LSLD datasets often exhibit significant class imbalance, with minority class samples representing only a small portion of the overall dataset. This imbalance leads to models biased toward the majority class, whereas overlooking the minority class. To address this problem, various data augmentation methods have been proposed, which can be categorized into transform-based, deep generative model-based, transfer learning-based, and physical model-based approaches.

(a) LSLD based on transform techniques

1) The noise injection method enhances data generalization by superimposing random noise on the original data. The core idea is to simulate the uncertainty inherent in actual scenarios. The noise injection process is described as follows: (5)x^noise=xorg+ηnoise=(x1org+x1noise,⋯,xtorg+xtnoise)⊤where x^noise denotes the augmented data after noise injection, xorg denotes the original data, and ηnoise denotes the random noise. Gaussian noise is commonly used in existing research [21–26,104]. The noise intensity often requires dynamic adjustment based on physical constraints. To this end, references [28,29] employed signal-to-noise ratios to regulate the noise level, as shown below: (6a)ρSNR=10log10(1+φ) (6b)x^noise=xorg+10ρSNR/10ηnoisewhere ρ{SNR} denotes the signal-to-noise ratio, and η denotes the noise level.

2) The geometric scaling method generates new samples by adjusting the amplitude or time scale of the original data. This method aims to simulate variations in equipment operating conditions through techniques such as amplitude scaling and time scaling.

• The amplitude scaling is defined as follows: (7)x^amp=βamp⋅xorg=(βamp⋅x1org,⋯,βamp⋅xtorg)⊤where x^amp denotes the augmented data after amplitude scaling, and βamp denotes the amplitude scaling factor. This method simulates variations in sensor range or fluctuations in equipment load [22,25–27]. The scaling factor must be restricted within the operational limits of the equipment or sensor to ensure realistic data generation.

• The time scaling is defined as follows: (8a)x^time=ftime(xorg) (8b)x^itime=x⌊i/βtime⌋org⋅(1−δi)+x⌊i/βtime⌋+1org⋅δiwhere x^time represents the augmented data after time scaling, βtime is the time scaling factor, ⌊⋅⌋ represents the floor function, and δi=i/βtime−⌊i/βtime⌋. Time scaling simulates variations in the equipment’s operating rate while preserving essential physical characteristics [22,24,25]. This technique improves the generalization of the model under variable conditions and equipment speeds.

3) The zero-masking method generates new samples by randomly setting continuous segments of the original data to zero, thereby simulating sensor failures or communication interruptions [21,22,26–29]. The process is defined as follows: (9a)x^trunc=ltrunc⊙xorg=(x1org,⋯,xλ−1org,0,⋯,0,xλ+L+1org,⋯,xtorg)⊤ (9b)litrunc={0, i∈[λ,λ+L)1, others, λ∈[0,t−L)where λ is the truncation start position, L is the truncation length, ltrunc is the binary masking vector, and ⊙ denotes the Hadamard product.

4) The time-shift method generates new samples by shifting the original time series data to the left or right, thus simulating different time delays [21,24–26,30]. Given an original sample xorg=(x1org,…,xtorg)⊤, the shifted samples are defined as: (10a)x^right=(0,⋯,0,x1org,x2org,⋯,xt−korg)⊤ (10b)x^left=(xk+1org,xk+2org,⋯,xtorg,0,⋯,0)⊤where x^right and x^left denote the right-shifted and left-shifted samples, respectively, and k is the shift length.

5) The flip method reverses the order of the time series data to improve the model’s generalization capability [26,27]. Given an original sample xorg=(x1org,…,xtorg)⊤, the flipped sample is defined as: (11)xflip=(xtorg,xt−1org,…,x1org)⊤

Transform-based data augmentation methods apply various transformation strategies within the bounds of physical constraints to generate new samples and expand the dataset. Although these techniques improve the diversity of the time series data, their applicability is typically limited to specific scenarios. Therefore, it remains challenging to establish a unified framework for their general application.

(b) LSLD based on deep generative models

Deep generative models based on neural networks have emerged as a prominent direction in data augmentation research [105]. These methods aim to learn the latent data distribution using neural architectures, generating new samples through sampling from them. Among the various models, the variational autoencoder (VAE) and the generative adversarial network (GAN) are two representative and widely used methods, as illustrated in Fig. 9. Moreover, several studies [106] have transformed one-dimensional (1-D) signals into two-dimensional (2-D) images using techniques such as the fast Fourier transform [107], the continuous wavelet transform [108,109], and the Gramian angular field [110]. These transformations enable diffusion models, originally developed for image data, to be applied to LSLD scenarios. The details of three deep generative models are presented in Table 2.

1) VAEs can learn the original data distribution and generate new samples by sampling the latent space [31]. To improve the performance of VAEs for industrial data, existing studies have primarily focused on optimizing the objective function and network architectures.

• In terms of objective functions, Dixit et al. [32] proposed a fault diagnosis framework based on a modified conditional VAE with centroid loss, which effectively enhances the generation of virtual samples. Zhao et al. [33] developed a modified Wasserstein autoencoder that integrates a squeeze-and-excitation attention mechanism and utilizes the sliced Wasserstein distance along with a gradient penalty to improve sample similarity. In another study, Zhao et al. [34] further improved the representational capacity of VAEs by introducing a Gaussian mixture prior and optimizing it via the expectation-maximization algorithm.

• In terms of network architectures, Karamti et al. [35] introduced a stacked VAE framework for multi-fault machinery identification, in which a VAE is used for data augmentation and two sparse autoencoders are used for feature extraction. Han et al. [114] proposed a VAE incorporating long short-term memory units to capture temporal dependencies and generate realistic virtual samples. To address the limitations of non-Gaussian signals and sparse fault data, Luo et al. [115] developed a mixture network VAE by integrating Gaussian mixture models and Weibull distributions. Wang et al. [116] combined the noise robustness of VAE with the expressive capacity of convolutional neural networks (CNNs) to enhance diagnostic precision and robustness. Zhang et al. [117] proposed a multi-scale dilated variational convolutional autoencoder, which integrates a multi-scale dilated convolutional attention mechanism and a graph convolution module to capture both hierarchical features and inter-sensor dependencies. For CNN structure, Zeng et al. [118] introduced an attention-guided hierarchical wavelet CNN that integrates multi-layer wavelet decomposition, a time-frequency attention module, and gradient-weight fusion to simultaneously suppress noise and highlight fault-relevant frequency bands.

2) The training process of GANs is formulated as a minimax game between two neural networks: the generator and the discriminator. The overall performance of GANs is influenced by the structures of both components. Moreover, the stability and convergence of the training process depend on the design of loss functions. Consequently, existing studies have focused on three aspects: objective functions, generator structures, and discriminator structures. The development trend of these studies is summarized in Fig. 10.

• Regarding objective functions, the Wasserstein distance has been used as a replacement for the Kullback-Leibler (KL) divergence, effectively alleviating problems such as gradient vanishing and mode collapse [36]. Based on this advancement, [37] integrated the Wasserstein distance with a hierarchical feature matching loss to constrain local category similarity, thus improving the quality and validity of generated samples. Subsequently, reference [38] introduced a gradient penalty term to further stabilize the training process and reduce the risk of mode collapse. Moreover, studies such as [39,40] incorporated the Wasserstein distance and the gradient penalty into the time series GAN and the auxiliary classifier GAN, respectively, resulting in improved model convergence.

• Regarding generator structures, Pan et al. [119] enhanced the generator by incorporating an additional sequence into the condition, allowing the generation of more abundant samples. Zareapoor et al. [120] proposed a minority oversampling GAN, where the generator learns a mixture data distribution to generate samples from the minority class. Zhang et al. [121] introduced an adaptive decoupling strategy in the generator, which adjusts each intermediate output using labels to prevent mode collapse and improve sample diversity. Xu et al. [122] employed multiple generators to generate virtual samples from the real data distribution, thus increasing sample variety and reducing the risk of mode collapse. Ren et al. [123] improved the generator through pre-training based on the majority classes, followed by fine-tuning with anchor samples. This approach preserves the learned distribution from pre-training while ensuring that the generated samples remain close to real ones. Huo et al. [124] incorporated a residual mixed self-attention module into the generator to effectively extract time- and frequency-domain features. Chen et al. [125] utilized a serial CNN-Transformer architecture as the foundation of the generator to capture long-range dependencies and improve understanding of local features.

• Regarding discriminator structures, Wang et al. [126] proposed a GAN framework where the generator generates new samples to expend the dataset, and a stacked denoising autoencoder serves as the discriminator to extract robust fault-related features, thereby enhancing the model’s fault classification capability. Ding et al. [127] introduced a multi-discriminator structure designed to learn data distributions associated with different health states, improving performance through a semi-supervised learning strategy.

3) The denoising diffusion probabilistic model (DDPM) is a representative diffusion-based generative model widely applied in data augmentation, as illustrated in Fig. 11. Reference [41] integrated a classifier-free denoising diffusion implicit model with multiclass contrastive learning to tackle challenges posed by limited fault samples and variable operating conditions. Reference [42] combined DDPM with a physical simulation model to generate diverse fault samples. In addition, reference [43] proposed a DDPM-based method that generates high-fidelity images, effectively enhancing the diversity of feature representations. Reference [44] introduced a lightweight DDPM variant that incorporates multi-dconv head transposed attention, reducing computational complexity while improving the model’s ability to capture fine local details.

(c) LSLD based on transfer learning

Transfer learning-based methods generate target domain data through cross-domain knowledge transfer. These methods utilize the distribution and feature relationships of the source domain to generate new samples in the target domain that conform to actual operating conditions. Recently, Li et al. [128] proposed a cross-domain augmentation method, which employs convex combinations of data and feature-label pairs to create an augmented domain. He et al. [45] introduced an attention-based cross-domain adaptive GAN that applies attention mechanisms for adaptive feature selection and incorporates correlation alignment regularization to enable transferable data augmentation between varying machining parameters. Ge et al. [46] developed a transfer learning method based on multiple mixed augmentations, which combines auto-augment-driven dynamic strategies with multi-stage data mixing. This method enhances cross-condition fault diagnosis by adaptively extracting transferable features from limited vibration data. Jian et al. [47] proposed an auxiliary classifier GAN (ACGAN) to generate open-set samples. This method used the multi-domain mixup to expand fault representation diversity across operating conditions, while ACGAN generates targeted open-set data, as illustrated in Fig. 12. Mu et al. [48] proposed a task-oriented meta-learning network based on the Theil index, incorporating a gradient calibration strategy to address the challenge of cross-domain fault diagnosis in rotating machinery under extremely limited sample conditions. This method integrates a fine-grained feature learner, a task-specific Theil index, and a gradient calibration mechanism to enhance model generalization and stability across diverse industrial scenarios. Wang et al. [49] proposed a dynamic collaborative adversarial domain adaptation network that adaptively adjusts the generator and adversarial components to enable unsupervised fault diagnosis of rotating machinery across multiple source domains, without requiring labeled data from the target domain.

(d) LSLD Data augmentation methods based on physical models

Physical model-based methods simulate the behavior of actual systems by establishing mathematical models. These methods are generally divided into two categories: numerical simulations and digital twins. Numerical simulations reproduce the dynamic behavior of physical systems through mathematical equations and computational techniques, while digital twins create real-time, dynamic digital representations of physical entities via virtual-real mapping, as illustrated in Fig. 13. Current research primarily focuses on developing physical models for rotating machinery, such as gears, bearings, and rotors, due to their well-defined physical characteristics and mechanical principles, which facilitate the construction of numerical simulations and digital twin models [129,130].

1) Regarding numerical simulations, references [50,51] developed simulation models of rotors and propellers to generate virtual fault samples, thus alleviating data imbalance problems in neural network-based fault diagnosis.

2) Regarding digital twins, Cai et al. [52] developed a digital twin model of a triplex pump to generate diverse fault data. Qin et al. [53] proposed a data augmentation method based on digital twin for rolling bearings to address class imbalance in fault diagnosis. Zhang et al. [54] constructed a digital twin-based rolling bearing model that integrates virtual modeling with transformer-based discrepancy learning and multi-loss alignment. Ming et al. [55] introduced a virtual-physical component fusion method that incorporates frequency-adaptive filtering and a feature fusion self-attention network to align subdomain features and effectively alleviate distribution discrepancies in imbalanced bearing fault diagnosis.

In this context, various data augmentation methods have been developed to address the challenges of LSLD data. Transform technique-based methods quickly generate new samples through simple signal transformations, offering high computational efficiency and strong interpretability. However, they often lack data diversity and struggle to provide a unified, generalizable framework. Deep generative model-based methods learn the data distribution to generate complex virtual samples, improving model generalization. Despite these benefits, such methods have high training complexity and unstable generation quality. Transfer learning-based methods use cross-domain knowledge transfer to alleviate data scarcity, enhancing adaptability under different operational conditions. However, their effectiveness is sensitive to domain discrepancies and feature alignment. Physical model-based methods expand datasets with strong physical interpretability and high fidelity but involve high modeling costs and offer limited adaptability.

The sparsity introduced by the CoD often causes overfitting in data-driven models. Therefore, data augmentation methods for SSHD data need to address the challenges associated with high dimensionality during sample generation processes. In this regard, statistical technique-based, deep generative model-based, and transfer learning-based approaches have been developed to effectively augment SSHD data and improve model generalization.

(a) SSHD based on statistical techniques

Statistical technique-based methods aim to learn the latent distribution of the original dataset and generate new samples, using statistical models in sparse areas, as illustrated in Fig. 14. However, directly applying traditional statistical methods to high-dimensional data poses significant challenges due to CoD. Feature engineering serves as an effective solution to alleviate these limitations by reducing dimensionality and highlighting relevant features.

Feature selection methods aim to obtain a subset of features that are most relevant to the target variables [56–58]. In contrast, feature extraction methods transform the original high-dimensional data into a low-dimensional space to capture essential patterns and reduce redundancy [59]. These techniques are effective in capturing nonlinear characteristics and enhancing low-dimensional representations. For example, reference [60] employed locally linear embedding to map high-dimensional data into a lower-dimensional space, which facilitated the identification of sparse areas for virtual sample generation (VSG). Similarly, reference [17] applied Isomap to embed data in a two-dimensional space, enabling a sparse area detection and interpolation-based augmentation. In [61], t-SNE was used for feature extraction and then virtual samples were generated by interpolation in the reduced space. In addition, reference [62] used discriminant locality preserving projection to handle high-dimensional data and introduced Monte Carlo sampling for VSG. Reference [63] proposed detecting sparse regions based on projection point spacing and then generated new samples based on midpoint and radial basis function interpolation. Finally, reference [131] utilized singular value decomposition to extract principal features, thus effectively addressing the challenge of high dimensionality.

In addition, multi-model methods have been explored to address the complexities associated with SSHD data. For example, reference [64] proposed a co-training strategy that detects sparse areas and fills them through interpolation. This method dynamically selects virtual samples based on model performance, improving the quality and relevance of the generated data. In another study, reference [65] introduced an adaptive data augmentation approach using generalized correntropy, which effectively utilizes α-order statistics to learn relationships between samples. This method is suited for scenarios such as high dimensionality, non-Gaussian noise, and data uncertainty.

(b) SSHD based on deep generative models

High-dimensional process data exhibit more complex and diverse characteristics compared to low-dimensional data, which presents significant challenges for training deep generative models. Therefore, the development and training of such models for SSHD data should be designed for specific tasks.

1) VAEs learn the low-dimensional latent distribution of the original data and generate new samples accordingly. To address information loss during layer-wise pretraining, Yuan et al. [66] generated virtual samples at each layer through linear interpolation between adjacent samples, thereby improving the training of stacked autoencoders. Jiang et al. [67] introduced a causality-informed VAE to alleviate data sparsity and high dimensionality in just-in-time learning for soft sensor applications. Tian et al. [4] developed an adaptive loss function and utilized kernel mean matching to assign weights to virtual samples, thus improving data quality and learning performance. Moreover, Xu et al. [68,69] integrated VAE with SMOTE and Glow models to stabilize the training process and enhance sample generation.

2) Training GANs with SSHD data often faces significant challenges, such as gradient vanishing and mode collapse. To address them, references [70,71] introduced a gradient penalty into the Wasserstein GAN (WGAN-GP), which stabilizes the training process and enhances the quality of generated samples. Based on this foundation, reference [72] incorporated a deep neural network regressor into a conditional WGAN-GP framework to address the small sample problem in soft sensor. Subsequently, Jiang et al. [73] extended this architecture to a multi-generator setting, enabling more effective handling of complex data imbalances. To address the any-shot learning problem in industrial fault diagnosis, Zhuo et al. [74] proposed a data augmentation method that integrates auxiliary fault attribute information with GANs. Moreover, Cui et al. [132] introduced fuzzy set theory into the GAN framework, which effectively alleviates uncertainty and high-dimensional challenges in data augmentation. Liu et al. [133] developed a robust VAE to constrain the generator’s sampling space, while density-based spatial clustering of applications with noise clustering was introduced to guide the generation process. Finally, Zhang et al. [134] incorporated regression loss assistance into a conditional StyleGAN to generate high-quality virtual samples.

(c) SSHD based on transfer learning

Recently, transfer learning has gained attention as a promising solution for SSHD data. Ren et al. [75] proposed a low-rank joint domain adaptation network, which enhances the effectiveness of industrial small-sample datasets by extracting discriminative features and aligning cross-domain sample distributions. Furthermore, Zhu et al. [76] introduced a data self-generating transfer learning framework designed for chemical process fault diagnosis under limited sample conditions. Li et al. [77] developed the dual adversarial and contrastive network, a data augmentation method that combines adversarial and contrastive learning. It is designed to generate virtual samples while simultaneously extracting domain-invariant feature representations from single-modality data.

In this context, various data augmentation methods have been developed to address the challenges associated with SSHD data. Statistical technique-based methods typically perform dimensionality reduction before VSG and employ interpolation in reduced spaces. These methods are valued for their simplicity and interpretability but struggle to capture the characteristics inherent in high-dimensional data. Deep generative model-based methods, on the other hand, utilize neural networks to extract features and learn latent distributions to generate virtual samples. Transfer learning-based methods address domain discrepancies by transferring knowledge from source to target domains, incorporating adversarial or contrastive learning mechanisms to generate domain-invariant features and samples. In summary, these methods alleviate overfitting risks and improve model robustness in SSHD scenarios.

SSLD data present challenges due to their limited latent information, which causes underfitting and hinders model performance. Data augmentation methods based on statistical techniques and deep generative models have been introduced to address the above problems and expand the size of the dataset.

(a) SSLD based on statistical techniques

In statistical techniques, data augmentation methods are generally classified into distributional assumption-based and interpolation-based approaches, depending on their generative principles. These studies are summarized in Table 3.

1) Distributional assumption-based methods fit a probability distribution to existing data and then generate new samples by sampling from the fitted distribution. The low dimensionality of SSLD data facilitates the modeling of their distributional characteristics. For example, in [78], the generalized internalized kernel density estimation was used to incorporate time-dependent data properties and address the challenges of knowledge acquisition in early-stage manufacturing systems. Reference [79] introduced a maximal p-value method based on the two-parameter Weibull distribution to generate virtual samples and evaluate product lifetime performance in scenarios with limited datasets. In [80], the small Johnson data transformation was proposed to normalize small datasets for effective VSG.

Information diffusion theory [136], which integrates distributional assumptions with fuzzy set theory, has been applied to estimate feature expansion ranges. Based on this, the mega-trend diffusion (MTD) method introduces an asymmetric diffusion mechanism to generate virtual samples [135], as illustrated in Fig. 15a. Liu et al. [81] further improved the MTD method by integrating PSO, allowing an accurate prediction of forming forces in single-point incremental forming processes under the conditions of limited data. To address challenges in supervised and unsupervised learning with small samples, Sivakumar et al. [82] proposed a modified version of MTD, termed k-Nearest Neighbor MTD.

2) Interpolation-based methods represent another prominent direction in data augmentation research. These techniques generate new samples based on relationships between existing data points, using linear or nonlinear interpolation.

• A representative linear approach is the synthetic minority over-sampling technique (SMOTE) [137], which interpolates between samples from minority class, as illustrated in Fig. 15b. To address limitations of standard SMOTE, Mathew et al. [83] proposed weighted kernel-based SMOTE, which performs oversampling in the feature space of support vector machines to better preserve minority class characteristics. Soltanzadeh et al. [84] introduced range-controlled SMOTE to alleviate class overlap near decision boundaries. Li et al. [85] proposed multi-vector stochastic exploration oversampling, which generates diverse virtual samples through randomized direction and scaling vectors. Khan [138] proposed a balanced weighted extreme learning machine (WELM) that integrates k-fold sampling strategies, such as SMOTE and TomekLinks, with a cost-sensitive WELM framework to reduce data complexity and improve minority class classification in imbalanced datasets.

• For nonlinear interpolation, He et al. [16] developed the nonlinear interpolation VSG method, integrated with neural networks to improve the accuracy of energy prediction. In another study, Zhu et al. [86] utilized a distance-based criterion to detect sparse data regions and employed Kriging interpolation to generate virtual samples. Similarly, Song et al. [87] proposed data augmentation with weighted interpolation for the VSG method, which improves sample quality for soft sensing models.

(b) SSLD based on deep generative models

For classification tasks, deep generative models play a crucial role in expanding minority class datasets to alleviate class imbalance and improve model performance. In [88], an augmented generator was introduced to produce high-quality virtual samples representing abnormal states. This approach incorporated augmented filter layers and batching techniques to enhance the diversity and reliability of the generated data. Then, reference [89] integrated a multi-head attention mechanism into a GAN framework, effectively improving the sample quality and modeling performance. Furthermore, reference [90] proposed two augmentation strategies, individual-based and concatenation-based, which enhanced the representation of minority classes and effectively balanced the dataset.

For regression tasks, deep generative models must generate virtual input-output pairs that maintain their mapping relationship. In [91], Gaussian noise was injected into the features extracted by a target-relevant autoencoder to generate informative input-output sample pairs. In [92], labels were incorporated as conditional information in a GAN framework, allowing the generation of labeled samples specifically designed to populate the sparse regions of the feature space. Then, reference [93] integrated GAN with vine copula regression to address the challenge of limited labeled data in complex chemical processes. In [94], a supervised variational autoencoder and a WGAN-GP were combined to generate high-quality labeled samples, thus improving the accuracy and robustness of soft sensor models. In [95], the TimeCVAE model incorporated time-dependent features and conditional information into the virtual sample generation process, improving the performance of time series prediction.

In this context, various data augmentation methods have been proposed for SSLD data. Statistical technique-based methods emphasize simplicity and computational efficiency. These methods are well-suited to low-dimensional data, where they can effectively infer latent information. However, they often face limitations when modeling complex, high-dimensional relationships. In contrast, deep generative model-based methods aim to learn latent data distributions through neural networks. Although these models are highly expressive and capable of capturing complex patterns, they require careful hyperparameter tuning to ensure training stability. Consequently, statistical methods provide greater interpretability and reliability in low-dimensional scenarios, whereas deep generative models demonstrate superior adaptability and performance.

High dimensionality in industrial datasets often results from multi-source data fusion across different processes, causing massive and complex datasets. However, these high-dimensional datasets contain substantial noise and redundant features. As a result, the research focus should shift from achieving ‘sufficient data size’ to ensuring ‘sufficient data quality and scenario coverage’. In this context, deep generative model-based and physical model-based data augmentation methods have been developed to address these challenges.

(a) LSHD based on deep generative models

For LSHD data, developing effective deep generative models remains a challenging task due to data sparsity and variability. Recent studies have addressed these limitations through task-specific model designs. For example, reference [96] proposed three data-level augmentation strategies:

a VAE-based augmentation approach that reconstructs virtual samples by learning latent feature distributions;

In [97], a complementary classifier was incorporated into a basic GAN framework to augment minority class and improve classification accuracy through collaborative adversarial training. Then, reference [98] proposed a data augmentation framework that integrates cooperative and competitive strategies. The cooperative strategy utilizes cross-training and parallel training among multiple GANs to improve diversity and efficiency. In contrast, the competitive strategy introduces a filtering mechanism to retain only high-quality generated samples. Zaman et al. [99] developed a comprehensive framework combining preprocessing, GAN-based augmentation, automated feature selection, and predictive modeling to alleviate class imbalance and enhance generalization to unseen data. To support privacy-preserving learning, Feng et al. [100] integrated CVAE and GAN to generate balanced virtual samples in federated learning environments and used geometric median aggregation to ensure privacy preservation. Furthermore, Zhang et al. [101] employed Isomap for dimensionality reduction while preserving critical data structures and used CGAN to generate additional fault samples.

Furthermore, for high-precision quality prediction tasks, the availability of sufficient training samples is critical. To address this, reference [139] proposed a two-phase framework that first introduced TimeGAN to generate temporally consistent data to impute missing values and then used minimal gated unit to enable efficient quality prediction with reduced computational complexity.

(b) LSHD based on physical models

Huang et al. [102] proposed an edge-intelligent digital twin framework that utilizes edge-cloud collaboration and real-time data processing to identify early-stage faults in automation systems. In another study, Krespach et al. [103] introduced a hybrid data augmentation method that combines historical data with virtual data generated from a digital twin. This method addresses the limitations of traditional data-driven predictive control by generating virtual samples that cover previously unexplored operational conditions.

In this context, data augmentation methods for LSHD data have been preliminarily explored. Among them, deep generative model-based methods effectively address the challenges of high-dimensional complexity and class imbalance by learning latent data distributions. However, these methods often involve complex training procedures, high computational costs, and risks of pattern collapse. In contrast, physical model-based methods generate high-quality samples grounded in physical mechanisms, enabling the capture of complex dynamic behaviors and improving prediction reliability. However, their applicability to unseen operating conditions remains limited due to strict model structures and high modeling overhead.

The chemical process involves the transformation of raw materials into final products through chemical and physical operations. It is a critical industry where accurate process monitoring and key performance index prediction are essential. Numerous studies have focused on these tasks using simulated and actual datasets. A typical benchmark is the Tennessee Eastman (TE) platform, a simulation system modeled on an actual chemical reaction process [140,141], as shown in Fig. 16. The TE dataset includes 11 manipulated and 41 measured variables across various operating conditions, as it is suitable for evaluating data augmentation methods. Beyond simulations, researchers have also investigated actual industrial scenarios, such as purified terephthalic acid and ethylene production processes, to assess the practical effectiveness of proposed data augmentation methods.

1) To alleviate the class imbalance in TE datasets, various data augmentation methods have been proposed. For example, Jiang et al. [73] developed a multi-generator framework to overcome the limitations of single-generator models in handling complex imbalance scenarios. Zhuo et al. [74] introduced auxiliary fault attributes into a GAN framework, enabling VSG for rare and unseen faults, thus supporting zero-shot and few-shot fault diagnosis. Furthermore, Xu et al. [68] proposed a deep encoder-decoder network integrated with SMOTE to balance data distribution, complemented by an ensemble classifier to improve diagnostic accuracy. Transfer learning has also gained advances in augmented TE datasets. Ren et al. [75] extracted simplified and accurate fault features from multiple conditions to augment data under the current condition. Similarly, Zhu et al. [76] utilized adversarial learning to transfer features of real samples for VSG, paired with a model-based transfer learning approach to improve robustness against low-quality generated samples. In addition, Li et al. [77] addressed single-source domain generalization by generating virtual samples and learning domain-invariant feature representations, thereby enabling cross-mode fault diagnosis.

2) In purified terephthalic acid (PTA) processes, the conductivity of the solvent dehydration tower is a key quality index, so the development of accurate soft sensors is important. However, due to the SSHD characteristics of the PTA data, multiple learning techniques such as locally linear embedding [60], Isomap [17], and t-SNE [61] have been widely employed to reduce dimensionality and identify sparse areas to generate samples based on interpolation. For example, reference [63] utilized projection point spacing in the low-dimensional feature space to detect sparsity and then generated virtual samples using midpoint and radial basis function interpolation. Similarly, reference [131] applied the singular value decomposition to extract the principal features and expand datasets and generated the corresponding output values using the gradient boosting decision tree. In another study, reference [64] generated virtual samples through interpolation and used co-trained KNN regressors to generate the associated outputs. Furthermore, Xu et al. [69] proposed a Gaussian VAE-based VSG method, which employs improved least squares regression to generate high-quality virtual outputs.

3) In ethylene production processes, accurate energy analysis is vital to optimize industrial operations. However, the limited data on energy consumption pose significant challenges. To address this, data augmentation techniques have been widely utilized. For example, reference [16] applied interpolation within the hidden layers of neural networks to generate virtual input samples, while employing the Moore-Penrose generalized inverse to estimate the corresponding output values. Similarly, reference [86] utilized dimension-wise interpolation via Kriging to create feasible virtual samples in sparse regions, thus improving the prediction accuracy of soft sensors. In addition, several studies have focused on deep generative models to further improve the quality of generated samples. Reference [91] introduced Gaussian noise into latent features extracted from a target-relevant VAE, generating informative input-output pairs. Meanwhile, in [92], a combination of local outlier factor and K-means clustering was used to identify sparse regions for input generation, and a CGAN was used to generate the corresponding output values. These approaches significantly improved the performance of data-driven soft sensor models under limited data.

Rotating machinery includes mechanical systems that perform energy conversion or transmission through rotational motion. As essential components in modern industrial systems, their health monitoring and fault diagnosis are crucial to ensuring operational safety and reliability [142,143]. To facilitate theoretical research and validation of diagnostic methods, several benchmark datasets have been developed. Among them, the Case Western Reserve University (CWRU) bearing dataset is one of the most widely utilized in the diagnosis of rotating machinery faults [144], as illustrated in Fig. 17. This dataset includes signals collected under normal conditions as well as various fault scenarios, such as 12k and 48k drive-end bearing faults and fan-end bearing faults. Similarly, the University of Connecticut (UoC) gearbox dataset provides experimental data from a two-stage gearbox system equipped with interchangeable gears [145]. It involves a series of conditions, including healthy operation, missing teeth, root cracks, spalling, and chipped teeth. Due to the limited availability of fault data, numerous data augmentation methods have been proposed to expand datasets and enhance robustness and generalization.

The data augmentation methods based on transform technique generate new samples by applying various transformations to original vibration signals. Common techniques include noise injection, geometric scaling, zero-masking, time-shifting, and signal flipping [21–30,104]. In recent years, deep generative models have become a major focus in machine fault diagnosis. Among these, VAEs employ an encoder to extract distributional representations and a decoder to reconstruct low-dimensional latent variables [31–35,114–117]. GANs use adversarial learning between a generator and a discriminator to generate high-quality virtual samples [36–40,119–127]. Based on these, the conversion of 1-D vibration signals into two-dimensional 2-D images has enabled the use of more advanced deep learning algorithms. Recently, diffusion models based on Markov chain processes have emerged as an effective generative framework that offers stable training and high-quality image generation [41–44]. Meanwhile, transfer learning techniques use data and knowledge from a source domain to generate more diverse and informative samples for a target domain [45–49,128]. Finally, several studies have introduced numerical simulations [50,51] and digital twin models [52–55], which are based on physical laws and mechanistic knowledge, to expand datasets and improve model generalization capabilities.

The MSWI process has emerged as the primary approach to managing MSW, due to its advantages in innocuity, reduction, and reuse. The MSWI process comprises several stages: feeding, combustion, heat exchange, gas cleaning, and emissions [5,146], as illustrated in Fig. 18. Among emission by-products, dioxins (DXNs) represent a critical environmental pollutant because of their high toxicity and persistence. However, DXNs are difficult to detect online, resulting in a limited number of effective samples with true values [5]. Moreover, since DXN emissions are influenced by the whole MSWI process, the DXN data are high-dimensional. Therefore, data augmentation methods have become essential for enabling data-driven modeling of DXN emissions.

For high-dimensional DXN data, dimensionality reduction is a critical preprocessing step. In [56,58,147,148], features were selected based on domain knowledge and expert experience to reduce data complexity. In [132], the random forest algorithm was introduced to achieve a subset of features relevant to DXN emissions by modifying the model performance. In [59], the principal component analysis was applied to extract low-dimensional features that capture the dominant variance in the data.

Based on this basic, reference [56] improved the MTD method by expanding the range of input features and employing equal-interval interpolation to generate virtual input-output pairs. Similarly, reference [58] used MTD and interpolation techniques to generate virtual samples and further enhanced the PSO algorithm to select high-quality samples. Then, reference [147] used a multi-objective PSO approach to identify optimal virtual samples for soft sensor modeling, balancing model accuracy with minimal sample selection. Reference [59] generated virtual samples along independent principal components using kernel density estimation and obtained virtual outputs through mapping models. Furthermore, reference [148] integrated the active learning mechanism with GAN to generate virtual samples, building an accurate and robust DXN warning model. In reference [132], fuzzy set theory was incorporated into the GAN framework to address uncertainty and facilitate the generation of high-quality samples.

Building on this foundation, the applications of data augmentation techniques in representative industrial processes are summarized in Table 4. From a managerial perspective, practitioners can make informed decisions regarding model deployment by applying data augmentation techniques to specific data regimes and industrial conditions, as well as guiding data acquisition planning and digital strategies. This survey is intended to serve as a comprehensive reference for mapping data-related challenges to suitable data augmentation strategies within a range of industrial constraints.