A multi-scale adaptive dual forecast model applied to remaining useful life prediction of air turbine starter bearing

Abstract

The air turbine starter (ATS) is a critical device for starting aircraft engines, in which rolling bearings play a core role. Predicting the remaining useful life (RUL) of rolling bearings is challenged by complex dependencies. Feature extraction, as a crucial step for RUL prediction, reveals the operational state of the bearings. To comprehensively extract the operational characteristics of rolling bearings, this paper introduces the multi-scale adaptive dual forecast (MADF) model. First, the model employs a multi-scale adaptive router to dynamically select optimal patch scales. Subsequently, these scales are used to segment the data into different temporal slices. For each scale, the model applies a dual forecast block to capture global and local temporal dependencies. Finally, a multi-scale aggregator integrates information across different bearing scales, further refining the multi-scale modeling process to predict the RUL. The experimental results demonstrate that the proposed method achieves reductions in mean absolute error (MAE) by 15.76%, 24.79%, 33.08%, 39.83%, and 49.15%; root mean square error (RMSE) by 16.01%, 29.64%, 29.97%, 38.72%, and 47.92%; and score by 21.01%, 34.52%, 41.29%, 43.61%, and 47.52% compared to the multi-resolution time-series transformer (MTST), patch time-series transformer (PatchTST), TimesNet, Scaleformer, and convolutional neural network (CNN) models, respectively. The effectiveness of the proposed method is validated through experimental data from the RUL prediction of civil aircraft-bearing components, showcasing its ability to achieve high-precision predictions of bearing RUL.

Keywords

Remaining useful life bearing patch multi-scale adaptive

Introduction

The air turbine starter (ATS) serves as the core starting device in the engine system of civil aircraft, as shown in Figure 1. The ATS converts compressed air pressure into mechanical force, sufficient to increase the speed of the engine to the required starting speed (Guo and Ji, 2023). Consequently, preventing ATS failures is of paramount importance. Prognostics and health management (PHM) enhances the reliability and utilization efficiency of mechanical equipment through monitoring and analyzing data, with remaining useful life (RUL) prediction being its foundation and core (Zhang et al., 2023; Zheng et al., 2024). Among the various components of the ATS, rolling bearings are a crucial fundamental part (Guo et al., 2024). Therefore, accurately predicting the RUL of rolling bearings is essential to ensure the reliable operation of the ATS.

Currently, techniques for estimating the RUL of bearings are primarily categorized into model-based and data-driven methods (Li et al., 2023; Nimri et al., 2023). The former relies excessively on extensive prior knowledge of the equipment, leading to increased experimental costs and complexity (Ferreira and Gonçalves, 2022). In contrast, data-driven methods exploit hidden degradation patterns within sensor data for bearing RUL prediction (Wu et al., 2023b). This method is less dependent on operating conditions, offering broader applicability in real-world scenarios. In RUL prediction, effective feature extraction enables the model to identify potential degradation information from complex bearing data, directly impacting the accuracy of the predictions. In recent years, deep neural networks (DNNs), as a key data-driven technology, have demonstrated remarkable potential in bearing RUL prediction due to their powerful feature learning capabilities. Among these, convolutional neural networks (CNNs) (Jiang et al., 2024; Yin et al., 2024) excel at extracting local degradation features from data, while long short-term memory networks (LSTMs) (He et al., 2024b; Xue et al., 2023) are adept at capturing long-term dependencies in temporal data. However, the bearing signals collected from multiple sensors often exhibit nonlinear and highly correlated characteristics, which, when combined, form complex multivariate time series. This makes the extraction and analysis of in-depth features more challenging (He et al., 2024a). Traditional DNNs encounter the issue of incomplete feature extraction when processing such data, as they struggle to effectively capture both local and global features within a single model. Consequently, they are unable to comprehensively encapsulate the operational characteristics of the bearing across its entire life cycle.

To address this issue, patch time-series transformer (PatchTST) (Nie et al., 2023) proposes an innovative feature extraction method—patch segmentation. Specifically, the model segments the original time series into subsequence-level patches based on a patch size, retaining local semantic information, and further models the temporal dependencies between patches to capture global features. This approach improves the identification of complex patterns within the time series, leading to better prediction performance. Therefore, applying this patch segmentation technique to bearing data effectively mitigates the challenges posed by multiple feature data in complex systems. To fully learn temporal information within patches, we leverage the temporal convolutional network (TCN) (Peng et al., 2023), which employs dilated convolution to flexibly adjust the receptive field so as to capture local details from time series. To further enhance global modeling capabilities, the efficient channel attention network (ECA-net) (Waghumbare et al., 2024) is employed to capture key features between patches. This network introduces an adaptive channel attention mechanism that dynamically adjusts channel weights to achieve robust adaptability across various input data. Despite PatchTST’s excellent performance in feature modeling through the patch segmentation method, it primarily focuses on feature analysis at a single scale. Bearing degradation data exhibit diverse varying patterns across different time scales (Ying and Lu, 2024). These include local changes (e.g. short-term vibrations caused by small cracks on the bearing surface) and global changes (e.g. long-term degradation trends resulting from prolonged operation and wear accumulation). This single-scale analysis may lead to the model overlooking certain temporal patterns, limiting its ability to capture multi-scale features in the bearing degradation process and ultimately affecting the accuracy of RUL prediction.

So, considering information at multiple time scales is crucial for accurately capturing the dynamic degradation process of bearings. In recent years, various advanced methods based on multi-scale feature extraction have been proposed. For example, the multi-resolution time-series transformer (MTST) (Zhang et al., 2024) adopts a multi-branch architecture to extract features at different time scales. Pyraformer (Liu et al., 2022) introduces a multi-layer pyramid attention module (PAM) that progressively extracts features at different resolutions, from high to low. Scaleformer (Shabani et al., 2023) utilizes multi-frequency down-sampling and a multi-layer network architecture for seamless multi-scale information integration. Thus, combining the patch segmentation method of PatchTST with multi-scale feature extraction techniques enables data segmentation based on different patch sizes, allowing the model to better adapt to variations in the bearing over multiple time scales. The patch size determines the granularity of each time segment, and, based on these segmented patches, the model can extract and integrate features at varying temporal granularities. This approach can more comprehensively capture the multi-hierarchical structure of the bearing data and further improve RUL prediction accuracy. However, these multi-scale feature modeling approaches typically rely on manually defined scale partitioning rules, such as fixed window sizes (e.g. MTST) or predefined resolution hierarchies (e.g. Pyraformer and Scaleformer). Bearing data collected from different sensors often intertwines with complex periodic and trend patterns. Simple scale partitioning may overlook the temporal correlations within the bearing data and the interactions between different scales, potentially failing to adapt to the degradation changes of bearings over different periods. This partitioning approach could disrupt the underlying structure of the data, thereby limiting the model’s ability to accurately capture dynamic characteristics.

It is necessary to conduct an analysis of the inherent laws of time series to establish more flexible and reasonable scale partitioning strategies. Time-series decomposition offers an effective method for mining the intrinsic characteristics of time series. By leveraging underlying patterns and prior knowledge to decompose complex time series into trend and seasonal components, disentangling mixed temporal patterns. Models like Autoformer (Wu et al., 2021) and frequency enhanced decomposition transformer (FEDformer) (Zhou et al., 2022) employ additive time-series decomposition to sequentially extract trend and periodic components of the time series, thereby gaining a more accurate understanding of the dynamic properties of the data and enabling models to better balance long-term trends and short-term fluctuations. Building on this, an adaptive selection mechanism has been developed to further enhance the flexibility of time-series decomposition. This mechanism dynamically selects appropriate scales for modeling based on the intrinsic properties of bearing data, thereby eliminating the cumbersome process of manually adjusting scale partitions.

To further expand and deepen the research, this paper innovatively combines an adaptive selection mechanism equipped with time-series decomposition technology and multi-scale patch feature modeling methods, proposing the multi-scale adaptive dual forecast (MADF) model. This model dynamically extracts and integrates time-series features across multiple scales, capturing both local and global trend characteristics, thereby improving the accuracy and reliability of RUL prediction for air turbine starter rolling bearings. The primary innovations of this work are as follows:

We propose a multi-scale adaptive feature extraction strategy. It utilizes a multi-scale adaptive router with temporal decomposition to select optimal patch sizes. After the bearing data is divided into various patch types, a multi-scale aggregator combines the temporal correlations of these segments. This approach flexibly accommodates data at different scales, effectively captures multi-scale features, and better represents variable patterns in bearing.

We propose a multi-scale patch dual forecast block. This module integrates TCN with an improved efficient channel attention (IECA) mechanism to capture both intra-patch and inter-patch temporal dependencies. It can efficiently extract local and global information from multivariate bearing data while revealing the intrinsic relationships among variables.

Related works

TCN network

The TCN network is an efficient algorithm specifically designed for processing time-series data. It primarily consists of dilated causal convolution layers and residual connections.

Dilated causal convolution, as the core module of TCN, combines the strengths of causal and dilated convolutions, ensuring that the output at time $T$ depends only on the current and previous inputs, while adding a fixed step between every two adjacent filter taps to expand the receptive field. For one-dimensional time-series input $Y \in R^{T}$ and a filter $K : {0, 1, . . ., β - 1} \to R$ , where $β$ is the size of the convolution kernel. The dilated causal convolution results $F_{DCC} (s)$ for the sth element are defined as

F_{DCC} (s) = (Y *_{d} K) (s) = \sum_{i = 0}^{β - 1} K (i) \cdot Y_{s - d \cdot i}

(1)

where * denotes the convolution, $d$ denotes the dilation factor and increases exponentially by 2 with the depth of the network layers, $K (i)$ denotes the ith weight of the filter. $Y_{s - d \cdot i}$ accounts for the value of the input sequence $Y$ at time step $s - d \cdot i$ .

The residual connection in the TCN is composed of dilated causal convolution layers and rectified linear unit (ReLU) activation functions. The residual connection enhances gradient propagation by directly linking inputs and outputs of the previous layer. The network further incorporates weight normalization and dropout layers for regularization, mitigating overfitting. In addition, a $1 \times 1$ convolution operation is introduced for dimensionality reduction, ensuring consistency between the input and output channels.

Efficient channel attention network

ECA-Net is a new type of channel attention mechanism, as illustrated in Figure 2. Its core module—efficient channel attention (ECA)—ingeniously employs an adaptive 1D convolutional layer to capture inter-channel dependencies. The ECA module involves three steps. (1) Global average pooling (GAP) averages the spatial dimensions (height $\tilde{h}$ and width $w$ ) of the input feature map $χ_{in} \in R^{C \times \tilde{h} \times w}$ generating a global feature vector $χ_{GAP} \in R^{C \times 1 \times 1}$ . (2) Adaptive 1D convolution simulates cross-channel interaction, with an adaptive kernel size $α$ determining the range of these interactions. In group convolution, a fixed number of groups applies long-range (short-range) convolutions to high-dimensional (low-dimensional) channels. Thus, $α$ is proportional to the channel count $C$ , and since $C$ is typically a power of 2, a nonlinear mapping is used to better capture their relationship

C = φ (α) = 2^{(γ \times α - b)} \to α = ψ (C) = | \frac{\log_{2} (C)}{γ} + {\frac{b}{γ} |}_{odd}

(2)

χ = Conv 1 D (χ_{GAP}, α)

(3)

where $b$ and $γ$ are hyperparameters (typically $b = 1$ , $γ = 2$ ), $| \cdot |_{odd}$ indicates the nearest odd number. (3) A ReLU function is applied to learn relative weighting factors for each channel, subsequently weighting the input features to produce a weighted output $χ_{out} \in R^{C \times \tilde{h} \times w}$

χ_{out} = ReLU (χ) \otimes χ_{in}

(4)

where ⊗ denotes element-wise multiplication.

Methodology

Multi-scale adaptive dual forecast model

ATS bearing data pre-processing

For monitoring the condition of the ATS bearing, H sensors are concurrently deployed to collect multi-dimensional feature values of bearings under various operating conditions, serving as the input data for the model. In this context, the value of H is set to 3, indicating that three types of sensors are used for signal acquisition: a vibration acceleration sensor, a torque sensor, and a pressure sensor. For the bearing data collected by each individual sensor is denoted as $X^{(i)} = {x_{1}^{(i)}, x_{2}^{(i)}, . . ., x_{k}^{(i)}, . . ., x_{L}^{(i)}}$ , where $i = [1, H = 3]$ denotes the entire set of sensor types, $L$ denotes the bearing data length (time dimension), which is generated by repeated sampling every 10 seconds for each sensor, with each sampling lasting 0.1 second, until bearing failure occurs. The number of sampling points in each sampling set for each different sensor is determined by its sampling frequency. Suppose the sampling frequency is $f_{(s)}$ , the number of sample points can be calculated as ( $0.1 s \times f_{(s)}$ ). The sampling frequency of the vibration acceleration signal and torque signal is 20.48 kHz. The sampling frequency of the pressure signal is 10 kHz. For example, the accelerometer generates 2048 data points, which are then aggregated into a single time point $x_{k}^{(i)}$ , where $x_{k}^{(i)}$ denotes the sampling value of the sensor $i$ at time $k$ , yielding the final comprehensive bearing sensor data $X = {X^{(1)}, X^{(2)}, . . ., X^{(H)}}$ . Subsequently, the bearing input data $X$ undergoes a reversible instance normalization (RevIN) process, which first normalizes the input sequences and then denormalizes the model output sequences to solve the time-series forecasting problems against distribution shift (Kim et al., 2022). Following this, the processed data $Z_{out}^{0}$ is fed into the multi-scale forecasting (MSF) block.

The overall model architecture

The proposed MADF model aims to capture temporal dependencies by extracting features at different time scales, thereby enabling accurate prediction of the RUL of bearings. The MADF model consists of three MSF modules stacked in a residual connection manner. For the jth layer of MADF ( $j = [1, 2, 3]$ ), the input is $Z_{out}^{j - 1} \in R^{L \times H}$ and the process can be formalized as

Z_{out}^{j} = MSFBlock (Z_{out}^{j - 1}) + Z_{out}^{j - 1}

(5)

The structure and process within the MSF block are illustrated in Figure 3. First, the multi-scale adaptive router equipped with a time-series decomposition module is proposed, which adaptively selects specific patch sizes by dynamically analyzing the trend information of the input bearing data. Next, the bearing data is non-overlappingly segmented into multiple time patches based on the chosen patch size $p$ . Then, the dual forecast block integrates TCN with an IECA mechanism, which extracts the local temporal dependencies within each patch and the long-range dependencies across different patches, enabling comprehensive capture of the complex dynamic features in the time series. Finally, the multi-scale aggregator integrates feature information from different patch scales to enhance the model’s representational capacity of multi-scale information. The final outputs $Z_{out}^{3}$ from stacked MSF blocks are denormalized and passed into a multi-layer perceptron (MLP) for prediction, yielding the bearing RUL result.

MSF block

The MSF block entails three components: (1) multi-scale adaptive router; (2) multi-scale patch dual forecast block; and (3) multi-scale aggregator. The following sections provide a detailed description of these three key modules.

Multi-scale adaptive router

We have designed a multi-scale adaptive router to dynamically select the optimal patch sizes, where the adaptive router incorporates a time-series decomposition module, which integrates periodic decomposition and multi-kernel trend decomposition operations.

To eliminate the interference of external factors such as external temperature variations and load fluctuations on bearing data, it is essential to analyze the periodic characteristics of the data. The fast Fourier transform (FFT) (Ma and Dan, 2023) layer is employed to convert the time-domain signal $Z_{out}^{j - 1} = {Z^{(1)}, Z^{(2)}, Z^{(3)}}$ , with the vibration acceleration signal $Z^{(1)}$ , torque signal $Z^{(2)}$ , and pressure signal $Z^{(3)}$ individually transformed into frequency-domain signals. Considering the sparsity of the frequency domain (Wu et al., 2023a), the amplitudes of the three frequency-domain components are calculated and sorted in descending order. From each, the $Top - q_{f}$ frequency components with the largest amplitudes are selected for the inverse fast Fourier transform (IFFT) to obtain the periodic representation $X_{Period} \in R^{L \times H}$

A = Sort (Amp (FFT (Z_{out}^{j - 1}))), (f_{[1, q_{f}]}, e_{[1, q_{f}]}, o_{[1, q_{f}]}) = argTop - q_{f} (A)

(6)

X_{Period} = IFFT (f_{[1, q_{f}]}, e_{[1, q_{f}]}, o_{[1, q_{f}]}), {\begin{matrix} f_{[1, q_{f}]} = (f_{1}, f_{2}, . . ., f_{q_{f}}) \\ e_{[1, q_{f}]} = (e_{1}, e_{2}, . . ., e_{q_{f}}) \\ o_{[1, q_{f}]} = (o_{1}, o_{2}, . . ., o_{q_{f}}) \end{matrix}

(7)

Here, $A$ denotes the calculated amplitude of each frequency, $Amp (\cdot)$ denotes the calculation of amplitude values, $(f_{1}, f_{2}, . . ., f_{q_{f}})$ , $(e_{1}, e_{2}, . . ., e_{q_{f}})$ , and $(o_{1}, o_{2}, . . ., o_{q_{f}})$ , respectively denote the $Top - q_{f}$ dominant frequencies extracted from the bearing vibration acceleration, torque, and pressure signals. The specific process of periodic decomposition is shown in Figure 4(a).

The traditional moving average method highlights trends by first applying padding operations to the data boundaries, followed by calculating the average of data within a fixed window. However, due to its reliance on a single window for moving average, it often fails to accurately capture the complex trend variations in the bearing operating condition (Huang and Liu, 2024). To overcome this limitation, an innovative multi-kernel trend decomposition module is proposed, as shown in Figure 4(b). The core of this module lies in a set of 1D moving average kernels with varying sizes ${θ_{1}, θ_{2}, . . ., θ_{τ}}$ , which are utilized to extract multi-scale trend components $F \in R^{τ \times L \times H}$ from the residual portion after periodic decomposition $X_{Res} = X - X_{Period}$ . In this study, we propose recommended values based on extensive simulation experiments; thus, we set $τ = 4$ , with the specific kernel sizes defined as follows: $θ_{1} = 4$ , $θ_{2} = 8$ , $θ_{3} = 12$ , $θ_{4} = 16$ . The exact settings should be adjusted according to the specific circumstances. To further optimize the extraction of trend components, the channel attention mechanism (CAM) (Sanghyun et al., 2018) is employed to dynamically evaluate the weights of different trend components. By performing a weighted summation of the extracted multiple trend components, a weighted representation $X_{F} \in R^{τ \times L \times H}$ is obtained. Subsequently, all trend components are summed along the channel dimension to form the final trend representation $X_{T} \in R^{L \times H}$ . This enables the model to better adapt to and represent the various trend features within the bearing. The specific computation steps are as follows

F^{(i)} = MA (X_{Res})_{θ_{i}}, F = [F^{(1)}, F^{(2)}, . . ., F^{(i)}, . . ., F^{(τ)}], i \in [1, τ]

(8)

X_{F} = CAM (F) = σ (W_{1} (W_{0} (F_{avg}^{τ}))) + W_{1} (W_{0} (F_{\max}^{τ}))) \otimes F

(9)

where $F^{(i)}$ denotes the trend result calculated using a moving average kernel of size $θ_{i}$ , $MA (\cdot)_{θ_{i}}$ denotes distinct average pooling functions. $σ$ denotes the sigmoid function, the MLP weights, $W_{0} \in R^{τ / r \times τ}$ and $W_{1} \in R^{τ \times τ / r}$ , are shared for both inputs, $F_{avg}^{τ}$ , and $F_{\max}^{τ}$ denote average-pooled features and max-pooled features, respectively. Through the linear mapping, the final trend representation $X_{T}$ obtained is transformed and merged along the time dimension to yield the time-series decomposition result $X_{trans} \in R^{H}$ .

To determine the optimal patch sizes, an adaptive router is employed to dynamically compute the route weights ${r_{1}, r_{2}, . . ., r_{M}}$ (priorities) of M patch sizes ${p_{1}, p_{2}, . . ., p_{M}}$ in response to the time-series decomposed result $X_{trans}$ . Specifically, the algorithm first calculates the load of multi-scale patches by assigning weights to them and introduces learnable noise perturbations during the generation of routing weight $R (X_{trans}) = {r_{1}, r_{2}, . . ., r_{M}}$ to increase the randomness and prevent premature convergence to local optima. Next, the router adjusts the participation of each patch based on its routing weight, ensuring that the load discrepancies among patches are minimized. By optimizing the load distribution, the algorithm achieves balanced utilization of multi-scale information, preventing over-reliance on specific scales due to excessive load or the omission of key scales due to insufficient load. Furthermore, properly distributing the patch load mitigates the risk of excessively large or small gradients during training, thereby promoting a more stable training process. Meanwhile, to achieve sparsity in the routing process and select the optimal patch sizes, a strategy is adopted where the $Top - q$ weight values are selected from the generated M route weights ${r_{1}, r_{2}, . . ., r_{M}}$ , while the remaining weights are set to zero, resulting in the final route weight representation $\tilde{R} (X_{trans}) \in R^{M}$ . The specific generation process is as follows

R (X_{trans}) = softmax (X_{trans} W_{R} + ς \cdot Softplus (X_{trans} W_{noise})), ς ~ η (0, 1)

(10)

\tilde{R} (X_{trans}) = {\begin{matrix} r_{i}, if r_{i} is among the Top - q weights \\ 0, otherwise \end{matrix}

(11)

where $R (\cdot)$ represents the whole routing function, $W_{R}$ and $W_{noise} \in R^{H \times M}$ are learnable parameters used to generate the weight coefficients, $H$ denotes the feature dimension of $X_{trans}$ . $M$ corresponds to the number of patch size values ${p_{1}, p_{2}, . . ., p_{M}}$ . $ς$ represents a small constant to prevent the noise term from being zero, thereby ensuring the stability of noise generation. $Softplus$ is a smooth approximation of the ReLU function, ensuring that the noise term is always non-negative.

Multi-scale patch dual forecast block and multi-scale aggregator

This multi-scale patch dual forecast block collaborates with a multi-scale adaptive router, dynamically adjusting the multi-scale modeling process based on the varying time scale options of the bearing data. This block consists of two main components: patch segmentation and a dual forecast block.

The MADF model consists of three MSF blocks stacked together via residual connections. Considering the multi-scale characteristics and temporal dependencies and aiming to achieve progressive learning of information across multiple layers at different scales. Each MSF module corresponds to a set $P_{i}$ containing M distinct patch sizes, with M set to 4. Specifically, the values of the patch size sets for each MSF module are defined as follows: $P_{1} = {16, 24, 32, 96}$ , $P_{2} = {12, 16, 24, 32}$ , and $P_{3} = {4, 8, 12, 16}$ . The patch sizes ( $P_{1}$ , $P_{2}$ , and $P_{3}$ ) we designed are specifically tailored for bearing time-series data, adhering to a global-to-local strategy. Based on our experience, for long-sequence data, the patch size is typically smaller than 100. Therefore, we set 96 as the largest patch size and use it as a reference point for gradual reduction. We sequentially select patch sizes of 32, 24, and 16 to effectively capture the long-time degradation trend of the bearing. Since the patch size is positively correlated with the granularity of the features, to extract meaningful local features, the smallest patch size for local trends should preferably be no less than 4. Thus, we adopt 4 as a reference point and gradually enlarge the patch size to 8, 12, and 16, respectively, forming the patch set $P_{3}$ to capture local vibration variations. To strike a balance between the overall trend and local variations, we introduce an intermediate patch size set $P_{2}$ between the two ends ( $P_{1}$ and $P_{3}$ ). This set consists of representative mid-scale values 12, 16, 24, and 32 selected from $P_{1}$ and $P_{3}$ , which serves as a complement by capturing between global and local features, thereby enabling a more comprehensive representation of the bearing degradation process. In each MSF layer, the repeated patch sizes (e.g. 32 in $P_{2}$ and 12 in $P_{3}$ ) preserve key scale information across different layers, allowing this information to be repeatedly learned and propagated in deeper layers. To further enhance the model’s adaptability and its sensitivity to bearing condition variations, the MADF model integrates adaptive routers into different MSF layers. This adaptive mechanism dynamically selects the appropriate patch sizes for subsequent feature processing based on the characteristics of the input data. For a particular layer of the MSF block, a predefined set of patch size values, denoted as $P_{i} = {p_{1}, p_{2}, . . ., p_{M}}$ , is established, where $p_{i}$ denotes patch length ( $p_{i}$ time steps). Based on one of the patch sizes $p$ selected by the multi-scale adaptive router, the input multivariate bearing data $X \in R^{L \times H}$ is segmented into $N$ non-overlapping patches ( $N = L / p$ ), which reduces the dimensionality of data processing and the computational load. The segmented set of patches is represented as $X_{P} = {X^{1}, X^{2}, . . ., X^{N}}$ , where $X^{i} \in R^{p \times H}$ .

Based on the patch set $X_{P}$ obtained from patch segmentation, a dual forecast block is constructed to deeply mine the temporal dependencies within the degradation evolution of bearing data. This block integrates TCN to capture the local correlations within each patch and the interrelationships between different features of the bearing while employing an IECA mechanism to capture the global correlations between different patches.

Intra-patch

When dealing with multivariate bearing data, TCN integrates various features from the bearing data into a unified 1D feature representation through sharing the weights of the convolutional kernels. This not only effectively captures the common inherent temporal dependencies across the different sensor signals but also simultaneously reveals the potential correlations between these signals.

For the selected patch size $p$ , a set of patches ${X^{1}, X^{2}, . . ., X^{N}}$ is identified. A TCN equipped with three layers of dilated causal convolutional layers is utilized to capture the local dependencies within each patch. The specific operation steps are illustrated in Figure 5(a). First, for a single patch $X^{i} \in R^{p \times H}$ within this set, a padding operation of 3, 6, and 12 is performed sequentially to ensure that the input and output have the same temporal dimension. In addition, during the convolution process, 16, 64, and 128 one-dimensional convolution kernels are employed across the layers, with kernel size set to 7 and dilation factors set to 1, 2, and 4. Next, the resulting multiple 1D data from each layer of convolution are concatenated along the feature dimension as input to the next layer. Finally, weight normalization, the ReLU activation function, and dropout are applied to the result of the third layer to obtain the output $X_{intra}^{i} \in R^{p \times 128}$ .

To ensure dimensional consistency for subsequent processing, a 1D convolutional transformation is applied along the feature dimension to obtain the final output $X_{intra}^{i} \in R^{p \times H}$ for $X^{i}$ . After repeating the above operations for all patches, the local outputs from all the patches are concatenated to obtain the final internal output $X_{intra} \in R^{N \times p \times H}$ for the entire patch set $X_{P}$

X_{intra} = Concat (X_{intra}^{1}, X_{intra}^{2}, . . ., X_{intra}^{N})

(12)

Inter-patch

In the ECA module, relying solely on global average pooling to aggregate features from each patch proves insufficient for effectively capturing key bearing data patterns. Inspired by the CAM module, an enhanced approach is proposed by integrating both global average pooling and global max pooling. This parallel capture of dependencies between different patches from both the average and salient feature dimensions enhances the channel attention mechanism’s ability to focus on critical features. For bearing degradation systems, this method offers more flexibility and retains critical characteristic information of bearings across different operating stages, thereby improving prediction performance. Consequently, an improved ECA module, termed IECA, is introduced, as illustrated in Figure 5(b). For the patch set $X_{P}$ from a patch size $p$ segmentation, a 3D processing is performed to acquire 3D data $X_{P} \in R^{N \times p \times H}$ . After processing through the IECA module, a weighted global output $X_{i n t e r} \in R^{N \times p \times H}$ is obtained.

To comprehensively consider both the local correlations within patches and the global correlations between patches, we perform an element-wise summation operation along the corresponding dimensions between the internal output $X_{intra} \in R^{N \times p \times H}$ of patches and the global output $X_{inter} \in R^{N \times p \times H}$ between patches. This results in the final output $X_{out} \in R^{N \times p \times H}$ of the dual forecast block, further enhancing the model’s ability to comprehensively analyze the data

X_{out} = X_{intra} + X_{inter}

(13)

The multi-scale aggregator enhances the model’s ability to capture multi-scale dynamic patterns by integrating bearing features from different scales. Initially, the $Top - q$ different optimal scales ${X_{out}^{1}, . . ., X_{out}^{q}}$ from the multi-scale patch dual forecast block require calibration; the 3D output tensors of each scale are reshaped into 2D matrices ${\hat{X}}_{out}^{i} \in R^{L \times H}$ to accommodate the aggregation process. Subsequently, the multi-scale outputs are aggregated with weighted summation based on specific route weights $\tilde{R} (X_{trans})$ computed by the multi-scale adaptive router, yielding the final output $Z_{out}^{j - 1} \in R^{L \times H}$ of the MSF block

Z_{out}^{j - 1} = \sum_{i = 1}^{M} Δ (\tilde{R} (X_{trans})_{i} > 0) R (X_{trans})_{i} {\hat{X}}_{out}^{i}

(14)

where $Δ (\cdot)$ is a decision function that outputs 1 when $\tilde{R} (X_{trans})_{i} > 0$ , indicating that patch segmentation and dual prediction should proceed; whereas, when $\tilde{R} (X_{trans})_{i} = 0$ , it outputs 0, indicating that the current patch size should be ignored.

Experiments

This section employs a test platform for civil aircraft-bearing components to acquire sensor signals from experimental bearings. Initially, a brief overview of the experimental bearing dataset is provided. Following that, the experimental setup and parameter settings are discussed, after which the dataset is used to evaluate the performance of the proposed prediction framework. Finally, ablation experiments and comparative experiments are conducted to validate the effectiveness of the improved methods.

Bearing data set

To verify the method proposed in this paper, we conducted accelerated fatigue experiments using the RUL prediction test bench for bearing components of civil aircraft. The hardware of the experimental platform consists primarily of a drive motor, gearbox, rolling bearings, connecting shafts, disk flywheels, pulleys, adjustable load couplings, and other components. As illustrated in Figure 6(a), three main sensors were employed to monitor the bearings throughout their lifespan, from normal operation to failure: the vibration acceleration sensor measures the vibration acceleration of the bearings during the experiment; the pressure sensor measures the magnitude of the applied radial force; and the torque sensor measures the working torque of the transmission system. Bearing failures are primarily categorized into pitting and wear. Pitting is caused by surface micro-fatigue induced by alternating stress, resulting in the formation of pits or the detachment of small particles. Wear typically occurs prior to pitting, primarily due to inadequate lubrication or the ingress of contaminants. Due to the unidirectional nature of gear transmission, the side subjected to greater stress is more prone to wear. Figure 6(b) presents a photograph of deteriorated bearings, providing a visual representation of their degradation. The experimental dataset includes vibration acceleration, torque, and pressure data. Table 1 presents the sampling settings of the three sensors, while Table 2 lists the two operating conditions. Under the first operating condition (speed: 1400 RPM), experimental data is collected from four bearings (training bearing 1_1, bearing 1_4, and test bearing 1_2 and bearing 1_3). In the second operating condition (speed: 1500 RPM), experimental data is gathered from three bearings (training bearing 2_1, bearing 2_2, and test bearing 2_3).

Table 1.

Sampling settings of vibration acceleration, torque, and pressure.

Sensors	Vibration acceleration	Torque	Pressure
Sample frequency	20.48 kHz	20.48 kHz	10 kHz
Sample time	0.1 second	0.1 second	0.1 second
Sample points	2048	2048	1000
Sample interval	10 seconds	10 seconds	10 seconds

Table 2.

Representation of bearing data under different operating conditions.

Working conditions	Condition 1		Condition 2
Rotationspeed	1400 rmp		1500 rmp
Radial tension	9 kN		8 kN
Training set	Bearing1_1	Bearing1_4	Bearing2_1	Bearing2_2
Full life(10 seconds)	672	864	1440	2496
Testing set	Bearing1_2	Bearing1_3	Bearing2_3
Full life(10 seconds)	480	288	576

Experimental setup and parameter settings

The configuration of the computer used in the laboratory is as follows: The processor is an Intel(R) Xeon(R) W-2133 CPU running at 3.60 GHz with 32 GB of RAM onboard. The operating system is Windows 10 (64-bit), and the programming simulation environment is Python 3.7.16.

Experimental setup

In the prediction of bearing remaining useful life, we integrated existing domain knowledge to extract features from sensor data and design the model. We replace each minute of sensor data with corresponding statistical features to serve as new input for predicting the bearings’ RUL. These features include peak-to-peak value, standard deviation, and root mean square (RMS), among others. By analyzing the monotonicity and trends of these features, we identified the most suitable ones for training and testing, ultimately leading to the selection of the RMS value. The related parameters of the MADF model are shown in Table 3. The number of training epochs is set to 300, and the initial learning rate is 0.004. When the number of training epochs reaches 220, the learning rate changes to 0.0008. The loss function is mean squared error (MSE), and the optimization algorithm used is Adam.

Table 3.

Related parameters of the MADF model.

Related parameters
Training epoch	300
Initial learning rate	0.004
Change training epoch	220
Change learning rate	0.0008
Loss function	MSE
Optimizer	Adam

To validate the accuracy and robustness of the proposed model, three sets of evaluation metrics are introduced to assess the differences between the predicted RUL and the actual RUL: mean absolute error (MAE), root mean square error (RMSE), and score. The formulas for the three indicators are

MAE = \frac{1}{n} \sum_{i = 1}^{n} | R_{i}^{pre} - R_{i}^{act} |

(15)

RMSE = \sqrt{\frac{1}{n} \sum_{i}^{n} {(R_{i}^{pre} - R_{i}^{act})}^{2}}

(16)

Score = {\begin{matrix} \sum_{i = 1}^{n} (e^{- \frac{R_{i}^{pre} - R_{i}^{act}}{13}} - 1), R_{i}^{pre} - R_{i}^{act} < 0 \\ \sum_{i = 1}^{n} (e^{\frac{R_{i}^{pre} - R_{i}^{act}}{10}} - 1), R_{i}^{pre} - R_{i}^{act} \geq 0 \end{matrix}

(17)

where $n$ is the number of testing units, $R_{i}^{pre}$ represents the predicted value of RUL, $R_{i}^{act}$ represents the actual label value of RUL. $R_{i}^{pre} - R_{i}^{act} < 0$ represents an early prediction, $R_{i}^{pre} - R_{i}^{act} \geq 0$ represents a late prediction. The smaller the values of MAE, RMSE, and score, the better the fit of the predictions. However, there exists a subtle difference among the three indicators. MAE is used to reflect the true errors. RMSE is sensitive to the larger errors in the samples, it can be used to detect the larger and less common errors in the samples. The score penalizes the late prediction more than the early prediction so that it can effectively prevent the missed report caused by the late prediction results rather than the real results.

Parameter settings

We selected Bearing1_2, Bearing1_3, and Bearing2_3 to validate the advantages of multi-scale features in RUL prediction. We varied the number of optimal patch sizes q (q = 1, q = 2, q = 3, and q = 4) selected by the adaptive router to compare the MAE, RMSE, and score values corresponding to different q values. The experiment results are shown in Table 4. Compared to the single segmentation scale (q = 1), multi-scale patch segmentation (q = 2, 3, 4) for feature extraction yields better prediction performance. Furthermore, the MADF model performs best when q = 2, with MAE reduced by 13.82% and 21.70% compared to q = 3 and q = 4, RMSE reduced by 12.21% and 21.64%, and score reduced by 13.48% and 23.32%, respectively. Notably, as q increases to 3 and 4, the model’s prediction performance deteriorates, indicating that more scales do not necessarily lead to better prediction results. This demonstrates that the adaptive routing function’s selection of key scales can enhance prediction accuracy to some extent. Therefore, in the following experiments, we selected a segmentation scale of q = 2 as one of the experimental conditions for investigation.

Table 4.

The prediction accuracy varies with q.

Models	q = 1			q = 2			q = 3			q = 4
	MAE	RMSE	Score	MAE	RMSE	Score	MAE	RMSE	Score	MAE	RMSE	Score
Bearing1_2	0.0524	0.0678	1.982	0.0424	0.0535	1.615	0.0464	0.0564	1.785	0.0503	0.0594	1.912
Bearing1_3	0.0529	0.0660	1.238	0.0342	0.0426	0.838	0.0397	0.0510	0.959	0.0460	0.0581	1.047
Bearing2_3	0.0505	0.0613	2.378	0.0337	0.0420	1.551	0.0420	0.0499	1.885	0.0446	0.0585	2.263
Average	0.0519	0.0650	1.866	0.0368	0.0460	1.335	0.0427	0.0524	1.543	0.0470	0.0587	1.741

The bold values in Table 4 demonstrate that setting the number of optimal patch sizes (q) to 2 yields the lowest MAE, RMSE, and Score, thereby indicating the model's highest prediction accuracy.

Based on the above experimental results, we determined that the optimal number of patch sizes for the adaptive router selection is q = 2. Since the model is composed of three stacked MSF modules, we visualized the loss curves for both the training and testing sets under two operating conditions to validate the training performance of the model. In Condition 1, bearing 1_1 was used as the training set, and bearing 1_3 as the testing set; in Condition 2, bearing 2_1 was used for training, and bearing 2_3 for testing. The model was trained for 100 epochs with a learning rate of 0.04. As shown in Figure 7(a) and (b), under the configuration of three stacked MSF modules, the loss curves for both the training and testing sets converge, with only a small difference between them. This indicates that the model demonstrates good stability across different operating conditions. Furthermore, the smooth loss curves without significant fluctuations suggest that the model does not suffer from noticeable overfitting or underfitting during the training process. This further validates the effectiveness of the MADF model in multi-scale feature modeling.

Performance evaluation, ablation experiments, and comparative experiments

Performance evaluation

Taking bearing 2_3 as an example, Figure 8 illustrates the whole lifetime degradation trend curves of the bearing. The data collected from three sensors (vibration acceleration, torque, and pressure) demonstrates that between time steps 0 and 4000 seconds, the RMS values of the three signal curves remain within a stable range, indicating that the bearing operates in good condition during this period without significant wear. However, at 4230 seconds, the RMS values of the three signals exceed the critical threshold for the first time, indicating that the bearing has malfunctioned. Subsequently, the signal values continue to rise, with increasing fluctuation amplitude, reflecting the progressive deterioration of the bearing’s performance and an accelerating degradation rate. Finally, at 5760 seconds, the bearing completely fails and ceases operation. In summary, the bearing exhibits relatively stable performance during the early operational stage, while performance degradation primarily emerges in the mid-to-late stages, showing an accelerating deterioration trend that ultimately leads to failure.

During the mid-to-late phase of bearing operation, degradation feature signals collected by the three types of sensors are particularly obvious, facilitating more effective feature extraction and learning by the model. Therefore, initiating prediction of bearing data from the mid- stage of the bearing’s lifecycle is more conducive to accurately capturing degradation patterns, thereby improving the reliability and practicality of RUL estimation. To this end, we further present the RUL prediction results for the test bearings (Bearing 1_2, Bearing 1_3, and Bearing 2_3) in the mid-to-late stages from the validation dataset, as shown in Figure 9(a)–(c). When evaluating the performance of these three bearings, the MAE for bearing 1_2 is 0.0266, and the RMSE is 0.0316; for bearing 1_3, the MAE is 0.0221, and the RMSE is 0.0276; for bearing 2_3, the MAE is 0.0239, and the RMSE is 0.0297. These results indicate that the proposed method demonstrates superior performance and high precision in predicting the RUL of bearing components, effectively capturing the degradation characteristics of the bearings, thereby providing reliable support for practical applications.

Ablation experiment

To evaluate the effectiveness of the proposed MADF model, an ablation study of the proposed approach is performed. Specifically, we replaced the multi-kernel trend decomposition module with a single-kernel trend decomposition module and removed the time-series decomposition, inter-patch (IECA), and intra-patch (TCN) modules, respectively, to verify the importance of each component to the model’s performance. The experimental results can be found in Table 5 and Figure 10(a) and (b).

Table 5.

Evaluation metrics (RMSE and score) in the ablation experiment.

Models	Proposed Method		W/O Time-series Decomp		Single-kernel trend decomp		W/O Inter (IECA)		W/O Intra (TCN)
	RMSE	Score	RMSE	Score	RMSE	Score	RMSE	Score	RMSE	Score
Bearing1_2	0.0577	1.794	0.0651	1.991	0.0598	1.952	0.0742	2.358	0.0788	2.592
Bearing1_3	0.0490	0.916	0.0691	1.307	0.0586	1.178	0.0754	1.458	0.0853	1.612
Bearing2_3	0.0444	1.599	0.0662	2.402	0.0611	2.344	0.0814	3.167	0.0843	3.290
Average	0.0504	1.436	0.0668	1.900	0.0598	1.825	0.0770	2.328	0.0828	2.498

The bold values in Table 5 indicate that the proposed method achieves the lowest RMSE and Score compared to other ablation models, further confirming the superior prediction accuracy of the MSDF model.

As shown in Figure 10, the proposed MADF model has the minimum error range of RMSE and score. Table 5 highlights the unique contribution of each module. Initially, when the time-series decomposition (W/O Time-series Decomp) module is removed, the RMSE increases by 24.55% and the score by 24.42% compared to the proposed method. This indicates that time-series decomposition effectively captures the temporal dynamics of the bearing data, facilitating the identification of appropriate patch sizes for combinations. Next, replacing the multi-kernel trend decomposition with single-kernel trend decomposition results in a 15.72% increase in RMSE and a 21.32% increase in score compared to the proposed model. However, when compared to the W/O Time-series Decomp setting, the RMSE decreases by 10.48% and the score by 3.95%. This indicates that multi-kernel trend decomposition effectively improves the model’s predictive accuracy and plays a more significant role in time-series decomposition than its single-kernel trend decomposition. Furthermore, removing the inter-patch (IECA) and intra-patch (TCN) modules leads to a considerable performance decline. The RMSE increases by 34.55% and 39.13%, while the score rises by 38.32% and 42.51%, respectively. This indicates that TCN excels at capturing local patterns within individual patches, aiding in the characterization of fine-grained degradation features. Meanwhile, IECA identifies broader global patterns across patches, effectively enhancing the model’s overall representational capacity.

Comparative experiments

In this section, we focus on the prediction of the RUL of the tested bearings (Bearing1_2, Bearing 1_3, and Bearing 2_3). Through a series of comparative experiments, we validate the superiority of the proposed method. Figure 11(a)–(c) display the prediction results of the proposed method compared to various baseline models, including MTST, PatchTST, TimesNet, Scaleformer, and convolutional neural network (CNN) models. Among them, CNN represents a conventional deep learning network; Scaleformer represents a more advanced network capable of multi-scale feature extraction; and TimesNet has demonstrated outstanding performance in recent time-series forecasting tasks. PatchTST introduces an innovative patch segmentation mechanism, which effectively captures local patterns in time series and enhances the model’s ability to capture long-term dependencies. MTST further improves upon PatchTST by incorporating a multi-branch structure for patch segmentation, thereby enhancing the model’s multi-scale feature representation and prediction performance. Table 6 provides detailed values for the three evaluation metrics, further quantifying the performance of the MADF model in the RUL prediction task.

Table 6.

Evaluation metrics results for the proposed method and other models.

Models	Index	Bearing1_2	Bearing1_3	Bearing2_3	Average
Proposed Method	MAE	0.0362	0.0351	0.0361	0.0358
	RMSE	0.0451	0.0457	0.0446	0.0451
	Score	1.350	0.808	1.656	1.271
MTST	MAE	0.0380	0.0464	0.0431	0.0425
	RMSE	0.0487	0.0547	0.0576	0.0537
	Score	1.541	1.045	2.242	1.609
PatchTST	MAE	0.0450	0.0484	0.0494	0.0476
	RMSE	0.0600	0.0701	0.0622	0.0641
	Score	2.039	1.218	2.565	1.941
TimesNet	MAE	0.0527	0.0540	0.0539	0.0535
	RMSE	0.0633	0.0659	0.0639	0.0644
	Score	2.281	1.358	2.856	2.165
Scaleformer	MAE	0.0673	0.0563	0.0550	0.0595
	RMSE	0.0772	0.0757	0.0679	0.0736
	Score	2.507	1.361	2.894	2.254
CNN	MAE	0.0692	0.0687	0.0734	0.0704
	RMSE	0.0807	0.0851	0.0941	0.0866
	Score	2.592	1.532	3.143	2.422

The bold values in Table 6 indicate that the proposed method achieves the lowest MAE, RMSE, and Score compared to other baseline models, further confirming the superior prediction accuracy of the MSDF model.

In all experimental models, we consistently used a learning rate setting of 0.0008. By analyzing the average evaluation metrics for the three test bearings, we found that, compared to the MTST, PatchTST, TimesNet, Scaleformer, and CNN models, the proposed method attained reductions in MAE by 15.76%, 24.79%, 33.08%, 39.83%, and 49.15%, respectively; RMSE by 16.01%, 29.64%, 29.97%, 38.72%, and 47.92%; and score by 21.01%, 34.52%, 41.29%, 43.61%, and 47.52%. The results of various evaluation metrics indicate that the proposed method outperforms other models in terms of performance. A deeper analysis reveals that, in comparison with the traditional deep learning network CNN, Scaleformer’s multi-scale feature extraction mechanism offers certain advantages in capturing variations across different temporal scales in bearing operation data. In comparison with TimesNet, the patching mechanism introduced by PatchTST provides greater innovation and effectiveness in bearing feature extraction. Thus, MTST enhances the representation of complex multi-scale features by combining patch division with a multi-branch structure, making it more suitable for modeling bearing degradation trends. The proposed MADF model exhibits superior performance improvements, highlighting the effectiveness of the adaptive router equipped with time-series decomposition in multi-scale modeling. This module flexibly allocates patch scales across different time scales for feature modeling, enabling more accurate prediction of the bearing degradation process.

Conclusion

In this paper, we propose a model for predicting the RUL of bearings, referred to as the MADF model. The innovation of this model lies in the introduction of patches with multiple different scales. By excluding the interference of the external environment, the adaptive routing mechanism dynamically selects and aggregates specific scale features based on the bearing degradation trend, thereby further enhancing the accuracy and performance of predictions. The model implements a dual forecast mechanism on the segmented data to capture the temporal dependencies both within different patches and between them. By integrating multi-scale features and an adaptive routing strategy, the MADF model not only strengthens the model’s adaptability to complex data features but also effectively provides more reliable support for bearing health management.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China under Grant No. 62173331, in part by the Tianjin Science and Technology Plan Project under Grant No. 23JCYBJC00060, in part by Civil Aviation Safety Operation Guarantee Capability Construction Project under Grant 202412, in part by the Project of Aviation Science Foundation under Grant 2019ZD067007, and in part by the Special Program of Talents Development for Excellent Youth Scholars in Tianjin under Grant TJTZJH-QNBJRC-2-19.

ORCID iDs

Runxia Guo

Xin Wei

Data availability statement

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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