Design of a transfer learning fault identification model based on a Markov field and an improved DarkNet

Abstract

To address the challenge of small-sample fault diagnosis for mechanical equipment operating under complex and variable working conditions, this study proposes a comprehensive transfer learning-based diagnostic framework and systematically verifies its effectiveness. First, raw vibration signals are preprocessed to extract 120 statistical features from each sample. These feature vectors are regarded as one-dimensional sequences and transformed into two-dimensional Markov transition field (MTF) images, thereby preserving both temporal transition information and global structural characteristics. The generated MTF images are then fed into a pretrained DarkNet19 network for deep feature extraction. To further enhance feature representation, a gated recurrent unit (GRU) and a multi-head self-attention (MSA) mechanism are incorporated to capture temporal dependencies and emphasize discriminative fault features. In addition, an improved sparrow search algorithm (TLSSA), integrating a t-distribution mutation strategy and Lévy flight, is introduced to adaptively optimize network hyperparameters, improving both convergence efficiency and diagnostic accuracy. The proposed framework is comprehensively evaluated through four aspects: (1) validation of the TLSSA optimization algorithm using benchmark functions; (2) comparative experiments with representative fault diagnosis models to verify the superiority of the proposed framework; (3) ablation studies to quantify the contributions of MTF visualization, transfer learning, GRU, MSA, and TLSSA; and (4) applicability validation on the publicly available Southeast University (SEU) gearbox dataset to evaluate the generalization capability of the proposed framework across different datasets and operating conditions. Experimental results show that the proposed model achieves an overall accuracy of 99.375% on the industrial wind farm dataset and 99.17% on the SEU public dataset, while maintaining rapid convergence, low overfitting, and excellent classification stability. These results demonstrate that the proposed MTF visualization-DarkNet transfer learning framework exhibits strong cross-dataset adaptability, robust generalization capability, and promising engineering potential for intelligent fault diagnosis under complex operating conditions.

1 Introduction

With the rapid advancement of industrial intelligence, system faults caused by abnormal equipment operating conditions have attracted increasing research attention (Luo et al., 2023). Transfer learning-based fault identification methods not only improve the adaptability of diagnostic models in data-scarce scenarios but also significantly reduce reliance on large volumes of labeled data, offering substantial theoretical significance and promising engineering application potential (Wang S. et al., 2020). Although the dataset used in this study contains 800 samples in total, each fault category includes only 100 labeled samples. In the context of industrial fault diagnosis, acquiring labeled fault data is costly and time-consuming. Therefore, this study adopts the concept of small-sample learning, which focuses on learning robust diagnostic models from limited labeled samples rather than the classical N-way K-shot-few-shot learning paradigm commonly used in computer vision. Accordingly, “few-shot learning” refers to practical industrial small-sample fault diagnosis under limited labeled data rather than the strict meta-learning setting.

The core task of transfer learning in fault diagnosis is to effectively migrate knowledge learned from a source domain to a target domain, enabling accurate mapping of fault features and recognition of operating states across different working conditions or equipment types: that is, leveraging existing diagnostic experience to identify fault modes of new or heterogeneous machinery (Xu et al., 2022). Traditional diagnostic approaches often suffer from discrepancies in data distributions and limited model generalization, whereas transfer learning provides a powerful solution through domain adaptation and feature reuse mechanisms, thereby enhancing diagnostic robustness under complex working conditions (Li and Ma, 2023). Pattern recognition aims to enable machines to automatically or semi-automatically identify structured information embedded within data and determine the corresponding category (Yu et al., 2023). Modern intelligent pattern recognition technologies, driven by the integration of traditional methodologies and advanced artificial intelligence techniques, further enable machines not only to “see” or “hear” data but also to understand and respond intelligently to the complex patterns contained within it (Gao et al., 2025). These capabilities have become fundamental to ensuring the safe and reliable operation of industrial systems (Jagadeesha et al., 2023).

Current transfer learning strategies mainly involve deep-network-based feature transfer and adversarial domain adaptation approaches (Gill et al., 2023; Zhang L. et al., 2024). Meanwhile, the Markov random field (MRF), as a graphical model capable of capturing complex dependencies among random variables, can effectively represent spatial correlations and contextual constraints among fault features (Junling, 2024). Integrating MRFs with transfer learning is expected to more precisely characterize the joint distribution and local interactions of equipment operating states during knowledge transfer, thereby further enhancing the accuracy and robustness of cross-domain fault identification.

Transfer learning techniques have achieved remarkable success in image recognition and natural language processing. Given the similarity between visual/textual data and mechanical vibration signals in terms of transferable deep feature distributions, these techniques have been increasingly applied to mechanical fault diagnosis (Mahalle et al., 2023). In bearing fault identification, models trained on source domain data typically serve as the foundation, while domain adaptation strategies are employed to reduce distributional discrepancies between the source and target domains and enhance model adaptability under new operating conditions (Sun et al., 2025).

In most existing transfer learning-based diagnostic studies, deep networks are used to automatically extract and migrate discriminative features prior to model construction (Prabhakar and Lee, 2022). Asutkar and Tallur (2023) proposed a deep transfer learning strategy that fine-tunes low-level convolutional layers and transfers high-level fully connected parameters. This method can achieve high diagnostic accuracy even with low-resolution sensors and very limited labeled samples. However, its performance is sensitive to the selection of fine-tuned layers, and its generalization capability under complex cross-device scenarios remains uncertain. Wang et al. (2025) developed CTDAN, a cross-device bearing fault diagnosis method that integrates multi-kernel MMD, covariance alignment, and class-induced fine alignment, along with trajectory transfer to facilitate feature sharing. This approach significantly improves cross-machine diagnostic accuracy. Tian et al. (2023) introduced an open-set, multi-source domain adaptation method that combines unknown fault detection with multi-source mutual supervision via a weighted adversarial framework based on multi-classifier mobility metrics. The method effectively recognizes unseen faults and outperforms traditional domain adaptation approaches when new fault types emerge. However, its performance depends on the relevance of the source domains, and unknown fault detection accuracy is influenced by threshold selection. Wei et al. (2025b) proposed a single-source domain generalization method based on discrete wavelet convolution and cross-contrast perturbation, which expands the feature distribution under constraints of noise suppression, semantic preservation, and instance similarity, improving the generalization of bearing and other device fault diagnosis in unseen domains. Song et al. (2022) presented a simulation-driven deep transfer learning approach that constructs a planetary gearbox dynamic model to generate simulated fault data and integrates a 1D convolutional neural network (CNN), attention mechanism, and domain adaptation to learn domain-invariant features for transferring diagnosis from simulation to real equipment. This method effectively mitigates the scarcity of labeled real-world data and achieves high diagnostic accuracy, although its performance still depends on the gap between simulation and reality and suffers from relatively complex network architecture and limited training efficiency. Lu and Li (2023) proposed a method combining thermal imaging with a hybrid deep learning model, using temperature field modeling to generate fault images and integrating CNN-based transfer learning with a deep belief network. This approach reduces training time to one-fifth of that required by traditional CNNs and achieves more than 97% simulated fault recognition accuracy, but it relies heavily on the accuracy of thermal imaging and temperature field modeling. Wei et al. (2025a) further introduced a semi-supervised domain generalization method guided by domain knowledge, which enhances pseudo-label quality and learns domain-invariant features through pseudo-label weighting, adaptive thresholding, channel feature expansion, and domain archetype alignment, thereby substantially improving cross-domain bearing diagnosis performance.

Despite these advances, existing studies still face limitations in terms of generalization across complex working conditions, robustness of feature representations, and optimization efficiency (Keshun et al., 2025b). To address these challenges, this study integrates deep feature extraction, attention mechanisms, and intelligent optimization algorithms within a unified transfer learning framework to develop a more robust and adaptive bearing fault identification model, thereby enhancing diagnostic reliability and applicability across multi-condition and multi-equipment scenarios (Keshun et al., 2025a).

In summary, an Markov transition field (MTF)-TLSSA-DarkNet-gated recurrent unit (GRU)-multi-head self-attention (MSA) fusion fault recognition model is constructed in this study. The main body of the model adopts the DarkNet19 model for target detection and migration learning as the framework of the fault recognition model. At the same time, GRU and MSA are added to improve the feature grasping ability of the model. For data preprocessing, the Markov field is used to transform the one-dimensional waveform into an image. For the hyperparameters in the DarkNet19-GRU-MSA model, the self-adaptive parameters are used to transform the one-dimensional waveform into an image. The TLSSA is optimized by configuring hyperparameters in the DarkNet19-GRU-MSA model, the adaptive t-distribution, and the Lévy flight-improved sparrow search method. Finally, the reliability and accuracy of the method are verified by experiments (Zhao et al., 2025). Unlike domain adaptation studies that transfer knowledge across different operating conditions, this work adopts parameter transfer learning by fine-tuning a DarkNet19 model pretrained on ImageNet. Therefore, the transfer learning investigated in this study refers to visual feature transfer from a large-scale natural image dataset to MTF-based fault images.

2 Theoretical foundation

2.1 DarkNet19 model

DarkNet19 is an efficient deep convolutional neural network backbone architecture whose core design philosophy is to pursue faster computation while maintaining higher accuracy for real-time processing. The name of the network derives from the fact that it contains 19 convolutional layers, and it discards the parameter-heavy fully connected layers of traditional CNNs in favor of global average pooling for classification, a change that greatly reduces the number of model parameters and effectively prevents overfitting (Loey et al., 2021). As shown in Figure 1, in terms of overall structure, DarkNet19 progressively reduces the spatial resolution from 224 × 224 to 7 × 7 by downsampling five times through a convolutional or max-pooling layer with a step size of 2, while continuously increasing the number of channels to 1024 to extract rich features from low-level edges to high-level semantics in a hierarchical manner. Finally, the network compresses each feature graph into a single feature value through global average pooling, which is fed directly into the softmax classifier to output the results. This careful design has enabled DarkNet19 to achieve an excellent balance of performance and speed on datasets such as ImageNet and has been the cornerstone of many subsequent advanced target detection algorithms such as YOLOv2/v3. In fault identification tasks, its powerful feature extraction capability makes it an excellent base model for migration learning applications.

FIGURE 1

2.2 Markov transfer field

The MTF is an effective method for converting one-dimensional time-series data into a two-dimensional image representation; its core idea is to construct an image that can preserve the temporal dynamic features by capturing the transfer probabilities between states in a time series (Andersen et al., 2023). The method first discretizes the original time series into quartiles, dividing the continuous data into Q state intervals, and maps each data point to the corresponding state symbol according to its numerical magnitude. Subsequently, a first-order Markov transfer matrix is computed to describe the probability distribution of transitions between states (Donnelly et al., 2024). Finally, an image representation with spatiotemporal characteristics is generated by organizing the state-transfer probabilities between any two time points in the time series into a two-dimensional matrix (Zhang S. et al., 2024).

In this study, the number of state divisions Q is a critical parameter that influences the resolution of the state discretization and the quality of the generated MTF images. A small Q may insufficiently represent dynamic transitions, while an excessively large Q may introduce noise and increase computational complexity. The quantization level Q was optimized using the TLSSA within the search range {4, 8, 12}. A larger Q preserves more transition details but also increases texture complexity and computational cost, whereas a smaller Q may lose discriminative information due to coarse quantization. According to the optimization results, Q = 8 provides the best balance between feature representation and classification performance.

To clarify the data-processing workflow, the raw vibration signals collected from the wind turbine monitoring system were first preprocessed through denoising and normalization. Subsequently, 120 statistical features, including time-domain, frequency-domain, and statistical descriptors, were extracted from each signal segment. These 120-dimensional feature vectors were then treated as one-dimensional sequences and used directly as input to the MTF. The MTF algorithm converted the sequential relationships among the feature values into two-dimensional transition probability matrices, thereby generating image representations suitable for feature extraction by the pretrained DarkNet19 network. Therefore, the 120-dimensional statistical features directly participate in the MTF transformation rather than being extracted after image generation.

2.3 Convolutional neural network

A convolutional neural network (CNN) is a deep learning architecture dedicated to processing data with lattice topology (e.g., images, time-series signals). Its core design idea is to efficiently learn a hierarchical feature representation of the data through local connectivity, weight sharing, and spatial upsampling (Tian et al., 2022). A typical CNN consists of input, convolutional, activation, pooling, and fully connected layers stacked sequentially to form an end-to-end feature extraction and classification pipeline (Van et al., 2019). In the convolutional layer, multiple learnable filters (convolutional kernels) are slid over the input data to extract the underlying feature patterns through the local receptive fields, the mathematical operation of which can be expressed as the following equation:where X denotes the input feature map, W is the convolution kernel, b is the bias term, Y represents the output feature map, C is the number of input channels, and M×N denotes the kernel size.

By alternately stacking multiple convolution-activation-pooling modules, the CNN progressively learns a complex hierarchy of features ranging from low-level edges and corner points to mid-level textures and parts to high-level semantic objects (Tokarev et al., 2021). At the end of the network, the fully connected layer maps the learned high-level features to the sample labeling space to complete the classification or regression task. In the field of fault diagnosis, when one-dimensional time-series data, such as vibration signals, are transformed into images by techniques such as Markov transfer fields, the CNN can automatically learn the spatial patterns related to faults in them, such as shock components or harmonic structures, so as to construct robust and accurate intelligent diagnostic models (Sarpotdar, 2022). Its parameter-sharing mechanism dramatically reduces model complexity, and its local connectivity property fits the local correlation of fault features in the signal, making CNNs a powerful tool for processing image-based fault data (Sarıgül et al., 2019).

2.4 Multiple attention mechanisms

The multi-head attention mechanism is a core component of the Transformer architecture that enhances the model’s ability to capture different aspects of dependencies in information by running multiple independent attention units in parallel (Liu et al., 2022). The core idea is to map the input sequence to multiple subspaces, compute the attention independently in each subspace, and finally fuse the information from all subspaces to obtain a richer feature representation (Zhang et al., 2023). The mechanism first passes the input sequence through three different sets of learnable weight matrices to generate the query, key, and value components, respectively. The computation of each attention head is based on the scaled dot product attention equation:where Q, K, and V represent the query, key, and value matrix, respectively; is the dimension of the key vector; and the scaling factor is used to prevent the gradient from disappearing due to the large dot product result. The formula first calculates the similarity between the query and the keys, normalizes it to the attention weights using the “softmax” function, and then applies these weights to the value matrix for weighted summation. The complete multi-attention computation consists of the following steps:where h denotes the number of attention heads; , , is the projection matrix corresponding to the i-th attention head; and is the output matrix that projects the splicing result back to the original dimension. With this design, each attention head can learn to attend to different types of relational patterns in the input sequence, such as local dependencies, global dependencies, or semantic associations.

2.5 An improved sparrow search algorithm TLSSA

The sparrow search algorithm based on t-distribution and Lévy flight improvement (TLSSA) is an intelligent optimization method that enhances the global optimization performance of the original sparrow search algorithm by fusing two important mathematical mechanisms (Hong et al., 2022). The algorithm addresses the shortcomings of the original sparrow search algorithm in solving high-dimensional complex problems, such as easy premature convergence and insufficient population diversity. It organically combines the adaptive t-distribution variational operator with the Lévy flight strategy to make the algorithm achieve a better balance between exploration and exploitation (Xue and Shen, 2020).

In terms of t-distribution variation, the algorithm uses the property that the t-distribution varies with the degrees-of-freedom parameter to adaptively adjust the search strategy. Its position update formula is expressed aswhere and denote the position of an individual before and after mutation, respectively; is a t-distribution random variable with the degree of freedom as the current number of iterations; and is a step-size control factor. The key of this formula is that the shape of the t-distribution changes dynamically with the degree of freedom: at the beginning of the iteration when the degree of freedom is small, the t-distribution has a heavy tail, which is conducive to generating a large variation in step size and enhancing the ability of global exploration; with the increase in the degree of freedom of the iteration, the t-distribution gradually approaches a Gaussian distribution, and the variation tends to be locally fine-tuned to search for the realization of the transition from the global exploration to the local development of the adaptive transition.

In terms of the Lévy flight mechanism, the algorithm adopts a heavy-tailed probability distribution to simulate the intermittent movement pattern in nature, and its step size generation formula is:where and are mutually independent normally distributed random variables; is a scale parameter, and is a gamma function. This mechanism enables the algorithm to effectively jump out of the local optima and explore the solution space more comprehensively by generating a search pattern that combines many short moves with occasional long jumps (Keshun et al., 2024b).

As shown in Figure 2, the two mechanisms work together in the algorithm: the t-distribution variation mainly supports the local fine search for ordinary individuals in the population, while the Lévy flights are responsible for the emergent global exploration of the optimal individuals. This dual-strategy architecture allows TLSSA to significantly improve the robustness of solving complex optimization problems while maintaining the convergence speed. In fault diagnosis applications, the algorithm can effectively optimize the hyperparameter combinations of deep learning models and provide reliable support for building high-performance fault identification systems. In this study, the objective of TLSSA is to automatically determine the optimal hyperparameter combination for the proposed fault diagnosis model. Specifically, the classification accuracy on the validation set is adopted as the optimization criterion. During each iteration, TLSSA generates a candidate hyperparameter set, trains the corresponding network, and evaluates its classification performance on the validation dataset. The fitness value is defined as 1 − Accuracy, thereby transforming the maximization of classification accuracy into a minimization problem. The hyperparameter combination yielding the minimum fitness value is selected as the optimal solution for the final model training.

FIGURE 2

2.6 MTF-TLSSA-DarkNet-GRU-MSA migration learning fault recognition procedures

This study proposes a comprehensive solution for mechanical fault recognition under complex and variable operating conditions. The core objective is to construct an intelligent diagnostic system that achieves high accuracy, strong generalization capability, and efficient computation by integrating multiple advanced technologies. Specifically, the model first employs the MTF to convert the extracted 120-dimensional statistical feature sequences, rather than the raw vibration waveforms, into two-dimensional images, thereby transforming temporal features and state-transition dynamics into spatial texture patterns. This conversion provides richer structural information and a more expressive data foundation for subsequent deep learning-based feature extraction.

Building upon this representation, the model utilizes a pretrained DarkNet19 network as the backbone feature extractor. Through transfer learning, the strong visual feature extraction capability of DarkNet19 is effectively adapted to the fault diagnosis scenario. This not only enhances feature quality but also reduces dependence on large amounts of labeled data and significantly shortens the training time.

To further strengthen the modeling of feature relationships, GRUs and a MSA mechanism are incorporated into the architecture (Keshun et al., 2025c). The GRU module captures temporal dependencies inherent in vibration signals, while the MSA mechanism adaptively assigns importance weights to different feature channels, enabling precise emphasis on critical fault-related information. To address the challenge of manually selecting hyperparameters, an improved sparrow search algorithm (TLSSA) is introduced for automatic optimization. By integrating adaptive t-distribution mutation and Lévy flight strategies, the algorithm balances global exploration and local exploitation, significantly improving the efficiency and accuracy of hyperparameter search and ensuring an optimal model configuration.

As shown in Figure 3, the proposed framework featuring temporal signal visualization, transfer learning-based feature extraction, attention-enhanced representation learning, and intelligent hyperparameter optimization achieves accurate, robust, and efficient mechanical fault identification through a coordinated and innovative multi-technology pathway. Specifically, DarkNet19 pretrained on the ImageNet dataset was employed as the backbone network to initialize the feature extraction module. During transfer learning, the first three convolutional blocks were frozen to preserve the generic low-level visual features learned from the source domain, while the remaining convolutional blocks together with the GRU, MSA, and fully connected classification layers were fine-tuned using the target-domain MTF images generated from the wind turbine monitoring data. The entire network was optimized using the Adam optimizer, with an initial learning rate of 3.2 × 10⁻⁴, allowing the pretrained backbone to gradually adapt to the target-domain fault characteristics while maintaining stable convergence. In this framework, the ImageNet dataset serves as the source domain, providing generic visual representations through large-scale image classification, whereas the MTF images generated from wind turbine vibration signals constitute the target domain. This transfer learning strategy enables the model to effectively leverage prior visual knowledge while adapting to the characteristics of wind turbine fault diagnosis.

FIGURE 3

3 Experimental validation

3.1 Experimental setup

The experimental data were collected from the condition monitoring system of a utility-scale wind farm located in Guangxi Province, China. The monitored equipment consisted of horizontal-axis variable-speed wind turbines operating under normal grid-connected conditions. During the monitoring period, the turbines operated in typical industrial environments with varying wind speeds and loads, allowing the system to capture representative fault information. The condition monitoring system was equipped with multiple sensors, including triaxial vibration sensors, temperature sensors, and rotational speed sensors, which continuously recorded the operating status of the drivetrain and rotating components. The monitoring system synchronized data from different sensors and segmented the data into fixed-length samples before feature extraction. To ensure consistency and robustness, all collected signals were preprocessed through noise reduction, normalization, and feature extraction. Finally, 120 statistical features, including time-domain descriptors, frequency-domain characteristics, and statistical indicators, were extracted from the multi-sensor signals to construct the final dataset used in this study.

The experimental environment was configured as follows: the software platform was MATLAB 24.2.0.2712019 (R2024b), and the hardware platform consisted of a Lenovo Legion R9000P 2021H laptop equipped with an AMD Ryzen 7 5800H processor with Radeon Graphics and an NVIDIA GeForce RTX 3060 Laptop GPU (6 GB VRAM).

To ensure clarity, consistency, and reproducibility of the experimental setup, all experiments in this study were conducted using a high-dimensional numerical fault dataset obtained from the operational condition monitoring system of a wind farm. The dataset underwent engineering feature extraction and standardization procedures and contains 800 samples, each consisting of 120 consecutive numerical features and one labeled fault category. The feature values range from −1.366 to 1.407, and no missing values are present, which aligns with common preprocessing practices in industrial monitoring scenarios.

The dataset encompasses eight representative mechanical fault types, including wear, imbalance, looseness, and various abnormal operating conditions. Each category contains 100 samples, resulting in a balanced dataset that supports fair and unbiased model evaluation. The 120-dimensional feature vector captures multi-source monitoring information from vibration, temperature, and rotational speed signals, including time-domain, frequency-domain, and statistical descriptors. These features constitute high-dimensional, small-sample engineering data rather than raw vibration waveforms or image signals.

All experiments were performed on this unified dataset, with samples randomly divided into training and testing sets in an 8:2 ratio to maintain the stability of model training and the reliability of evaluation results. By standardizing the data source, structure, and task configuration, this study ensures a clean, reproducible experimental environment and provides a solid foundation for subsequent model performance verification.

To further clarify the data acquisition and feature extraction process, the original monitoring data were collected through a wind turbine condition monitoring system equipped with multiple sensors, including vibration, temperature, and rotational speed sensors. The raw time-series signals were continuously sampled under stable operating conditions and segmented into fixed-length windows. For each sample, a set of 120 statistical and signal-processing-based features was extracted from the raw signals. These features include time-domain indicators (e.g., mean, standard deviation, skewness, and kurtosis), frequency-domain characteristics (e.g., spectral energy and dominant frequency components), and statistical descriptors derived from multi-sensor measurements. All features were subsequently normalized to ensure consistency and stability during model training. This feature engineering process transforms raw sensor signals into structured high-dimensional representations that serve as the input for subsequent MTF-based image transformation and deep learning analysis. Note that the proposed deep learning model does not directly use the tabular features for classification. Instead, the extracted 120-dimensional feature vectors are transformed into MTF images, which serve as the actual input to DarkNet19.

The 120 features are statistical descriptors extracted from raw vibration, temperature, and rotational speed signals. These feature vectors are subsequently converted into MTF images before being fed into the proposed deep learning framework. To ensure a fair comparison, all deep learning baseline models were implemented under the same experimental environment, including the same training/testing split, optimizer, batch size, and stopping criteria. For ResNet50, the learning rate was initialized according to the commonly adopted setting for transfer learning and was further adjusted through preliminary validation experiments to achieve stable convergence. Because the objective of this study is to evaluate the effectiveness of the proposed framework rather than exhaustively optimize each baseline model, identical training protocols were adopted whenever possible to minimize the influence of external factors.

3.2 Algorithm validity verification

To evaluate the effectiveness of the improved TLSSA algorithm, eight widely used benchmark test functions reported by Li et al. (2022) were selected for numerical experiments. These functions encompass unimodal, multimodal, and fixed-dimension multimodal categories, thereby enabling a comprehensive assessment of the optimization performance. The TLSSA algorithm was compared in detail with three representative heuristic optimization algorithms: the basic sparrow search algorithm (SSA), particle swarm optimization (PSO), and the gray wolf optimizer (GWO). To ensure fairness and accuracy in the numerical evaluation, the parameter settings for all algorithms were kept consistent. Specifically, the population size was set to 30, and the maximum number of iterations was fixed at 500. Each algorithm was executed independently 20 times on every benchmark function, and the optimal solution obtained from each run was recorded to assess algorithmic stability and convergence characteristics. The experimental results are presented in Figure 4.

FIGURE 4

As shown in Figure 4 and summarized in Table 1, the TLSSA algorithm consistently outperforms the three benchmark algorithms across all test functions. It demonstrates superior optimization accuracy on both unimodal and multimodal functions and achieves the optimal fitness value within approximately 300 iterations. A comparison of the vertical axes across the subplots reveals that the fitness values obtained by TLSSA are substantially lower, often by several orders of magnitude, than those produced by SSA, PSO, and GWO.

Moreover, TLSSA exhibits steeper convergence curves and requires fewer iterations to reach near-optimal regions, indicating a markedly faster convergence rate. In addition to improved convergence behavior, the algorithm requires less computational time for parameter optimization while maintaining higher solution precision. Overall, the results confirm that the TLSSA algorithm achieves both faster convergence and higher optimization accuracy than the reference algorithms.

3.3 Model parameter setting

Note that the selection of key parameters, such as the MTF state division number (Q) and image resolution, is not arbitrary but determined through a combination of prior empirical knowledge and adaptive optimization. In particular, the TLSSA algorithm searches for optimal parameter combinations within predefined ranges, ensuring that the selected configuration balances model complexity, computational efficiency, and diagnostic performance. This strategy improves the reproducibility and robustness of the proposed method while reducing the dependency on manual parameter tuning.

The key hyperparameter configurations of the MTF-TLSSA-DarkNet-GRU-MSA model are determined to ensure stable training, efficient convergence, and optimal diagnostic performance, based on the adaptive optimization capability of the improved sparrow search algorithm (TLSSA) and the characteristics of the small-sample fault diagnosis task. All hyperparameters are obtained by iteratively optimizing TLSSA within the preset search space, balancing the model complexity and computational cost. The specific parameters are shown in Table 2.

Figure 5 shows the iterative convergence characteristics of the model based on the above parameter configurations during the training process. Figure 5 shows the trend of the loss values of the training and validation sets with the number of iterations, where the blue curve represents the training loss, and the orange curve represents the validation loss. In the early stage of model training (0–2000 iterations), the training loss and validation loss decrease rapidly, indicating that the migration features of DarkNet19 and MTF visualization features are effectively integrated, and the GRU and MSA mechanisms begin to capture timing dependencies and critical fault features. In 2000–6780 iterations, the loss values tend to level off, and the validation loss stabilizes below 0.02 without obvious oscillation, indicating that the TLSSA optimization of the superlosses has good convergence with the number of iterations. This indicates that the hyperparameters of TLSSA optimization are reasonably configured, and the model is not overfitted or underfitted. After 6780 iterations, the training loss drops to 0.012, and the validation loss drops to 0.018. The model reaches a converged and stable state, and training is stopped (the early stop mechanism is not triggered, and there is no stagnation in three rounds due to a continuous drop in validation loss). The gap between the training loss and validation loss is always less than 0.01, indicating that the model generalization error is small, and TLSSA has no obvious shock. The difference between the training loss and the validation loss is always less than 0.01, indicating that the model generalization error is small, and the optimization of hyperparameters by TLSSA effectively suppresses overfitting, which is suitable for small-sample fault diagnosis scenarios.

FIGURE 5

The parameter configuration and the iteration curve together verify the validity of model training: the optimized hyperparameters of TLSSA enable the model to converge quickly within a limited number of iterations, while maintaining excellent generalization ability, which provides a guarantee for the reliability of the subsequent experimental results.

3.4 Analysis of results

As shown in Figure 6, the fault diagnosis model developed in this study demonstrates excellent classification performance on the test set. The overall test accuracy reaches 99.375%, indicating that the model can effectively distinguish between different fault categories. A comparison of the predicted labels with the ground truth reveals that, among the 140 consecutive test samples (indices 20–160), only a very small number of samples are misclassified, while the vast majority show complete consistency.

FIGURE 6

The strong overlap between the predicted trajectories and the true labels further confirms that the proposed time-series visualization method and transfer learning framework successfully extract discriminative fault features, thereby enhancing the model’s recognition capability. Meanwhile, the few isolated misclassification points suggest that the model remains robust even under complex working conditions. These results collectively verify the effectiveness of the proposed method for fault diagnosis and demonstrate its promising potential for practical engineering applications.

The evaluation results based on the confusion matrix further demonstrate that the proposed fault diagnosis model achieves outstanding performance across the eight-class classification task. As shown in Figure 7, the model achieves perfect classification for all samples (20 per class) from Classes 1 to 7. For Class 8, 19 of 20 samples are correctly identified, with only a single sample misclassified as Class 7. This minor deviation results in an overall test accuracy of 99.375%, fully confirming the model’s effectiveness for multiclass fault identification.

FIGURE 7

The slight confusion between Classes 7 and 8 may stem from the intrinsic similarity of their fault characteristics under certain operating conditions, providing a valuable insight for future model refinement. Overall, the confusion matrix clearly illustrates the model’s strong discriminative capability across different fault categories, offering robust evidence of its reliability and applicability in real-world industrial scenarios. The confusion between Classes 7 and 8 can be attributed to their similar vibration characteristics. Both fault types produce comparable energy distributions and partially overlapping frequency-domain components, leading to similar transition patterns in the generated MTF images. Consequently, the extracted deep features exhibit some overlap, resulting in occasional misclassification.

The fault diagnosis model proposed in this study exhibits strong convergence behavior during training, as visualized by an error–iteration curve plotted in a polar coordinate system. As shown in Figure 8, the blue spiral trajectory demonstrates a clear inward convergence trend from the outer rings of the polar plot toward its center, while the orange markers highlight the sequential reduction in training error throughout the optimization process. As the number of iterations increases to 6780, the training error decreases steadily and smoothly, ultimately converging to a low-error region near the plot center.

FIGURE 8

This smooth and monotonic convergence trajectory confirms both the efficiency of the optimization algorithm in navigating the parameter space and the rationality of the model architecture. Moreover, the absence of oscillations or divergence during training provides strong evidence of the model’s stability and reliability. The polar coordinate representation further emphasizes the cyclic nature of the error-reduction process and intuitively captures the continuous improvement of model performance through the progressive shortening of the radial distance.

The fault diagnosis model developed in this study demonstrates outstanding convergence behavior during training, and its performance evolution is visualized using an accuracy–iteration curve plotted in polar coordinates. As shown in Figure 9, the training accuracy exhibits a clear and systematic improvement trend throughout the optimization process. In the early stage (0–200 iterations), the accuracy increases rapidly from the baseline level, reflecting the model’s strong capability to quickly capture essential data features. During the mid-iteration phase (200–700 iterations), accuracy continues to improve steadily, with the trajectory gradually contracting toward the center of the polar plot, indicating progressive refinement of the model parameters. Once the number of iterations exceeds 700, the accuracy curve converges into the central region of the plot, corresponding to an accuracy level approaching 100%. The significantly reduced fluctuations in this final stage further demonstrate the stability of the training process and the reliability of the convergence.

FIGURE 9

This characteristic trajectory of continuously shrinking from the outer rings toward the center provides an intuitive visualization of the model’s performance improvement and highlights the effective search capability of the optimization algorithm within the parameter space. Notably, the accuracy curve exhibits neither sharp oscillations nor divergence at any point, confirming that the adopted training strategy ensures a stable learning process. From a dynamic perspective, the convergence characteristics objectively validate the soundness of the model architecture and training methodology, providing a solid foundation for its reliable deployment in real industrial applications.

In order to further verify the superiority of the proposed MTF-TLSSA-DarkNet-GRU-MSA model, six mainstream models in the field of machinery fault diagnosis are selected as comparison objects, as shown in Table 3. Fair comparison experiments are conducted under the same dataset, hardware environment, and evaluation criteria, focusing on differences in diagnostic accuracy, convergence efficiency, and generalization ability of the models. All the comparison models adopt the same dataset division (training set: test set = 8:2), loss function (cross entropy), and early stopping strategy (no improvement in the accuracy of the validation set for three rounds to stop the training) to ensure the fairness of the experiments.

Before presenting the comparison results, note that the conventional machine learning baseline (SVM) and the deep learning models employ different input representations. The SVM classifier operates directly on the extracted 120-dimensional statistical feature vectors, whereas the deep learning models use the MTF images generated from the same feature vectors. Although the input modalities differ, both approaches are derived from the identical raw vibration data. This comparison evaluates the relative effectiveness of conventional feature-based learning and image-based deep feature learning under a unified data source.

Table 4 summarizes the comparative experimental results, demonstrating that the proposed MTF-TLSSA-DarkNet19-GRU-MSA model substantially outperforms all baseline methods on the test set. The model achieves an accuracy of 99.375%, exceeding that of the traditional SVM by 21.25 percentage points, the basic CNN by 10 percentage points, and even surpassing the state-of-the-art transfer learning method CTDAN by 1.25 percentage points.

Regarding misclassification performance, the proposed model records only one misclassified sample in Class 8, whereas all comparative models exhibit two or more misclassifications. This result confirms that the MTF-based temporal image transformation and the MSA-driven key-feature focusing mechanism significantly enhance the model’s ability to distinguish highly similar fault patterns.

In terms of convergence efficiency, the model converges within 6780 iterations, representing a 14.8% and 16.8% reduction compared with CNN-LSTM and CTDAN, respectively. This improvement can be attributed to the efficient hyperparameter optimization capability of the TLSSA algorithm. Meanwhile, the total training time (42.5 min) remains lower than that of ResNet50 (55.9 min) and CTDAN (62.3 min), illustrating that combining a lightweight backbone network and efficient feature processing effectively balances performance and computational cost.

With respect to generalization, the model achieves a generalization error of only 0.8%, far lower than that of SVM (8.7%), indicating that the transfer learning framework and TLSSA optimization effectively suppress overfitting. Moreover, compared with the DarkNet19-GRU model, introducing the MSA mechanism increases accuracy by 1.875% and reduces the generalization error by 0.7 percentage points, while TLSSA reduces the required convergence iterations by 540. These results verify the significant contributions of the multi-self-attention mechanism and the improved optimization algorithm in capturing spatiotemporal dependencies and enhancing training efficiency.

3.5 Analysis of ablation experiment results

In order to quantitatively evaluate the contribution of each core component and verify the necessity of their synergistic integration in the proposed model, ablation experiments are conducted on the MTF, the DarkNet19-GRU-MSA feature extraction architecture, and the improved sparrow search algorithm (TLSSA). This analysis further aims to identify redundant modules and confirm the effectiveness of each key technical component.

Following the single-variable principle, only one component is modified (removed or replaced) in each ablation setting, while all other experimental conditions, including dataset, hyperparameters, and training strategy, are kept strictly consistent to ensure fair comparison. Model performance is evaluated from three perspectives: classification accuracy, convergence behavior, and generalization error, enabling a comprehensive and objective assessment.

The complete framework is denoted as Model B (baseline: MTF + DarkNet19 + GRU + MSA + TLSSA). Based on this, seven ablation variants (Models A1–A7) are summarized in Table 5.

To ensure experimental reliability and comparability, all experiments are conducted under a unified setting: the dataset contains 800 samples across eight categories with an 8:2 training–testing split, and no additional data augmentation is applied. Unless otherwise specified, identical hyperparameter ranges are used for all models, including learning rate [1e−5, 1e−3], batch size [16, 32], GRU hidden units [64, 128], and MSA heads [4, 8]. All models are trained with the Adam optimizer and cross-entropy loss, a maximum of 6780 iterations, and an early stopping mechanism.

Each ablation variant is specifically designed to isolate the effect of a single component, including (i) removal of MTF for raw 1D input comparison, (ii) removal of GRU or MSA modules, (iii) replacement of TLSSA with the original SSA, (iv) disabling transfer/migration learning in DarkNet19, and (v) substitution of the backbone network with ResNet50. To reduce randomness, all experiments are independently repeated 20 times, and the final results are reported using averaged performance metrics to ensure statistical robustness and reliability. The detailed results are presented in Table 6.

The ablation results demonstrate that each core component of the proposed framework contributes positively to the overall diagnostic performance. As shown in Table 6, the baseline model (Model B) achieves the highest test accuracy of 99.375%, converges within 6780 iterations, and maintains the lowest generalization error of 0.8%, confirming the effectiveness of integrating MTF, the enhanced DarkNet19-GRU-MSA architecture, and the TLSSA optimization algorithm.

When the MTF module is removed (Model A1), test accuracy decreases significantly to 92.625%, while the generalization error increases to 2.3%, and the number of misclassified Class 8 samples increases from 1 to 4. These results indicate that the MTF transformation effectively converts one-dimensional vibration signals into discriminative image representations, providing richer structural information for subsequent deep feature extraction.

Removing only the GRU module (Model A2) reduces the classification accuracy to 97.114%. Although the model still converges efficiently, the performance degradation indicates that GRU effectively captures the temporal dependencies embedded in the extracted feature sequences, thereby improving fault discrimination. Likewise, removing only the MSA module (Model A3) decreases accuracy to 98.504%, demonstrating that the multi-head self-attention mechanism enhances feature representation by emphasizing informative regions while suppressing redundant information. Compared with Models A2 and A3, the complete framework achieves the highest accuracy, indicating that the GRU and MSA modules complement each other in improving diagnostic performance.

To further evaluate their combined contribution, both the GRU and MSA modules are simultaneously removed in Model A4, resulting in a further decrease in accuracy to 96.875%. This result confirms that temporal dependency modeling and attention-based feature enhancement jointly contribute to the proposed framework’s superior performance.

Replacing TLSSA with the original SSA (Model A5) decreases the accuracy to 97.500% and increases the convergence iterations from 6780 to 8240, demonstrating that the proposed optimization strategy accelerates convergence and improves hyperparameter optimization efficiency. Removing the transfer learning strategy (Model A6) further reduces the accuracy to 94.375% and increases the generalization error to 3.1%, confirming the effectiveness of transfer learning in improving feature extraction under limited training samples. Finally, replacing the DarkNet19 backbone with ResNet50 (Model A7) increases the training time from 42.5 min to 55.9 min while achieving a slightly lower accuracy (98.125%), indicating that the improved DarkNet19 backbone provides a better balance between computational efficiency and classification performance for the proposed fault diagnosis task.

Overall, the ablation study systematically verifies the effectiveness of each component and demonstrates that integrating MTF, GRU, MSA, transfer learning, and TLSSA yields the best overall diagnostic performance.

3.6 Applicability analysis

To further verify the applicability and generalization capability of the proposed MTF-TLSSA-DarkNet-GRU-MSA framework, additional experiments were conducted on the publicly available Southeast University (SEU) gearbox dataset. Unlike the proprietary wind farm dataset used in the previous experiments, the SEU dataset is widely adopted as a benchmark for intelligent fault diagnosis research, enabling fair comparison and improving the reproducibility of the proposed method.

The SEU dataset was collected from a Drivetrain Dynamics Simulator (DDS) under two different operating conditions (20 Hz–0 V and 30 Hz–2 V). It contains multi-channel vibration signals acquired from gearbox and bearing components, with a sampling frequency of 5120 Hz. The gearbox dataset includes five operating states, namely one healthy condition and four representative fault conditions, including root crack, chipped tooth, missing tooth, and gear wear. Each fault category contains 1000 training samples and an equal number of test samples, providing sufficient diversity for evaluating model robustness under varying operating conditions.

Using the same network architecture and hyperparameter configuration as those employed for the wind farm dataset, the proposed framework achieved an overall classification accuracy of 99.17% on the SEU gearbox dataset. As summarized in Table 7, the precision, recall, and F1-score of all five categories exceeded 98%, while both the macro-average and weighted-average metrics remained above 99%, indicating that the proposed model maintains consistently high recognition performance across different fault categories without obvious class bias. The excellent classification results demonstrate that the proposed framework can extract highly discriminative fault representations even when the data source, operating conditions, and fault characteristics differ substantially from those of the original dataset.

The confusion matrix shown in Figure 10 further illustrates the excellent classification capability of the proposed method. Most samples are concentrated along the diagonal, whereas only a few isolated misclassifications appear in the off-diagonal positions, indicating that different fault categories are effectively separated with minimal inter-class confusion. This result confirms the robustness of the proposed MTF visualization and transfer learning feature extraction framework under cross-dataset conditions.

FIGURE 10

The prediction comparison presented in Figure 11 shows that the predicted labels almost completely overlap with the corresponding ground-truth labels throughout the entire testing sequence. Only a very limited number of samples are misclassified, which is consistent with the overall accuracy of 99.17%. The absence of systematic prediction deviations demonstrates the stability and reliability of the proposed model under different operating conditions.

FIGURE 11

To further investigate the discriminative capability of the learned feature representations, the t-SNE visualization of the original vibration signals is shown in Figure 12. Before feature learning, samples belonging to different fault categories exhibit severe overlap in the embedded feature space, and no obvious decision boundaries can be observed. This phenomenon indicates that raw vibration signals contain complex nonlinear characteristics and substantial inter-class similarity, making direct classification difficult. After processing with the proposed MTF visualization and transfer learning framework, these hidden discriminative characteristics are effectively extracted, leading to significantly improved class separability and ultimately achieving highly accurate fault identification.

FIGURE 12

Overall, the experimental results obtained on the publicly available SEU gearbox dataset demonstrate that the proposed MTF-TLSSA-DarkNet-GRU-MSA framework is not limited to the proprietary wind farm dataset. Instead, it exhibits strong adaptability and generalization capability across different datasets, operating conditions, and fault types, thereby providing convincing evidence for the universality of the proposed intelligent fault diagnosis framework.

4 Discussion and future work

Although the MTF-TLSSA-DarkNet-GRU-MSA fusion model proposed in this study achieves a high classification accuracy of 99.375%, demonstrating excellent convergence and stability, there is still significant potential to further enhance its applicability in real-world industrial settings. As with many deep learning models in complex fault diagnosis tasks, the primary challenge lies in limited data sources, computational demands, and interpretability.

First, like the challenges faced by deep learning models in the Internet of Things (IoT), the experiments in this study are based on data from a single monitoring system on the same wind turbine, which limits the scope of the model’s generalization ability. To overcome this limitation, future work could expand the dataset by integrating heterogeneous data from different equipment types, manufacturers, and operating conditions. This would enable more comprehensive cross-equipment validation and provide a more rigorous assessment of the model’s performance in industrial environments that often exhibit sensor inconsistencies, structural variations, and environmental disturbances. Expanding the dataset will enhance the model’s robustness and provide a stronger foundation for its deployment at scale.

Second, similar to the challenges encountered with vibration signal-based models in rolling bearing fault diagnosis, the current model primarily relies on high-dimensional numerical features derived from vibration signals. Although techniques like MTF-based image transformation and attention mechanisms have improved the feature representation, relying on a single sensing modality may still limit the model’s ability to fully characterize equipment conditions, especially in complex industrial scenarios. To address this, future research should explore multi-sensor and multimodal data fusion strategies. By integrating vibration, temperature, acoustic emission, rotational speed, and current data, the model can capture cross-modal correlations in the equipment’s operating state. Implementing decision-level fusion or end-to-end multimodal deep learning frameworks would improve the model’s ability to detect weak faults, compound faults, and early degradation behaviors (Huo et al., 2023).

In line with recent studies in rolling bearing fault diagnosis, integrating multiple sensing modalities could significantly enhance diagnostic capabilities by providing a richer representation of the equipment’s health state. Additionally, improving the interpretability of the model remains a key goal. Just as fusing quadratic neural networks (QNNs) with bidirectional long short-term memory (Bi-LSTM) networks has improved both the speed and accuracy of fault diagnosis in rolling bearings (Wang X. et al., 2020), our future work will benefit from further advancing the interpretability of the model. Visualization techniques like attention mapping of the QNN and hidden state analysis in Bi-LSTM (Keshun et al., 2024a), as applied in recent work, could be explored to gain a better understanding of the model’s decision-making process, ultimately improving the transparency of our health monitoring system (Wu et al., 2020).

Although this work focuses on an industrial small-sample diagnosis scenario, future studies will further investigate more challenging few-shot settings (e.g., 5-shot or 10-shot) together with meta-learning frameworks. In summary, expanding data sources and incorporating multimodal sensing fusion are key directions for improving the engineering applicability and generalization capability of the proposed method. These advancements will contribute to a more intelligent, stable, and interpretable health monitoring system, ultimately improving the industrial robustness of the model and supporting large-scale deployment in real-world environments. Although this work adopted pretrained transfer learning, cross-condition transfer diagnosis will be investigated in future work.

5 Conclusion

In this study, an innovative fusion-based fault recognition model integrating Markov fields with an enhanced DarkNet architecture is proposed. By leveraging multi-technology synergies, the method achieves substantial improvements in diagnostic performance under complex working conditions. First, one-dimensional vibration time-series signals are transformed into two-dimensional image representations through Markov transfer fields, providing a more expressive and structurally informative input form for deep learning. Building upon this, the pretrained DarkNet19 network is adapted to the fault diagnosis domain via a transfer learning strategy, effectively addressing the challenge of model training in small-sample scenarios.

To further enhance feature representation, the model incorporates gated GRUs and a MSA mechanism, forming a dual-enhancement architecture capable of capturing spatiotemporal dependencies and emphasizing critical fault-related information. In addition, an improved sparrow search algorithm (TLSSA) is introduced for hyperparameter optimization. By integrating adaptive t-distribution variation and Lévy flight strategies, TLSSA significantly enhances search efficiency and ensures optimal model configuration.

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