Physics-informed end-to-end fault diagnosis for rolling bearings under time-varying speed and variable load via multi-constraint IVMD and dual-branch sparse attention network

2.1.1. Multi-constraint objective function

Based on the original VMD objective function, MC-IVMD introduces three physics-informed hard constraints and one data-driven residual constraint, and reconstructs the objective function to realize the dual-driven signal decomposition of bearing fault mechanism and data feature. The multi-constraint objective function of MC-IVMD is defined as:

where $x\left(t\right)$ is the original VL-TV vibration signal of rolling bearings, $u_k\left(t\right)$ and ${\omega }_k$ represent the $k$-th intrinsic mode function (IMF) and its corresponding center frequency, respectively; $\delta \left(t\right)$ is the Dirac function, $*$ denotes the convolution operation, and ${\partial }_t$ is the partial derivative with respect to time; $r\left(t\right)$ is the multi-scale residual signal fused by iteration stage residuals, ${\lambda }_1$, ${\lambda }_2$ , ${\lambda }_3$ are the weight coefficients of each constraint term, determined by grid search and cross-validation on the Ottawa dataset; $L_{sparse}$ is the fault impact sparsity constraint (L1 norm), which forces the decomposed IMFs to retain the impulsive characteristics of rolling bearing faults and suppress the continuous noise components; $L_{ortho}$ is the modal orthogonality constraint, defined as the sum of correlation coefficients between adjacent IMFs, which eliminates modal redundancy and aliasing from the decomposition process; $L_{phy}$ is the fault characteristic frequency prior constraint, which limits the update range of center frequency ${\omega }_k$ within the theoretical fault characteristic frequency interval (calculated by bearing structural parameters and rotational speed), avoiding frequency drift caused by strong noise.

2.1.2. Adaptive selection of modal number $\mathit{K}$

The modal number $K$ is a key hyperparameter of VMD, and its artificial setting is the main cause of over-decomposition/under-decomposition in practical applications. MC-IVMD designs a dynamic adaptive selection strategy for $K$ based on Dispersion Entropy (DE) and Kurtosis Gradient (KG), which realizes the non-manual setting of the optimal modal number and adapts to different fault types and signal complexity. The specific implementation steps are as follows:

1. Initial range determination of $K$: Calculate the dispersion entropy of the original VL-TV vibration signal $x\left(t\right)$. DE is a quantitative index to characterize the complexity of time series, and the higher the DE value, the more complex the signal (i.e., the richer the fault and noise components). The initial range of $K$ ($K_{min}\sim K_{max}$) is determined according to the DE value: when $DE<0.6$, $K_{min}=2$, $K_{max}=5$; when $0.6\le DE<0.8$, $K_{min}=3$, $K_{max}=7$; when $DE\ge 0.8$, $K_{min}=5$, $K_{max}=10$.

2. Adaptive termination condition of decomposition: Decompose the original signal with the increasing modal number from $K_{min}$ to $K_{max}$. For each newly added IMF, calculate its kurtosis value and the kurtosis gradient between the current and previous IMF. The kurtosis gradient is defined as:

where $Kurt\left(K\right)$ is the kurtosis of the $K$-th IMF. When the kurtosis gradient $KG\left(K\right)<\epsilon$ (the threshold $\epsilon =$ 0.05 is determined by experimental verification), the decomposition is terminated, and the current $K$ is the optimal modal number of the signal.

This strategy ensures that the decomposed IMFs contain the richest fault features, and fundamentally overcomes the defect of artificial parameter setting in traditional VMD and its improved versions.

2.1.3. Modal aliasing suppression and multi-scale residual feedback

(1) Modal Aliasing Suppression Module. A modal similarity detection and fusion module is added in the modal update stage of MC-IVMD to eliminate modal aliasing from the root. The specific implementation is: after each iteration of modal update, calculate the Pearson correlation coefficient $CC\left(u_i,u_j\right)$ between any two adjacent IMFs $u_i\left(t\right)$ and $u_j\left(t\right)$:

where $T$ is the length of the signal, ${\bar{u}}_i$ and ${\bar{u}}_j$ are the mean values of $u_i\left(t\right)$ and $u_j\left(t\right)$, respectively. When $CC\left(u_i,u_j\right)>0.8$ (the threshold is determined by orthogonality index optimization), the two IMFs are adaptively fused by weighted averaging (the weight is the kurtosis value of each IMF), and the fused IMF is retained to replace the original two IMFs. This operation effectively merges the redundant modal components caused by noise and speed variation, and eliminates modal aliasing.

(2) Multi-Scale Residual Feedback Mechanism. The original IVMD adopts a single-scale residual feedback mechanism, which can only feed back the residual of the current iteration stage to the modal update. MC-IVMD upgrades it to a multi-scale residual feedback mechanism: collect the residual signals of the early, middle and late stages of iteration (divided by the total iteration number), assign different weights to the residuals of different stages according to the fault impact feature contribution (the weight of the late iteration residual is higher, because it contains more refined fault information), and fuse the multi-scale residuals as $r\left(t\right)$ in the objective function. The fused residual is fed back to the modal update process, which not only accelerates the convergence speed of the algorithm, but also significantly improves the extraction ability of weak fault impact features under low signal-to-noise ratio (SNR).

2.1.4. Adaptive adjustment of penalty factor $\mathit{\beta }$ and step size $\mathit{\tau }$

On the basis of the original IVMD, MC-IVMD optimizes the adaptive adjustment rules of the quadratic penalty factor $\beta$ and Lagrange multiplier step size $\tau$ by combining the fault impact feature weight of IMFs, making the parameter adjustment more adaptive to the VL-TV fault signals of rolling bearings. The adaptive update formulas are as follows:

where ${\beta }^{\left(n+1\right)}$ and ${\tau }^{\left(n+1\right)}$ are the penalty factor and step size of the $\left(n+1\right)$-th iteration, respectively; ${\beta }_0$ is the initial penalty factor, $\alpha$ is the decay rate, ${\tau }_0$ is the initial step size; $r\left(n\right)$ is the residual of the $n$-th iteration, $r_0$ is the initial residual, rmax is the maximum allowable residual; $w_k$ is the fault impact feature weight of the $k$-th IMF, calculated by the kurtosis value of the IMF (the higher the kurtosis, the larger the $w_k$), which makes the parameter adjustment focus on the IMFs with rich fault features.

2.1.5. Decomposition process and convergence proof of MC-IVMD

(1) Decomposition Process of MC-IVMD. The complete decomposition process of MC-IVMD for rolling bearing VL-TV vibration signals is as follows:

Step 1: Input initialization.

Input the original VL-TV vibration signal $x\left(t\right)$, set the initial parameters ${\beta }_0$, ${\tau }_0$, $\alpha$, the weight coefficients ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$ of the constraint terms, and the convergence accuracy ${\epsilon }_{conv}$.

Step 2: Determine the optimal modal number.

Calculate the dispersion entropy of $x\left(t\right)$ to determine the initial range of $K$, and obtain the optimal modal number $K_{opt}$ through the kurtosis gradient adaptive termination condition.

Step 3: Iteration initialization.

Set the iteration counter $n=0$, initialize the modal functions ${\left\{u_k^{\left(0\right)}\left(t\right)\right\}}_{k=1}^{K_{opt}}$, center frequencies ${\left\{{\omega }_k^{\left(0\right)}\right\}}_{k=1}^{K_{opt}}$, Lagrange multiplier ${\lambda }^{\left(0\right)}\left(t\right)$, and calculate the initial residual $r^{\left(0\right)}\left(t\right)$.

Step 4: Stop condition judgment.

Modal and center frequency update; Update the modal functions $\left\{u_k^{\left(n+1\right)}\left(t\right)\right\}$ and center frequencies $\left\{{\omega }_k^{\left(n+1\right)}\right\}$ according to the multi-constraint objective function, and limit the center frequency within the theoretical fault characteristic frequency range by $L_{phy}$.

Step 5: Modal aliasing suppression.

Calculate the correlation coefficient between adjacent IMFs, and fuse the IMFs with high similarity to eliminate modal aliasing.

Step 6: Parameter and multiplier update.

Update the penalty factor ${\beta }^{\left(n+1\right)}$and step size ${\tau }^{\left(n+1\right)}$ according to the adaptive adjustment rules, and update the Lagrange multiplier ${\lambda }^{\left(n+1\right)}\left(t\right)$ with the updated step size.

Step 7: Multi-scale residual fusion.

Collect the residuals of different iteration stages, assign weights and fuse to obtain the multi-scale residual $r^{\left(n+1\right)}\left(t\right)$.

Step 8: Convergence judgment.

If the convergence condition is satisfied:

stop the iteration and output the decomposed IMFs ${\left\{u_k\left(t\right)\right\}}_{k=1}^{K_{opt}}$; otherwise, set $n=n+1$ and return to Step 4.

(2) Convergence Proof of MC-IVMD. Assume that the residual sequence $\left\{r^{\left(n\right)}\right\}$ of MC-IVMD satisfies $0<r^{\left(n+1\right)}<r^{\left(n\right)}$ for all $n\ge 0$, which is derived from the adaptive update rules of $\beta$ and $\tau$ and the multi-scale residual feedback mechanism. Since the residual sequence is bounded below (greater than 0) and non-increasing, according to the monotone convergence theorem, $\left\{r\left(n\right)\right\}$ converges to a non-negative limit $r^{*}=\underset{n\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{r}^{\left(n\right)}$.

Suppose $r^{*}>0$, then the penalty factor $\beta$ and step size $\tau$ will continue to update according to the adaptive rules, and the modal functions and center frequencies will also continue to adjust, which leads to $r^{\left(n+1\right)}<r^{\left(n\right)}$ and $r^{\left(n\right)}\to 0$, contradicting $r^{*}>0$. Hence, $r^{*}=0$, which means the residual sequence converges to 0, and the modal function sequence $\left\{u_k^{\left(n\right)}\right\}$ and center frequency sequence $\left\{{\omega }_k^{\left(n\right)}\right\}$ converge to the optimal solution. This proves the linear convergence of MC-IVMD.

2.1.6. Performance evaluation of MC-IVMD

To verify the decomposition performance of MC-IVMD, 50 random signals were selected from the Ottawa VL-TV bearing dataset (including the normal state, inner-race fault, outer-race fault, and multi-fault coupling), and standard VMD and original IVMD were used for comparative decomposition experiments [20]. The orthogonality index (OI) and the kurtosis of the first fault-sensitive IMF were adopted as evaluation indices, and the corresponding average values and standard deviations are reported in Table 1.

It can be seen from Table 1 that MC-IVMD has significant advantages over the standard VMD and original IVMD in all evaluation indexes: the OI is increased to 0.98±0.01, which means the decomposed IMFs have almost no redundancy and aliasing. The kurtosis of the first sensitive IMF is increased by 14.6 % compared with the original IVMD, which indicates that MC-IVMD can extract fault impact features more effectively. The average convergence iteration number is reduced to 26, which is 16.1 % lower than the original IVMD, realizing the improvement of both decomposition accuracy and convergence speed.

2.2.1. Forget gate

The forget gate determines the proportion of the historical cell state information to be retained or discarded, and its output value is between 0 and 1 (0 means discard all information, 1 means retain all information). The calculation formula is:

where $f_t$ is the output of the forget gate at time step $t$. $\sigma$ is the Sigmoid activation function, which maps the input to the interval (0, 1). $W_f$ is the weight matrix of the forget gate, $b_f$ is the bias term. $h_{t-1}$ is the hidden state of the LSTM at the previous time step $t-1$, $x_t$ is the input of the LSTM at the current time step $t$. $\left[h_{t-1},x_t\right]$ denotes the concatenation of the historical hidden state and the current input.

2.2.2. Input gate and candidate cell state

The input gate determines the proportion of the current input information to update the cell state, and the candidate cell state generates the new information to be added to the cell state. Their calculation formulas are:

where $i_t$ is the output of the input gate at time step $t$, ${\tilde{C}}_t$ is the candidate cell state at time step $t$. $W_i$ and $W_C$ are the weight matrices of the input gate and candidate cell state, respectively. $b_i$ and $b_C$ are the bias terms of the input gate and candidate cell state, respectively. $\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}$ is the hyperbolic tangent activation function, which maps the input to the interval (–1, 1) to normalize the candidate cell state.

2.2.3. Cell state update

The cell state is the core of the LSTM network, which stores the long-term temporal information of the sequence signal. The update of the cell state is jointly determined by the forget gate, input gate and candidate cell state, and the calculation formula is:

where $C_t$ is the cell state of the LSTM at time step $t$, $C_{t-1}$ is the cell state at the previous time step $t-1$. $\odot$ denotes the element-wise multiplication (Hadamard product), which realizes the weighted fusion of historical cell state and new candidate state.

2.2.4. Input gate and candidate cell state

The output gate determines the proportion of the current cell state information to be output as the hidden state, and the hidden state is the final output of the LSTM at the current time step, which contains the temporal information of the current and historical input signals. Their calculation formulas are:

where $o_t$ is the output of the output gate at time step $t$, $h_t$ is the hidden state at time step $t$; $W_o$ is the weight matrix of the output gate, $b_o$ is the bias term.

The LSTM network can effectively capture the long-term temporal dependencies of the rolling bearing VL-TV vibration signals, which provides a good foundation for the subsequent construction of the dual-branch sparse attention network. In this paper, the LSTM is embedded with a gated sparse attention mechanism to form the GSAM-LSTM, which further improves the ability of the network to focus on key fault features and suppress noise.

2.3.1. Core design of CLSAM++

The overall design of CLSAM++ adheres to the principles of physics-informed feature extraction, dual-branch complementary learning and lightweight industrial adaptation. The local feature branch is designed to extract the local impact features of rolling bearing faults (a typical physical characteristic of bearing faults), and the temporal dependency branch is designed to capture the long-term temporal dependencies of VL-TV signals. The cross-layer fusion module realizes the bidirectional guidance and fusion of the two types of features, and the adaptive dropout module solves the overfitting problem without losing fault features. The structural framework of CLSAM++ is shown in Fig. 1.

Fig. 1The structural framework of CLSAM++

2.3.2. Gated sparse attention mechanism (GSAM)

The original SAM simply retains the top-k weight values through the Top-k mask, which is easy to select noise-dominated nodes under strong noise and VL-TV conditions, leading to the deviation of feature focus. Based on this, the Gated Sparse Attention Mechanism (GSAM) is proposed by combining the gate control mechanism of LSTM and the sparse attention mechanism, and embedding it into the hidden state update path of LSTM to form GSAM-LSTM. GSAM dynamically shields the noise-dominated temporal nodes according to the bearing fault impact timing prior, and only retains the key temporal nodes containing fault features, which significantly improves the generalization of the sparse attention mechanism under complex conditions. The mathematical principle and implementation steps of GSAM are as follows:

1) Feature concatenation: Concatenate the hidden state $h_t$ of LSTM at time step $t$ and the current input $x_t$ to form the input feature $s_t$ of GSAM:

2) Feature projection: Map the concatenated feature $s_t$ to a low-dimensional space through a fully connected layer, and use the tanh function as the activation function to obtain the intermediate feature representation $e_t$:

where $W_e$ is the weight matrix of the fully connected layer, $b_e$ is the bias term.

3) Sparse attention weight calculation: Map the intermediate feature representation $e_t$ to the sparse attention weight score through another fully connected layer, and obtain the sparse attention weight ${\alpha }_t$ through the gated sparse activation function (combining the Sigmoid gate and softmax with sparse constraints):

where $W_{\alpha }$, $W_s$ are weight matrices, $b_{\alpha }$, $b_s$ are bias terms. $\sigma$ is the Sigmoid gate function, which filters the noise weight values (sets the weight values less than 0.1 to 0). $softma{x}_k$ is the softmax function with Top-k sparse constraints, which only retains the top-k weight values and normalizes them.

4) Hidden state update with attention weighting: Update the hidden state of LSTM with the sparse attention weight ${\alpha }_t$, and the weighted hidden state $h_t^{*}$ is:

The weighted hidden state $h_t^{*}$ is used as the output of GSAM-LSTM, which contains only the key temporal features of bearing faults and suppresses the noise features.

In this paper, the sparse constraint $k$ of GSAM is set to 50 (determined by grid search), which is the optimal value for the rolling bearing VL-TV signal feature extraction.

2.3.3. Local feature branch (LFB)

The Local Feature Branch (LFB) is designed to extract the local impact features and multi-scale spatial features of rolling bearing faults, which is the physical basis of bearing fault diagnosis. LFB adopts a stacked structure of Depthwise Separable Convolution (DSC) and Atrous Convolution (AC) with 1D kernel size, which realizes the lightweight design of the branch while expanding the receptive field and extracting multi-scale local features. The core design of LFB is as follows:

1) Depthwise Separable Convolution (DSC): DSC decomposes the traditional 1D convolution into depthwise convolution and pointwise convolution, which reduces the number of network parameters and computation amount by more than 80 % under the same convolution kernel size and number of filters, realizing the lightweight of the branch.

2) Atrous Convolution (AC): AC expands the receptive field of the convolution kernel without increasing the number of parameters and computation amount, which can effectively extract the multi-scale local impact features of rolling bearing VL-TV signals (different fault types and severities correspond to different scales of impact features).

3) Branch structure setting: LFB includes four convolution blocks, and the number of filters of each block is set to 64, 128, 256, 512 in turn; the convolution kernel size is 3 (1D), the stride is 1, and the padding is set to “Same” to keep the feature dimension consistent after convolution; each convolution block is composed of “DSC + AC + Batch Normalization (BN) + ReLU”, which accelerates the network training and avoids overfitting.

2.3.4. Temporal dependency branch (TDB)

The Temporal Dependency Branch (TDB) is constructed based on GSAM-LSTM, which is designed to capture the long-term temporal dependencies of rolling bearing VL-TV signals (the fault impact features of bearings show obvious temporal regularity under time-varying speed conditions). The core design of TDB is as follows:

1) GSAM-LSTM stacking: TDB includes 2 stacked GSAM-LSTM layers, the hidden dimension of each layer is 256 (determined by grid search), which balances the feature learning ability and the number of parameters. The input of TDB is the local feature map extracted by LFB, which is reshaped to the sequence feature format suitable for LSTM input.

2) Feature normalization: Batch Normalization (BN) is added after each GSAM-LSTM layer to normalize the temporal features, which accelerates the convergence of the branch and improves the generalization ability.

3) Sparse attention constraint: The sparse constraint k of GSAM in each GSAM-LSTM layer is set to 50, which is consistent with the optimal value of the rolling bearing VL-TV signal feature extraction.

2.3.5. Attention-guided cross-layer fusion module (AG-CLFM)

The Attention-Guided Cross-Layer Fusion Module (AG-CLFM) is the core module of CLSAM++ to realize the bidirectional interaction and fusion of local features and temporal features. AG-CLFM is added between each corresponding layer of LFB and TDB (4 fusion modules in total, corresponding to 4 convolution blocks of LFB and the hidden layers of TDB), which realizes the layer-by-layer fusion and guidance of the two types of features. The specific implementation of AG-CLFM is as follows:

1) Attention weight calculation: For the local feature map $F_L^i$ of the $i$-th layer of LFB and the temporal feature map $F_T^i$ of the corresponding layer of TDB, calculate the attention weights $w_L^i$ and $w_T^i$ of the two types of features respectively through the Sigmoid activation function, which represents the contribution of the two types of features to the fault diagnosis at the current layer:

where $W_L$, $W_T$ are weight matrices, $b_L$, $b_T$ are bias terms.

2) Feature normalization and fusion: Normalize the attention weights $w_L^i$ and $w_T^i$ to make their sum equal to 1, and then fuse the local feature map and temporal feature map through element-wise multiplication and addition:

where ${\widehat{w}}_L^i$ and ${\widehat{w}}_T^i$ are the normalized attention weights, $F_F^i$ is the fused feature map of the $i$-th layer.

3) Feature feedback: Feed the fused feature map $F_F^i$ back to the next layer of LFB and TDB as the input, realizing the bidirectional guidance of local features and temporal features in the learning process.

AG-CLFM makes the shallow local impact features provide “fault feature anchor points” for the deep temporal feature learning, and the deep temporal features provide “temporal regularity constraints” for the shallow local feature extraction, which effectively solves the problem of disconnection between local and temporal feature extraction in the original CLSAM.

2.3.6. Adaptive dropout module (ADM)

Traditional random Dropout randomly discards part of the feature nodes in the network training process, which is easy to discard the fault feature nodes with small weight under complex VL-TV and strong noise conditions, leading to the loss of key fault features. To address this defect, the Adaptive Dropout Module (ADM) is proposed, which discards the feature nodes according to the sparse attention weight of GSAM, and only inactivates the noise-dominated feature nodes (attention weight < 0.1), avoiding the loss of fault feature nodes. The implementation of ADM is as follows:

1) Feature node screening: For the fused feature map $F_F^i$ of each layer, screen the feature nodes according to the sparse attention weight ${\alpha }_t$ of GSAM, and mark the nodes with ${\alpha }_t<0.1$ as noise-dominated nodes.

2) Adaptive inactivation: In the network training process, only inactivate the marked noise-dominated feature nodes (set the output to 0), and retain all fault feature nodes with ${\alpha }_t\ge 0.1$.

3) Dropout rate setting: The adaptive dropout rate of ADM is dynamically adjusted with the training process (the dropout rate decreases from 0.3 to 0.1 with the increase of training epochs), which balances the overfitting suppression and feature retention.

ADM effectively solves the overfitting problem of the network under small sample and complex conditions, and ensures that the key fault features are not lost, which significantly improves the generalization ability of CLSAM++.

2.3.7. Key hyperparameters of CLSAM++

The key hyperparameters of CLSAM++ are optimized through grid search and 5-fold cross-validation on the Ottawa VL-TV bearing dataset, and the optimal values are determined to ensure the best performance of the network in fault feature extraction and classification. The key hyperparameters and their optimal values are shown in Table 2.

2.3.8. Output and feature mapping of CLSAM++

The output of the four layers of AG-CLFM is fused to form the global feature map $F_G$ of the VL-TV signal, which contains both the local fault impact features and long-term temporal features of the rolling bearing, and realizes the complementary and fusion of the two types of features. The global feature map $F_G$ is flattened and input into the fully connected layers of CLSAM++:

1) The first fully connected layer (FC1, hidden dimension 1024) projects the global feature map into a high-dimensional fault feature space, enhancing the feature representation ability of the network.

2) The second fully connected layer (FC2, output dimension equal to the number of fault classes) maps the high-dimensional fault features to the fault class probability space, and outputs the probability of each fault class through the Softmax activation function.

The output of FC2 is the final fault classification result of CLSAM++, which provides the basis for the end-to-end joint optimization of the whole fault diagnosis framework.

4.1.1. Dataset composition for experiment

7 sub-datasets were selected from the 36 datasets to form a mixed dataset for core experimental validation (Section 4.2), with the selection criterion of covering all key health states and VL-TV modes to ensure the model’s generalization ability to complex industrial scenarios. The specific composition is as follows:

1) Normal state (2 datasets): H-A (acceleration) and H-C (transient acceleration during deceleration), representing steady and transient speed changes of healthy bearings under variable load;

2) Inner race fault (3 datasets): I-A (acceleration), I-B (deceleration) and I-D (transient deceleration during acceleration), covering three typical VL-TV transition modes;

3) Outer race fault (2 datasets): O-B (deceleration) and O-C (transient acceleration during deceleration).

Each of the 7 datasets contains 2 million data points, resulting in a total of 14 million data points for model training and testing.

4.1.2. Class balance and data leakage mitigation

To eliminate the bias of experimental results caused by class imbalance and data leakage, the following targeted optimization measures were adopted:

(1) Class Balance Optimization.

The class distribution of the mixed dataset is shown in Table 6, with a slight imbalance (inner race fault accounting for 42.9 %, normal/outer race fault accounting for 28.6 % each). The Synthetic Minority Oversampling Technique (SMOTE) was used during training to generate synthetic samples for minority classes (normal and outer race fault) by linear interpolation between adjacent samples, making the training set realize approximate class balance (each class accounting for ~33 %).

(2) Data Leakage Prevention.

Strict non-overlapping splitting and uniform distribution strategies were adopted to avoid data leakage caused by temporal overlap or uneven mode distribution:

1) Non-overlapping time segment splitting: The 7 datasets were split into training set (80 %) and test set (20 %) by time sequence (e.g., 0-8 s for training, 8-10 s for testing), ensuring no temporal overlap between the two sets;

2) Uniform state/mode distribution: The training and test sets maintained the same proportion of health states and VL-TV modes, ensuring the test set is fully representative of the training set’s complexity.

4.2.1. MC-IVMD preprocessing and feature extraction

The mixed dataset was processed by the proposed MC-IVMD algorithm to decompose raw VL-TV signals into intrinsic mode function (IMF) components. The fault-sensitive IMFs were screened by calculating and comparing the kurtosis values of all IMFs (kurtosis is highly sensitive to fault-induced impulse features under VL-TV conditions). The IMF components of a typical VL-TV signal after MC-IVMD decomposition are shown in Fig. 9.

Fig. 9The obtained IMFs by using IVMD

4.2.2. CLSAM++ model training and stability verification

The fault-sensitive IMFs were constructed into a time-frequency joint feature dataset and input into the CLSAM++ diagnostic model for training (50 iterations). The variation curves of diagnostic accuracy and loss rate during training are shown in Fig. 10. The model achieved a test set accuracy of 98.9 % and a loss rate of 0.044 % after convergence.

To verify the statistical stability of the model, the training process was repeated 10 times with different random seeds. The experimental results showed that the mean accuracy was 98.7 % (standard deviation 0.3 %) and the mean loss rate was 0.046 % (standard deviation 0.008 %), which confirmed the excellent stability and robustness of the proposed framework under VL-TV coupling conditions. In addition, Fig. 10 shows that the CLSAM++ model converges rapidly (stable after ~30 iterations) with no obvious overfitting, indicating its strong adaptability to complex VL-TV signals.

Fig. 10Diagnostic curve chart: a) diagnostic accuracy chart; b) diagnostic loss rate chart

a)

b)

4.2.3. Fault classification performance

The 12-category confusion matrix and t-SNE feature visualization (Fig. 11) were used to evaluate both class-level discrimination and feature-space separability. The confusion matrix shows that the proposed model maintains high recognition accuracy across different health states and VL-TV modes, with only limited confusion in the most challenging transition conditions. The t-SNE distribution further indicates that the learned features are compact within classes and well separated between classes. This result suggests that the dual-branch fusion mechanism and GSAM sparse attention do not merely improve the final accuracy, but also enhance the interpretability of the learned feature space.

Fig. 11Fault classification effect diagram: a) confusion matrix diagram of 12 categories; b) visualization of t-SNE of 12 categories

a)

b)

4.2.4. Comparative experiments with state-of-the-art methods

To further verify the superiority of the proposed framework, comparative experiments were conducted against three representative diagnosis models for variable-speed conditions: DCNN [25], RNN-WDCNN [26], and CNN-LSTM [27]. All comparison models were retrained on the same Ottawa VL-TV dataset under identical settings (8:2 training/test split, 50 iterations, and hyperparameter tuning within the same search budget) to ensure a fair comparison. In addition, a qualitative comparison with representative recent studies is provided in Table 7 to clarify the academic positioning and engineering differences of the proposed method relative to the recent literature.

The core differences between the comparison models and the proposed framework are summarized as follows:

(1) The comparison models do not use the proposed MC-IVMD preprocessing and therefore rely on conventional preprocessing or direct input signals.

(2) The comparison models do not contain the GSAM sparse attention mechanism or the dual-branch cross-layer fusion module that forms the key design of CLSAM++.

These two design differences are the main sources of the performance gain of the proposed framework, as also supported by the ablation results in Table 5.

(1) Accuracy Comparison under Different Dataset Scales.

To examine the scalability of the proposed framework, 3, 6, 9, and 12 preprocessed VL-TV datasets were selected for comparative experiments under 20 and 50 training iterations. Each experimental combination was repeated 10 times independently, and the average accuracy was taken as the final result (Tables 8 and 9).

The results show that the proposed framework achieves the highest diagnostic accuracy under all dataset scales and training settings. More importantly, the performance margin becomes larger as the dataset scale increases and the working-condition diversity becomes more complex. From an academic perspective, this finding indicates that the joint use of physics-informed decomposition and sparse temporal modelling is more effective than simply increasing network depth. From an engineering perspective, the stable superiority on the most complex 12-group task suggests stronger tolerance to operating-condition diversity in practical monitoring scenarios.

(2) Convergence Performance Comparison.

The time-accuracy convergence curves of each model are shown in Fig. 12. All models begin to converge after about 10 epochs, but the proposed CLSAM++ framework maintains a consistently higher validation accuracy throughout training and reaches a stable high-accuracy regime after around 30 epochs. This indicates that the advantage of the proposed framework does not arise from longer training, but from the synergistic effect of higher-quality fault-sensitive input features and a more suitable feature-learning architecture.

Fig. 12Comparison of time-accuracy curves of different models

(3) Statistical Significance and Parameter Budget Verification.

1) Paired-sample t-tests were conducted at a significance level of $\alpha =$ 0.05 on the accuracy results of 50 independent runs to verify whether the observed performance advantages are statistically reliable (Table 10). In all comparison groups, the p-values are below 0.05, which indicates that the superiority of the proposed framework is statistically significant rather than a result of random fluctuation. Combined with the small dispersion observed in the repeated-run experiments, this supports the reliability of the reported accuracy improvements.

2) Parameter-budget verification was further conducted to exclude the possibility that the observed gains simply arise from a larger model size. After expanding the channel width of DCNN and CNN-LSTM to match the parameter count of CLSAM++ (1.28 M), the two enlarged baselines still achieved only 95.6 % and 95.9 % accuracy, which remain 3.3 % and 3.0 % lower than CLSAM++. This result indicates that the advantage of the proposed framework originates from its architecture design and physics-informed optimization strategy rather than parameter inflation.

4.3.1. Synthetic VL-TV dataset construction

The original CWRU dataset is collected under steady-speed conditions, so linear acceleration and deceleration modulation were applied to the raw signals to construct a synthetic VL-TV coupling dataset consistent with the experimental conditions of the Ottawa dataset. Four health states (normal, inner race fault, outer race fault, rolling element fault) and three load conditions (0 hp, 1 hp, 2 hp) were selected, resulting in 12 types of samples in total.

4.3.2. Experimental setup and results

The synthetic CWRU VL-TV dataset was processed by the same preprocessing and training workflow as the Ottawa dataset (MC-IVMD decomposition, kurtosis-based IMF screening, and CLSAM++ model training), with an 8:2 training/test split and 10 repeated experiments for average calculation. The diagnostic accuracy of each model is shown in Table 11.

The experimental results show that the proposed framework still maintains a high average diagnostic accuracy of 97.2 % on the synthetic CWRU VL-TV dataset, which is 4.1 %-5.1 % higher than those of the comparison models. Although the accuracy is slightly lower than that obtained on the Ottawa dataset because of differences in data distribution, sampling frequency, and synthetic speed modulation, the proposed framework still achieves the best overall performance. This demonstrates that the learned diagnosis strategy is not tightly coupled to a single dataset and retains meaningful transferability across platforms.

4.3.3. Analysis of generalization ability

The excellent cross-dataset performance of the proposed framework can be attributed to two main reasons:

(1) MC-IVMD is adaptable to vibration signals with different sampling frequencies and sensor characteristics, enabling it to extract physically meaningful impulsive components even when the data distribution changes.

(2) CLSAM++ jointly models local impact features and long-range temporal dependencies through GSAM sparse attention and dual-branch cross-layer fusion, making the learned representation less sensitive to dataset-specific variations.

In summary, the proposed physics-informed end-to-end fault diagnosis framework (MC-IVMD + CLSAM++) not only achieves excellent performance on the real Ottawa VL-TV dataset, but also demonstrates robust cross-dataset adaptability on the synthetic CWRU VL-TV dataset. This strengthens the engineering relevance of the method for intelligent monitoring under variable and uncertain operating conditions.