Review Reports - A Deep Equilibrium Model for Remaining Useful Life Estimation of Aircraft Engines

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript proposed a DEM to predict the Remaining Useful Life of turbofan jet engines. However, there are several areas that require improvement:
1. The literature review in Chapter 1 comes across as merely a compilation of references. To strengthen the chapter, it would be beneficial to weave a stronger logical narrative between the cited studies. This would enhance the structural integrity and foster a more cohesive understanding of the field.
2. Section 2.1 outlines several forward propagation methods, yet it falls short in explaining why these methods were chosen for different situations. A more in-depth exploration of the selection criteria and the specific scenarios where each method is most effective would greatly improve this section.
3. Regarding the backward propagation method in Section 2.2, it is ambiguous whether this is a novel contribution. If it is original, a comprehensive explanation of its inner workings and benefits is necessary to highlight its value. If not, proper citation of existing work is required to acknowledge its origins and provide context for readers.
4. When introducing the DE layer architecture in Chapter 3, the absence of a high - level overview at the beginning makes it difficult for readers to grasp the entire structure. Including a general schematic diagram would provide a visual aid, making the architecture easier to understand and follow.
5. The flow within Chapter 3 is somewhat disjointed, which disrupts the reading experience. Restructuring this chapter into clearly defined subsections would create a more logical progression of ideas and improve overall clarity.
6. In Section 4.3, the reasoning behind the parameter settings is not sufficiently explained. Providing a solid justification or citing relevant studies to back up these settings would strengthen the credibility and reliability of the experimental design.
7. Including an introduction to the PHM08 evaluation metrics would be highly valuable. This would give readers a clear understanding of the performance criteria used to assess the model, allowing for a more accurate interpretation of the results.
8. A discussion on the computational cost of the proposed model is essential. This should include both the configuration cost and time overhead, as it would give readers a more complete picture of the model's practicality and feasibility in real - world applications.
9. The main text positions the proposed method as a deep - equilibrium - based RUL prediction model, but the conclusion narrows its application to turbofan jet engines. The authors need to clarify the model's scope. If it is specifically designed for turbofan engines, this should be explicitly stated. If it is intended to be a general - purpose RUL prediction model, then the discussion should include its generalization ability across different systems to demonstrate its broader applicability.

Author Response

We sincerely thank the reviewer for the comprehensive and insightful review. We have carefully considered all suggestions and believe that addressing them has significantly improved the manuscript.

Comments 1: The literature review in Chapter 1 comes across as merely a compilation of references. To strengthen the chapter, it would be beneficial to weave a stronger logical narrative between the cited studies. This would enhance the structural integrity and foster a more cohesive understanding of the field.

Response 1: Thank you for your valuable feedback. To address your comment, we revised Chapter 1 by adding guiding phrases at the beginning of two paragraphs to help improve the flow and strengthen the connections between the cited studies. These changes aim to create a more logical and cohesive narrative in the literature review. Specifically, we added the phrase “Several studies have employed conventional machine learning methods to represent the degradation process and estimate the RUL.” at the beginning of the fourth paragraph (line 44), and “Deep learning models are sufficient for accurately predicting remaining useful life (RUL), having as compelling advantage the learning of complex hierarchical features directly from raw sensor data.”(line 60) at the beginning of the fifth paragraph.

Comments 2: Section 2.1 outlines several forward propagation methods, yet it falls short in explaining why these methods were chosen for different situations. A more in-depth exploration of the selection criteria and the specific scenarios where each method is most effective would greatly improve this section.

Response 2: Thank you for the comment. We have revised Section 2.1 to provide a clearer explanation of the reasoning behind the selection of each forward propagation method. Specifically, we now describe the scenarios in which each method is most effective. These changes are highlighted in red in the revised manuscript (page 6).

Comments 3: Regarding the backward propagation method in Section 2.2, it is ambiguous whether this is a novel contribution. If it is original, a comprehensive explanation of its inner workings and benefits is necessary to highlight its value. If not, proper citation of existing work is required to acknowledge its origins and provide context for readers.

Response 3: Thank you for your observation. We acknowledge the ambiguity in the original version of Section 2.2. In the revised manuscript, we have updated the introductory paragraph to clarify that the backward propagation method is based on existing work. We have also added the appropriate citation to properly acknowledge its origin and provide readers with relevant context (page 6, line 250).

The updated paragraph is:

The forward pass of DE models requires iterative solvers, making conventional back-propagation infeasible due to the increased storing demand of the intermediate gradients, generated during the process.

To overcome the problem, gradients are computed using the Implicit Function Theorem (IFT), a technique that avoids the unrolling of the iterative process \citep{DBLP:journals/corr/abs-1909-01377}.

Comments 4: When introducing the DE layer architecture in Chapter 3, the absence of a high-level overview at the beginning makes it difficult for readers to grasp the entire structure. Including a general schematic diagram would provide a visual aid, making the architecture easier to understand and follow.

Response 4: Thank you for the suggestion. In response, Figure 2 (previously Figure 4, page 8) which illustrates the overall architecture of the framework, has been moved to the beginning of Chapter 3 to provide a clearer high-level overview. Furthermore, a new introductory paragraph was added to accompany the figure, improving the overall readability (page8, line 279). Also, the caption of Figure 4 was updated.

The new introductory paragraph is:

Figure 2 provides a visual diagram of the architecture of the proposed framework for RUL estimation. The equilibrium model (DE block) is consisting of two key components: a convolutional block and a Dual-Input Interconnection mechanism. Additionally, a Monte Carlo Dropout Feed forward Neural Network is incorporated as the final block, improving both the performance and reliability of the framework.

Comments 5: The flow within Chapter 3 is somewhat disjointed, which disrupts the reading experience. Restructuring this chapter into clearly defined subsections would create a more logical progression of ideas and improve overall clarity.

Response 5: Thank you for your valuable feedback on the structure of Chapter 3. We appreciate your suggestion and agree that better organization would enhance clarity and the overall reading experience. In response, we have restructured Chapter 3 by introducing two subsections titled “Deep Equilibrium Block” (starts at page 8) and “Monte-Carlo Dropout Feed-Forward Neural Network” (starts at page 9).

Comments 6: In Section 4.3, the reasoning behind the parameter settings is not sufficiently explained. Providing a solid justification or citing relevant studies to back up these settings would strengthen the credibility and reliability of the experimental design.

Response 6: To clarify the selection process of key hyper-parameters, we have added a brief paragraph in Section 4.3 of the manuscript (page 13, line 448).

Inserted paragraph in section 4.3: The hyper-parameters (learning rate, batch size, dropout ratio, number of convolutional filters, attention projection dimension) were selected by a combination of trial-and-error experimentation and reference to values commonly reported in related literature. A limited manual tuning process was employed on a validation split from the training data, where the performance metric of RMSE was monitored. A full grid search of the hyper-parameters can further improve the performance of the RUL estimation framework but in this study was not conducted.

Comments 7: Including an introduction to the PHM08 evaluation metrics would be highly valuable. This would give readers a clear understanding of the performance criteria used to assess the model, allowing for a more accurate interpretation of the results.

Response 7: The manuscript has been updated to include an introduction to the PHM08 evaluation metrics (page 14, line 458). This addition provides readers with a clear understanding of the performance criteria used to assess the model, enabling a more accurate interpretation of the results.

Comments 8: A discussion on the computational cost of the proposed model is essential. This should include both the configuration cost and time overhead, as it would give readers a more complete picture of the model's practicality and feasibility in real-world applications.

Response 8: We thank the reviewer for highlighting the importance of discussing the model’s computational cost. In response, we have added a subsection titled “Computational Cost and Time Overhead” of the revised manuscript (page 15, line 518, subsection 4.8).

Added subsection: Computational Cost and Time Overhead

To evaluate the practicality and feasibility in real-world applications, we present an analysis of the computation requirements of the proposed Bayesian Deep Equilibrium framework in the FD001 sub-dataset of CMAPSS. The model's training was conducted in a system with an NVIDIA GeForce GTX 1060 with 6 GB, utilizing a batch size of 256 over 35 epochs. The sequence window is set to 15, building a model with 21,955 trainable parameters, making it lightweight and suitable even for low-resource environments. The training time interval was 17 minutes, which corresponds to an average of 29.5 seconds per epoch.

Also, a computational analysis of the inference process is provided. To estimate predictive uncertainty during inference, we use 100 Monte Carlo forward passes in the Bayesian framework. This increased around 25% the overhead during inference. However, the response time for each engine is 0.008 seconds, making the model suitable for real-time industrial applications.

Comments 9: The main text positions the proposed method as a deep-equilibrium-based RUL prediction model, but the conclusion narrows its application to turbofan jet engines. The authors need to clarify the model's scope. If it is specifically designed for turbofan engines, this should be explicitly stated. If it is intended to be a general-purpose RUL prediction model, then the discussion should include its generalization ability across different systems to demonstrate its broader applicability.

Response 9: Thank you for your feedback. We have updated the title to "A Deep Equilibrium Model for Remaining Useful Life Estimation of Aircraft Engines" to better describe the used dataset in our experimental evaluation. However, we would like to point out that the proposed Deep Equilibrium Model is not restricted to aircraft engines and can be applied to any complex system since it is designed in a general manner. Sequential sensor data from the complicated system is the only requirement.

New text in line 162, page 4: Also, the proposed DEM can be applied on any complex system since it has a general purpose design, provided that multivariate sequential sensor data is available.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

1. Insufficient Novelty and Positioning

The manuscript heavily leans on established Deep Equilibrium Model (DEM) literature (e.g. Bai et al. [25]) and simply applies DEM to RUL estimation without clear conceptual innovation. The claimed “novel Dual-Input Interconnection Attention Block” (pp. 3–4) is just a straightforward attention mechanism grafted onto DEM; this is incremental at best, not a substantive advance.
There is no discussion of why standard Transformer or graph-based equilibrium layers would fail here, nor any theoretical justification that DEM offers unique benefits for RUL beyond what existing implicit models already demonstrate.

2. Overstated Performance Claims

Table 3 reports modest RMSE gains (e.g. from 11.62 to 10.06 on FD001). Framing these as “state-of-the-art” is hyperbolic—improvements are within typical variance of deep models and no statistical tests are provided.
The conclusions repeatedly assert “superior efficiency and stability,” yet runtime or convergence speed comparisons against baselines are absent. Without wall-clock or iteration-count benchmarks versus competing methods, claims of efficiency remain unsubstantiated.

3. Experimental Design Flaws

Ablation Study Missing: There is no ablation isolating the impact of the DEM core, the dual-input attention, GroupNorm, or calibration. It is impossible to tell which component actually drives the performance gains.
Calibration Evaluation Is Superficial: Reporting only coverage percentages (Table 5) without calibration error metrics (ECE/MCE) or reliability diagrams prevents assessing the true quality of uncertainty estimates.
Limited Dataset Scope: Only the CMAPSS benchmark is used. No real-world or cross-domain validation is provided, undermining claims of general applicability.

4. Methodological Opacity and Reproducibility Issues

Hyperparameter Details Are Vague: Key settings (e.g. number of Anderson iterations, convergence tolerance, grouping strategy for calibration) are given without rationale or sensitivity analysis.
No Code or Seed Disclosure: No link to code, random seeds, or hardware specs is provided, making reproduction practically impossible.

5. Mathematical and Presentation Weaknesses

Equation numbering is inconsistent (e.g. references to “Eq.” without numbering in the text around pp. 1–2). Variable definitions (e.g. dimensions in Eq. (14)) are incomplete, forcing readers to guess.
The writing is often convoluted (“implicit layers often refer to: i) … ii) … iii) …” on p. 2) and riddled with grammatical errors, reducing clarity. Figures are low-contrast and captions lack sufficient detail to stand alone.

6. Lack of Critical Discussion

The paper does not acknowledge known downsides of DEMs, such as potential instability when Lipschitz conditions are violated, or the practical difficulties of tuning spectral norms.
There is no failure analysis—where does the model break down? What types of engine profiles or sensor noise significantly degrade performance?

Author Response

We sincerely thank the reviewer for the comprehensive and insightful review. We have carefully considered all suggestions and believe that addressing them has significantly improved the manuscript.

Comments 1: Insufficient Novelty and Positioning:

The manuscript heavily leans on established Deep Equilibrium Model (DEM) literature (e.g. Bai et al. [25]) and simply applies DEM to RUL estimation without clear conceptual innovation. The claimed “novel Dual-Input Interconnection Attention Block” (pp. 3–4) is just a straightforward attention mechanism grafted onto DEM; this is incremental at best, not a substantive advance.

There is no discussion of why standard Transformer or graph-based equilibrium layers would fail here, nor any theoretical justification that DEM offers unique benefits for RUL beyond what existing implicit models already demonstrate.

Response 1: We thank the reviewer for this insightful observation.

We fully agree that a clearer presentation of our architectural innovations within the Deep Equilibrium framework is essential for strengthening the manuscript. To address this, we have revised the relevant paragraph on page 3 (starts at line 117) and updated the contribution statements within the Introduction section to explicitly highlight these distinctions (page 4, line 170), with all changes marked in red text.

Revised paragraph in the Introduction section: The Deep Equilibrium (DE) block integrates convolutional operations and a novel attention-based Dual-Input Interconnection mechanism, created specifically for implicit deep models on multivariate time-series data. The convolutional component extracts local spatial and temporal patterns from raw sensor inputs, producing an input feature map representing short-term dependencies. The input feature mapping and a latent representation vector, which encodes the internal health state of the system, are dynamically processed by the Dual-Input Interconnection mechanism. This allows the model to perform a cross-attention-like operation, where the input mapping is projected as keys and values and the health state as queries in a shared embedding space. So, the latent state is adaptively updated based on the most relevant observed patterns of the input. The equilibrium state is used as a health indicator for system monitoring since it captures long-term degradation patterns and local sensor behavior. So, the DE block allows for a highly expressive and memory-efficient representation that captures the dynamics of the underlying system by iteratively updating the latent state until convergence. The architecture differs from conventional attention-based architectures in DE models, where self-attention is applied only to the latent representation vector and the input vector mapping is used as a residual connection.

Revised statement on contributions: The core element of the Deep Equilibrium Model is a novel Dual-Input Interconnection Attention Block, which enables iterative and adaptive updates of the latent degradation representation by jointly processing the internal health state and the spatio-temporal features extracted from convolutional blocks. Unlike standard Transformer self-attention mechanisms used in DE frameworks, which typically operate only on the latent representation and incorporate input features as a static residual, the proposed attention-based block performs a cross-attention-like interaction between two distinct inputs. This design enhances the model’s ability to capture complex degradation dynamics, leading to a more expressive and context-aware health representation.

Comments 2: Overstated Performance Claims:

Table 3 reports modest RMSE gains (e.g. from 11.62 to 10.06 on FD001). Framing these as “state-of-the-art” is hyperbolic—improvements are within typical variance of deep models and no statistical tests are provided.
The conclusions repeatedly assert “superior efficiency and stability,” yet runtime or convergence speed comparisons against baselines are absent. Without wall-clock or iteration-count benchmarks versus competing methods, claims of efficiency remain unsubstantiated.

Response 2: We thank the reviewer for the helpful comments.

In response, we have revised the manuscript to avoid overstatements, replacing terms such as “state-of-the-art” and “superior efficiency” with more appropriate phrases like “competitive performance” and “consistent improvements.”
We also clarified that the efficiency of the proposed model refers specifically to its fast convergence behavior, compact parameter size, and low inference latency. To support this, Section 4.8 (“Computational Cost & Time Overhead”) now includes detailed information on training time, model size and prediction speed. For example, the model achieves a per-engine inference time of 0.008 seconds even with 100 Monte Carlo passes, demonstrating its suitability for real-time applications.

Revised text in Abstract (page 1, line 16):
Simulation experiments on the widely used NASA Turbofan Jet Engine Data Set show consistent improvements, with the proposed framework offering a competitive alternative for RUL prediction under diverse conditions.

Revised text in Conclusions and Future Work (page 17, line 547):
The proposed framework achieves consistent performance improvements across different CMAPSS sub-datasets, particularly in complex scenarios involving multiple fault modes and operating conditions. These results suggest that the method is a competitive and reliable alternative among recent approaches.

Section 4.8 – Computational Cost and Time Overhead (new section, page 15, line 518):
To evaluate the practicality and feasibility in real-world applications, we present an analysis of the computation requirements of the proposed Bayesian Deep Equilibrium framework in the FD001 sub-dataset of CMAPSS. The model's training was conducted on a system with an NVIDIA GeForce GTX 1060 (6 GB), using a batch size of 256 over 35 epochs. The sequence window was set to 15, resulting in a model with 21,955 trainable parameters—lightweight and suitable even for low-resource environments. The total training time was 17 minutes, averaging 29.5 seconds per epoch. For inference, we use 100 Monte Carlo forward passes to estimate uncertainty, which adds about 25% overhead. Nevertheless, the per-engine response time is 0.008 seconds, confirming the model’s suitability for real-time industrial applications.

Comments 3: Experimental Design Flaws

Ablation Study Missing: There is no ablation isolating the impact of the DEM core, the dual-input attention, GroupNorm, or calibration. It is impossible to tell which component actually drives the performance gains.
Calibration Evaluation Is Superficial: Reporting only coverage percentages (Table 5) without calibration error metrics (ECE/MCE) or reliability diagrams prevents assessing the true quality of uncertainty estimates.
Limited Dataset Scope: Only the CMAPSS benchmark is used. No real-world or cross-domain validation is provided, undermining claims of general applicability.

Response 3: We thank the reviewer for the thoughtful feedback.

While we acknowledge the absence of a full ablation study, we include a comparison in Table 4 (page 16) between the complete model and a version without Monte Carlo Dropout. This shows that the core performance improvements are driven primarily by the Deep Equilibrium Model (DEM) architecture, since disabling Monte Carlo Dropout results in only minor changes to RMSE and PHM08 scores.
The strength of the DEM lies in its ability to model the nonlinear degradation present in multivariate time-series data like CMAPSS. Its equilibrium-based structure allows it to capture both long-term and short-term sensor variations through an iterative process, without the need for deep, stacked layers. This architectural choice is well suited to the nature of the problem and is the main reason behind the observed performance gains.
While we recognize that calibration metrics such as ECE and MCE or reliability diagrams are valuable tools, our primary goal in this study was to demonstrate practical improvements in uncertainty quantification through a calibration approach based on Gaussian Mixture clustering.
In particular, the presented coverage percentages (Table 5, page 17) demonstrate how well our suggested calibration improves the alignment of real data within the estimated uncertainty interval.
Coverage percentages inherently measure calibration performance in a meaningful and intuitive way, particularly in prognostics contexts where interval-based reliability directly informs decision-making. Given this direct interpretability and practical relevance, we believe additional calibration error metrics, although beneficial in certain theoretical contexts, are not strictly necessary for validating the effectiveness of our approach within this particular application.
We agree that validation using diverse datasets, particularly from real-world or cross-domain applications, would indeed strengthen the generalization claims of our method. However, our primary objective in this study was to introduce and thoroughly demonstrate the effectiveness of our proposed Deep Equilibrium Model within the widely accepted CMAPSS benchmark, which remains a critical baseline for prognostics and health management tasks due to its complexity and diversity in operational conditions.

Comments 4: Methodological Opacity and Reproducibility Issues:

Hyperparameter Details Are Vague: Key settings (e.g. number of Anderson iterations, convergence tolerance, grouping strategy for calibration) are given without rationale or sensitivity analysis.
No Code or Seed Disclosure: No link to code, random seeds, or hardware specs is provided, making reproduction practically impossible.

Response 4: We thank the reviewer for raising this important concern regarding methodological transparency and reproducibility.

We have updated Section 4.3 and Section 4.8 to clarify key hyper-parameter choices. Specifically, we now explain that the number of Anderson iterations (set to 200) and the convergence tolerance (10^{-4}) were chosen based on stability observed during preliminary experiments, balancing convergence accuracy with computational cost. Similarly, the grouping strategy used in the calibration step (based on Gaussian Mixture Modeling over prediction mean and standard deviation) was selected to reflect variability in model confidence across different operating conditions.
To support reproducibility, we have also added the full hardware specification (NVIDIA GTX 1060, 6 GB VRAM), optimizer settings, and training schedule in Section 4.8 (page 15, line 518). Additionally, we are preparing to release our source code, including training scripts and random seeds, upon acceptance.

Added text in Section 4.3 (page 13, line 448):

The hyper-parameters (learning rate, batch size, dropout ratio, number of convolutional filters, attention projection dimension) were selected by a combination of trial-and-error experimentation and reference to values commonly reported in related literature. A limited manual tuning process was employed on a validation split from the training data, where the performance metric of RMSE was monitored. A full grid search of the hyper-parameters can further improve the performance of the RUL estimation framework but in this study was not conducted.

Comments 5: Mathematical and Presentation Weaknesses

Equation numbering is inconsistent (e.g. references to “Eq.” without numbering in the text around pp. 1–2). Variable definitions (e.g. dimensions in Eq. (14)) are incomplete, forcing readers to guess.
The writing is often convoluted (“implicit layers often refer to: i) … ii) … iii) …” on p. 2) and riddled with grammatical errors, reducing clarity. Figures are low-contrast and captions lack sufficient detail to stand alone.

Response 5: We thank the reviewer for pointing out these presentation and clarity issues.

We have revised the manuscript to ensure all numbered equations are explicitly referenced in the main text, improving the logical flow.
Additionally, we clarified variable definitions in Equation (14), including the dimensions of the attention matrices and input features, to eliminate ambiguity (page 8, line 302).
Additionally, we enhanced picture captions and contrast for easier reading and rewrote a few sections for grammatical clarity.

Comments 6: Lack of Critical Discussion

The paper does not acknowledge known downsides of DEMs, such as potential instability when Lipschitz conditions are violated, or the practical difficulties of tuning spectral norms.There is no failure analysis—where does the model break down? What types of engine profiles or sensor noise significantly degrade performance?

Response 6: We appreciate the reviewer’s valuable comments highlighting the need for deeper analysis of the DEM framework's robustness.

In response, we have expanded the manuscript by including a subsection (Failure Analysis on the CMAPSS Dataset) to discuss conditions where our model's performance decreases, particularly emphasizing scenarios involving irregular degradation patterns, varying operating conditions and sensor anomalies within the CMAPSS dataset. Additionally, in the discussion section, we acknowledge these identified limitations as valuable opportunities for future research, suggesting advanced strategies for improving model performance in challenging operational scenarios (such as those encountered in CMAPSS FD002 and FD004). We believe these revisions comprehensively address your concerns and enhance the overall critical discussion in our paper.

Section 4.9 - Failure Analysis on the CMAPSS Dataset (new section, page 16, line 532):

It is crucial to provide deeper insights into the robustness of the DE model observed within the CMAPSS dataset and analyze the specific conditions where its predictive performance decreased.

Observing the performance across the four CMAPSS sub-datasets, we notice increased sensitivity and higher error rates for datasets with multiple operating conditions. The model has difficulties with engines that show sudden degradation patterns or irregular sensor responses in different operating modes. Furthermore, sensor noise and sudden measurement anomalies are commonly encountered in realistic aircraft engine operations, disrupting the equilibrium convergence process and resulting in significant prediction inaccuracies. The model’s reliance on strict Lipschitz constraints for stability further contributes to sensitivity since small deviations can considerably affect performance, underscoring the need for careful parameter initialization, regularization, and normalization strategies in practical applications.

New text in Conclusions and Future Work (page 17, line 558):

Also, the limitations identified through the failure analysis of the CMAPSS dataset present valuable opportunities for future research aimed at enhancing the reliability of the DEM framework. Future work could examine advanced approaches to handle irregular or rapid engine degradation scenarios frequently encountered in datasets such as CMAPSS FD002 and FD004.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Reviewer Comments

Comment 1: The title should be adjusted. At the end it should include something like “A Deep Equilibrium Model for Remaining Useful Life for NASA Turbofan Jet Engine Data” otherwise the title of the manuscript misleads readers to think that this model can be applied to any remaining useful life data which is not refelected in the provided abstract or throughout the manuscript.

Comment 2: The following statement by the authors are not necessarily true “However, conventional ML methods use statistical features to represent the degradation process, while DL frameworks automatically learn complex hierarchical features from raw sensor data.” Deep learning frameworks are favoured by many researchers to be used however not all deep learning frameworks can automatically learn complex features from raw sensor. It really depends on the framework, dataset being worked on and design of the network architecture. The authors should consider rephrasing this sentence.

Comment 3: The followings sentence in the introduction has no context to the techniques used the authors should expand on what the technique does. “So, the extracted features using the Empirical Mode Decomposition Technique are integrated into a Bayesian-optimized Random Forest Model to estimate the RUL of bearing”, what does empirical mode decomposition technique do and how does it integrate into bayesian optimized random forest model in order to estimate RUL of a bearing?

Comment 4: As an example the authors should expand on “where the model dynamically adapts through the application of an unscented Kalman filter” how each of these mentioned models in the introduction and filters are used to determine RUL of specific machine component for the NASA dataset. Otherwise this is a weak literature review to state a sentence for each method or filter to say this has been done. Doesn’t show the understanding of each method in the literature review. The literature review should lead the readers to understand that there is a need to propose a new framework for RUL. Simply stating different methods back to back doesn’t portray this.

Comment 5: The following statement requires a reference or references “Most neural frameworks designed for RUL estimation rely on explicit layers, where the output is produced directly from the input based on mathematical formulations defined by their architecture”

Comment 6: On line 133, “The contributions of the research paper follow::” there is an extra : please remove

Comment 7: Figure 1 should be placed after the first mention in text for ease of readability. Additionally each terminology should be explained in text explanation sounds like the figure title and doesn’t provide any benefit. How does the second parts unfolding process allow to reach convergence what differentiates your framework from existing DE frameworks? I fail to see this, the authors should highlight what makes their framework unique compared to existing DE frameworks. The authors should in the introduction provide a deeper understanding of the different available DE frameworks and lead to their proposed framework and clearly show the difference between them.

Comment 8: In the introduction no DE framework architectures were discussed but later on architecture of DE is provided the authors should provide architectures of existing research related to DE.

Comment 9: The authors state that Figure 2 has “three convolutional blocks are used” however in the figure there are only 2 convolutional present? Also why is there a convolutional block as the architecture title is the architecture called a convolutional block?

Comment 10: How is the adaptive process described in equation 15 and shown in figure 3 expand on this. What does a dual interconnection block do? What dies zk to two linear block that lead to Q and K then to MatMul mean? Also what is MatMul? The architecture of the figure 3 should be explained in depth for understanding.

Comment 11: Is Figure 4 suppose to be addition of figure 2 and figure 3 color coded blocks if so this should be explained and for the previous figures this should be mentioned the color coding and the main title of convolutional block. And why is the name convolutional block selected as a title additionally figure 2 seems to have an input drawn leading to further confusion since the convolutional block alredy has an input signal present then in figure 4 another input signal is fed in leading the readers to believe there would be two series of input signals fed in. I recommend removing the input signal from the figure two convolutional block unless this was the intended purpose or take it outside the color coded block like the one show in in dual input interconnection block to stay consistent with drawings.

Comment 12: Figure 5 should have axis labels. And what does this indicate?

Comment 13: From the sound of the entire manuscript the authors seem to propose the Deep Equilibrium Models archiecture yet compare their results to non DEM models specifically looking at table 3. From this aspect the comparison doesn’t seem to be fair instead authors should compare to existing DE models or make another table with DE models existing and their proposed DE model to show that it outperforms all instead of existing deep learning models.

Author Response

Comments 1: The title should be adjusted. At the end it should include something like “A Deep Equilibrium Model for Remaining Useful Life for NASA Turbofan Jet Engine Data” otherwise the title of the manuscript misleads readers to think that this model can be applied to any remaining useful life data which is not refelected in the provided abstract or throughout the manuscript.

Response 1: Thank you for your feedback. We have updated the title to "A Deep Equilibrium Model for Remaining Useful Life Estimation of Aircraft Engines" to better describe the used dataset in our experimental evaluation. However, we would like to point out that the proposed Deep Equilibrium Model is not restricted to aircraft engines and can be applied to any complex system since it is designed in a general manner. Sequential sensor data from the complicated system is the only requirement.

New text (page 4, line 162): Also, the proposed DEM can be applied on any complex system since it has a general purpose design, provided that multivariate sequential sensor data is available.

Comments 2: The following statement by the authors are not necessarily true “However, conventional ML methods use statistical features to represent the degradation process, while DL frameworks automatically learn complex hierarchical features from raw sensor data.” Deep learning frameworks are favoured by many researchers to be used however not all deep learning frameworks can automatically learn complex features from raw sensor. It really depends on the framework, dataset being worked on and design of the network architecture. The authors should consider rephrasing this sentence.

Response 2: Thank you for your feedback. We agree with your point about the statement concerning deep learning frameworks and their feature learning capabilities. We have updated the text to present a more accurate understanding.

Original text: However, conventional ML methods use statistical features to represent the degradation process, while DL frameworks automatically learn complex hierarchical features from raw sensor data \citep{electronics11071125}.

Revised text (page 2, line 37): Conventional ML methods typically rely on handcrafted statistical features to represent the degradation process, whereas many advanced DL frameworks are designed to learn complex hierarchical features directly from raw sensor data, often reducing the need for extensive manual feature engineering. However, the specific framework, the characteristics of the dataset and the designed architecture of the network influence the effectiveness of the automatic feature extraction \citep{electronics11071125}.

Comments 3: The followings sentence in the introduction has no context to the techniques used the authors should expand on what the technique does. “So, the extracted features using the Empirical Mode Decomposition Technique are integrated into a Bayesian-optimized Random Forest Model to estimate the RUL of bearing”, what does empirical mode decomposition technique do and how does it integrate into bayesian optimized random forest model in order to estimate RUL of a bearing?

Response 3: Thank you for your feedback. We have revised the sentence to clarify the use of Empirical Mode Decomposition (EMD) and its capability to extract robust degradation features from non-linear and non-stationary bearing signals. These features then serve as inputs for the Random Forest (RF) model, which is further optimized using Bayesian optimization to enhance RUL prediction accuracy.

Original text: So, the extracted features using the Empirical Mode Decomposition Technique are integrated into a Bayesian-optimized Random Forest Model to estimate the RUL of bearing in \citep{9889166}.

Revised text (page 2, line 45): In a study, Empirical Mode Decomposition (EMD) effectively analyzes non-linear and non-stationary data to extract robust features. These features are then inserted into a Random Forest (RF) model, which is optimized through Bayesian optimization for superior RUL prediction \citep{9889166}.

Comments 4: As an example the authors should expand on “where the model dynamically adapts through the application of an unscented Kalman filter” how each of these mentioned models in the introduction and filters are used to determine RUL of specific machine component for the NASA dataset. Otherwise this is a weak literature review to state a sentence for each method or filter to say this has been done. Doesn’t show the understanding of each method in the literature review. The literature review should lead the readers to understand that there is a need to propose a new framework for RUL. Simply stating different methods back to back doesn’t portray this.

Response 4: Thank you for your valuable feedback. We have revised the literature review to address your concern about providing more context and depth for the methods discussed.

Original text: In a different research study, advanced hybrid metrics are combined with a robust logistic regression model to predict the remaining useful life (RUL) index, where the model dynamically adapts through the application of an unscented Kalman filter (UKF) \cite{machines11020163}.

Revised text (page 2, line 48): Also, to achieve adaptability and real-time capability of RUL prediction, another research effort uses the Unscented Kalman Filter (UKF) to recursively update the degradation parameters within a logistic regression model \cite{machines11020163}. While this method offers advantages in online adaptability, it can be sensitive to the initial state estimation and the underlying assumptions about the system's dynamics.

Comments 5: The following statement requires a reference or references “Most neural frameworks designed for RUL estimation rely on explicit layers, where the output is produced directly from the input based on mathematical formulations defined by their architecture”

Response 5: Thank you for your valuable feedback. The statement is directly supported by the discussion in the preceding paragraph, where it is clear that most neural approaches for RUL prediction are based on explicit layers. The text has been revised for clarity.

Original text: Most neural frameworks designed for RUL estimation rely on explicit layers, where the output is produced directly from the input based on mathematical formulations defined by their architecture.

Revised text (page 2, line 80): As discussed in the architectures above, most neural frameworks for RUL estimation use explicit layers to directly produce output from input based on their mathematical formulations.

Comments 6: Comment 6: On line 133, “The contributions of the research paper follow::” there is an extra : please remove

Response 6: Thank you for observing that. The extra colon on line 165 (previously line 133) has been removed.

Comments 7: Figure 1 should be placed after the first mention in text for ease of readability. Additionally each terminology should be explained in text explanation sounds like the figure title and doesn’t provide any benefit. How does the second parts unfolding process allow to reach convergence what differentiates your framework from existing DE frameworks? I fail to see this, the authors should highlight what makes their framework unique compared to existing DE frameworks. The authors should in the introduction provide a deeper understanding of the different available DE frameworks and lead to their proposed framework and clearly show the difference between them.

Response 7: Thank you for your insightful comment. Figure 1 is already positioned close to its first appearance in the manuscript, in Section 2, where the main concepts of Deep Equilibrium models are introduced. Also, it acts as an introductory illustration of the generic architecture and convergence behavior of a DE layer and provide a high-level theoretical overview for readers who may be unfamiliar with the inner workings of equilibrium models.

The novelty of our framework lies in the internal design and functional mechanisms presented in Figure 2 and Figure 4. Figure 2 presents the architecture of the proposed Deep Equilibrium Model (DEM) for RUL prediction, including all key components. Figure 4 illustrates our Dual-Input Interconnection Block, a key innovation in our study. Unlike conventional DE approaches that use self-attention within the latent representation only, our model introduces a novel cross-attention-like mechanism that adaptively processes input feature maps with the latent state representation, enhancing modeling health state representation capability.

To clearly distinguish our architectural innovations within the DE framework, we have revised the relevant paragraph on page 3 of the Introduction and updated the contribution statements accordingly (indicated in red in the manuscript).

Revised paragraph in the Introduction section (page 3, line 117): The Deep Equilibrium (DE) block integrates convolutional operations and a novel attention-based Dual-Input Interconnection mechanism, created specifically for implicit deep models on multivariate time-series data. The convolutional component extracts local spatial and temporal patterns from raw sensor inputs, producing an input feature map representing short-term dependencies. The input feature mapping and a latent representation vector, which encodes the internal health state of the system, are dynamically processed by the Dual-Input Interconnection mechanism. This allows the model to perform a cross-attention-like operation, where the input mapping is projected as keys and values and the health state as queries in a shared embedding space. So, the latent state is adaptively updated based on the most relevant observed patterns of the input. The equilibrium state is used as a health indicator for system monitoring since it captures long-term degradation patterns and local sensor behavior. So, the DE block allows for a highly expressive and memory-efficient representation that captures the dynamics of the underlying system by iteratively updating the latent state until convergence. The architecture differs from conventional attention-based architectures in DE models, where self-attention is applied only to the latent representation vector and the input vector mapping is used as a residual connection.

Revised statement on contributions (page 4, line 170): The core element of the Deep Equilibrium Model is a novel Dual-Input Interconnection Attention Block, which enables iterative and adaptive updates of the latent degradation representation by jointly processing the internal health state and the spatio-temporal features extracted from convolutional blocks. Unlike standard Transformer self-attention mechanisms used in DE frameworks, which typically operate only on the latent representation and incorporate input features as a static residual, the proposed attention-based block performs a cross-attention-like interaction between two distinct inputs. This design enhances the model’s ability to capture complex degradation dynamics, leading to a more expressive and context-aware health representation.

Comments 8: In the introduction no DE framework architectures were discussed but later on architecture of DE is provided the authors should provide architectures of existing research related to DE.

Response 8: Thank you for your valuable feedback. We agree that the introduction should better present the existing Deep Equilibrium (DE) frameworks and their architectures. To address this, we have revised the Introduction section to include a discussion of DE architectures found in the literature.

Revised Text (page 3, line 86):

Implicit layers often refer to:

Neural Ordinary Differential Equations (Neural ODEs), where the model learns a function that represent the evolving dynamics of the hidden state over time. This continuous transformation is solved using numerical integration methods. Because of their ability to share parameters across time, these models are memory efficient \citep{DBLP:journals/corr/abs-1806-07366}.
Optimization-based Implicit Layers define their output as the outcome of a mathematical optimization problem embedded within the neural architecture. Gradients are estimated using the implicit differentiation theorem. The model learns not only from data but also ensures to satisfy certain mathematical properties or constraints \citep{DBLP:journals/corr/abs-1910-12430}.
Deep Equilibrium Models (DEMs) where the hidden representation is computed as the equilibrium of a fixed point equation. This fixed point is computed through iterative root-finding methods, allowing the model to represent arbitrarily deep computations without explicitly stacking layers, while using constant memory during training \citep{DBLP:journals/corr/abs-1909-01377}.

While these frameworks provide valuable approaches for implicit modeling, they often lack internal expressive mechanisms for modeling multivariate spatio-temporal dependencies in the time-series sensor data of complex systems.

Comments 9: The authors state that Figure 2 has “three convolutional blocks are used” however in the figure there are only 2 convolutional present? Also why is there a convolutional block as the architecture title is the architecture called a convolutional block?

Response 9: Thank you for your careful observation. The figure 3 (previously Figure 2) has been updated to accurately reflect the internal structure of the convolutional component, which consists of three convolutional layers, each followed by a ReLU activation function. Also, the manuscript text and figure caption have been revised to clarify the process and resolve any inconsistencies (page 8).

Comments 10: How is the adaptive process described in equation 15 and shown in figure 3 expand on this. What does a dual interconnection block do? What dies zk to two linear block that lead to Q and K then to MatMul mean? Also what is MatMul? The architecture of the figure 3 should be explained in depth for understanding.

Response 10: Thank you for this valuable and detailed comment.

To address your suggestions, we have revised the manuscript to expand the explanation of the adaptive process described in Equation 15, particularly focusing on the function and role of the Dual-Input Interconnection Block (pages 8, 9). Also, we have inserted a caption for Figure 4 (page 10, previously Figure 3) that clearly explains each component, including the MatMul operation, which refers to matrix multiplication used in attention mechanism.

Comments 11: Is Figure 4 suppose to be addition of figure 2 and figure 3 color coded blocks if so this should be explained and for the previous figures this should be mentioned the color coding and the main title of convolutional block. And why is the name convolutional block selected as a title additionally figure 2 seems to have an input drawn leading to further confusion since the convolutional block alredy has an input signal present then in figure 4 another input signal is fed in leading the readers to believe there would be two series of input signals fed in. I recommend removing the input signal from the figure two convolutional block unless this was the intended purpose or take it outside the color coded block like the one show in in dual input interconnection block to stay consistent with drawings

Response 11: Thank you for your insightful feedback.

We have revised the manuscript and figures to address the suggestions. It is now clearly stated that Figure 2 (previously Figure 4) combines the components shown separately in Figures 3 (previously Figure 2) and 4 (previously Figure 3), illustrating the full internal structure of the model. Also, the captions in the figures have been updated.

Comments 12: Figure 5 should have axis labels. And what does this indicate?

Response 12: Thank you for your comment. The caption for Figure 5 (page 13) has been updated to include axis descriptions and clarify its purpose. The x-axis represents time steps (cycles), and the y-axis represents normalized sensor values. The figure shows both the original and smoothed signals for selected sensors from the FD001 dataset, highlighting the degradation trends used for model analysis.

Comments 13: From the sound of the entire manuscript the authors seem to propose the Deep Equilibrium Models architecture yet compare their results to non DEM models specifically looking at table 3. From this aspect the comparison doesn’t seem to be fair instead authors should compare to existing DE models or make another table with DE models existing and their proposed DE model to show that it outperforms all instead of existing deep learning models.

Response 13: Thank you for your feedback.

To address this concern, we have updated Table 3 (highlighted in red text, page 16) to include an implicit model (Neural ODE), which represents the class of Deep Equilibrium based approaches. To the best of our knowledge, Neural ODE is the only implicit model previously applied to RUL prediction.

Additionally, we have revised the manuscript to explicitly mention this comparison. The following sentence has been added to the paragraph discussing model performance (page 14, line 474):

“An interesting comparison is with the Neural ODE model, which represents implicit deep learning approaches. We observe that our proposed DEM consistently outperforms Neural ODE, demonstrating its superior ability to capture degradation patterns in RUL prediction.”

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

This manuscript proposes a novel deep learning framework based on Deep Equilibrium (DE) Models for Remaining Useful Life (RUL) prediction, leveraging an implicit modeling approach that uses a fixed-point solver to estimate system dynamics instead of a fixed number of layers. The authors incorporate a dual-input interconnection attention mechanism to fuse latent and convolutional features, and apply a calibrated Monte Carlo dropout strategy to quantify predictive uncertainty. The approach is rigorously evaluated on the CMAPSS dataset and demonstrates competitive performance compared to several state-of-the-art baselines. The paper is technically sound, well-structured, and presents several methodological innovations, including uncertainty calibration with GMM-based grouping, making it a valuable contribution to the PHM and deep learning communities.

That said, the manuscript can be further improved in the following aspects:

(1) While the use of Deep Equilibrium models is well motivated, the paper should more clearly differentiate its architectural innovations (e.g., dual-input attention) from existing DE variants. This will help better situate the contribution in context.

(2) Section 4.2: The feature selection using Random Forest is sound, while it is recommended to briefly discuss whether other feature selection algorithms were considered, and whether they could potentially lead to better performance.

(3) Section 4.3: It seems that the authors did not describe how key hyperparameters (e.g., learning rate) were selected. Were these values tuned using grid search, trial-and-error, or based on previous studies?

(4) While the technical content is strong, several sentences would benefit from language refinement for clarity and fluency. Here are some examples, “So, the extracted features using …” in line 38, and “…, something that is very important for real PHM applications” in line 414.

Author Response

Comments 1: While the use of Deep Equilibrium models is well motivated, the paper should more clearly differentiate its architectural innovations (e.g., dual-input attention) from existing DE variants. This will help better situate the contribution in context.

Response 1: We appreciate this observation and agree that more clearly distinguishing our architectural innovations within the DE framework would strengthen the manuscript. To address this, we have revised the relevant paragraph on page 3 and updated the contribution statements in the Introduction section accordingly (red text in manuscript).

Revised paragraph in the Introduction section (page 3, line 117): The Deep Equilibrium (DE) block integrates convolutional operations and a novel attention-based Dual-Input Interconnection mechanism, created specifically for implicit deep models on multivariate time-series data. The convolutional component extracts local spatial and temporal patterns from raw sensor inputs, producing an input feature map representing short-term dependencies. The input feature mapping and a latent representation vector, which encodes the internal health state of the system, are dynamically processed by the Dual-Input Interconnection mechanism. This allows the model to perform a cross-attention-like operation, where the input mapping is projected as keys and values and the health state as queries in a shared embedding space. So, the latent state is adaptively updated based on the most relevant observed patterns of the input. The equilibrium state is used as a health indicator for system monitoring since it captures long-term degradation patterns and local sensor behavior. So, the DE block allows for a highly expressive and memory-efficient representation that captures the dynamics of the underlying system by iteratively updating the latent state until convergence. The architecture differs from conventional attention-based architectures in DE models, where self-attention is applied only to the latent representation vector and the input vector mapping is used as a residual connection.

Revised statement on contributions (page 4, line 170): The core element of the Deep Equilibrium Model is a novel Dual-Input Interconnection Attention Block, which enables iterative and adaptive updates of the latent degradation representation by jointly processing the internal health state and the spatio-temporal features extracted from convolutional blocks. Unlike standard Transformer self-attention mechanisms used in DE frameworks, which typically operate only on the latent representation and incorporate input features as a static residual, the proposed attention-based block performs a cross-attention-like interaction between two distinct inputs. This design enhances the model’s ability to capture complex degradation dynamics, leading to a more expressive and context-aware health representation.

Comments 2: Section 4.2: The feature selection using Random Forest is sound, while it is recommended to briefly discuss whether other feature selection algorithms were considered, and whether they could potentially lead to better performance.

Response 2: To address the valuable comment, the following text has been inserted in the section #4.2 of the manuscript.

Inserted text in section 4.2 (page 12, line 403): This method is selected due to its robustness and ability to handle non-linear relationships. Other feature selection methods, such as mutual information or recursive feature elimination (RFE) could also be applied, leading to alternative feature subsets and influencing model performance. A comparative analysis of feature selection strategies is left as future work, to explore whether such alternatives could present improvements in RUL estimation accuracy, especially under varying operational conditions.

Comments 3: Section 4.3: It seems that the authors did not describe how key hyperparameters (e.g., learning rate) were selected. Were these values tuned using grid search, trial-and-error, or based on previous studies?

Response 3: To clarify the selection process of key hyperparameters, we have added a brief paragraph in Section 4.3 of the manuscript.

Inserted paragraph in section 4.3 (page 13, line 448): The hyper-parameters (learning rate, batch size, dropout ratio, number of convolutional filters, attention projection dimension) were selected by a combination of trial-and-error experimentation and reference to values commonly reported in related literature. A limited manual tuning process was employed on a validation split from the training data, where the performance metric of RMSE was monitored. A full grid search of the hyper-parameters can further improve the performance of the RUL estimation framework but in this study was not conducted.

Comments 4: While the technical content is strong, several sentences would benefit from language refinement for clarity and fluency. Here are some examples, “So, the extracted features using …” in line 38, and “…, something that is very important for real PHM applications” in line 414.

Response 4: We agree that the particular phrases could be revised for improving clarity and fluency.

Initial text: So, the extracted features using the Empirical Mode Decomposition Technique are integrated into a Bayesian-optimized Random Forest Model to estimate the RUL of bearing in \citep{9889166}.

Revised text (page 2, line 45): In a study, Empirical Mode Decomposition (EMD) effectively analyzes non-linear and non-stationary data to extract robust features. These features are then inserted into a Random Forest (RF) model, which is optimized through Bayesian optimization for superior RUL prediction \citep{9889166}.

Initial text: Therefore, the application of MC dropout enhances the model’s predictive accuracy and provides an reliability tool simultaneously, a significant advantage for real-world Prognostics and Health Management (PHM) applications.

Revised text (page 14, line 492): The application of MC dropout not only enhances the model's predictive accuracy but also provides a crucial reliability tool, which is highly valuable for real-world PHM applications.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have made revisions to the state variables in the manuscript as required. Overall, the revised article is now essentially in acceptable condition for publication. However, there are still a few minor modifications that need to be addressed before final acceptance:
1. It is recommended to add a subsection on convolution blocks in Chapter 3 to enhance the completeness of the chapter.
2. The paper explains the selection of hyper-parameters and clarifies that this is not the focus of the current study. However, no references are provided to support these choices. It is recommended that the authors include relevant literature or theoretical justification to enhance the persuasiveness and scientific rigor of the explanation.

Author Response

Comments 1: It is recommended to add a subsection on convolution blocks in Chapter 3 to enhance the completeness of the chapter.

Response 1: We thank the reviewer for the helpful suggestion to add a subsection on convolutional blocks in Chapter 3. While we did not introduce a separate subsection—given that Convolutional Neural Networks (CNNs) are widely known and well-established in the deep learning community—we have incorporated two additional paragraphs within the relevant section (red text in page 8, line 285).

These paragraphs focus specifically on the use of 1D convolutional layers for time series data, providing both an intuitive explanation and a formal mathematical formulation of the operation.

We also included a reference to support this explanation (‘Krichen, M. Convolutional Neural Networks: A Survey. Computers 2023, 12, 151. https://doi.org/10.3390/computers12080151’).

We hope this addresses the reviewer’s concern by enhancing the clarity and completeness of the model description without interrupting the structural flow of the chapter.

Comments 2: The paper explains the selection of hyper-parameters and clarifies that this is not the focus of the current study. However, no references are provided to support these choices. It is recommended that the authors include relevant literature or theoretical justification to enhance the persuasiveness and scientific rigor of the explanation.

Response 2: We appreciate the reviewer’s observation regarding the lack of references supporting our hyper-parameter choices. In response, we have revised the manuscript to include citations to relevant literature that justify the selected values for key hyper-parameters such as the learning rate, batch size, dropout ratio, and convolutional architecture [1-3,12,17] (Page 14, line 463). We believe that these additions enhance the scientific rigor and transparency of our methodological explanation.

Thank you for this helpful suggestion.

[1] Zio, E. Prognostics and Health Management (PHM): Where are we and where do we (need to) go in theory and practice. Reliability Engineering & System Safety 2022, 218, 108119.

https://doi.org/10.1016/j.ress.2021.108119

[2] Kumar, S.; Raj, K.K.; Cirrincione, M.; Cirrincione, G.; Franzitta, V.; Kumar, R.R. A Comprehensive Review of Remaining Useful Life Estimation Approaches for Rotating Machinery. Energies 2024, 17, 5538. https://doi.org/10.3390/en17225538

[3] Ferreira, C.; Gonçalves, G. Remaining Useful Life prediction and challenges: A literature review on the use of Machine Learning Methods. Journal of Manufacturing Systems 2022, 63, 550–562.

https://doi.org/https://doi.org/10.1016/j.jmsy.2022.05.010.

[12]Muneer, A.; Taib, S.M.; Naseer, S.; Ali, R.F.; Aziz, I.A. Data-Driven Deep Learning-Based Attention Mechanism for Remaining Useful Life Prediction: Case Study Application to Turbofan Engine Analysis. Electronics 2021, 10. https://doi.org/10.3390/electronics10202453.

[17] Li, H.; Zhao, W.; Zhang, Y.; Zio, E. Remaining useful life prediction using multi-scale deep convolutional neural network. Applied Soft Computing 2020, 89, 106113. https://doi.org/10.1016/j.asoc.2020.106113