Method for Remaining Useful Life Prediction of Turbofan Engines Combining Adam Optimization-Based Self-Attention Mechanism with Temporal Convolutional Networks

Abstract

Conducting the remaining useful life (RUL) prediction for an aircraft engines is of significant importance in enhancing aircraft operation safety and formulating reasonable maintenance plans. Addressing the issue of low prediction model accuracy due to traditional neural networks’ inability to fully extract key features, this paper proposes an engine RUL prediction model based on the adaptive moment estimation (Adam) optimized self-attention mechanism–temporal convolutional network (SAM-TCN) neural network. Firstly, the raw data monitored by sensors are normalized, and RUL labels are set. A sliding window is utilized for overlapping sampling of the data, capturing more temporal features while eliminating data dimensionality. Secondly, the SAM-TCN neural network prediction model is constructed. The temporal convolutional network (TCN) neural network is used to capture the temporal dependency between data, solving the mapping relationship of engine degradation characteristics. A self-attention mechanism (SAM) is employed to adaptively assign different weight contributions to different input features. In the experiments, the root mean square error (RMSE) values on four datasets are 11.50, 16.45, 11.62, and 15.47 respectively. These values indicate further reduction in errors compared to methods reported in other literature. Finally, the SAM-TCN prediction model is optimized using the Adam optimizer to improve the training effectiveness and convergence speed of the model. Experimental results demonstrate that the proposed method can effectively learn feature data, with prediction accuracy superior to other models.

1. Introduction

The turbofan engine is the primary power source for an aircraft. However, as the engine accumulates operating time, its internal components gradually suffer from factors such as wear, corrosion, and fatigue, leading to performance degradation and shortened lifespan. Conducting thorough maintenance checks on engines has become an essential aspect ensuring their healthy operation. Prognostics and health management (PHM) [1] technology is a predictive maintenance strategy that adapts to conditions. It aims to timely detect potential fault indicators through real-time monitoring, analysis, and evaluation of equipment’s operational status and to implement preventive maintenance measures, thereby enhancing equipment safety and operational efficiency. The RUL [2] prediction for turbofan engines refers to assessing and predicting the engine’s future remaining operational lifespan by analyzing its operating status, historical monitoring data, and other information using techniques such as mathematical models, statistical methods, or machine learning algorithms.

RUL prediction technology is a important step in implementing PHM, and therefore, predicting the RUL of engines holds significant research significance and practical value in ensuring flight safety and reducing operational costs for an aircraft. Despite the rapid development of RUL prediction technology in the aviation industry, it still faces several challenges, such as difficulty in extracting effective features from vast amounts of engine raw monitoring data and issues with low accuracy in predicting engine remaining lifespan. In conclusion, integrating advanced domain knowledge with innovative technologies to continually improve the practicality of engine RUL prediction remains a key challenge in the field of aviation maintenance.

Currently, engine RUL prediction methods include empirical methods, physics-based failure model methods, and data-driven methods [3]. Prediction methods based on physical failure models require utilizing the interactions among internal physical processes and components of the engine to establish mathematical models. However, constructing the physical model presents significant challenges [4]. Empirical prediction methods [5] rely on expert industry knowledge and involve analyzing the engine’s operational status and fault data to generate predictions. While simpler, these methods often yield lower accuracy in RUL prediction. Data-driven prediction methods [6] primarily rely on extensive raw engine monitoring data. They use advanced computer algorithms to uncover patterns in the data and construct predictive models for RUL estimation, achieving higher prediction accuracy. Compared to the other two methods, data-driven approaches are more versatile.

Although early machine learning methods such as random forest [7], support vector machine [8] and Markov models [9] have shown good capabilities in data processing, their relatively simple structures make it difficult to fully explore the hidden relationships among data, resulting in lower prediction accuracy. In contrast, deep learning methods can learn abstract feature representations of data, which give them an advantage in handling long-term data dependencies and complex dynamic patterns.

In the realm of deep learning methods, Caceres and others [10] have proposed a recurrent neural network (RNN) prediction method based on probabilistic Bayesian principles to address the uncertainty factors in RUL prediction. Babu et al. [11] were the first to apply convolutional neural network (CNN) methods to RUL prediction problems, enabling automatic learning from raw signals. Zhang et al. [12] developed a bidirectional long short-term memory (LSTM) neural network prediction model to assess engine remaining life. LYU et al. [13] used 1D-CNN to map multiple state variables’ long-term sequences directly to the engine’s remaining life, achieving more accurate RUL values. Although the aforementioned neural network learning methods have been extensively researched in engine remaining life prediction, certain limitations persist. For instance, RNN neural networks exhibit poor feature extraction ability, CNN neural networks fall short in handling time series data, and LSTM neural networks show low predictive accuracy under complex operating conditions. Considering these shortcomings, researchers [14] have introduced a TCN network. This neural network architecture is capable of effectively capturing local patterns in time series data. Huang [15] leveraged a multi-head probability sparse self-attention mechanism in deep learning to enhance data relevance and combined it with causal convolutions in TCN neural networks to boost the model’s learning capabilities. Song et al. [16] designed a MCA-DTCN network lifespan prediction model to extract key features from engine data. These research findings indicate that TCN prediction methods offer advantages in engine life prediction. However, considering that different sensor monitoring data may impact engine remaining life to varying degrees, it is essential to thoroughly investigate these issues during model training to prevent a decrease in prediction accuracy. Hence, when utilizing TCN neural network modeling, continuous optimization and improvement of the TCN neural network structure are necessary to enhance predictive performance.

To better address the above issue, this paper constructs a turbine engine RUL prediction method based on an Adam-optimized SAM-TCN. This approach aims to accurately capture significant feature data contributions to address the issue of low RUL prediction accuracy. The main contributions of this paper are as follows.

Considering that turbofan engine data used for experiments are multivariate time series, TCN neural networks are employed in this study to handle these temporal data sequences effectively, thereby avoiding the problems of gradient vanishing and slow convergence speed typically encountered in recurrent neural networks during training. Simultaneously, the SAM method is utilized to capture the impact magnitude of each feature sequence on the engine’s RUL, enabling the prediction model to focus more on useful feature data and enhancing its capability to extract key features. Additionally, the Adam optimization method is introduced to globally optimize the prediction model, thereby improving its training effectiveness and accuracy.

Experimental verification and analysis are conducted on the C-MAPSS dataset for turbofan engines (FD001-FD004 datasets). The RMSE values obtained are 11.50, 16.45, 11.62, and 15.47, respectively, and the Score values are 225.32, 1136.27, 259.79, and 1365.40, respectively. The results of these two evaluation metrics demonstrate that the proposed method in this paper achieves lower prediction errors and better performance compared to others in the literature, effectively validating the efficacy of the proposed approach.

2. Background

2.1. Self-Attention

The SAM is primarily used for handling complex data. The SAM can capture the relationships between the overall feature sequence and the current moment’s features, assigning different attention weights to input data features [17]. Compared to the attention mechanism, SAM not only swiftly captures internal feature relationships in data, thus reducing dependence on external information, but also allows the model to attend to all positions within the input sequence simultaneously during processing.

The three crucial vectors comprising the self-attention mechanism are $Q_i$, $K_i$, and $V_i$ vectors. These vectors are obtained through linear transformations of the input data $X=\left\{x_1,x_2,\cdots ,x_i\right\}$ using corresponding learned weight matrices $W_q$, $W_k$, and $W_v$.

To represent the attention level between each pair of input sequence elements $x_i$ and $x_j$, attention scores need to be computed. These scores are initially calculated as the dot product of the $Q_i$ and the $K_i$, which is divided by a scaling factor. According to reference [18], the calculation formula is as follows:

$$score(Q_i,K_i)={\displaystyle \frac{Q_i{K}_j}{\sqrt{d_k}}}$$

(1)

In the equation, $d_K$ denotes the dimensionality of the K vector.

To normalize the computed attention scores using function softmax—transforming them into values between 0 and 1 such that their sum equals 1—we obtain attention weight values. The calculation formula is

$$w_{ij}=softmax\left(score\left(Q_i,K_i\right)\right)=softmax\left({\displaystyle \frac{Q_i{K}_j}{\sqrt{d_k}}}\right)$$

(2)

In the equation, $W_{ij}$ represents the attention weight.

We calculate the final output result:

$$z_i=\sum _{j=1}^n{w}_{ij}{V}_j$$

(3)

In the equation, $Z_i$ represents the final output result.

The principle structure diagram of self attention mechanism is shown in Figure 1 [19]. Introducing self-attention mechanisms in neural networks effectively addresses the issue of information overload in predictive models [20]. Self-attention allows the model to dynamically adjust attention distributions when processing input sequences, thereby reducing reliance on the entire sequence and improving computational efficiency.

2.2. Temporal Convolutional Network

Researchers introduced the TCN into their networks to address the inefficiency of CNNs in handling sequential data. This neural network not only inherits CNN’s powerful feature extraction capabilities but also leverages recurrent neural networks’ ability to store historical information. The TCN network can efficiently process input sequences of any length, effectively improving the efficiency and accuracy of prediction models.

Causal convolution predicts the output data $x_{t+1}$ at time $t+1$ by learning input information $\left\{x_1,x_2,\cdots ,x_t\right\}$ up to time t, effectively preventing future information leakage and addressing differences in input and output time steps during model training. When processing long-sequence input data, stacking more layers of networks is necessary, increasing the network’s complexity. To tackle these issues, dilated convolution is introduced to improve operational efficiency. Dilated convolution expands the receptive field by inserting zero elements between convolutional kernel elements. The dilated convolution of input data $X=\left\{x_1,x_2,\cdots ,x_t\right\}$ at time t is formulated as [21]

$$\left(F{\ast }_{d}X\right)x_t=\sum _{k=1}^K{f}_k{x}_{t-(K-k)d}$$

(4)

In the formula, $F(f_1,f_2,\cdots ,f_k)$ represents the filter; K is the size of the convolution kernel; d represents the dilation factor.

To effectively mitigate issues such as gradient explosion and vanishing gradients in deep networks, the residual neural network structure was introduced to learn residual errors of feature mappings across layers, thereby enhancing neural network stability. Residual connections combine with identity mapping functions to transmit information across layers in the network. The computational formula is

$$o=Activation(x+F(x\left)\right)$$

(5)

In the formula, x represents the input information; Activation represents the activation function; $F\left(x\right)$ represents the output after residual connection.

The structure of the temporal convolutional network is shown in Figure 2 [22]. The output from the dilated causal convolutions is added to the input using 1 × 1 convolution operations. Weight normalization accelerates neural network convergence and improves training speed.

2.3. Adam Optimization Algorithm

The Adam algorithm [23] is a stochastic optimization algorithm designed to accelerate the optimization process and improve algorithm performance. It adjusts the learning rate adaptively to optimize the parameters of neural networks, demonstrating excellent performance in large-scale data and noisy environments.

Let the parameters at the current step t be ${\theta }_t$, the gradient of ${\theta }_t$ at the current step be $g_t$, the first moment estimate be $m_t$, and the second moment estimate be $v_t$. Their respective formulas are as follows [24]:

$$m_t={\beta }_1{m}_{t-1}+\left(1-{\beta }_1\right)g_1$$

(6)

$$v_t={\beta }_2{v}_{t-1}+\left(1-{\beta }_2\right)g_t^2$$

(7)

In the equation, ${\beta }_1$ and ${\beta }_2$ represent adjustable exponential decay rates.

$${\widehat{m}}_t={\displaystyle \frac{m_t}{1-{\beta }_1^t}}$$

(8)

$${\widehat{v}}_t={\displaystyle \frac{v_t}{1-{\beta }_2^t}}$$

(9)

We update the model parameters ${\theta }_t$:

$${\theta }_t={\theta }_{t-1}-\alpha {\displaystyle \frac{{\widehat{m}}_t}{\sqrt{{\widehat{v}}_t}+\epsilon }}$$

(10)

In the equation, $\alpha$ denotes the learning rate; $\epsilon$ is a very small numerical value.

3. Related Work

Research on RUL prediction of turbofan engines is a crucial topic in the field of flight safety. Accurately predicting the engine’s remaining lifespan can assist airlines in optimizing maintenance schedules and reducing unscheduled maintenance incidents. Currently, machine learning and artificial intelligence technologies are predominantly used in the research of turbofan engine RUL prediction, supported by advanced data collection and sensor technologies.

In existing studies, Qiao et al. [25] discussed the data and results from RUL prediction experiments on turbofan engines, finding that indirect mapping approaches yield better results despite being more challenging and time-consuming to implement. Song et al. [26], considering the high dimensionality and volume of engine monitoring data, designed a bidirectional long short-term memory network for RUL prediction. Li et al. [27] respectively utilized CNN and LSTM networks to extract spatial features and perform data fusion, and they employed the SAM method to obtain feature weights. Zhen et al. [28] proposed a TCN-attention model for oil well production prediction to overcome issues with traditional neural networks such as poor data processing effects and gradient vanishing. Although this method mitigates the shortcomings of traditional neural networks, its prediction accuracy and overall performance still require improvement.

To address these challenges effectively, this study introduces a self-attention mechanism into the temporal convolutional network to obtain different feature weights and utilizes the Adam optimization algorithm to enhance the overall performance of the prediction model, significantly improving prediction accuracy.

4. The Proposed Method

4.1. SAM-TCN

To address the issue where TCN neural networks fail to sufficiently extract relationships between input data during model training, thereby resulting in poor performance, this paper introduces a SAM based on the TCN. The SAM method assigns varying attention weights to input features, thereby enhancing the TCN neural network’s feature extraction capabilities. Figure 3 shows the model structure diagram of the self-attention mechanism and temporal convolutional network. Using the Q and K vectors, attention scores are computed, normalized using the $softmax$ function, and then multiplied by the V vector values for each data point before summing them up. The output data are activated using the ReLU activation function to obtain feature channel data weighted by different attention weights. These new feature data are then input into the TCN neural network, where dilated causal convolutions extract features from the inputted new features and produce the final output features.

4.2. The Adam-SAM-TCN Remaining Useful Life Prediction Model

Figure 4 shows the overall structure of the Adam SAM TCN prediction model. The specific process of the RUL prediction method is as follows:

(1)

From the 21-dimension raw data of datasets FD001 to FD004, 14 dimensions showing significant degradation trends were selected. These data were normalized and labeled with RUL, and initial values for sliding windows were set.

(2)

The SAM attention mechanism was introduced into the TCN to assign higher weights to input data that have a greater impact on the engine’s remaining useful life, thereby obtaining new feature data. We input these new features into the TCN network for training.

(3)

After initializing the parameters of the Adam optimizer, the parameters of the constructed SAM-TCN network were optimized to obtain the optimal parameters.

(4)

Utilizing the optimized Adam-SAM-TCN prediction model, the RUL prediction for the turbofan engine was conducted on the four datasets. The predicted results were analyzed and evaluated based on evaluation metrics.

The parameter settings of the proposed Adam-SAM-TCN neural network prediction model are shown in

Table 1. Configuration of Adam-SAM-TCN model structure parameters.

Structure	Specific Parameters
Input	$W_1=40, W_2=40, W_3=50, W_4=50$
SAM layer	$in=40, out=40, stride=1, softmax$
TCN layer cov11	$kerne{l}_{-}size=3, stride=1, padding=2, ReLU, Dropout(p=0.2)$
TCN layer cov21	$kerne{l}_{-}size=3, stride=1, padding=2, ReLU, Dropout(p=0.2)$
Adam layer	$lr=0.01, betas=(0.9,0.999), eps=1e-8$

. Here, W represents the sliding window size, in denotes the number of input channels, $out$ indicates the number of channels produced by convolution, $stride$ represents the step size, $Softmax$ and $ReLU$ are activation functions, $kerne{l}_{-}size$ denotes the size of the convolutional kernel, and $padding$ refers to the padding value on both sides of the input.

5. Experiment

5.1. Experimental Data

Validated using C-MAPSS data developed by the NASA’s Ames Research Center, we simulated the degradation of various components such as fan, turbine, and compressor in a turbofan engine under 9000 pounds of thrust. Parameters recorded include pressure, temperature, and rotational speed. This dataset is known as the C-MAPSS data [29]. Based on different engine operational conditions, this dataset was divided into four subsets. Among them, FD001 and FD003 contain single operational condition data, while FD002 and FD004 contain multi-operational condition data.

Table 2. C-MAPSS dataset information.

Dataset	FD001	FD002	FD003	FD004
Number of Training Samples	100	260	100	249
Number of Test Samples	100	259	100	248
Total Length of Training Set	20,630	53,758	24,719	61,248
Total Length of Test Set	13,095	33,990	16,595	41,213
Operating Conditions	1	6	1	6
Fault Mode	1	1	2	2

shows the descriptive information of the data, where fault mode 1 denotes a high-pressure compressor fault, and fault mode 2 denotes both high-pressure compressor and fan faults. The training set comprises data covering the entire lifecycle of the engine. The test set consists of monitoring data from early normal operation to a specific point before failure occurs.

The dataset includes data from three operational condition settings and 21 sensor measurements. Specific descriptions are provided in

Table 3. Description of operational settings and sensor monitoring parameters.

No.	Parameters	Unit
1	Setting_1	–
2	Setting_2	–
3	Setting_3	–
4	$T_2$	°R
5	$T_{24}$	°R
6	$T_{30}$	°R
7	$T_{50}$	°R
8	$P_2$	psia
9	$P_{15}$	psia
10	$P_{30}$	psia
11	$N_f$	rpm
12	$N_c$	rpm
13	$e_{pr}$	–
14	Ps30	psia
15	Phi	$P_{ps/psi}$
16	$N_{Rf}$	rpm
17	$N_{Rc}$	rpm
18	BPR	–
19	$B_{fa}$	–
20	Bleed	–
21	$N_f$_dmd	rpm
22	$N{R}_f$_dmd	rpm
23	$W_{31}$	lbm/s
24	$W_{32}$	lbm/s

. Among them, °R denotes Rankine temperature units; $psia$ represents pressure units in pounds per square inch absolute $(lbf-/i{n}^2)$; $rpm$ stands for rotational speed in revolutions per minute; $lbm/s$ indicates flow rate in pounds per second.

5.2. Evaluation Metrics for Predicted Results

The definition of RUL for a turbofan engine is the remaining usable cycles from the current operating time of the engine until complete performance degradation and failure. The specific expression for RUL of a turbofan engine is [30]

$$T_{RUL}\left(t\right)=t_f-t_c\mid t_f\ge t_c$$

(11)

In the expression, $t_f$ denotes the time of engine degradation failure, and $t_c$ represents the current operating time of the engine.

This study used two evaluation indicators to analyze the prediction results. $RMSE$ considers both the magnitude and direction of prediction errors, making it very useful for evaluating the overall performance of models [31]. The Score metric imposes greater penalties for subsequent fault predictions. The formulas for these two evaluation metrics are as follows:

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}\sum _i^n{\left({\widehat{y}}_i-y_i\right)}^2}$$

(12)

$$Score=\left\{\begin{array}{c}{\sum }_{i=1}^n(e^{\frac{-({\widehat{y}}_i-y_i)}{13}}-1), {\widehat{y}}_i-y_i<0\hfill \\ {\sum }_{i=1}^n(e^{\frac{{\widehat{y}}_i-y_i}{10}}-1), {\widehat{y}}_i-y_i>0\hfill \end{array}\right.,$$

(13)

In the above formulas, $y_i$ represents the actual RUL value of the ith engine, ${\widehat{y}}_i$ represents the predicted RUL value of the ith engine, and n denotes the number of engines.

5.3. Experimental Result

5.3.1. Data Processing

Feature Selection

The C-MAPSS dataset includes a total of 21 monitored parameters collected by sensors. The trends of raw monitoring parameters from a single engine are illustrated in Figure 5. The trends of $T_2$, $P_2$, $P_{15}$, $epr$, $B_{fa}$, $N_{fdmd}$, and $N{R}_{fdmd}$ are not significant, indicating that these parameters have minimal influence on the remaining life of the engine. To reduce data redundancy, these seven parameters were removed. On the other hand, $T_{24}$, $T_{30}$, $T_{50}$, $P_{30}$, $N_f$, $N_c$, $P{s}_{30}$, $phi$, $N_{Rf}$, $N_{Rc}$, $BPR$, $Bleed$, $W_{31}$, and $W_{32}$ show clear monotonic trends with increasing engine operation cycles, suggesting a certain correlation with the engine degradation process. Therefore, these 14 monitored parameters were selected as input variables for the prediction model.

Data Normalization and RUL Labeling

Since the experimental data from CMAPSS consists of multiple parameters monitored by different sensors, these parameters exhibit varying units and significantly different scales. To mitigate the errors caused by dimensional discrepancies during model training, this study employed the MinMax scaling method to normalize the 14 monitored parameters in the dataset. The specific formula is as follows [32]:

$$x_i^{\prime }={\displaystyle \frac{x_i-x_i^{min}}{x_i^{max}-x_i^{min}}}$$

(14)

In the equation, $x_i$ and $x_i^{\prime }$ respectively denote the monitor data and the data after normalization of the ith monitored parameter from the sensor; $x_i^{max}$ and $x_i^{min}$ represent the maximum and minimum datas of the ith monitored parameter from the sensor.

Figure 6 illustrates the effects of normalization on the 14 types of monitoring data for a single engine.

The paper employed piecewise linear functions to describe the RUL of turbofan engines, as shown in Figure 7. In the initial stages of engine operation, wear was minimal, and the RUL remained stable. As the engine reached a certain degradation threshold, performance started to decline, causing the RUL to gradually decrease until reaching its end of life. The formula for calculating the segmented linear degradation curve of the engine RUL is as follows [33]:

$$RUL=\left\{\begin{array}{c}RU{L}_{\mathrm{init}}, RU{L}_i\ge RU{L}_{\mathrm{init}}\hfill \\ RU{L}_i, RU{L}_i<RU{L}_{\mathrm{init}}\hfill \end{array}\right.$$

(15)

In the equation, $RU{L}_{\mathrm{init}}$ represents the initial degradation threshold of the engine, and $RU{L}_i$ denotes the RUL of the engine in the ith cycle period.

5.3.2. Predicted RUL Results

Different lengths of sliding window settings capture data patterns and features at different scales. Appropriate sliding window settings can enhance the overall accuracy of prediction models. To determine the optimal window length and maximize the performance of the prediction model, this study experimented with sliding window sizes of 10, 20, 30, 40, 50, and 60 for comparison of the experimental results. Figure 8 illustrates the $RMSE$ values under different sliding window sizes across four datasets. As seen in Figure 8, the $RMSE$ values of the model were minimized when the sliding window size was 40 for FD001 and FD002 and when it was 50 for FD003 and FD004.

Figure 9 shows the change in loss function throughout the entire iteration process during training of the SAM-TCN neural network. As seen in Figure 9, it is evident that across all four datasets that as the number of iterations increased, the predictive performance of the SAM-TCN neural network steadily improved. The objective loss function continuously decreased and eventually stabilized.

To explore the performance of the proposed Adam-SAM-TCN method across different datasets, the results are shown in Figure 10. In the figure, the horizontal axis represents the engine number, and the vertical axis represents the RUL of the engines. The solid blue line indicates the actual RUL values of the engines, while the dashed red line represents the predicted RUL results using the method proposed in this paper. As seen in Figure 10, it can be observed that the blue solid line fits closely with the red dashed line, indicating a small prediction error and hence a good predictive performance. The predictions for the single operating condition datasets FD001 and FD003 are closer to the actual values, with smaller prediction errors. In contrast, datasets FD002 and FD004, collected under multiple operating conditions, exhibited more complexity and posed greater challenges for prediction, resulting in lower prediction accuracy. Overall, the proposed method demonstrates a generally close alignment between the predicted results and actual values across all datasets.

To better analyze the predictive performance of the proposed method on individual engines, RUL predictions were conducted on the engines numbered 20, 185, 1, and 111 from the test set across four datasets. The predicted results are shown in Figure 11, where the actual RUL values of the engines closely align with the predictions, with the majority of predicted values falling within the 95% confidence interval. This indicates that the proposed method demonstrates effective predictive performance.

Figure 12 shows the distribution of prediction errors for the remaining useful life of engines on four datasets. The horizontal axis represents the size of prediction error. The vertical axis shows the frequency of occurrences within each prediction error interval. As seen in Figure 12, it is evident that the majority of prediction errors across all four datasets fell within the range of [−25, 25], indicating that the proposed method provides fairly accurate predictions of engine RUL.

6. Discussion

6.1. Experimental Results Dicussion

This paper proposed a hybrid network structure using Adam optimization and SAM-TCN for predicting the RUL of engines. It utilized a SAM to measure the contributions of different features to the engine’s remaining life. The results were evaluated using comprehensive performance metrics, with the RMSE and Score on four datasets reported as follows: 11.50, 16.45, 11.62, and 15.47 for the RMSE and 225.32, 1136.27, 259.79, and 1365.40 for the Score. From these metrics, it is evident that the proposed method performed well on the FD001 and FD003 datasets with smaller errors. This is primarily because the other two datasets contain more noise and complex operating conditions, making it challenging for the prediction model to capture these variations and resulting in less satisfactory performance. However, overall, the proposed model achieved good predictive accuracy. To comprehensively assess the prediction effectiveness of the proposed method, comparisons and analyses with methods from other literature are conducted in the next section.

6.2. Comparative Study

To better analyze the strengths and weaknesses of the proposed method, the Adam-SAM-TCN method was compared with recent methods. The RMSE and Score values of these methods on the dataset are presented in

Table 4. Comparison of RMSE values of other RUL prediction methods on the C-MAPSS dataset.

Method	FD001	FD002	FD003	FD004	Year
LSTM [34]	14.18	25.25	12.79	27.63	2021
CNN-BGRU-SA [35]	13.88	17.25	14.85	19.39	2022
TaFCN [36]	13.99	17.06	12.01	19.79	2022
Multi-attention-TCN [37]	13.25	19.57	13.43	21.69	2022
RCNN-Abi-LSTM [38]	12.98	19.16	13.24	22.29	2023
GATA-TCN [39]	12.80	17.61	13.16	21.04	2024
Adam-SAM-TCN	11.50	16.45	11.62	15.47	2024

and

Table 5. Comparison of Score values of other RUL prediction methods on the C-MAPSS dataset.

Method	FD001	FD002	FD003	FD004	Year
CNN-BGRU-SA [35]	248	1140	295	1840	2022
TaFCN [36]	336	1946	251	3671	2022
Multi-attention-TCN [37]	235	1655	239	2415	2022
RCNN-Abi-LSTM [38]	258	2980	246	3795	2023
GATA-TCN [39]	234.31	1361.23	290.63	2303.42	2024
Adam-SAM-TCN	225.32	1136.27	259.79	1365.40	2024

. As seen in Table 4, it can be observed that all prediction methods performed better on the datasets FD001 and FD003 compared to the datasets FD002 and FD004. Compared to the LSTM [34], CNN-BGRU-SA [35], TaFCN [36], Multi-attention-TCN [37], RCNN-Abi-LSTM [38], and GATA-TCN [39] methods proposed in other studies, the Adam-SAM-TCN method achieved the smallest RMSE values, indicating superior predictive performance in engine life prediction.

As seen in Table 5, it can be observed that, unlike the RMSE values in Table 4, the method from reference [37] achieved the lowest Score on the FD003 dataset. This is mainly due to Score’s penalties for overestimation and underestimation of the RUL. Apart from the FD003 dataset, the prediction method Adam-SAM-TCN obtained smaller Score values compared to the methods from other literature sources.

7. Conclusions

To address the challenges in predicting the RUL for turbofan engines under multi-operational conditions, characterized by difficulties and low accuracy, this paper proposed an Adam-optimized SAM-TCN neural network for engine RUL prediction. After experimental verification, the following conclusions have been obtained:

(1)

The Adam-SAM-TCN method introduces an SAM module on top of the TCN neural network for prediction modeling. This neural network structure enables the model to focus on local patterns during training while effectively capturing global patterns in data.

(2)

The method utilizes the Adam to optimize the prediction model, which adaptively adjusts the learning rate based on historical gradient information. The algorithm swiftly and accurately minimizes the loss function, thereby effectively enhancing the training effectiveness and generalization capability of the neural network.

(3)

By conducting training and testing on four datasets, experimental results show that the RMSE values for this paper came out to 11.50, 16.45, 11.62, and 15.47 respectively. Compared to existing prediction models, the evaluation metrics of the proposed method were consistently lower, demonstrating the effectiveness of the Adam-SAM-TCN prediction model proposed in this study.