2.1 Convolutional neural networks

Due to the temporal nature of rolling bearing vibration signals, one-dimensional convolutional neural networks (1D-CNNs) are effective in capturing key time-domain features while reducing computational cost. Mirela et al. [31] noted that CNNs are capable of processing various data types-including time-series, images, and videos—making them suitable for vibration signal analysis.

Traditional methods such as time-domain statistics, STFT, and WT depend on manual feature design, limiting adaptability. In contrast, CNNs can automatically learn discriminative features through deep architectures. Guo et al. [32] demonstrated that CNNs reduce manual intervention and outperform fully connected networks by leveraging local receptive fields and weight sharing to lower complexity.

1. (1) Convolutional Layer

Xu et al. [33] emphasized that convolutional layers extract local patterns through shared-weight kernels using a sliding window, enabling progressive abstraction with fewer parameters. Zhang et al. [34] further noted that this hierarchical representation enhances computational efficiency and feature expressiveness. The 1D convolution process is illustrated in Fig 1.

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Fig 1. Diagram of one-dimensional data convolution operation.

The convolution kernel slides over the input data and performs weighted summation at each position to generate the output.

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In a convolutional neural network, convolution operations are performed on the input data using convolutional kernels, as shown in Equation (1):

(1)

In this equations,

represents the input features of the -th layer,

denotes the convolution kernel weights of that layer,

represents the convolution operation,

is the bias term,

is the output feature map after convolution.

During convolution, the kernel operates on local regions, which may result in the loss of edge information. Alrasheedi et al. [35] noted that padding is commonly used to mitigate this issue by extending the input at its boundaries. Among various techniques, zero padding is the most widely adopted. As Iftekharuddin et al. [36] pointed out, it preserves the input dimensions by adding zeros around the edges. A schematic of zero padding is shown in Fig 2.

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Fig 2. Zero Padding operation diagram.

Zeros are added at both ends of one-dimensional data to maintain output length during convolution with kernel weights [1–3], preserving edge features.

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1. (2) Batch Normalization (BN Layer)

Bilal et al. [37] described batch normalization (BN) as a technique that accelerates training and improves model stability, particularly in deeper networks where shifting activation distributions can lead to unstable gradients and slow convergence.

BN normalizes the inputs of each mini-batch by adjusting their mean and variance, followed by a learnable scaling and shifting operation to retain representational flexibility. This process mitigates vanishing or exploding gradients and enhances both convergence speed and generalization.

In convolutional neural networks (CNNs), the BN layer is typically applied after convolution. During training, it computes batch-wise statistics; during inference, it uses the mean and variance estimated during training for normalization.

Overall, BN is a powerful technique that significantly accelerates neural network training and enhances model performance by stabilizing gradient updates. The computation process of BN is described by Equations (2), (3), and (4), where the mean and variance of all samples in a mini-batch are calculated as follows:

(2)

Normalized Data:

(3)

Linear Transformation:

(4)

In the equations above, and are learnable parameters used to restore the model’s representation capability.

The Batch Normalization (BN) layer significantly improves convergence speed, stabilizes training, and enhances generalization performance. It is commonly positioned after convolutional or fully connected layers, although its placement may vary depending on the specific network design.

1. (3) Activation Layer

Convolution operations are inherently linear, limiting their capacity to model complex patterns. To address this, activation functions introduce nonlinearity, enabling the network to learn more expressive feature representations through nonlinear transformations of the input.

An effective activation function should be nonlinear, differentiable, computationally efficient, and conducive to fast convergence. Its mathematical form is presented in Equation (5).

(5)

Common activation functions include Sigmoid, Tanh, ReLU, and their variants:

1. 1) Sigmoid Function

The Sigmoid function is calculated as shown in Equation (6).

(6)

Elfwing et al. [38] reported that the Sigmoid activation function outputs values in the range [0, 1] but suffers from gradient saturation when inputs are extreme, causing vanishing gradients that hinder deep network training. They also noted its derivative is computationally expensive, limiting its use. The function’s schematic is shown in Fig 3(a).

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Fig 3. Sigmoid and Tanh activation function diagrams.

(a) The sigmoid function, a bounded and differentiable curve that maps real-valued inputs into the range (0, 1). (b) The tanh function, which transforms inputs into the range (−1, 1), offering zero-centered output. Both functions exhibit sigmoid-like shapes and are widely used to introduce non-linearity in neural networks.

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1. 2) Tanh Function

The Tanh function is calculated as shown in Equation (7).

(7)

The Tanh function outputs values in the range and, unlike Sigmoid, is zero-centered, which facilitates optimization convergence. This centralized output benefits data processing. However, it also suffers from gradient saturation, causing vanishing gradients, and its derivative remains computationally complex. The function’s schematic is shown in Fig 3(b).

1. 3) ReLU (Rectified Linear Unit) Function

The ReLU (Rectified Linear Unit) function is defined mathematically as shown in Equation (8).

(8)

The ReLU activation function is a piecewise linear function that maintains a linear output for positive inputs and exhibits a non-saturating property, effectively alleviating the vanishing gradient problem.

ReLU’s sparse activation characteristic enhances the model’s learning ability. However, if the network’s weights and learning rate are not set properly, ReLU may cause some neurons to get stuck in the negative half-region, preventing them from activation—a phenomenon known as the dying ReLU problem.

The function’s schematic diagram is shown in Fig 4(a).

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Fig 4. Relu and PRelu activation function diagram.

(a) ReLU function, which outputs the input if positive, else zero. (b) PReLU function, a generalization of ReLU with a learned slope for negative inputs.

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1. 4) PReLU (Parametric ReLU) Function

The Parametric ReLU (PReLU) function is mathematically defined as shown in Equation (9):

(9)

In the formula, is a learnable parameter.

Parametric ReLU (PReLU) extends ReLU by introducing a learnable parameter γ for the negative input region. For the ith neuron, negative inputs are scaled by this parameter instead of being set to zero.

This parametric form enables the network to adaptively learn the activation shape during training, improving expressiveness and generalization. PReLU is particularly effective for handling diverse data distributions by learning a more suitable activation function. The function’s schematic is shown in Fig 4(b).

1. (4) Pooling Layer

Pooling layers are commonly used in convolutional neural networks to reduce dimensionality and compress input data while retaining essential features. Typically applied after convolutional layers, pooling helps eliminate redundant information and highlights salient features, enhancing the network’s ability to learn.

Two main types of pooling are max pooling, which selects the maximum value within a region, and average pooling, which computes the average value. Their corresponding formulas are shown in Equations (10) and (11), with the operation illustrated in Fig 5.

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Fig 5. One-dimensional data pooling operation diagram.

Max pooling selects the maximum value from each sliding window. Average pooling computes the mean value within each window. Both operations reduce data dimensionality while preserving key features.

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(10)

(11)

The pooling layer slides a fixed-size window over the input with a set stride. Within each window, it aggregates data into a single value, typically using max pooling or average pooling depending on the method applied.

1. (5) Fully Connected Layer

The fully connected (FC) layer, positioned at the final stage of a convolutional neural network, transforms extracted features into the class space of input samples. The output is computed via the Softmax function, as shown in Equations (12) and (13). A schematic of the FC layer operation is presented in Fig 6.

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Fig 6. Diagram of fully connected layer operation for one-dimensional data.

Illustrates how flattened one-dimensional data is processed through a fully connected neural network layer. Each node in the fully connected layer receives input from all nodes in the previous layer, transforming the data through learned weights to produce the output.

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(12)

(13)

In these equations:

represents the output of the fully connected layer.

denotes the Softmax output, which represents the probability of the input belonging to class .

represents the total number of classes.

Features from the final pooling layer are first flattened into a one-dimensional vector before being input to the fully connected (FC) layer. In this layer, the output is produced by a weighted sum of all inputs, followed by an activation function. The FC layer integrates local features extracted by convolutional and pooling layers, forming a fully connected network structure.

2.2 Propagation process of convolutional neural networks

Morsy et al. [39] and Yamashita et al. [40] stated that convolutional neural networks (CNNs) rely on forward propagation and backpropagation to optimize network parameters, enabling effective feature extraction and classification. These processes are crucial for refining parameters and improving the model’s ability to distinguish fault patterns in rolling bearing diagnosis. Forward propagation passes input data through the network layers to compute feature representations and predictions, while backpropagation updates parameters based on error feedback to enhance training effectiveness.

In rolling bearing vibration analysis, signals contain rich fault-related information. CNNs utilize multi-layer convolutions to extract meaningful features and suppress redundancy, thereby improving diagnostic accuracy.

The key steps of forward propagation include:

1. 1. Convolutional Layer: Extracts local features via convolution, described by Equation (14).

(14)

In these equation, represents the input data, and denotes the convolution kernel.

1. 2. Activation Function: Performs nonlinear mapping to enhance feature representation capability. Common activation functions are expressed as shown in Equation (15).

(15)

1. 3. Pooling Layer: Reduces data dimensions, decreases computational complexity, and retains key features. A commonly used pooling method is max pooling, with its computation formula shown in Equation (16).

(16)

1. 4. Fully Connected Layer: Maps the extracted features to fault categories and computes the final output using Softmax, with its computation formula shown in Equation (17).

(17)

In these equation, represents the output of the fully connected layer, and denotes the number of categories.

Backpropagation is a method used to compute the gradient of the loss function with respect to each weight and bias, enabling the update of network parameters. The core steps are as follows.

1. 1. Error Calculation: The difference between the predicted value and the actual value is measured by the loss function, with the specific computation formula shown in Equation (18).

(18)

1. 2. Error Backpropagation: The error is propagated backward through the network from the output layer to the input layer, gradually computing the gradient of each parameter. The specific computation formula is shown in Equation (19).

(19)

1. 3. Parameter Update: The stochastic gradient descent (SGD) algorithm is used to adjust the weights based on the computed gradients to minimize the loss. The specific computation formula is shown in Equation (20).

(20)

In the equation, represents the learning rate.

Forward propagation and backpropagation are fundamental to CNN training. Forward propagation computes outputs, allowing the model to predict data categories, while backpropagation adjusts parameters to enhance accuracy. Through iterative interaction, the CNN progressively improves its fault diagnosis performance. These processes are illustrated in Fig 7.

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Fig 7. Convolutional neural network forward and backpropagation processes.

The figure illustrates the forward propagation from input through convolutional, pooling, normalization, activation, and fully connected layers to produce predictions, followed by backpropagation that computes loss and updates weights across the network.

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2.3 One-dimensional convolutional neural network model

Convolutional neural networks (CNNs) are primarily designed for image processing using two-dimensional (2D) convolutional architectures to extract spatial features. To apply 2D CNNs to one-dimensional (1D) vibration signals, researchers often reshape 1D data into 2D matrices or transform signals into time-frequency representations via wavelet or fast Fourier transforms (FFT).

While these approaches improve fault diagnosis, they incur high computational cost and complexity. In contrast, 1D CNNs offer lower computational overhead and training time. The traditional 1D CNN architecture is illustrated in Fig 8.

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Fig 8. Traditional one-dimensional convolutional neural network model.

The model processes raw one-dimensional signals through sequential convolution and pooling layers for feature extraction, followed by a fully connected layer for final output prediction.

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One-dimensional convolutional neural networks (1D-CNNs) are designed for time series data and can be viewed as simplified versions of two-dimensional CNNs (2D-CNNs). While 2D-CNN kernels typically have dimensions of , 1D-CNN kernels reduce the height dimension, resulting in kernels.

The operation of 1D-CNNs parallels that of 2D-CNNs. Since rolling bearing vibration signals are one-dimensional time series, 1D-CNNs are better suited for fault diagnosis in this context.

2.4 Parallel one-dimensional convolutional neural network model

Although one-dimensional convolutional neural networks (1D-CNNs) effectively process time series signals, they rely solely on time-domain information, limiting feature extraction. Li et al. [41] proposed a parallel 1D-CNN (P1D-CNN) model to incorporate frequency-domain information more comprehensively.

This model employs two channels with convolution kernels of different sizes in the first layer. Subsequent layers use smaller kernels, enabling deeper architectures, enhancing local feature capture, and reducing training time. Sun et al. [42] noted that larger kernels have wider receptive fields, making them better suited for extracting global features.

The receptive field denotes the input region influencing each output neuron in a CNN layer. For a 1D convolutional layer with kernels of width , each neuron processes a local segment of length . Fig 9 illustrates this concept.

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Fig 9. Schematic diagram of receptive field.

Depicts how the receptive field expands hierarchically across layers, with each neuron processing increasingly broader regions of the input through connections to previous layers.

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Compared to single-channel 1D-CNNs, the parallel 1D-CNN (P1D-CNN) effectively extracts both time- and frequency-domain features from vibration signals while accelerating network training. The architecture of the P1D-CNN is shown in Fig 10.

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Fig 10. Parallel one-dimensional convolutional neural network model.

Shows two separate convolution and pooling pathways with distinct kernel sizes processing the same input. The outputs merge at the fully connected layer, enabling multi-scale feature extraction.

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2.5 Improved parallel one-dimensional convolutional neural network model

2.5.1 Gated recurrent unit.

Hochreiter et al. [43] proposed the Long Short-Term Memory (LSTM) network in 1997.However, due to the limitations of computing power and dataset size at the time, it did not gain widespread application. In 2014, Cho et al. [44]introduced the Gated Recurrent Unit (GRU), an improved recurrent neural network (RNN) architecture.

GRU effectively addresses the vanishing gradient problem in traditional RNNs and reduces computational complexity by incorporating two gating mechanisms: the reset gate and the update gate These gate vectors determine which information should be retained or discarded, allowing the model to capture long-term dependencies in time series data more effectively.

At time step t the reset gate and the update gate are computed using the following equations, as shown in Equations (21) and (22):

(21)(22)

In these equations, represents the Sigmoid activation function, while and are learnable weight matrices. denotes the hidden state from the previous time step, and represents the current input features.

The reset gate determines whether the network should ignore previous state information, while the update gate controls the degree to which new information is accepted. By regulating the interaction between the current input and the previous hidden state, the reset gate helps compute the candidate hidden state , which is formulated as shown in Equation (23):

(23)

The final hidden state is computed using the update gate and the candidate state , as shown in Equation (24):

(24)

GRU addresses the vanishing gradient issue in traditional RNNs by using reset and update gates to regulate information flow. The reset gate controls the extent to which previous state information is ignored, while the update gate determines how much new information is retained. This gating structure allows the network to dynamically manage information, preserving key features and improving time series representation.

In rolling bearing fault diagnosis, where vibration signals exhibit complex temporal patterns and long-term dependencies, GRU effectively captures these characteristics, enhancing diagnostic performance.

2.5.2 Attention mechanism.

Bahdanau et al. [45] introduced the attention mechanism, which enables models to focus selectively on important parts of the input, rather than treating all inputs equally. This mechanism effectively mitigates performance degradation caused by long-range dependencies in traditional recurrent neural networks. Unlike fixed-weight or fully connected methods, the attention mechanism dynamically learns weights, reducing parameter count and improving both efficiency and generalization.

Woo et al. [46] proposed the Convolutional Block Attention Module (CBAM), a lightweight attention module that applies attention sequentially along spatial and channel dimensions. This study adopts CBAM due to its effectiveness in enhancing feature representation. The CBAM module operates in two steps, with formulas given in Equations (25) and (26), and its schematic is shown in Fig 11.

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Fig 11. CBAM attention mechanism.

Depicts the sequence of modules: input, channel attention, spatial attention, and output. CBAM efficiently enhances network performance while reducing parameters and computation, and can be integrated into various networks.

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1. 1. Channel Attention (ChannelAttention): The input feature F undergoes max pooling and average pooling operations, producing the max-pooled feature and average-pooled feature These features are concatenated and passed through a shared fully connected layer, generating the channel attention weight , with its computation process given in Equation (25).

(25)

In the equation, represents the Sigmoid activation function, and FC denotes the fully connected layer operation.

Then, the channel attention weight is element-wise multiplied with the input feature to obtain the adjusted feature . Its computation is given by Equation (26).

(26)

1. 2. Spatial Attention (Spatial Attention): The channel attention-adjusted feature undergoes max pooling and average pooling along the channel dimension, producing the spatial features and .These two features are concatenated and processed through a convolutional layer to generate the spatial attention weight . Its computation process is given by Equation (27).

(27)

The Sigmoid activation function is used to constrain the spatial attention weight within the range of 0–1.

Finally, the spatial attention weight is applied to the adjusted feature to obtain the final feature representation . Its computation is given by Equation (28).

(28)

The attention mechanism dynamically learns feature weights, automatically selecting the most relevant features from the input data and thereby enhancing the model’s feature selection capability. In rolling bearing fault diagnosis, vibration signals often contain substantial noise and redundant information. The attention mechanism effectively filters out these irrelevant components, significantly improving diagnostic accuracy. Moreover, by reducing redundant parameters, it enhances the model’s efficiency and generalization ability. This is particularly beneficial when processing complex vibration signals, as the attention mechanism helps the model adapt to varying data distributions and strengthens its overall generalization performance.

As shown in Fig 11, CBAM (Convolutional Block Attention Module) is a structure composed of input, channel attention module, spatial attention module, and output. This model not only saves parameters and reduces computational complexity but can also be flexibly embedded into existing network architectures for optimization. By leveraging channel attention and spatial attention mechanisms, CBAM can significantly enhance the network’s performance while maintaining computational efficiency.

2.5.3 Fault diagnosis process based on improved parallel one-dimensional convolutional neural network.

The parallel one-dimensional convolutional neural network (1D-CNN) is a deep convolutional neural network. As the network depth increases, it may suffer from the vanishing gradient problem. To address this issue, GRU (Gated Recurrent Unit) and an attention mechanism are introduced into the classifier module. GRU effectively mitigates the vanishing gradient and long-term dependency issues in recurrent neural networks by incorporating reset and update gates. However, when processing signals, GRU tends to forget information once the signal exceeds a certain length. To resolve this, an attention mechanism is introduced to compress the data, alleviating GRU’s forgetting problem and enabling the model to quickly pinpoint key information within a large dataset. The improved parallel one-dimensional convolutional neural network model structure is shown in Fig 12.

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Fig 12. Improved one-dimensional convolutional neural network model.

Features dual convolutional pathways with different kernel sizes for multi-scale feature extraction. Each pathway includes CBAM attention and GRU layers to enhance feature and temporal learning. Outputs are combined after global max pooling to generate refined representations.

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As shown in Fig 12, the improved parallel one-dimensional convolutional neural network (1D-CNN) model adopts a dual-channel convolutional kernel of different sizes to process vibration signals. The two channels separately extract the time-domain and frequency-domain features of the vibration signal, which not only accelerates the training speed of the network model but also fully utilizes the signal’s characteristic information. After extracting the vibration signal features through three convolutional and pooling layers, the GRU layer performs temporal modeling on the extracted features, learning the relationships and dynamic changes within the sequence. The model generates a large number of high-dimensional temporal features, which are often difficult to distinguish and interpret intuitively. By introducing an attention mechanism layer after the GRU layer, the model can extract important information, allowing it to quickly identify key points from a vast amount of data while filtering out irrelevant information, making the training process more efficient. Finally, a global max pooling layer replaces the fully connected layer, which reduces the number of parameters, improves computational efficiency, and lowers the risk of overfitting. The fault diagnosis process of the improved parallel 1D-CNN is illustrated in Fig 13.

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Fig 13. Improved one-dimensional convolutional neural network fault diagnosis flowchart.

The process includes data preprocessing, dataset splitting, model training and validation, and final testing to evaluate diagnostic accuracy.

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As shown in Fig 13, the fault diagnosis algorithm process is as follows:

1. Preprocessing the Vibration Signal:
   1. The original vibration signal is decomposed, and the relevant components are selected and reconstructed.
   2. The reconstructed signal is then divided into training, testing, and validation sets.
2. Building the Improved Parallel 1D-CNN Fault Diagnosis Model:
   1. A dual-channel multi-layer convolutional and pooling network is constructed.
   2. A Gated Recurrent Unit (GRU) is introduced to optimize the network model, reducing the risk of gradient explosion in deep networks.
   3. An attention mechanism is incorporated to address the forgetting issue that may arise in GRU during backpropagation.
   4. A global max pooling layer is used to prevent overfitting during training.
   5. Finally, a Softmax layer is applied for classification output, completing the fault diagnosis model setup.
3. Model Training and Validation:
   1. The training dataset is fed into the improved 1D-CNN model for forward propagation and loss function computation.
   2. The validation set is then input into the model for performance evaluation.
   3. The output results are compared with the ground truth, and the error is calculated.
   4. If the stopping criterion is met, the training terminates, and the model is saved.
   5. If the criterion is not met, error backpropagation updates the parameters of each network layer, and training continues until the termination condition is satisfied.
4. Model Testing and Evaluation:
   1. The testing dataset is fed into the trained model.
   2. The model outputs a confusion matrix for the test set to validate the diagnostic results.

3.1 Experimental verification and result analysis based on public datasets

To verify the effectiveness of the improved one-dimensional convolutional neural network (1D-CNN) algorithm proposed in this paper, the Case Western Reserve University (CWRU) bearing dataset was selected for experimental analysis. This dataset is a widely used benchmark in the field of rolling bearing fault diagnosis and can effectively evaluate the reliability and accuracy of mechanical fault diagnosis algorithms.

In the experiment, a two-horsepower motor was used for fault testing. Single-point faults of different diameters were introduced at various locations of the motor, and vibration signals were collected using sensors mounted on the motor housing. The detailed configuration of the fault testing platform is shown in Fig 14.

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Fig 14. Experimental platform for CWRU bearing vibration dataset.

Photograph of the experimental setup for collecting bearing vibration data. [Source: Case Western Reserve University Bearing Data Center].

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The CWRU dataset covers various fault types and operating conditions, including inner ring faults, outer ring faults, rolling element faults, and normal states. To comprehensively evaluate the performance of the model, the signal sampling frequency was set to 12 kHz, and the rotational speed was 1797 r/min. Data from 10 categories were selected, including one class of normal vibration signals and nine classes of fault vibration signals. The fault defect diameters were 0.1778 mm, 0.3556 mm, and 0.5334 mm. The dataset sample length was set to 1200, and the dataset was divided into training, testing, and validation sets in a ratio of 7:2:1. Table 1 presents the detailed classification of bearing data labels, including the bearing condition, fault diameter, number of samples, and corresponding class labels.

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Table 1. Classification of bearing data labels.

https://doi.org/10.1371/journal.pone.0327206.t001

Using TensorFlow-GPU (version 2.0.0) and Python (version 3.6.3), we executed and implemented the model. The computer processor (CPU) used was an Intel(R) Core(TM) i7-13620H, and the graphics card (GPU) was an NVIDIA GeForce RTX 4060.To better demonstrate the advantages of the model, we optimized the performance of the network model. The parameter metrics are shown in Table 2.

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Table 2. Model parameter indicators.

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The learning rate controls the size of parameter updates in each step, ensuring a more stable and reliable training process. Dropout randomly deactivates certain neuron activations during training, helping to prevent overfitting. The ReLU activation function is used to measure model performance and guide optimization, further mitigating overfitting. Batch size determines the number of samples used to update model parameters during iterative training, enhancing the model’s generalization ability and stability. Properly setting the number of iterations effectively trains a high-performance model with strong generalization capabilities.

The preprocessed CWRU public dataset underwent noise reduction before being divided into training, testing, and validation sets in a 7:2:1 ratio. The training set samples were fed into the improved parallel 1D-CNN model for training, where forward propagation was performed to compute the loss function. The validation set samples were then input into the model for validation. For comparison, single-channel large-kernel CNN, single-channel small-kernel CNN, and parallel 1D-CNN models were selected alongside the proposed algorithm in this chapter. The accuracy variation curves of the validation sets for each model during training are shown in Fig 15.

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Fig 15. Accuracy chart of each model validation set.

The improved parallel 1D-CNN model achieves higher accuracy and greater stability than other models. The inset zooms in on accuracy fluctuations in the later training iterations.

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From Fig 15, it can be observed that the accuracy variation curve of the dual-channel CNN network is more stable and achieves higher accuracy compared to the single-channel CNN network. Compared to the parallel 1D-CNN, the proposed algorithm in this chapter converges faster, reaching a stable accuracy after 20 training epochs, which verifies the model’s accuracy and stability. The loss variation curves of the validation sets for each model during training are shown in Fig 16.

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Fig 16. Loss rate graph of each model validation set.

The improved parallel one-dimensional CNN model achieves a faster decrease in loss and maintains higher stability compared to other models. The inset highlights detailed loss variations during the final training stages.

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The segmented test set samples were input into the trained improved parallel 1D-CNN fault diagnosis model for validation, and the resulting test set accuracy is shown in Fig 17(a). From the figure, the horizontal axis represents the number of actual labels, while the vertical axis represents the number of predicted labels. Based on the label correspondence, it can be observed that the proposed algorithm in this chapter achieves high accuracy on the test set. The same dataset samples were then tested using the parallel 1D-CNN, single-channel small-kernel CNN, and single-channel large-kernel CNN models. The corresponding results are shown in Figs 17(b), 17(c), and 17(d), respectively. By comparison, it is evident that the proposed model in this chapter achieves significantly higher accuracy in diagnosing the test set than the other models.

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Fig 17. Accuracy of test sets for each model, specifically.

The matrices show classification performance for four models: (a) Improved Parallel 1D-CNN, (b) Single-Channel Large-Kernel CNN, (c) Parallel 1D-CNN, and (d) Single-Channel Small-Kernel CNN. Each matrix presents the counts of correct and incorrect predictions across classes, enabling direct comparison of model accuracy.

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By organizing the confusion matrices mentioned above, the accuracy of different fault diagnosis models was obtained, as shown in Table 3. It can be seen that the proposed algorithm in this paper achieves an accuracy of 99.62%, which is higher than that of other models. Compared to the small-kernel CNN, the accuracy of the large-kernel CNN decreases. This is mainly because small kernels have fewer parameters, enabling better parameter sharing, making the model more compact, and reducing the risk of overfitting. The dual-channel network achieves higher accuracy than the single-channel network, confirming that the dual-channel structure can more comprehensively capture feature information, thereby enhancing the model’s recognition accuracy. The proposed algorithm in this chapter shows a significant improvement in accuracy compared to the dual-channel convolutional neural network model. This indicates that the proposed algorithm can fully utilize feature information, reduce training time while improving model accuracy, and verify the effectiveness of the model.

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Table 3. Accuracy of test set for different fault diagnosis models.

https://doi.org/10.1371/journal.pone.0327206.t003

To further verify the generalization ability of the proposed method, in addition to using the publicly available CWRU dataset, experiments were also conducted on the publicly available Southeast University dataset. This dataset includes a wider range of fault types and operating conditions, allowing for a more comprehensive evaluation of the model’s performance. The experimental results show that the proposed model achieves an average diagnostic accuracy of 99.62% on the CWRU dataset and an average diagnostic accuracy of 98.7% ± 0.3% (95% confidence interval) on the other public dataset. This represents a 5.2 percentage point improvement over the traditional CNN-GRU hybrid model, further demonstrating its effectiveness and generalization ability.

3.2 Experimental verification and result analysis based on measured vibration signals from a test bench

3.2.1 Performance comparison of representative conventional methods.

To verify the effectiveness of the proposed improved method, experimental analysis was conducted using vibration signal data measured from a test rig. The test rig was designed to simulate real operating conditions, collecting vibration signals under different operational states to provide real experimental data for algorithm validation. The bearing vibration dataset experimental platform is shown in Fig 18.

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Fig 18. The self-built bearing vibration data acquisition experimental platform.

The platform comprises a motor, bearing housing, and data acquisition system, designed to capture vibration signals from bearings under various operating conditions.

https://doi.org/10.1371/journal.pone.0327206.g018

Deep learning network models often require a large number of data samples for training. In this section, an overlapping sampling method is used for data augmentation. Data augmentation not only expands the dataset but also helps the model learn different representations of the data, reducing overfitting and improving the model’s adaptability to new data. The resampling schematic diagram is shown in Fig 19.

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Fig 19. Schematic diagram of resampling.

Shows the extraction of training samples from a signal using a defined sampling region and stride to generate distinct samples.

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The vibration signal data collected under four different conditions of rolling bearings were selected for the fault diagnosis experiment. The denoised and preprocessed vibration data were resampled and used as input samples for the fault diagnosis model. The dataset was divided into training, testing, and validation sets in a 7:2:1 ratio. The network was built using the same configuration and optimization approach as when validating the public dataset. The iterative curves of the model’s accuracy and loss rate are shown in Fig 20.

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Fig 20. Analog signal accuracy and loss rate.

Shows the changes in accuracy and loss over training iterations, reflecting the model’s learning progress and generalization performance.

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As shown in Fig 19, after 25 iterations of training, both the model’s accuracy and loss rate reach a stable state, indicating that the model effectively extracts features from the data samples. Moreover, the curve exhibits a high degree of fitting, suggesting that the model does not suffer from severe overfitting or underfitting, thereby confirming the accuracy of the proposed method. To further verify the superiority of the proposed algorithm in diesel engine fault diagnosis, the same three comparison methods from the previous section were selected as benchmark models. The accuracy and loss rate of the validation set during training for each model are shown in Figs 21 and 22, respectively.

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Fig 21. Accuracy chart of each model validation set.

The improved parallel 1D-CNN model achieves higher accuracy and greater stability compared to others. The inset highlights accuracy fluctuations in later training stages.

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Fig 22. Loss rate graph of each model validation set.

The improved parallel 1D-CNN model exhibits a faster loss decline and more stable performance than other models. The inset details loss fluctuations in later training stages.

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As shown in Figs 20 and 21, the curve of the single-channel 1D CNN is highly unstable, and its accuracy is lower than that of the dual-channel 1D CNN. The proposed algorithm demonstrates a faster convergence speed with minimal waveform fluctuations, indicating good stability. The test results of each model are presented in Table 4. The proposed fault diagnosis method achieves the highest diagnostic accuracy of 99.56%, which represents a 0.62% improvement over the parallel 1D CNN. Additionally, it shows a significant enhancement compared to the single-channel large-kernel CNN and the single-channel small-kernel CNN. These results confirm that the proposed algorithm has high fault identification accuracy, making it better suited for vibration signal monitoring.

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Table 4. Accuracy and loss values of each model.

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3.2.2 Comparative evaluation with recent state-of-the-art models.

To comprehensively assess the diagnostic performance and engineering applicability of the proposed model, we conducted a comparative study with five recently published and representative fault diagnosis approaches. These methods encompass diverse modeling paradigms, including probabilistic inference, image-based enhancement, adversarial learning, and interpretable hybrid architectures, ensuring a broad and robust benchmarking foundation.

The selected methods include:

Bayesformer: A Bayesian variational Transformer that models attention weights as stochastic distributions, enabling infinite ensemble learning and enhancing generalization capacity.

HMM-based health assessment model: Utilizes a hidden Markov process to model the stochastic degradation transitions of rolling bearings, suitable for sequential fault evolution tracking.

GADF + DSCM: Converts raw time-series signals into Gramian Angular Difference Field (GADF) images, which are then processed by a dynamic self-calibrated convolutional module (DSCM) to enhance spatial discriminability.

CWMS-GAN: A generative adversarial network that integrates continuous wavelet transform and multi-scale convolutional attention, designed to support fault diagnosis under limited-sample conditions.

QNN + BiLSTM: Combines quadratic neural networks (QNN) with bidirectional long short-term memory (BiLSTM) to enhance the nonlinear modeling and interpretability of dynamic signals.

All comparative methods were re-implemented according to the configurations reported in their respective publications. A summary of their architectural structures is as follows:

Bayesformer: Composed of three encoder blocks with Bayesian multi-head attention, a feedforward network with 256 units, and residual connections, followed by two fully connected classification layers.

HMM: Implements a three-state discrete hidden Markov model with symbol-based emissions, trained using the Baum-Welch algorithm and inferred via Viterbi decoding.

GADF + DSCM: Input signals are transformed into 64 × 64 GADF images and passed through two DSCM layers (each with 3 × 3 kernels and 32 filters), followed by global pooling and a dense output layer.

CWMS-GAN: The generator comprises three convolutional blocks with continuous wavelet transform kernels, while the discriminator consists of two convolutional layers (32 and 64 filters), an attention mechanism, and a softmax classification head.

QNN + BiLSTM: Contains two QNN layers with 128 quadratic neurons each, followed by two BiLSTM layers with 64 hidden units per direction, accompanied by dropout and softmax layers for classification.

Proposed Method: Employs a parallel dual-branch architecture. The primary branch consists of two 1D convolutional layers (32 and 64 filters), a GRU layer with 64 units, and an attention module. The auxiliary branch includes a shallow convolution layer, a pooling layer, and a fully connected layer. Features from both branches are concatenated and passed through two dense layers for final prediction.

All methods were trained on the same dataset with consistent experimental configurations wherever possible. The learning rate, batch size, number of epochs, and optimizer used for each model are summarized in Table 5.

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Table 5. Training configurations of the compared fault diagnosis methods.

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All input signals were standardized using Z-score normalization. Each sample consisted of a fixed length of 1200 data points. For the GADF + DSCM method, the one-dimensional signals were converted into two-dimensional Gramian Angular Difference Field (GADF) images as input. The output for all methods comprised six fault classes encoded using one-hot vectors. The classification tasks uniformly employed the cross-entropy loss function.

The experimental environment included an Intel(R) Core(TM) i7-13620H processor, an NVIDIA GeForce RTX 4060 GPU, Windows 11 operating system, Python 3.6.3 programming language, and TensorFlow-GPU 2.0.0 deep learning framework. The dataset utilized was the CWRU bearing dataset, encompassing fault categories such as normal, inner race fault, and outer race fault, among others. The data were partitioned into 70% training, 20% validation, and 10% testing subsets using stratified random sampling.

Three evaluation metrics were employed to comprehensively assess classification performance: Accuracy, representing the overall correct classification rate; Precision, indicating the model’s capability to avoid false positives; and F1-score, the harmonic mean of precision and recall, which balances fault detection and misclassification. All metrics were calculated as macro-averaged values to account for class imbalance.

Table 6 presents the performance of each method with respect to accuracy, precision, and F1-score. As shown in Table 6, the proposed method achieved accuracy, precision, and F1-score values of 99.68%, 99.65%, and 99.67%, respectively, significantly outperforming the comparative approaches. The improvements achieved by the proposed method are statistically significant (p < 0.05) according to a paired t-test over 30 repeated trials.

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Table 6. Performance comparison of different fault diagnosis methods in terms of Accuracy, Precision, and F1-score.

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Fig 23 illustrates a bar chart comparing the accuracy and F1-score of the six methods, clearly demonstrating the superior overall classification accuracy and stability of the proposed model. Fig 24 depicts a radar chart encompassing accuracy, precision, and F1-score, reflecting the comprehensiveness and balance of each model’s performance. Notably, the radar chart for the proposed method approximates a near-perfect circle, indicating consistent excellence across all evaluated metrics and highlighting its robustness and practical applicability in fault diagnosis. Table 7 compares the optimal application scenarios and limitations of each method, further highlighting the applicability of the proposed approach in multi-fault, noisy, and real-time contexts, while also noting its potential need for parameter tuning when deployed on lightweight devices.

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Table 7. Comparative analysis of application scenarios and limitations of different fault diagnosis methods.

https://doi.org/10.1371/journal.pone.0327206.t007

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Fig 23. Accuracy and F1-score comparison among six advanced models.

The proposed model consistently outperforms five recent methods across both metrics.

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Fig 24. Radar chart of accuracy, precision, and F1-score.

The chart highlights the superior performance of the proposed model across all three metrics.

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The proposed model demonstrates consistently superior performance across both metrics compared with five recent methods.

The proposed method exhibits a well-balanced and dominant performance across all key indicators.

The superiority of the proposed method can be attributed to several key factors: the dual-branch architecture effectively integrates time-domain and frequency-domain features, outperforming approaches based solely on GADF or pure time-series networks; the incorporation of GRU and attention modules enhances sequence learning capabilities by focusing on critical regions, thereby avoiding underfitting common in simpler models and overfitting associated with more complex ones; the attention mechanism significantly improves discriminative power under noisy conditions, demonstrating marked advantages over HMM and CWMS-GAN in low signal-to-noise ratio environments. Additionally, although CWMS-GAN excels in small-sample scenarios and Bayesformer exhibits strong generalization ability, both entail higher computational costs. The proposed method achieves a favorable balance between complexity and performance, making it well-suited for real-time deployment in industrial settings.