An effective strategy for online safety assessment is the guarantee and fundamental to ensuring the safe and stable operation of industrial systems. However, in increasingly dynamic and complex industrial systems, operational condition transitions and production state changes may cause the probability density function (PDF) to exhibit notable irregularities or even severe collapse. This renders the conventional methods for process safety index calculation no longer applicable, let alone conducting online safety assessments based on the process safety index. To solve the problem, a modified process safety index calculation method based on the Gaussian mixture variational autoencoder (GMVAE) is proposed in this article. This method, through the nonlinear capability of GMVAE, transforms the irregular PDFs into Gaussian-distributed PDFs in the latent space, and then denotes the probability within the safety region determined by the PDFs as the process safety index. Afterward, the process safety index calculated in the latent space is mapped back to the original feature space for safety assessment in real-world scenarios. Finally, the experimental case is designed on the actual industrial process to verify and validate the effectiveness and superiority of the proposed method.

Cross-condition fault diagnosis of planetary gearboxes based on domain generalization (DG) has gained significant attention recently. However, most existing methods mainly rely on data-driven approaches and neglect the physical characteristics of planetary gearboxes. Moreover, signals collected in real industrial environments are often corrupted by noise, which reduces the generalization ability of diagnostic models. To address these challenges, this article proposes an angular margin consistency-based physical Transformer (AMC-PT). First, the model incorporates a physical encoding layer that enhances fault-related modal characteristics and suppresses noise, thereby improving cross-condition diagnosis under noisy environments. Second, instead of extracting only domain-invariant features, the model employs an angular margin consistency loss to capture stable relationships between each fault category and the healthy category across conditions. Lastly, a two-step inference strategy is designed to refine predictions under target conditions. Experiments on 18 cross-domain fault diagnosis tasks demonstrate that the proposed AMC-PT significantly improves performance in noisy environments and exhibits strong generalization.

For fault diagnosis under unseen working conditions (WCs), it is crucial to extract general knowledge unrelated to data distribution from available source data and identify transferable discriminative features. However, WC-related information is often tightly coupled with health state (HS)-related information, making it difficult to directly distinguish their contributions, posing challenges to fault diagnosis. To address this issue, a novel approach named WC-decoupled and invariant-feature fusion transformer (WCD-IFFT) is proposed, which aims to minimize the impact of WCs by extracting transferable features closely related to HSs. Specifically, two key components are designed to decouple WC-related features from HS-related features: orthogonality separation and decouple loss. Additionally, to enrich the semantics of HS-related features, time-domain and Fourier phase features are mapped into a unified space and fused, combining the instantaneous changes of time-domain signals with frequency-domain distribution information to enhance the feature representation capability. Extensive experiments on cross-domain fault diagnosis tasks demonstrate the effectiveness of the proposed method.

Domain generalization methods have attracted increasing attention in fault diagnosis within the Industrial Internet of Things (IIoT), as they aim to transfer diagnostic knowledge from source conditions to unseen target conditions. However, the inherent time-varying characteristics of planetary gearboxes further intensify personalized bias induced by varying working conditions, thereby making diagnostic model generalization more challenging. To address this challenge, a Transformer-based domain generalization architecture is developed. First, the architecture leverages the global attention mechanism of the Transformer, incorporating new patch encoding and Token designs. Second, a multi-source domains adversarial learning method is employed to extract domain invariant features that are robust to condition shifts. Finally, a causal trend alignment loss is introduced. It reduces personalized bias across different working conditions by aligning the causal trends of the same fault class. Experiments demonstrate that the proposed architecture improves average diagnostic accuracy by 5.68% under cross-speed settings and 2.46% under cross-machine settings, outperforming state-of-the-art baselines.

In this article, we propose a feature distillation-based uniformity few-shot domain adaptation (FUFD), for cross-domain fault diagnosis with sample shortage. To address the the few-shot problem, a uniformity prototypical contrastive network is designed to improve the data sensitivity of the model. Compared to the vanilla prototypical network, the learned prototypes contain more information about fault classes by encoding semantic structure information into the feature space while dynamically estimating the distribution concentration around each class prototype. Uniformity and correlation principles are introduced to alleviate prototype collapse: the uniformity principle ensures balanced prototype distribution, while the correlation principle enhances the diversity and distinctiveness of prototypical features. In addition, a cross-domain feature distillation-based domain adaptation module is designed to address significant domain shift. This module softens the class-specific information to capture more domain-consistent information and avoid overfitting to source working condition. Finally, experiments and ablation studies on cross-domain bearing fault diagnosis tasks with limited samples validate the effectiveness of FUFD and its individual modules in enhancing few-shot cross-domain fault diagnosis performance.

A certainty and transferability guided few-shot domain adaptation network is proposed to address few-shot open-set cross-domain fault diagnosis in this article. The proposed method is composed of a feature extractor, a certainty-guided prototypical contrastive module and a transferability weighting domain adaptation module. The certainty-guided prototypical contrastive module based on samples informative importance is designed to enhance the data sensitivity with limited samples while achieving well class separation for open-set scenarios. The module infers informative importance of samples to guide method learn more effective representations. Meanwhile, correlation and uniformity principles are incorporated to alleviate prototype collapse. The transferability weighting domain adaptation module is designed to address great domain gaps and negative transfer caused by asymmetrical label spaces. The module quantifies sample transferability and down-weights the irrelevant samples based on their transferability scores. Experimental results on few-shot open-set cross-domain bearing fault diagnosis tasks demonstrated the superior and effectiveness of the proposed method.

Unsupervised domain adaptation is widely used for fault diagnosis under variable working conditions. However, loss oscillation and slow convergence, which are caused by the dynamically varying alignment of targets during domain adaptation, are ignored. Therefore, a Gaussian mixture variational based transformer domain adaptation (GMVTDA) fault diagnosis method is proposed. A feature extractor based on transformer layers is designed to capture long-term dependency information and local features. Subsequently, a domain alignment term is proposed to project the features learned from both working conditions into the common assistance distribution and make them follow the same distribution after the alignment process. Additionally, considering that fault diagnosis is a multiclassification process, a Gaussian mixture is utilized to build the common assistance distribution. Ultimately, the proposed GMVTDA is applied to bearing fault diagnosis under variable working conditions, and the experimental results prove its effectiveness.

Deep state space model (DSSM) are a temporal model that is actively researched. It combines deep neural networks with classic state space model (SSM) so that it can be used for prediction and identification of dynamic system. However, DSSM treats observation as multiple 1-D variables instead of one multidimensional variable, which does not make the most of correlation information in observation. On the other hand, DSSM cannot combine the mechanism of the system. Therefore, we propose a new model called mechanism-assisted deep state space model (MA-DSSM). We use the multidimensional time-varying SSM to describe the temporal structure of system observation. System observation is assumed to be a multidimensional random variable. Two recurrent neural networks are used to extract static and dynamic features of the system. These measures enhance the prediction performance of the model. In addition, we design conditional mask which can finely and flexibly combine prior knowledge in the SSM with DSSM. So that MA-DSSM can combine data with mechanism and further improve performance. This article introduces the detailed structure of MA-DSSM, the calculation of training loss and the process of prediction. Three numerical experiments are used to verify the performance of MA-DSSM. The experimental results show that MA-DSSM has the best prediction performance with the assistance of system mechanism. The prediction performance is also better than the comparison model without the assistance of system mechanism.

Aiming at the problem of colored noise in the signal, this article proposes a subspace frequency estimation approach under colored noise with application to fault diagnosis of motor rolling bearings. First, a nonlinear discrete-time system is described to generate colored noise. An extended I/O model with parameters of a nonlinear discrete-time system is given by the subspace method. Then, the gap metric-aided system order determination approach is developed for extended observability matrix identification. Then, the data-driven diagnostic observer parameter identification approach and the fast approximate power iterative subspace method are adopted to realize online monitoring for frequency change detection. Eventually, a data-driven design scheme of residual generator is proposed for the implementation of fault detection. The effectiveness of the proposed methods is verified for fault diagnosis performance through numerical simulations and the experimental measurements from the dynamic motor rolling bearing experiment rig.

Accurate bearing fault diagnosis is of great significance of the safety and reliability of rotary mechanical system. In practice, the sample proportion between faulty data and healthy data in rotating mechanical system is imbalanced. Furthermore, there are commonalities between the bearing fault detection, classification, and identification tasks. Based on these observations, this article proposes a novel integrated multitasking intelligent bearing fault diagnosis scheme with the aid of representation learning under imbalanced sample condition, which realizes bearing fault detection, classification, and unknown fault identification. Specifically, in the unsupervised condition, a bearing fault detection approach based on modified denoising autoencoder (DAE) with self-attention mechanism for bottleneck layer (MDAE-SAMB) is proposed in the integrated scheme, which only uses the healthy data for training. The self-attention mechanism is introduced into the neurons in the bottleneck layer, which can assign different weights to the neurons in the bottleneck layer. Moreover, the transfer learning based on representation learning is proposed for few-shot fault classification. Only a few fault samples are used for offline training, and high-accuracy online bearing fault classification is achieved. Finally, according to the known fault data, the unknown bearing faults can be effectively identified. A bearing dataset generated by rotor dynamics experiment rig (RDER) and a public bearing dataset demonstrates the applicability of the proposed integrated fault diagnosis scheme.

The application of deep learning-based rolling bearing fault diagnosis methods in high reliability scenarios is limited due to low transparency. In addition, the scaling up of the deep learning models, in order to improve the performance of rolling bearing fault diagnosis (RBFD), has led to difficulties in its application in low-resource scenarios. Based on these facts, a new neural network, restricted sparse networks (RSNs), is proposed in this article. First, a restricted sparse frequency-domain space (RSFDS) is proposed for the interpretable representation of rolling bearing fault features (RBFFs) based on the quadratic complex domain equation. Second, an interpretable multichannel fusion mechanism is designed to map RBFFs to RSFDS. Furthermore, a high-power feature extraction module is developed to extract RBFFs in an efficient and easy-to-understand manner. Finally, an end-to-end RBFD network is provided for high reliability and resource-constrained scenarios. The experimental results show that RSNs have favorable fault diagnosis accuracy performance that is parallel to the state-of-the-art methods. More importantly, the model size of the proposed network only accounts for 20%–30% of the conventional methods.