CONELPABO: composite networks learning via parallel Bayesian optimization to predict remaining useful life in predictive maintenance

ORIGINAL ARTICLE CONELPABO: composite networks learning via parallel Bayesian optimization to predict remaining useful life in predictive maintenance David Solı ´s-Martı ´n 1,2 •Juan Gala ´n-Pa ´ez 1,2 •Joaquı ´n Borrego-Dı ´az 1,2 Received: 13 March 2024 / Accepted: 3 January 2025 / Published online: 29 January 2025 The Author(s) 2025 Abstract Maintaining equipment and machinery in industries is imperative for maximizing operational efficiency and prolonging their lifespan. The adoption of predictive maintenance enhances resource allocation, productivity, and product quality by proactively identifying and addressing potential equipment anomalies through rigorous data analysis before they escalate into critical issues. Consequently, these measures strengthen market competitiveness and generate favorable economic outcomes. In many applications, sensors operate at high frequencies or capture data over extended periods. This work introduces CONELPABO (Composite Networks Learning via Parallel Bayesian Optimization), a framework for analyzing long time series data, particularly for predicting the remaining useful life of a system or component. It uses a divide-andconquer strategy to manage the exponential growth in the hyperparameter search space during Bayesian Optimization and to accelerate model training by 50%. Additionally, this strategy enables the training of deeper networks with limited resources. The usefulness of the framework is demonstrated through two case studies, in which it achieves state-of-the-art results, showing that CNN-CNN and RNN-RNN architectures are highly effective for long time-series data. These architectures outperform many existing approaches and challenge the common academic focus on CNN-RNN hybrids. Keywords Prognostics and health management Remaining useful life Deep learning Deep learning Bayesian optimization List of symbols XRaw signals dataset e XNormalized signals dataset Yk tRUL of a unit kat time t e Xk tNormalized signals at time tof a unit k LwLength of the sliding window e Xk tNormalized signals of a unit kbetween t Lwand t TULkTotal useful life of the unit k Ck tCycle number of the unit kat time t CbsConvolution block size NcbNumber of convolution blocks Neb;Neb Number of encoding and decoding Nrb Number of recurrent blocks of a RNN Nru Number recurrent units of a RNN layer drate Dilation rate of a convolution fcxNeurons in the x-th fully connected layer rconv;rfc;rout Activation function of a convolution layer, fully connected layer and the output layer of a network. KsKernel size of a convolution l1;l2Weights for L1and L2regularization lrLearning rate BsBatch size A jj Number of data attributes stride Step between windows applied to the raw input sw RUL prediction smoothing window size Juan Gala ´n-Pa ´ez and Joaquı ´n Borrego-Dı ´az have contributed equally to this work. &David Solı ´s-Martı ´n [email protected] Juan Gala ´n-Pa ´ez [email protected] Joaquı ´n Borrego-Dı ´az [email protected] 1 Departamento de Ciencias de la Computacio ´n e Inteligencia Artificial, Universidad de Sevilla, Sevilla, Spain 2 Datrik Ingelligence, S.A, Seville, Spain 123 Neural Computing and Applications (2025) 37:7423–7441 https://doi.org/10.1007/s00521-025-10995-z(0123456789().,-volV)(0123456789().,-volV) 1 Introduction Proper maintenance of industrial equipment and machinery is critical to ensure operating efficiency and extend their lifespan, resulting in lower replacement and repair costs and a long-term positive economic impact. Additionally, the availability of the system is a key factor to consider. Estimating the probability of failure can help prevent unexpected malfunctions by enabling timely maintenance interventions, thereby increasing equipment availability and maintaining the production flow [36]. The field of research has gained considerable attention with the emergence of the Industry 4.0 paradigm [58,65]. Predictive maintenance, based on continuous data monitoring and the utilization of advanced technologies such as vibration analysis or wear sensors, proactively identifies potential failures before they occur. Preventing unplanned downtime and optimizing the availability of productive assets, results in significant cost savings and, therefore, increased profitability. This approach not only avoids the substantial costs associated with machinery failure but also mitigates the intricate processes involved in handling system failures [58,64]. This also improves resource planning, productivity and product quality, and market competitiveness, resulting in a positive economic impact both internally and externally [47]. Estimating the remaining useful life (RUL) of a component or system is a critical task within Prognostics and Health Management (PHM), and involves analyzing the system’s behavior over time to predict its future reliability and degradation [46,54]. The main objective of RUL analysis is to determine when the system will fail or when a certain level of degradation will be reached. Solutions able to successfully estimate the RUL of a system can help industries mitigate losses caused by unplanned downtime and repairing expenses [20]. The main approaches for estimating the RUL of a system, in the field of PHM, could be classified as modelbased or data-driven. Model-based methods exploit physical and statistical modeling to design a degradation model that predicts the system’s degradation trend. Developing degradation models for highly complex systems is a challenging task as it requires an in-depth understanding of the physical characteristics of the components that are prone to failure, and the types of failures these may experience [73]. Furthermore, model-based methods rely on certain assumptions that can introduce biases, thus reducing their predictive performance. Data-driven methods have gained popularity due to their ability to leverage large volumes of data generated by modern sensors and systems (e.g., Internet of Things). This kind of method allows the processing of historical data to extract patterns, with the goal of detecting different degradation trends. Many approaches have been explored in the literature, including neural networks (NN) [57], deep learning (DL) [13], and ensemble methods based on decision trees [1,37]. Support vector machines (SVM) have been applied to predict the RUL of aircraft engines in studies such as [42] and [43]. To handle uncertainty, Bayesian networks [41] and fuzzy logic-based systems [6] have been proposed, leading to more robust RUL predictions. DL models are among the most popular and promising data-driven methods. Over the past decade, DL techniques have been widely utilized, particularly for complex tasks involving high-dimensional nonlinear data. DL has demonstrated remarkable success in diverse fields, such as Image Processing, Natural Language Processing and Signal Processing, among others. Consequently, it is not surprising that DL-based approaches have gained widespread use in PHM research. As previously noted, RUL prediction involves dealing with complex condition monitoring data from systems that can consist of multiple subsystems and failure types. This means that monitoring data will comprise multiple variables and dimensions, making it crucial for data-driven RUL prediction methods handle high-dimensional data. While DL algorithms are known for their ability to handle high-dimensional data, dimension reduction techniques are often employed to reduce computational complexity, and serve as regularization mechanisms. Such techniques can help to simplify and generalize the models used for RUL prediction, and thus, have become a common tool in the field. 1.1 Aim of the paper Many works propose network architectures without specifying how they derived those architectures. We emphasize the importance of a rigorous network architecture search to enhance the credibility of published research and reduce the risk of overfitting. However, this rigor comes with the disadvantage of requiring considerable time, particularly when validating the method across numerous architectures and with a cross-validation approach. Taking this into account, the goal of this work is not to outperform existing methods in predictive accuracy but to significantly reduce the time and memory resources required for training deep learning models, especially when a network architecture search is involved. The primary goal of this paper is to demonstrate the effectiveness of Parallel Bayesian Optimization (BO) in training composite networks and well-established methodologies, such as hybrid models (CNNs and RNNs), 7424 Neural Computing and Applications (2025) 37:7423–7441 123 to address a specific challenge: predicting the RUL using long data sequences. Additionally, we explore the use of various hybrid DL techniques in predicting the RUL of a system by addressing two distinct case studies. The concept of hybrid DL is extended to include composite structures, where combinations such as RNN-RNN or CNN-CNN are considered. Thus, this study is not limited solely to CNN-RNN hybrid models; instead, it examines all possible combinations of CNN and RNN architectures. To the best of our knowledge, such a complete study has not been previously conducted. To achieve this goal, a comprehensive framework was designed for analyzing very long data sequences, particularly time series data, in the context of RUL prediction. This framework integrates a meticulously designed crossvalidation and hyperparameter optimization process, leading to optimal models for analyzing such sequences. To this aim, a neural architecture search (NAS) approach enhanced by BO has been used. The innovation of this work is reflected in how we combine these elements to address challenges specific to the application domain. By using composite networks and strategically separating feature extraction and prediction components, we optimize resource utilization and make the BO search space more tractable. This approach reduces training time and achieves state-of-the-art (SOTA) results. The rest of the paper is structured as follows. The next section (Sect. 2) provides a brief description of the RUL problem and the main DL methods and techniques considered in this work. Section 3is devoted to describe the proposed framework CONELPABO, which aims to simplify the modeling process for this type of problem, through preprocessing techniques, adjusting the time window for RUL models, and exploiting other specific characteristics of the RUL challenge. Section 4introduces the case studies and their main features, and presents the results obtained by applying the proposed methodology on both. Section 5provides an analysis of the achievements of this work and highlights its strengths and weaknesses. The paper concludes with Sect. 6, providing some considerations about the conducted work. 1.2 Related work RNN and CNN have been widely used for predicting the RUL of machinery. For example, in [69,72,74] the authors successfully apply basic RNN models to predict RUL. In [45], a dual-channel LSTM architecture was utilized, incorporating momentum smoothing into the predictions. Bidirectional RNNs have been studied for RUL estimation in works like [21] and [32]. Other works have explored the application of CNNs to predict RUL. For example, [30] and [29] are noteworthy. [29] proposes a Multi-Scale CNN (MS-CNN), which consists of three multi-scale blocks (MS-BLOCKs) where convolution operations of three different sizes are applied in parallel to extract features at varying scales. Dimensionality reduction approaches have also been used to predict RUL. In [35,56,70], encoder-decoder networks were trained in an unsupervised manner. The encoder is then used to create a Health Index (HI) or to feed other models using a sliding window. Recently, researchers have been using combinations of DL architectures to leverage the unique advantages of each. One of the most common hybrid models consists of a CNN followed by a RNN. While the CNN focuses on extracting spatial features, the RNN exploits the temporal dependencies between time-series data points. These approaches usually divide time-series data into subsequences using a sliding window approach. Each subsequence is then designated as input to a model for feature extraction. Later, the extracted features from the first model are fed into a second model to accomplish the final task. Examples of these works include [5,8,27,48, 52,60,63,68]. In the context of BO, researchers have developed various methods to enhance its application in NAS. Much of the research in this domain has concentrated on designing innovative surrogate models [33,53,55,67] and encoding schemes for neural architectures [12,62,66]. The curse of dimensionality present in the BO search space have been approached too in some works. LaNAS[61] uses a hierarchical partitioning strategy to divide the search space into good or bad regions that contain networks with similar performance metrics and lead the search toward good regions. Outside the context of NAS, [23] and [17] propose decomposing the target function into additive structures. While this approach can be effective, it is limited to functions that are amenable to additive decomposition, restricting its applicability to a narrow subset of optimization problems. A more general approach is presented in [4], which addresses the optimization of composite functions of the form fðxÞ¼gðhðxÞÞ. In their framework, gis modeled as an expensive-to-evaluate black-box function, while hrepresents a computationally inexpensive function that can be evaluated or approximated efficiently. 1.3 Contributions In comparison with existing literature, our work diverges significantly in its focus and methodology. While many studies have explored hybrid network architectures, such as CNN-RNN combinations [5,8,27,48,52,60,63,68], Neural Computing and Applications (2025) 37:7423–7441 7425 123 there has been limited investigation into simpler architectures like CNN-CNN, RNN-RNN, or unconventional combinations such as RNN-CNN. Our analysis highlights that some specific tasks takes advantage of these architectures, challenging the prevailing emphasis on hybrid models in NAS. Regarding the BO process, our work addresses a similar problem to that explored in [4], which focuses on optimizing composite functions of the form fðxÞ¼gðhðxÞÞ. However, unlike [4], where the inner function his assumed to be cheap to evaluate, in our case, the ffunction is computationally expensive to evaluate. To address this challenge, we design a dual parallel BO process inspired by the additive decomposition methods proposed in [23]. This approach enables efficient exploration despite the computational demands associated with evaluating f. Moreover, by leveraging pre-computed embeddings, we significantly reduce the time required to evaluate the gfunction during the BO process, thus enhancing the overall efficiency of the framework. Moreover, unlike [4] and [23], which do not focus on NAS, our framework specifically applies BO techniques to NAS problems. This adaptation integrates a dual parallel BO process into the context of neural architecture optimization, enabling a novel contribution in the intersection of BO and NAS fields. To the best of our knowledge, no prior work has explored this or similar approaches within the NAS domain for RUL prediction tasks. The contributions of this work are summarized as follows: •The design of two parallel BO process, enabling efficient exploration of hyperparameters for network architectures. The methodology involves two learning stages: (i) learning an encoding from raw data, and (ii) using this encoding to train a final predictive model. Experimental results demonstrate SOTA performance on benchmark datasets, with a reduction in GPU memory consumption and a significant 50% reduction in training time. This enables the training of larger networks with limited resources and allows for a broader range of experiments during model search. •Empirical evidence showing that simpler architectures, such as CNN-CNN, can outperform more complex designs for specific datasets (e.g., N-CMAPSS), while RNN-RNN architectures excel in others (e.g., PRONOSTIA). These findings challenge conventional preferences for encoder-decoder or hybrid CNN-RNN approaches. •The publication of the source code of the framework and experimental settings, enhancing the reproducibility of the results and enabling future research. 2 Material and methods This section provides an overview of the methodologies and tools used in this work. First, the RUL estimation problem is formally defined, emphasizing the key metrics considered in this task. Next, the deep learning architectures utilized in the study are described, with a focus on Convolutional Neural Networks (CNN) and Recurrent Neural Networks (RNN). 2.1 RUL problem definition The main goal in RUL is to estimate the amount of time a piece of equipment will function effectively before needing to be replaced or overhauled. The problem at hand involves developing a model Fto predict the remaining useful life y. This problem can be expressed mathematically as the following optimization problem: arg min hX M i¼1 Lyi^ yi ðÞ ð1Þ where Mis the number of observations, Lis the loss function, yirepresents the actual RUL of the i-th observation, and ^ yirepresents the predicted RUL for the i-th observation, with ^ yi¼FhðÞ and hbeing the parameters of the model Fto be optimized. In this work, the loss function Lis defined as the mean-square error (MSE). MSE is sensitive to larger errors, which helps the model to focus on minimizing substantial prediction errors. This characteristic is particularly useful in applications where large deviations from the true value are critical and potentially more costly, like in RUL tasks. While the initial choice of MSE is motivated by its sensitivity to large errors, mathematical properties, and standard use in regression tasks, we acknowledge the importance of exploring alternative loss functions. Additionally, in this work, the mean-absolute error and the NASA scoring were computed for each experiment. The NASA scoring function (Ns,[51]) is defined as: Ns¼1 MX M i¼1 expðajyi^ yijÞ  1ð2Þ where aequals 1 13 when ^ yi\yand 1 10 otherwise. The NASA scoring function for RUL prediction is another relevant metric. It emphasizes the accuracy of the prediction near the end of the equipment life, which aligns closely with practical application needs. Incorporating this scoring function as a part of the evaluation metric could offer a more comprehensive assessment of the model performance in real-world scenarios. 7426 Neural Computing and Applications (2025) 37:7423–7441 123 Figure 1presents the metrics used in this work to provide a better understanding of their strengths and weaknesses. 2.2 Deep learning In this work, supervised DL methods are employed to address the mentioned RUL prediction problem. Therefore, the model Fis a deep neural network, and the parameters h to be optimized include the set of weights of the network, along with other additional hyperparameters (such as learning rate, batch size, window size, etc.). The network is fed with data units X2Rn;T, representing a multivariate time series input composed of nattributes with Tdata points each. Next, a concise overview of the different deep learning methods and architectures considered in this study is provided. 2.2.1 Convolutional neural networks (CNNs) CNNs[25] are a type of neural network known for their success in tasks like image processing. These networks use convolution and pooling operations to extract local features from the input, implementing parameter sharing and reducing computational requirements. Usually, CNNs can be divided into two parts. Firstly, the convolutional part focuses on extracting features. This part is formed by stacking convolutional layers and pooling layers, among other types of layers, such as batch normalization layers. The extracted features from the last layer of this first part are fed into the second part to make the final prediction. The second part typically consists of a few dense or fully connected layers. This architecture is depicted in Fig. 2A. In this work, the extension of the CNN, known as MSCNN [29], has been considered. This architecture includes, at the beginning of the network, a few blocks of three parallel convolution layers. Each of the convolutions has a different kernel size. The feature maps of the three convolution operations are collected and concatenated to be fed into the next block. This enables the extraction of features at multiple scales. This architecture is shown in Fig. 2B. 2.2.2 Recurrent neural networks (RNNs) RNNs[50] are designed to handle sequential data by incorporating memory mechanisms to capture temporal dependencies. LSTM[19] cells improve upon standard RNNs by managing long sequences and addressing the vanishing gradient problem through various gates that control information flow. GRU[10] cells offer a simpler alternative with fewer parameters, making them computationally efficient. Both RNN and LSTM/GRU architectures are used to capture time-dependent features and make predictions. Similar to CNNs, RNNs are typically divided into two parts. The first part consists of one or more recurrent layers that are adept at capturing temporal dependencies within sequential data. The next section of the network utilizes the features extracted by the recurrent layers to estimate the desired output. This architecture is illustrated in Fig. 2C. 3 CONELPABO framework This section introduces the CONELPABO framework and justifies the design decisions made to define it. Additionally, the experimental settings and parameters used to address each of the case studies are provided. 3.1 Modeling stages A typical machine learning task (in general terms, not specifically within the deep learning field) usually follows the following four phases: 1. Data preprocessing: In the data preprocessing phase, raw data is cleaned and transformed before training the models. The goal of this phase is to ensure that the data is prepared in a way that maximizes the performance of the models. 2. Feature engineering: This phase involves selecting features relevant for modeling, as well as creating new ones. Feature engineering usually requires a deep Fig. 1 MSE is suitable when it’s important to heavily penalize large deviations, but it can be overly influenced by outliers. MAE provides a straightforward interpretation and is robust to outliers, but treats all errors equally. NASA Scoring Metric offers a nuanced approach for RUL prediction by emphasizing the impact of errors, particularly late predictions, which is crucial in many engineering applications Neural Computing and Applications (2025) 37:7423–7441 7427 123 understanding of the data and problem at hand, and often can be an iterative process of testing and refining feature sets to achieve better results. 3. Model selection: In the model selection phase, the performance of different models and hyperparameter sets is evaluated and compared to select the best model. 4. Model validation: In this phase, the final model is trained on all pre-processed train data available, and evaluated using a test set, which was not used during any stage of the model development, to assess its performance. The methodology proposed in this work simplifies the previous list of phases by removing steps 2 and 4. Step 2 is no longer necessary since the networks are fed with the raw signal, and feature engineering is carried out by the network itself. Step 4 can also be excluded, as models generated through cross-validation can be used to make inferences by averaging their outputs. Using multiple models has the advantage that along with the average prediction, a confidence interval can be computed using the predictions of each model. The rest of the section covers various aspects related to the analysis of time series data. It begins by describing the process of data normalization, which involves transforming the data into a unified scale. Additionally, the section explains how the data unit is divided into multiple samples, each associated with a specific RUL target representing the remaining useful life. Then, the use of two models in a stacked manner to effectively process long time series is discussed. Finally, the hyperparameter optimization strategy will be outlined. 3.2 Data preprocessing Before training the model, the data were normalized and split as described below: •Data standardization. Normalization ensures that all features have a similar impact on the learning process and prevents any particular feature from dominating the training, which can help the model to converge faster. In this work, data is normalized using data standardization, by subtracting the mean from each feature and dividing by the standard deviation. This results in data having a mean of zero and a standard deviation of one: x0¼xl rð3Þ where x2Xis an attribute of the multivariate time series X. •Time window processing. The time series have to be partitioned using a sliding window across the standardized data (see Fig. 4). The width of the window Lw, will be optimized during the model selection phase (Sect. 3.5). Each data unit input Xk t(Eq. 4) is a sequence of the Lwmost recent data points of the normalized features e Xk, ending at time t. The corresponding ground-RUL label is denoted as yt. For each unit, TkLwinputs and labels are generated, where Tkis the total run-time of the unit in cycles. Xk t¼½e Xk tend Lw; :::; e Xk tend ð4Þ The RUL value associated with each window yk tis computed as TULkEk t, where TULkis the total Fig. 2 Schematic of CNN and RNN: Adepicts the common CNN divided into the convolutional part and the regression part. BShows an extension of the CNN with an additional part to extract multiscale features. CIllustrates an RNN composed of the first part focused on extracting timedependent features and the second part carrying out the prediction, with symbols indicating hyperparameters optimized during the model search process 7428 Neural Computing and Applications (2025) 37:7423–7441 123 number of life units in each data experiment k, and Ek tis the number of life units that have elapsed from the beginning of the data experiment at time t.Some studies, such as [29], often set a constant RUL at the beginning of each experiment. This is due to the assumption that there is not enough information initially available to capture the initial degradation process accurately. In this study, this procedure was not applied initially to avoid potential bias. 3.3 Cross-validation The large number of adjustable parameters existing in neural networks makes it challenging to find the combination of hyperparameters that yields the best performance without overfitting the training set. In addition, selecting the size of the sliding window is another critical aspect of the proposed methodology. In the literature, there exist various validation techniques that involve splitting data to avoid overfitting. The basic ‘‘single hold-out strategy’’ is usually not preferred as it can lead to overfitting on the hold-out set during model selection or hyperparameter optimization, resulting in poor generalization. In contrast, the k-fold cross-validation strategy has the disadvantage of decreasing the size of the validation set as kincreases (the number of folds), making it less representative of the entire dataset. In this context, a k-fold repeated random subsampling validation strategy is preferred. This strategy randomly divides the dataset into training and validation sets, and the process is repeated ktimes. During each repetition, a different random split is generated, ensuring that each sample is included in a validation set at least once. This approach provides a better representation of the entire dataset, and the average of the results obtained during the krepetitions is used to assess the performance of the model. Figure 3 illustrates the process of k-fold repeated random subsampling validation. In this work, k¼5 has been selected, and the chosen validation set size is approximately 20% of the experiment units for each dataset. The validation strategy presented in many works (e.g., [26,49], and [34]) consist in randomly selecting a certain percentage of data points as testing data, while the remainder is used as training data. The latter approach implies that testing and training data points are interleaved, providing the model with information about the testing data distribution, that is, including details about future data points, which may lead to overfitting, and therefore, lead to an unsafe model. In contrast, in the present work, to address such concerns, the division of training and validation sets is based on full units (such as bearings or flights), which constitutes a safer validation strategy. 3.4 Composite networks The architecture proposed in this work consists of two stacked models and a final moving average smoothing module, as shown in Fig. 4. The first Feature Extractor Network (FEN) aims to encode the raw signals, while the second one, Regression Network After (RNA), uses these encodings to predict the RUL. Both models, FEN and RNA, are trained in a supervised manner, with the input– output pairs being the time-windowed signals or encodings and the corresponding RUL labels, respectively. The goal of the encoding step is to transform the raw input signals into a lower-dimensional representation, capturing the most relevant information for predicting the RUL using a limited number of data points. This is achieved by training a model to encode the input signals into a set of features that preserve the important patterns while removing noise and other irrelevant information. The resulting encoding is then used as the input for the RUL prediction with the RNA model. The FEN model is trained to predict RUL, but after training, the last output layer is removed, and the previous layer is used to generate the encodings, which are lowdimensional representations of the inputs. Therefore, all the network architectures tested in this study, both CNN or RNN, have two fully connected layers on top to generate the final prediction of the RUL. This is illustrated in Fig. 4. In this work, RNNs and CNNs have been chosen based on their proven effectiveness for time-series analysis and sequence prediction tasks, which are central to our problem. RNNs are particularly well-suited for modeling temporal dependencies in sequential data, making them effective for capturing the dynamic nature of time-series data. CNNs, on the other hand, are highly effective in extracting local features and patterns, which can be beneficial in identifying important characteristics within long time-series sequences. We acknowledge that other architectures, such as transformers, have shown promising results in various Fig. 3 k-fold cross-validation strategy (left), and k-fold random sampling strategy (right) Neural Computing and Applications (2025) 37:7423–7441 7429 123 sequence modeling and time-series prediction tasks. Transformers, with their self-attention mechanisms, could potentially offer additional benefits in capturing long-range dependencies and improving model performance. However, due to limited time and resources for experimentation, this study is focused on RNNs and CNNs. In general, the choice of the architecture and hyperparameters will significantly impact the effectiveness of the RUL prediction models. Therefore, it is important to carefully evaluate and optimize these aspects of the model in order to achieve the best possible performance [14,22]. The following Sect. 3.5 describes the method for selecting hyperparameters considered in this work. The output of the networks was configured to use the ReLU activation function, since in RUL estimation negative values have no sense. Thus, constraining the output to be positive benefit the convergence of the learning process. Additionally, we introduce a moving average smoothing (MAS) module to generate the final RUL prediction. MAS improves RUL predictions by reducing noise and fluctuations in the predicted RUL values. With MAS, prediction stability is enhanced, and previous trends are retained, which provide valuable information for current time predictions. 3.5 Model selection Model selection refers to the process of choosing the most appropriate or optimal model from a set of candidate models for a given task or problem. It involves evaluating and comparing different models based on their performance, complexity, and other relevant criteria. In this work, the criterion used is the performance of the model obtained by cross-validation. Bayesian optimization (BO) is a technique that is used to optimize black-box functions that are expensive to evaluate. BO can be used as a model selection technique. In this case, the black-box function trains and evaluates a model [16,44]. The input is the set of hyperparameter values to be assessed in the current iteration, whereas the output of the function (the goal of the optimization process) is the model performance. BO works by building a surrogate model of the black-box function, which is updated after each evaluation (iteration). The surrogate model is used to decide the next set of hyperparameters to be Fig. 4 FEN-RNA composite networks. Firstly, the FEN model is trained to learn to predict the target, that is, the RUL of a certain system. Then, a feature set is extracted from the trained FEN model and used to train the RNA model using also the RUL as target. Finally, the prediction of the RNA passes through a moving average smoothing module to provide the final RUL prediction 7430 Neural Computing and Applications (2025) 37:7423–7441 123 evaluated, in a way that maximizes the expected improvement of the black-box function. In this work, two BO processes are utilized to optimize the hyperparameters, including those that define the network architecture. One challenge with BO is that the search space grows exponentially with the number of hyperparameters considered. This expansion in the search space requires a high number of iterations to find good local minima, which has proven to be extremely challenging [59]. To address this issue, a divide-and-conquer strategy is adopted. Instead of using only one BO process to optimize all the hyperparameters, two parallel BO processes are employed. The first BO process conducts the optimization of the FEN model, and the second one searches the best hyperparameters for the RNA-MAS composition, utilizing the last FEN model tested. During the execution of each model search iteration of the RNA-MAS model, the weights used in the last search iteration of the FEN are fixed, to avoid their modification during the training process of the RNA model. This process is described in detail in the Algorithm 1. Algorithm 1 Model search algorithm Each BO process is guided by different scores, which are used to select the next set of hyperparameters to be evaluated during the optimization process. For the BO process of the FEN, the score NFEN from the independently trained model is utilized. The score NFEN measures the contribution of the FEN to the final model. To guide the BO process of the RNA-NAS, the score NRNAMAS, obtained by training the RNA-NAS model by reusing the best-trained FEN, is used. It is important to note that while the BO process uses the NASA score, the network itself is trained using the MSE. To optimize the model, each BO process is executed for Niterations. In this work, Nwas set to 100. In the first 20 iterations, the input hyperparameters are selected randomly and used by the BO process to create the initial estimation of the hyperparameter search space. Subsequently, Bayesian rules are applied to select the most promising set of hyperparameters to be tested in the next iteration. This process continues until the completion of the desired number of iterations. Additionally, the early stopping technique is used as stopping criterion during the training of each model. This is a common technique used in Machine Learning to prevent overfitting. It involves monitoring the validation loss during training and stopping the training process when the validation loss stops improving, even if the training loss continues decreasing. The patience factor is a parameter that determines the number of epochs the model can continue training without any improvement in validation loss before stopping. In this work, the patience factor has been set to 8 epochs (that is, if in the last 8 epochs there is no improvement in the validation loss, the training process will be stopped). In addition, an adaptive learning rate is used, which is reduced by a factor of 0.1 if no improvement in the validation loss is observed during three training epochs. This technique accelerates the model convergence and prevents it from overshooting the minimum of the loss function. The architectures evaluated for the FEN, including CNN and RNN, are also candidate architectures in the RNA model selection phase. The four possible combinations, which are CNN-CNN, CNN-RNN, RNN-RNN and RNNCNN, have been analyzed. Table 1provides a summary of the parameter ranges used in the Bayesian optimization process for each of the architectures utilized. The majority of the hyperparameters correspond to the architecture of the network, except for the stride and sw. The stride hyperparameter defines the gap between data points when the sliding window is applied (see Fig. 4). The sw denotes the MAS window size used to apply smoothing for predicting the final RUL. 3.6 Methodology motivation Once the methodology and the framework have been introduced, this section will state the rationale behind the proposed approach. •Efficient resource utilization: Splitting the network into a feature extractor and a predictor allows training more complex models with limited GPU resources. This division reduces the computational load at each stage by decreasing the number of parameters in each network and the size of the input, enabling us to Neural Computing and Applications (2025) 37:7423–7441 7431 123 results obtained with other state-of-the-art methods, further validating the effectiveness of this approach. Additionally, classical machine learning methods, such as SVM and tree-based algorithms, were included in the comparison. The results highlight the superior performance of the neural network-based approach over these traditional methods, particularly due to the advantage of the network with automatic feature extraction. In contrast, the classical methods rely heavily on manual feature selection and extraction, which makes them less effective when dealing with large time-series datasets. The neural network models are able to learn representations directly from raw data, reducing the need for handcrafted feature engineering and the bias this could introduce, and giving them a significant edge in performance. The main advantage of the proposed method is the significant reduction in training time. By splitting the network into two components-one to generate embeddings and another to estimate the final RUL-this approach not only enables the precomputation of embeddings but also reduces the hyperparameter search space, making the BO process more tractable. These factors contribute to the superiority of this approach over others. Equations 5and 6provide a mathematical explanation of this issue. For example, when training the RNN-RNN network on the PRONOSTIA dataset, the end-to-end network takes approximately 720 s per epoch. In contrast, using the proposed approach, the FEN network takes 49 s per epoch, and the RNA network takes 220 s per epoch, for a total of roughly 269 s per epoch. It is also important to note that generating embeddings for the entire dataset with the FEN network takes around 5 s, distributed across the epoch. Thus, the total reduction in training time exceeds 50%. This time saving is particularly crucial in the context of model search, where numerous models need to be evaluated and tested. Regarding the network size, dividing the end-to-end network allows training larger models. It is noteworthy that, with an end-to-end network approach, the input dimension is significantly higher. For example, in the previous example, the end-to-end network has an input shape of (360x2x360), which consists of 360 subsequences, each with 360 data points and two features. With a float32 data representation and a batch size of 128, this configuration requires 0.1GB of GPU memory just for the input data. In contrast, the input shape of the FEN network is (360x2), and for the RNA network, it is (360x100). Therefore, the reduced input shape is another important achievement of the proposed approach in decreasing resource usage. The implications of these findings are significant for both the research and industrial communities, particularly for those with limited computational resources or time constraints. This approach is advantageous for organizations that cannot afford extensive computational resources or need to obtain results faster. With respect to the N-CMAPSS dataset used in this study, this dataset provides valuable insights and allows for rigorous testing of predictive models. However, it is essential to acknowledge that simulations, by their nature, are limited in their ability to capture the full complexity of real-world scenarios. Although our DNN models demonstrate high performance and agreement with the simulated data, this performance may not directly translate to realworld propulsion systems due to discrepancies between simulated and actual conditions. Therefore, while our findings provide a strong foundation and demonstrate the potential of the proposed approach, further validation with real-world data is necessary to confirm the generalizability and robustness of the models. The main weakness of the proposed approach is that the acquisition function used by the BO process for the FEN model is completely independent of the final FEN-RNA model. As a result, the hyperparameters selected by the BO process are likely to be suboptimal. Providing information about the contribution of the FEN model to the final FENRNA model could be an interesting improvement. This is a point that should be considered as an area for enhancement in future work. 6 Conclusions The framework presented in this work offers a robust approach for RUL estimation that does not require expert knowledge of the system under study. It establishes a robust model search and cross-validation strategy, mitigating the risk of overfitting. Applying Bayesian Optimization directly to the hyperparameters of both networks Table 6 Comparison of our best results in PRONOSTIA with other similar approaches in the literature and other machine learning algorithms. Bold value identify the best performance Model MAPE RMSE LSVM 39.19 ±11.03 0.31 ±00.24 SVM 1.0e3 ±2.0e3 0.16 ±00.01 RF 6.6e3 ±1.2e3 0.19 ±00.02 XGB 51.18 ±15.28 0.22 ±00.01 CNN-GRU [40] 42.34 0.19 Bi-GRU [9] 44.49 – CNN-LSTM [18] 46.32 – CONELPABO (ours) 35.27 ±21.03 0.13 ±00.02 7438 Neural Computing and Applications (2025) 37:7423–7441 123 would be intractable. Therefore, a parallel BO process has been designed to facilitate the convergence of the model search in a reduced number of iterations. A series of experiments were performed to compare several combinations of state-of-the-art network architectures, selecting the best-performing architectures to build the final solution. The process of obtaining the solution mainly involved two learning stages: first, an encoding of the raw data is learned and used as input for the second learning stage, which produces the final model capable of estimating RUL. The proposed methodology has achieved results that surpass the state-of-the-art ones. Remarkably, the results revealed that CNN-CNN, despite being one of the most basic architectures analyzed, exhibited outstanding performance with the N-CMAPSS dataset. Conversely, when dealing with the PRONOSTIA dataset, the RNNRNN architecture outperformed the rest of the configurations. We believe that these results are quite interesting to the research community, as previous works in the literature have mainly focused on CNN-RNN or encoder-decoder architectures. The proposed methodology not only achieves state-ofthe-art performance results but also reduces resource usage, including GPU memory and training time. This reduction allows training larger networks with fewer resources and increases the number of experiments that can be conducted during model search. To conclude, it is worth noting that this work has been developed with a focus on reproducible research. To this aim, the parameter ranges used for model selection have been detailed. In addition, the source code to train the models and validate results can be found in a GitHub repository. 2 7 Technical issues Python 3.9 was the primary programming language used in this work, and the framework was implemented using it. For network design and training, one of the latest versions of TensorFlow [2] and Keras [11] (i.e., Tensorflow 2.0 and Keras 2.3, respectively) were used. To optimize the hyperparameters of the deep learning model, we used Tune 1.6.0 [31], which is a scalable hyperparameter tuning library developed by Ray. The computational resources used for the experiments were a server with 32GB of RAM memory, and four threads. We also used two GTX 1080Ti GPUs to distribute the different network training tasks and speed up the training process. Acknowledgements This work has been supported by Grant PID2023-147198NB-I00 funded by MICIU/AEI/10.13039/ 501100011033 (Agencia Estatal de Investigacio ´n) and by FEDER, UE, and by the Ministry of Science and Education of Spain through the national program ‘‘Ayudas para contratos para la formacio ´nde investigadores en empresas (DIN2019-010887/AEI/10.13039/ 50110001103)’’, of State Programme of Science Research and Innovations 2017-2020. Funding Funding for open access publishing: Universidad de Sevilla/ CBUA. Spanish National Plan for Scientific and Technical Research and Innovation (MICIU/AEI/10.13039/501100011033, DIN2019010887/AEI/10.13039/50110001103). Data availability This work used the public benchmarks datasets N-CMAPSS [3] and PRONOSTIA [39]. Declarations Conflict of interest The authors declare that they have no Conflict of interest related to this work. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, 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 changes were made. 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/4.0/. References 1. Remaining Useful Life (RUL) Prediction of Rolling Element Bearing Using Random Forest and Gradient Boosting Technique, ASME Int. Mechanical Engineering Congress and Exposition, vol 13: Design, Reliability, Safety, and Risk (2018) 2. Abadi M, Agarwal A, Barham P, et al (2015) TensorFlow: largescale machine learning on heterogeneous systems. https://www. tensorflow.org/, software available from tensorflow.org 3. Arias Chao M, Kulkarni C, Goebel K et al (2021) Aircraft engine run-to-failure dataset under real flight conditions for prognostics and diagnostics. Data 6(1):5 4. Astudillo R, Frazier P (2019) Bayesian optimization of composite functions. In: International conference on machine learning, PMLR, pp 354–363 5. Azyus AF, Wijaya SK, Naved M (2023) Prediction of remaining useful life using the CNN-GRU network: a study on maintenance management. Softw Impacts 17:100535 6. Baraldi P, Di Maio F, Al-Dahidi S et al (2017) Prediction of industrial equipment remaining useful life by fuzzy similarity and belief function theory. Expert Syst Appl 83:226–241 7. Berghout T, Mouss MD, Mouss LH et al (2022) Prognet: a transferable deep network for aircraft engine damage propagation prognosis under real flight conditions. Aerospace 10(1):10 8. Borre ´A, Seman LO, Camponogara E et al (2023) Machine fault detection using a hybrid CNN-LSTM attention-based model. Sensors 23(9):4512 2 url anonymized. Neural Computing and Applications (2025) 37:7423–7441 7439 123 9. Chen Y, Peng G, Zhu Z et al (2020) A novel deep learning method based on attention mechanism for bearing remaining useful life prediction. Appl Soft Comput 86:105919 10. Cho K, Van Merrie ¨nboer B, Bahdanau D, et al (2014) On the properties of neural machine translation: encoder-decoder approaches. arXiv preprint arXiv:1409.1259 11. Chollet F et al (2015) Keras. https://github.com/fchollet/keras 12. Deng B, Yan J, Lin D (2017) Peephole: predicting network performance before training. arXiv preprint arXiv:1712.03351 13. Deutsch J, He D (2017) Using deep learning-based approach to predict remaining useful life of rotating components. IEEE Trans Syst Man Cybern Syst 48(1):11–20 14. Ding J, Tarokh V, Yang Y (2018) Model selection techniques: an overview. IEEE Signal Process Mag 35(6):16–34 15. Dong X, Huang J, Yang Y, et al (2017) More is less: a more complicated network with less inference complexity. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 5840–5848 16. Frazier PI (2018) A tutorial on bayesian optimization. arXiv preprint arXiv:1807.02811 17. Gardner J, Guo C, Weinberger K, et al (2017) Discovering and exploiting additive structure for bayesian optimization. In: Artificial intelligence and statistics, PMLR, pp 1311–1319 18. Hinchi AZ, Tkiouat M (2018) Rolling element bearing remaining useful life estimation based on a convolutional long-short-term memory network. Procedia Comput Sci 127:123–132 19. Hochreiter S, Schmidhuber J (1997) Long short-term memory. Neural Comput 9(8):1735–1780 20. Holmberg K, Komonen K, Oedewald P, et al (2004) Safety and reliability—technology review. No. BTUO43-031209 in VTT Research Report, VTT Technical Research Centre of Finland, Finland 21. Huang CG, Huang HZ, Li YF (2019) A bidirectional LSTM prognostics method under multiple operational conditions. IEEE Trans Ind Electron 66(11):8792–8802 22. Kadane JB, Lazar NA (2004) Methods and criteria for model selection. J Am Stat Assoc 99(465):279–290 23. Kandasamy K, Schneider J, Po ´czos B (2015) High dimensional Bayesian optimisation and bandits via additive models. In: International conference on machine learning, PMLR, pp 295–304 24. Leal AS, Berton L, de Castro Santos LC (2022) Learning methods for remaining useful life prediction in a turbofan engine. In: Anais do XIX Encontro Nacional de Intelige ˆncia Artificial e Computacional, SBC, pp 556–566 25. LeCun Y, Haffner P, Bottou L, et al (1999) Object recognition with gradient-based learning. In: Shape, contour and grouping in computer vision. Springer, pp 319–345 26. Lee JE, Jiang JR (2019) Time series multi-channel convolutional neural network for bearing remaining useful life estimation. In: 2019 IEEE Eurasia Conference on IOT. Communication and Engineering (ECICE), IEEE, pp 408–410 27. Lee K, Kim JK, Kim J, et al (2018) CNN and GRU combination scheme for bearing anomaly detection in rotating machinery health monitoring. In: 2018 1st IEEE international conference on knowledge innovation and invention (ICKII), pp 102–105, https://doi.org/10.1109/ICKII.2018.8569155 28. Li C, Sa ´nchez RV, Zurita G et al (2016) Fault diagnosis for rotating machinery using vibration measurement deep statistical feature learning. Sensors 16(6):895 29. Li H, Zhao W, Zhang Y et al (2020) Remaining useful life prediction using multi-scale deep convolutional neural network. Appl Soft Comput 89:106113 30. Li X, Ding Q, Sun JQ (2018) Remaining useful life estimation in prognostics using deep convolution neural networks. Reliab Eng Syst Saf 172:1–11 31. Liaw R, Liang E, Nishihara R, et al (2018) Tune: A research platform for distributed model selection and training. arXiv preprint arXiv:1807.05118 32. Liu C, Zhang Y, Sun J et al (2022) Stacked bidirectional LSTM RNN to evaluate the remaining useful life of supercapacitor. Int J Energy Res 46(3):3034–3043 33. Ma L, Cui J, Yang B (2019) Deep neural architecture search with deep graph Bayesian optimization. In: IEEE/WIC/ACM international conference on web intelligence, pp 500–507 34. Ma Q, Zhang M, Xu Y, et al (2021) Remaining useful life estimation for turbofan engine with transformer-based deep architecture. In: 2021 26th international conference on automation and computing (ICAC), IEEE, pp 1–6 35. Malhotra P, TV V, Ramakrishnan A, et al (2016) Multi-sensor prognostics using an unsupervised health index based on LSTM encoder-decoder. arXiv preprint arXiv:1608.06154 36. Mateus BC, Mendes M, Torres Farinha J et al (2023) Improved GRU prediction of paper pulp press variables using different preprocessing methods. Prod Manuf Res 11(1):2155263 37. Mathew V, Toby T, Singh V, et al (2017) Prediction of remaining useful lifetime (rul) of turbofan engine using machine learning. In: 2017 IEEE Int. Conf. on circuits and systems (ICCS), IEEE, pp 306–311 38. Meir Y, Sardi S, Hodassman S et al (2020) Power-law scaling to assist with key challenges in artificial intelligence. Sci Rep 10(1):19628 39. Nectoux P, Gouriveau R, Medjaher K, et al (2012) Pronostia: an experimental platform for bearings accelerated degradation tests. In: IEEE international conference on prognostics and health management, PHM’12., IEEE Catalog Number: CPF12PHMCDR, pp 1–8 40. Neto D, Petrella L, Henriques J et al (2023) A hybrid deep learning-based approach for rolling bearing fault prognostics. IFAC-PapersOnLine 56(2):6588–6593 41. Ng SS, Xing Y, Tsui KL (2014) A naive Bayes model for robust remaining useful life prediction of lithium-ion battery. Appl Energy 118:114–123 42. Nieto PG, Garcı ´a-Gonzalo E, Lasheras FS et al (2015) Hybrid PSO-SVM-based method for forecasting of the remaining useful life for aircraft engines and evaluation of its reliability. Reliab Eng Syst Saf 138:219–231 43. Ordo ´n ˜ez C, Lasheras FS, Roca-Pardinas J et al (2019) A hybrid ARIMA-SVM model for the study of the remaining useful life of aircraft engines. J Comput Appl Math 346:184–191 44. Pelikan M, Goldberg DE, Cantu ´-Paz E, et al (1999) Boa: The Bayesian optimization algorithm. In: Proceedings of the genetic and evolutionary computation conference GECCO-99, Citeseer, pp 525–532 45. Peng C, Wu J, Wang Q et al (2022) Remaining useful life prediction using dual-channel LSTM with time feature and its difference. Entropy. https://doi.org/10.3390/e24121818 46. Peng Y, Wang Y, Zi Y (2018) Switching state-space degradation model with recursive filter/smoother for prognostics of remaining useful life. IEEE Trans Ind Inf 15(2):822–832 47. Psarommatis F, May G, Azamfirei V (2023) Envisioning maintenance 5.0: Insights from a systematic literature review of industry 4.0 and a proposed framework. J Manuf Syst 68:376–379 48. Qiao H, Wang T, Wang P et al (2018) A time-distributed spatiotemporal feature learning method for machine health monitoring with multi-sensor time series. Sensors 18(9):2932 49. Ren L, Cui J, Sun Y et al (2017) Multi-bearing remaining useful life collaborative prediction: A deep learning approach. J Manuf Syst 43:248–256 50. Rumelhart DE, Hinton GE, Williams RJ, et al (1985) Learning internal representations by error propagation 7440 Neural Computing and Applications (2025) 37:7423–7441 123 51. Saxena A, Goebel K, Simon D, et al (2008) Damage propagation modeling for aircraft engine run-to-failure simulation. In: 2008 international conference on prognostics and health management, IEEE, pp 1–9 52. Shcherbakov M, Sai C (2022) A hybrid deep learning framework for intelligent predictive maintenance of cyber-physical systems. ACM Trans Cyber-Phys Syst(TCPS) 6(2):1–22 53. Shen Y, Li Y, Zheng J, et al (2023) Proxybo: Accelerating neural architecture search via bayesian optimization with zero-cost proxies. In: Proceedings of the AAAI conference on artificial intelligence, pp 9792–9801 54. Soualhi A, Lamraoui M, Elyousfi B et al (2022) PHM SURVEY: implementation of prognostic methods for monitoring industrial systems. Energies. https://doi.org/10.3390/en15196909 55. Springenberg JT, Klein A, Falkner S, et al (2016) Bayesian optimization with robust Bayesian neural networks. Adv Neural Inform Process Syst 29 56. Su C, Li L, Wen Z (2020) Remaining useful life prediction via a variational autoencoder and a time-window-based sequence neural network. Qual Reliab Eng Int 36(5):1639–1656 57. Tian Z, Wong L, Safaei N (2010) A neural network approach for remaining useful life prediction utilizing both failure and suspension histories. Mech Syst Signal Process 24(5):1542–1555 58. Urbani M, Petri D, Brunelli M et al (2020) Maintenance-Management in Light of Manufacturing 4.0. Springer International Publishing, Cham, chap 4:97–11. https://doi.org/10.1007/978-3030-46103-4_5 59. Verma E, Chakraborty S (2021) Uncertainty-aware labelled augmentations for high dimensional latent space bayesian optimization. In: NeurIPS 2021 workshop on deep generative models and downstream applications 60. Wahid A, Breslin JG, Intizar MA (2022) Prediction of machine failure in industry 4.0: a hybrid CNN-LSTM framework. Appl Sci 12(9):4221 61. Wang L, Xie S, Li T, et al (2019) Sample-efficient neural architecture search by learning action space. arXiv preprint arXiv:1906.06832 62. Wang L, Zhao Y, Jinnai Y, et al (2019) Alphax: exploring neural architectures with deep neural networks and monte carlo tree search. arXiv preprint arXiv:1903.11059 63. Wang T, Guo D, Sun XM (2022) Remaining useful life predictions for turbofan engine degradation based on concurrent semisupervised model. Neural Comput Appl 34(7):5151–5160 64. Webert H, Do ¨ß T, Kaupp L et al (2022) Fault handling in industry 4.0: definition, process and applications. Sensors. https://doi.org/ 10.3390/s22062205 65. Werbin ´ska-Wojciechowska S, Winiarska K (2023) Maintenance performance in the age of industry 4.0: a bibliometric performance analysis and a systematic literature review. Sensors. https://doi.org/10.3390/s23031409 66. White C, Ai R, Neiswanger W, et al (2019) Neural architecture search via Bayesian optimization with a neural network model. NeurIPS 67. White C, Neiswanger W, Savani Y (2021) Bananas: Bayesian optimization with neural architectures for neural architecture search. In: Proceedings of the AAAI conference on artificial intelligence, pp 10293–10301 68. Xia T, Song Y, Zheng Y et al (2020) An ensemble framework based on convolutional bi-directional LSTM with multiple time windows for remaining useful life estimation. Comput Ind 115:103182 69. Yousuf S, Khan SA, Khursheed S (2022) Remaining useful life (RUL) regression using long-short term memory (LSTM) networks. Microelectron Reliab 139:11477 70. Yu W, Kim IY, Mechefske C (2020) An improved similaritybased prognostic algorithm for RUL estimation using an RNN autoencoder scheme. Reliab Eng Syst Saf 199:106926 71. Yu X, Zang H (2008) Transfomer fault diagnosis based on rough sets theory and artificial neural networks. In: 2008 international conference on condition monitoring and diagnosis, IEEE, pp 1342–1345 72. Yuan M, Wu Y, Lin L (2016) Fault diagnosis and remaining useful life estimation of aero engine using lstm neural network. In: 2016 IEEE international conference on aircraft utility systems (AUS), IEEE, pp 135–140 73. Zhao Z, Liang B, Wang X et al (2017) Remaining useful life prediction of aircraft engine based on degradation pattern learning. Reliab Eng Syst Saf 164:74–83 74. Zheng S, Ristovski K, Farahat A, et al (2017) Long short-term memory network for remaining useful life estimation. In: 2017 IEEE Int. Conf. prognostics and health management (ICPHM), IEEE, pp 88–95 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Neural Computing and Applications (2025) 37:7423–7441 7441 123