using an improved variational inference. Variational inference introduces inducing points and inducing variables to sparsely approximate the covariance matrix of each layer’s GP. The function values of each GP layer are given by: 11 Based on this, the optimization objective of the variational distribution can be defined as the variational lower bound : 12 where , is the covariance matrix between all training points, and is the sparse covariance matrix based on the inducing points: 13 By maximizing the variational lower bound ,, the model can simultaneously optimize the inducing points and the attention weights M. The gradient optimization formula for the weight matrix M is: 14 The optimization gradient for the inducing points can be calculated using the following formula: 15 The proposed TCVDGP structure is shown in Fig. 5 , and the RUL prediction process is shown in Fig. 6 . In TCVDGP, the input consists of raw time-series data collected by sensors. After data preprocessing, these data are fed into the model for feature extraction and prediction. First, in the first layer of TCVDGP, TCN is used as a feature extractor to extract multi-level feature representations from the input time-series data. The TCN module effectively captures short-term and long-term dependencies in the data through one-dimensional convolution and pooling operations, and it can adapt to complex temporal patterns. Next, to further enhance the focus on key features, SAM is integrated into the TCN, dynamically adjusting features through channel weights and impulse weights. This attention mechanism adaptively amplifies the features related to the equipment’s health status and suppresses redundant or irrelevant information, thus improving the quality of the feature representation. Then, variational inference is used during the feature extraction process to perform some preliminary modeling and reasoning of the latent variables. Variational inference helps simplify the complex posterior distribution, making the feature extraction process more efficient. Fig. 5. Open in a new tab Structure of the TCVDGP method. Fig. 6. Open in a new tab RUL prediction process of TCVDGP. In the second layer, the attention weights of the features are dynamically computed using SAM, enabling the model to adaptively focus on key features and optimize feature representation. Next, by combining the RBF kernel function with the dynamically adjusted attention mechanism, the ability to model the correlation between data is further enhanced. Then, the Evidence Lower Bound (ELBO) objective function is used for optimization. By maximizing the ELBO, the model approximates the true posterior distribution. ELBO optimization not only effectively improves the model’s ability to handle complex nonlinear features, but also enhances the model’s inference efficiency in high-dimensional data by optimizing the variational distribution and inducing point parameters. Finally, a distribution containing the predicted mean and standard deviation is generated. The predicted mean represents the central tendency of the RUL, while the standard deviation reflects the uncertainty in the prediction. Based on this distribution, the RUL prediction value and its confidence interval are calculated, providing a quantified level of confidence for the prediction result. Uncertainty quantification TCVDGP uses variational inference to approximate the posterior distribution, which in turn generates the predicted mean and standard deviation , and constructs confidence intervals using these quantifications. The core objective of variational inference is to introduce an easily optimized variational distribution to approximate the real posterior distribution which is difficult to solve directly. This process is achieved by optimizing the Evidence Lower Bound (ELBO): 16 Given the variational distribution , the model maximizes the ELBO to better fit the observed data and minimizes the KL divergence term to make the variational distribution closer to the true posterior distribution . Ultimately, the predicted mean and variance are extracted from the optimized variational distribution for the model’s prediction and uncertainty quantification. After introducing TCN and SAM, the predicted mean in TCVDGP is: 17 The predicted variance is calculated using the dynamically weighted covariance matrix: 18 With the introduction of attention weights, , calculated based on the dynamically adjusted covariance matrix, enhances sensitivity to features and time steps. dynamically reflects the weights of key features and time steps, improving the precision of uncertainty quantification. The overall loss function is: 19 The 95% confidence interval, with the introduction of dynamic weights, is: The formula for the confidence interval width (PIW) is: 20 Through the dynamic adjustments of TCN and SAM, the predicted mean and variance better capture the time dependencies and importance of the input features, accurately representing the model’s uncertainty for specific features and time steps. RUL prediction performance evaluation metric To evaluate the performance of the RUL prediction model on the test data, a scoring function (Scoring Function, SCORE) is provided to measure the quality of the model 35 . The definition of SCORE is given in Eq. ( 21 ). 21 where is the total number of samples in the test set, and , where is the predicted RUL for test sample , and is the true RUL for test sample . Eq. ( 21 ) provides different penalty values when the model underestimates or overestimates the RUL. If the predicted RUL is less than the true RUL, the penalty is smaller because there is still time for maintenance, and no major system failure will occur. If the predicted RUL is greater than the true RUL, the penalty is larger, because in this case, the maintenance plan will be delayed, potentially leading to system failure. This asymmetric design enables the SCORE metric to distinguish between the different consequences of overestimation and underestimation, making it more consistent with practical maintenance safety requirements. 22 Eq. ( 22 ) represents the Root Mean Square Error (RMSE), which is commonly used to evaluate RUL predictions. RMSE imposes equal penalties on overestimation and underestimation, serving as a classical and intuitive metric for measuring prediction error, which can reflect the overall accuracy. Whether the predicted value is too high or too low, RMSE assigns the same penalty weight. Fig. 7 shows the difference between SCORE and RMSE. indicates that the predicted RUL is the same as the true RUL. For both SCORE and RMSE, the smaller the value, the better the result. Fig. 7. Open in a new tab Comparison of SCORE and RMSE ( = RUL’ RUL, where RUL’ is the predicted RUL, and RUL is the true RUL). Compared with traditional methods that rely solely on RMSE, the SCORE metric can more realistically characterize prediction risks in maintenance scenarios. In industrial equipment life prediction, overly optimistic estimates (i.e., overestimating the remaining life) are often more dangerous than conservative ones (i.e., underestimation); therefore, higher weights should be assigned to overestimation errors. In this study, both RMSE and SCORE are jointly employed — RMSE measures the prediction accuracy, while SCORE reflects the model’s capability to control risks related to maintenance safety and cost-effectiveness, thereby providing a more practical evaluation basis for wind turbine component health management. Mean Absolute Error (MAE) is a method for measuring prediction accuracy by calculating the average absolute difference between the predicted values and the actual values. Compared to RMSE, MAE is less sensitive to outliers, making it more fair in performance evaluation and less affected by outliers. The formula for calculating MAE is: 23 To further quantify the uncertainty prediction performance of the model, the Width of Interval (WI) metric is introduced to measure the width of the predictive confidence interval. For the -th test sample, the prediction interval width is defined as the difference between the upper and lower bounds of the prediction interval. The mean interval width is defined as: 24 where and denote the lower and upper bounds of the predictive confidence interval, respectively, and represents the number of test samples. A smaller WI value indicates a narrower prediction interval, implying that the model achieves higher uncertainty estimation accuracy while maintaining reliable prediction performance. Experimental verification Wind turbine planetary gearbox test rig dataset The wind turbine planetary gear dataset 36 was obtained from an accelerated life test of planetary gear faults in a wind turbine gearbox conducted independently. The dataset is derived from a single planetary gear’s full-life accelerated fatigue test. To ensure temporal consistency and prevent information leakage, the entire lifecycle data is split into two parts: the first 70% is used as the training set to learn the gear degradation patterns, and the remaining 30% is used as the test set to validate the model’s performance in predicting degradation in the later stages. This split aligns with the practical engineering scenario of RUL prediction, where existing observation data is used to forecast future degradation trends. In the data preprocessing step, the data from four sensor channels is normalized using the min–max method to the range of [-1, 1], as shown in Eq. ( 25 ). 25 where is the reading of the -th sensor for the -th sample, and represent the maximum and minimum values of the j -th sensor readings in the training dataset, and is the normalized value of . Prediction results and comparative analysis To further demonstrate the performance of the DGP and TCVDGP methods in predicting the RUL of the wind turbine planetary gears throughout the degradation process, Fig. 8 is shown. In the figure, the blue line represents the true RUL of the engine unit, the red line represents the predicted RUL, and the gray area represents the 95% confidence interval. Fig. 8. Open in a new tab ( a ) RUL prediction performance of DGP; ( b ) RUL prediction performance of TCVDGP. TCVDGP is trained using the Adam optimizer with a learning rate of 0.02, a batch size of 100, and 1200 training epochs. The kernel size of the temporal convolutional network is set to 2, and the dropout rate is 0.2. The deep Gaussian process is trained using variational inference, and the ELBO is adopted as the loss function. 1) Comparison of RUL Prediction between DGP and TCVDGP. The results are shown in Figs. 8 (a) and (b), where the predicted values of both methods closely match the actual RUL, effectively revealing their degradation trends. The accuracy of RUL prediction is higher in the later stages of degradation than in the early stages. As shown in Fig. 8 , the TCVDGP model outperforms the DGP model overall, particularly in the later RUL stages as indicated in Table 1 , demonstrating that the improved design significantly enhances prediction accuracy. Since a convolutional structure is introduced in time series modeling to extract local correlations, the training time of TCVDGP is slightly longer than that of DGP. However, this structure enables the model to make better use of sequential features, thereby achieving superior performance in prediction accuracy, uncertainty quantification, and the capture of complex degradation patterns. Table 1. Prediction performance of DGP and TCVDGP for RUL ≤ 15. Model Evaluation metrics RMSE SCORE MAE WI DGP 4.778 4.570 4.163 6.35 TCVDGP 3.532 4.102 2.782 4.23 Open in a new tab Although DGP can capture the overall degradation trend well during the prediction process, it shows oscillations of varying amplitudes in the early and mid stages, which ultimately reduces the prediction accuracy. To achieve more accurate RUL predictions, the TCVDGP method is applied. As seen in Fig. 8 (b), compared to DGP, TCVDGP significantly reduces the oscillations in the RUL prediction, leading to a substantial improvement in prediction accuracy. Additionally, the confidence interval of TCVDGP is narrower than that of DGP, further demonstrating the model’s reliability and stability. 2) Comparison of TCVDGP with Other RUL Prediction Models. To verify the superiority of the TCVDGP method, this section selects representative RUL prediction models, including TCN 37 , LSTM with attention 38 , Transformer 39 , Bi-level LSTM Scheme 40 , DGP 41 , and AGCN 42 , for experimental comparison. Specifically, TCN employs one-dimensional causal convolutions and residual connections to achieve long-term sequence feature extraction with high computational efficiency, but it shows limited adaptability to variable operating conditions. LSTM with attention enhances the capture of key degradation features by combining long short-term memory networks with an attention mechanism, making it suitable for non-stationary sequences, although it has relatively high training complexity. Transformer relies on self-attention to model global dependencies and excels at capturing long-range relationships, but it requires large-scale training data and is more sensitive to noise. The Bi-level LSTM Scheme adopts a hierarchical LSTM structure to decompose degradation trends at different time scales, improving prediction accuracy, but at the cost of a large number of parameters and high training overhead. DGP introduces a deep structure into the Gaussian process framework to enhance nonlinear modeling and uncertainty quantification capabilities, though its inference cost remains high for large-scale datasets. AGCN leverages graph convolutional networks to model inter-component relationships and incorporates attention mechanisms to enhance information propagation, but its model construction relies on prior topological knowledge. Summarizing the characteristics of these comparative methods in this way provides a clearer demonstration of the advantages of TCVDGP in balancing sequential modeling capability and uncertainty quantification. Fig. 9 shows the performance results of the seven models. It can be seen that TCVDGP has significantly lower RMAE, SCORE, and MAE metrics across all planetary gearbox datasets compared to other methods, demonstrating its excellent predictive ability. Furthermore, as observed in Fig. 10 (a), the prediction error distribution of TCVDGP is more concentrated around an error of 0, while Fig. 10 (b) shows that its RMSE is significantly lower than that of other models, with the average value being closer to 0. These results fully demonstrate the superiority of this method in improving RUL prediction accuracy. Fig. 9. Open in a new tab Comparison of RUL prediction performance between TCVDGP and comparison models. Fig. 10. Open in a new tab Comparison of different models. ( a ) Box plot of prediction errors with overlaid normal distribution; ( b ) Box plot of RMSE for different algorithms. To further demonstrate the advantages of the proposed TCVDGP model, Table 2 compares different methods in terms of feature extraction capability, attention mechanism, probabilistic modeling, and uncertainty quantification. Traditional deep learning methods, such as TCN, LSTM, and Transformer, can effectively extract temporal features but are deterministic models and cannot quantify prediction uncertainty. Bi-level attention LSTM and AGCN enhance feature representation by introducing attention mechanisms; however, they still lack probabilistic modeling capability. Although DGP can provide uncertainty estimation, its feature extraction capability is limited, making it difficult to directly handle complex high-dimensional temporal degradation data. Table 2. Comparison of feature extraction, probabilistic modeling, and uncertainty quantification capabilities of different models. Method Temporal feature extraction Attention mechanism Spatial/structural modeling Probabilistic modeling Uncertainty quantification Nonlinear modeling capability TCN Yes No No No No Yes LSTM Yes No No No No Yes Transformer Yes Yes No No No Yes Bi-level attention LSTM Yes Yes No No No Yes AGCN Yes Yes Yes No No Yes DGP No No No Yes Yes Yes TCVDGP (Proposed) Yes Yes Yes Yes Yes Yes Open in a new tab In contrast, the proposed TCVDGP model integrates a temporal convolutional network, temporal attention mechanism, and deep Gaussian process into a unified framework, enabling efficient temporal feature extraction, adaptive enhancement of critical degradation information, nonlinear probabilistic modeling, and uncertainty quantification. Therefore, the TCVDGP model achieves significant advantages in both prediction accuracy and prediction reliability. Wind farm gearbox dataset Test data acquisition To evaluate the applicability of the proposed method in real-world scenarios, monitoring data from the Hanjinqi Wind Farm in Inner Mongolia was used in the experiment. The wind farm has a total of 31 turbines, and the monitoring data from three turbines of the same model were selected for the experiment. The structure of the turbine drivetrain is shown in Fig. 11 . The dataset consists of vibration signal features from 12 measurement points: gearbox second-stage planetary radial, upper frame (vertical to the main shaft), main shaft rear bearing radial, and main shaft front bearing radial, among others. The wind turbine gearbox operates continuously until failure occurs. Since faults typically happen in the later operational stages, which are the critical periods for maintenance and repair, only the monitoring data from the later stages are selected for RUL prediction analysis. The dataset is divided into a training set and a testing set for model training and evaluation. All data are normalized to the range [-1,1] using the Min–Max method. Fig. 11. Open in a new tab Structural diagram of the wind turbine transmission system. Comparison of RUL testing and prediction methods To better demonstrate the performance of the DGP and TCVDGP methods in predicting the RUL of wind turbine gearboxes throughout the degradation process, Fig. 12 is shown. Fig. 12. Open in a new tab ( a ) DGP prediction performance of wind turbine gearbox 1–3; ( b ) DGP prediction performance of wind turbine gearbox 1–6; ( c ) DGP prediction performance of wind turbine gearbox 1–11. 1) Comparison of RUL Prediction between DGP and TCVDGP. The RUL predictions from the DGP and TCVDGP methods for the wind turbine gearbox closely match the actual RUL, achieving good results in RUL prediction. As shown in Figs. 12 (a)-(c), although DGP exhibits relatively high accuracy in its predictions, it shows oscillations of varying amplitudes throughout the entire time period. This volatility ultimately leads to a reduction in prediction accuracy. In contrast, the TCVDGP method also shows oscillations of varying amplitudes in the early stages, but over time, its predictions become more stable, as shown in Figs. 13 (a)-(c). This characteristic indicates that TCVDGP is better at adapting to the long-term trends in the data, mitigating the impact of fluctuations in the mid-to-late stages, thereby improving the stability and reliability of the predictions. Particularly in the later stages, accuracy is more critical because it directly affects the effectiveness of maintenance and decision-making. Experimental comparisons show that TCVDGP not only has potential in terms of accuracy but also performs better in prediction consistency and sustainability. TCVDGP demonstrates stronger practical value, especially in applications requiring continuous monitoring and management. Fig. 13. Open in a new tab TCVDGP prediction performance for gearboxes 1–3 ( a ), 1–6 ( b ), and 1–11 ( c ). There is a significant difference in the confidence interval width between the TCVDGP and DGP models. As shown in Figs. 12 and 13 and Table 2 , the confidence intervals of TCVDGP are generally more compact, reflecting higher sharpness. More importantly, while maintaining good coverage of the true RUL, TCVDGP can still substantially narrow its confidence intervals, indicating that it not only effectively captures the latent structures within the data but also achieves superior effectiveness in uncertainty estimation. This characteristic of ‘improving sharpness while maintaining calibration’ enables TCVDGP to demonstrate stronger representation and generalization capabilities in complex time series modeling. For practical applications such as RUL prediction and fault diagnosis, more compact and well-calibrated confidence intervals can provide more informative decision-making references, thereby enhancing the model’s practicality. 2) Comparison of TCVDGP with Other RUL Prediction Models. The experimental results for RMSE, SCORE, and MAE are shown in Figs. 14 (a), (b), and (c). As seen in Fig. 14 , TCVDGP outperforms DGP in prediction performance, and DGP also performs relatively well in RUL prediction. However, overall, TCVDGP performs the best. The results indicate that the introduction of TCN and SAM significantly enhanced the performance of TCVDGP, improving its ability to capture time-dependent features and increasing the accuracy of RUL predictions. Fig. 14. Open in a new tab Comparison of different models. ( a )–( c ) Performance comparison for gearboxes 1–3, 1–6, and 1–11, respectively. ( d ) Prediction error of different models for gearbox 1–3. Additionally, as shown in Fig. 14 (d), the TCVDGP method demonstrates a clear advantage in terms of prediction error range and distribution. Compared to other models, TCVDGP has the narrowest error distribution range, indicating that its prediction results are more stable. Extreme errors are fewer, and the median and mean of the errors are closer to 0, showing higher prediction accuracy. Furthermore, the error distribution of TCVDGP is more concentrated, indicating that its predictions have smaller fluctuations across different samples and are less affected by data noise. In contrast, other models (such as TCN and LSTM with attention) exhibit wider error ranges, with greater uncertainty. While DGP and Transformer perform well in prediction, their stability and prediction accuracy still fall short compared to TCVDGP. This further validates that the TCVDGP method demonstrates excellent robustness in RUL prediction Table 3 . Table 3. Comparison of WI metric values between DGP and TCVDGP model prediction results. Model 1–3 1–6 1–11 DGP 10.52 10.51 9.60 TCVDGP 3.21 3.62 3.05 Open in a new tab Interpretability analysis In the proposed algorithm, the TCVDGP method is used to analyze the output of the hidden layer and study its statistical correlation with the internal parameters of the gearbox. This section focuses on analyzing four features: gearbox second-stage planetary radial (channel 1), upper frame (vertical to the main shaft) (channel 9), main shaft rear bearing radial (channel 11), and main shaft front bearing radial (channel 12). These features comprehensively reflect the internal operating state of the gearbox, fault propagation characteristics, and overall system health. Using the interpretability of TCVDGP, the correlation between these features and the predicted RUL is deeply investigated. It is worth noting that this method has good generalizability and is also applicable to the analysis of other internal parameters. To quantify the correlation, the Spearman correlation coefficient is used, with the following formula: 26 where: is the Spearman correlation coefficient, is the rank difference for each pair of observations, i.e., the difference in the rankings of variables for the -th observation, is the number of observations. The Spearman correlation coefficient ranges from [-1, 1], where 1 represents a perfect positive correlation, -1 represents a perfect negative correlation, and 0 represents no correlation. The entire operating cycle of wind turbine gearboxes for turbines 1–3 is divided into three stages: early [0, 30], mid [31, 50], and late [51, 75]. The mean and standard deviation of the correlation between features at each stage and the hidden layer output are shown in Fig. 15 . To focus on the strength of the correlation between the hidden layer output and input features, the correlation coefficients are calculated by taking their absolute values before computing the mean and standard deviation. Fig. 15. Open in a new tab Mean and standard deviation of correlation coefficients between features and hidden layer outputs of gearbox 1–3. The results show that the mean correlation coefficient between the input features and the hidden layer output in the early stage is relatively small, with a larger standard deviation. As seen in the results of Fig. 13 (a), the accuracy of RUL prediction in the early stage is relatively low, and the predictions exhibit more significant fluctuations. In contrast, the mean correlation coefficient in the mid and late stages is significantly higher, with a smaller standard deviation. Fig. 13 (a) further illustrates that, in these stages, the RUL prediction accuracy is higher, and the predictions are more stable. Conclusion This paper proposes a wind turbine gearbox health monitoring method based on TCVDGP, providing innovative solutions for varying operating conditions and uncertainty quantification. To address the varying operating conditions, a TCN module is integrated within the TCVDGP framework, significantly enhancing the model’s adaptability to complex time-series data and changing operational conditions by capturing both short- and long-term temporal dependencies. Additionally, an innovative SAM algorithm is proposed to dynamically adjust feature weights, improving the extraction of discriminative features and effectively suppressing redundant information. Concerning uncertainty quantification, this paper deeply integrates TCN and SAM into the DGP, optimizing the DGP’s capability to model complex features. Moreover, by constructing confidence intervals, the proposed method accurately characterizes the uncertainty associated with prediction outcomes. The proposed TCVDGP method was compared with the conventional DGP approach through ablation experiments on a self-collected planetary gear dataset and an industrial wind turbine gearbox dataset. Experimental results indicate that the TCVDGP method significantly outperforms the traditional DGP in terms of accuracy for RUL prediction, and also demonstrates superior performance in uncertainty quantification. Furthermore, comparative experiments against seven classical methods and state-of-the-art approaches proposed within the past three years further validate the superiority of the TCVDGP method, particularly highlighting its exceptional performance under complex operating conditions. Through interpretability analysis of the TCVDGP model, this paper further explores the underlying reasons for variations in RUL prediction accuracy at different stages (early, middle, and late). By analyzing the correlation coefficients between the outputs of the model’s hidden layers and internal gearbox parameters, the study explains differences in prediction performance across various stages. Experimental results show that the TCVDGP model maintains high prediction accuracy throughout all stages, particularly demonstrating sensitivity to changes in health status during middle and late stages. This enables effective capturing of the health evolution trends of wind turbine gearboxes. These results indicate that by focusing on key features and dynamically adjusting the weights of important variables, the proposed model achieves comprehensive and precise predictive capabilities, even under complex operational conditions. The TCVDGP method enhances the accuracy and reliability of RUL predictions for wind turbine gearboxes, thus optimizing maintenance strategies, reducing operational costs, and facilitating efficient operations and maintenance within the wind power industry. By accurately forecasting the health status of wind turbine gearboxes, this method significantly contributes to the advancement of sustainable energy, providing robust support toward achieving global carbon neutrality goals. Acknowledgements We acknowledge financial support from the National Natural Science Foundation of China (No. 62303065) and the “Qinxin Talent” Cultivation Program of Beijing Information Science and Technology University (No. QXTCPC202120). Author contributions Shuo Cui: Writing – original draft, Validation, Software, Methodology, Formal analysis. Xiuli Liu: Methodology, Writing – review & editing, Formal analysis, Conceptualization, Funding acquisition. Guoxin Wu: Supervision, Conceptualization, Visualization, Investigation. Wan Qiao: Visualization, Investigation. Jianyu Liu: Investigation, Formal analysis. Xuanyu Chen: Investigation, Conceptualization. Funding National Natural Science Foundation of China,No.62303065,“Qinxin Talent” Cultivation Program of Beijing Information Science and Technology University,No.QXTCPC202120. Data availability The data that support the findings of this study are available on request from the corresponding author, XL.L, upon reasonable request. Declaration Competing interests The authors declare no competing interests. Footnotes Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. References 1. Song, K. et al. Data-driven health state estimation and remaining useful life prediction of fuel cells. Renew. Energy 227 , 120491 (2024). [ Google Scholar ] 2. Wang, F., Kebede, G. A., Lo, S. & Woldegiorgis, B. H. An embedding layer-based quantum long short-term memory model with transfer learning for proton exchange membrane fuel stack remaining useful life prediction. Energy 308 , 133054 (2024). [ Google Scholar ] 3. Liu, L., Song, X. & Zhou, Z. Aircraft engine remaining useful life estimation via a double attention-based data-driven architecture. Reliab. Eng. Syst. Saf. 221 , 108330 (2022). [ Google Scholar ] 4. Merainani, B., Laddada, S., Bechhoefer, E., Chikh, M. A. A. & Benazzouz, D. An integrated methodology for estimating the remaining useful life of high-speed wind turbine shaft bearings with limited samples. Renew. Energy 182 , 1141–1151 (2022). [ Google Scholar ] 5. Hu, Y. et al. A prediction method for the real-time remaining useful life of wind turbine bearings based on the Wiener process. Renew. Energy 127 , 452–460 (2018). [ Google Scholar ] 6. Li, K. et al. Health state monitoring and predicting of proton exchange membrane fuel cells: A review. J. Power Sources 612 , 234828 (2024). [ Google Scholar ] 7. Song, W., Liu, H. & Zio, E. Long-range dependence and heavy tail characteristics for remaining useful life prediction in rolling bearing degradation. Appl. Math. Model. 102 , 268–284 (2022). [ Google Scholar ] 8. Xu, Z., Zhang, Y. & Miao, Q. An attention-based multi-scale temporal convolutional network for remaining useful life prediction. Reliab. Eng. Syst. Saf. 250 , 110288 (2024). [ Google Scholar ] 9. Znidaric, L., Gradisar, Z. & Juricic, D. Predicting the remaining useful life of solid oxide fuel cell systems using adaptive trend models of health indicators. Energies 17 (11), 2729 (2024). [ Google Scholar ] 10. Chang, Z. et al. Prediction of remaining useful life and recycling node of lithium-ion batteries based on a hybrid method of LSTM and LightGBM. Energy Sources Part A-Recovery Util. Environ. Eff. 46 (1), 1–13 (2024). [ Google Scholar ] 11. Ma, H., Wang, J., Han, B., Zhang, Z. & Bao, H. Temporal convolutional network with soft threshold and contractile self-attention mechanism for remaining useful life prediction of rolling bearings. Meas. Sci. Technol. 35 (12), 126004 (2024). [ Google Scholar ] 12. Li, W., Liu, D., Wang, X., Li, Y. & Cui, L. An integrated dual-scale similarity-based method for bearing remaining useful life prediction. Reliab. Eng. Syst. Saf. 256 , 110787 (2025). [ Google Scholar ] 13. Liu, S. et al. A two-stage remaining useful life prediction method based on adaptive feature metric and graph spatiotemporal attention rule learning. Reliab. Eng. Syst. Saf. 257 , 110802 (2025). [ Google Scholar ] 14. Wu, J. et al. Temporal multi-resolution hypergraph attention network for remaining useful life prediction of rolling bearings. Reliab. Eng. Syst. Saf. 247 , 110143 (2024). [ Google Scholar ] 15. Zhang, M., He, C., Huang, C. & Yang, J. A weighted time embedding transformer network for remaining useful life prediction of rolling bearing. Reliab. Eng. Syst. Saf. 251 , 110399 (2024). [ Google Scholar ] 16. Niazi, S. G., Huang, T., Zhou, H., Bai, S. & Huang, H. Multi-scale time series analysis using TT- ConvLSTM technique for bearing remaining useful life prediction. Mech. Syst. Signal Proc. 206 , 110888 (2024). [ Google Scholar ] 17. Ni, Q. et al. Data-driven bearing health management using a novel multi-scale fused feature and gated recurrent unit. Reliab. Eng. Syst. Saf. 242 , 109753 (2024). [ Google Scholar ] 18. Yang, X. et al. Remaining useful life prediction under multiple operating conditions based on a novel dual-layer temporal convolutional network. IEEE Sens. J. 25 (1), 1900–1911 (2025). [ Google Scholar ] 19. Zhang, T. et al. A novel multivariable prognostic approach for PEMFC degradation and remaining useful life prediction using random forest and temporal convolutional network. Appl. Energy 10.1016/j.apenergy.2025.125569 (2025). [ Google Scholar ] 20. Wang, Z. et al. A remaining useful life prediction framework with adaptive dynamic feedback. Mech. Syst. Signal Proc. 218 , 111595 (2024). [ Google Scholar ] 21. Song, W., Chen, D., Zio, E., Yan, W. & Cai, F. An adaptive generalized cauchy model for remaining useful life prediction of wind turbine gearboxes with long-range dependence. Fractal Fract. 10.3390/fractalfract6100576 (2022). [ Google Scholar ] 22. Qin, Y., Chen, D., Xiang, S. & Zhu, C. Gated dual attention unit neural networks for remaining useful life prediction of rolling bearings. IEEE Trans. Ind. Inform. 9 (17), 6438–6447 (2020). [ Google Scholar ] 23. Cao, W., Meng, Z., Li, J., Wu, J. & Fan, F. A remaining useful life prediction method for rolling bearing based on TCN-transformer. IEEE Trans. Instrum. Meas. 10.1109/TIM.2024.3502878 (2025). [ Google Scholar ] 24. Peng, P., Li, Y. & Guo, Z. High-performance remaining useful life prediction for aeroengine based on combining health states and trajectory similarity. Eng. Appl. Artif. Intell. 10.1016/j.engappai.2024.109799 (2025). [ Google Scholar ] 25. Shang, X. et al. A multi-task learning approach with meta auxiliary generation network for remaining useful life estimation. IEEE Internet Things J. 11 , 334–344 (2024). [ Google Scholar ] 26. Zhang, Y. et al. Remaining useful life prediction using multi-objective evolutionary deep networks for electro-mechanical actuator. IEEE Trans. Instrum. Meas. 10.1109/TIM.2025.3588985 (2025). [ Google Scholar ] 27. Liu, C. et al. Intelligent RUL prediction method of cutting tools based on GRU-LSTM. J. Braz. Soc. Mech. Sci. Eng. 47 (6), 278 (2025). [ Google Scholar ] 28. Xie, S. et al. Incremental contrast hybrid model for online remaining useful life prediction with uncertainty quantification in machines. IEEE Trans. Ind. Inform. 20 (12), 14308–14320 (2024). [ Google Scholar ] 29. Ding, Z., Qin, Q., Zhang, Y. & Lin, Y. An uncertainty quantification and calibration framework for RUL prediction and accuracy improvement. IEEE Trans. Instrum. Meas. 10.1109/TIM.2024.3485392 (2024). [ Google Scholar ] 30. Zhan, Y., Kong, Z., Wang, Z., Jin, X. & Xu, Z. Remaining useful life prediction with uncertainty quantification based on multi-distribution fusion structure. Reliab. Eng. Syst. Saf. 10.1016/j.ress.2024.110383 (2024). [ Google Scholar ] 31. Peng, W., Ye, Z. & Chen, N. Bayesian deep-learning-based health prognostics toward prognostics uncertainty. IEEE Trans. Ind. Electron. 67 (3), 2283–2293 (2020). [ Google Scholar ] 32. Mitici, M., de Pater, I., Barros, A. & Zeng, Z. Dynamic predictive maintenance for multiple components using data-driven probabilistic RUL prognostics: The case of turbofan engines. Reliab. Eng. Syst. Saf. 10.1016/j.ress.2023.109199 (2023). [ Google Scholar ] 33. Hernandez-Munoz, G., Villacampa-Calvo, C. Hernandez-Lobato, D. Deep gaussian processes using expectation propagation and monte carlo methods. Machine learning and knowledge discovery in databases, ECML PKDD 2020, PT III 12459 , 479–494 (2021). 34. Huang, J. et al. Fault diagnosis of planetary gearboxes under variable operating conditions based on AWM-TCN. Comput. Electr. Eng. 119 , 109520 (2024). [ Google Scholar ] 35. Heimes, F. O. Recurrent neural networks for remaining useful life estimation. 2008 International conference on prognostics and health management (PHM) , 59–64 (2008). 36. Liu, X. et al. Remaining useful life prediction with uncertainty quantification for rotating machinery: A method based on explainable variational deep gaussian process. Neurocomputing 638 , 130232 (2025). [ Google Scholar ] 37. Ji, W., Cheng, J. Chen, Y. Remaining useful life prediction for mechanical equipment based on temporal convolutional network. Proceedings of 2019 14th ieee international conference on electronic measurement & instruments (ICEMI) , 1192–1199 (2019). 38. Chen, Z. et al. Machine remaining useful life prediction via an attention-based deep learning approach. IEEE Trans. Ind. Electron. 68 (3), 2521–2531 (2021). [ Google Scholar ] 39. Gao, H., Li, Y., Zhao, Y. & Song, Y. Dual channel feature attention-based approach for RUL prediction considering the spatiotemporal difference of multisensor data. IEEE Sens. J. 23 (8), 8514–8525 (2023). [ Google Scholar ] 40. Song, T., Liu, C., Wu, R., Jin, Y. & Jiang, D. A hierarchical scheme for remaining useful life prediction with long short-term memory networks. Neurocomputing 487 , 22–33 (2022). [ Google Scholar ] 41. Wu, R. et al. The research of civil aero-engine remaining useful life estimation based on gaussian process. In International Congress and Workshop on Industrial AI 2021 (ed. Ahmadi, A.) 12–23 (Springer International Publishing AG, 2022). [ Google Scholar ] 42. Eke, I. & Kara, A. Remaining useful life prediction of machinery equipment via deep learning approach based on separable CNN and bi-LSTM (Springer Nature Singapore, International Symposium on Intelligent Manufacturing and Service Systems. Singapore, 2023). [ Google Scholar ] Associated Data This section collects any data citations, data availability statements, or supplementary materials included in this article. Data Availability Statement The data that support the findings of this study are available on request from the corresponding author, XL.L, upon reasonable request. Articles from Scientific Reports are provided here courtesy of Nature Publishing Group ACTIONS View on publisher site PDF (4.9 MB) Cite Collections Permalink RESOURCES Similar articles Cited by other articles Links to NCBI Databases Follow NCBI NCBI on X (formerly known as Twitter) NCBI on Facebook NCBI on LinkedIn NCBI on GitHub NCBI RSS feed Connect with NLM NLM on X (formerly known as Twitter) NLM on Facebook NLM on YouTube National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894 Web Policies FOIA HHS Vulnerability Disclosure Help Accessibility Careers NLM NIH HHS USA.gov Back to Top

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