Intelligent machine health monitoring and fault diagnosis are becoming increasingly important for modern manufacturing industries. Current fault diagnosis approaches mostly depend on expert-designed features for building prediction models. In this paper, we proposed IDSCNN, a novel bearing fault diagnosis algorithm based on ensemble deep convolutional neural networks and an improved Dempster–Shafer theory based evidence fusion. The convolutional neural networks take the root mean square (RMS) maps from the FFT (Fast Fourier Transformation) features of the vibration signals from two sensors as inputs. The improved D-S evidence theory is implemented via distance matrix from evidences and modified Gini Index. Extensive evaluations of the IDSCNN on the Case Western Reserve Dataset showed that our IDSCNN algorithm can achieve better fault diagnosis performance than existing machine learning methods by fusing complementary or conflicting evidences from different models and sensors and adapting to different load conditions.

Bearings running state affects the normal operation of mechanical equipment. It is of great theoretical and practical value to carry out bearing fault diagnosis. In bearing fault diagnosis research, the extraction and selection of fault features can help improving the accuracy of bearing fault diagnosis. However, these researches suffer from the following weaknesses. (1) High dimension of the selected features. (2) Uncertainty of single sensor for data sampling. Therefore, in this paper, a feature selection feedback network (FSFN) is proposed to overcome the first weakness. At the same time, we proposed an improved Dempster-Shafer (IDS) evidence theory fusion method based on the kappa coefficient to deal with the second weakness. Extensive evaluations of the proposed method on the CUT-2 experimental platform dataset showed that FSFN can not only reduce the dimension of the final selected feature without decreasing the diagnostic accuracy but also shorten the time of feature selection. Moreover, compared with the existing DS evidence theory fusion method, IDS can achieve higher average fusion precision and improve the accuracy and reliability of bearing fault diagnosis.

A convolutional neural network (CNN) has been used to successfully realize end-to-end bearing fault diagnosis due to its powerful feature extraction ability. However, the CNN is prone to focus on local information, ignoring the relationship between the whole and the part of the signal due to its unique structure. In addition, it extracts some fault features with poor robustness under noisy environment. A novel diagnosis model based on feature fusion and feature selection, GL-mRMR-SVM, is proposed to address this problem in this paper. First, the model combines the global features in the time-domain and frequency-domain of the raw data with the local features extracted by CNN to make full use of the signal information and overcome the weakness of traditional CNNs neglecting the overall signal. Then, the max-relevance min-redundancy (mRMR) algorithm is used to automatically extract the discriminative features from the fused features without any prior knowledge. Finally, the extracted discriminative features are input into the SVM for training and output the fault recognition results. The proposed GL-mRMR-SVM model was evaluated through experiments on bearing data of Case Western Reserve University (CWRU) and CUT-2 platform. The experimental results show that the proposed method is more effective than other intelligent diagnosis methods.

Effective feature selection can help improve the classification performance in bearing fault diagnosis. This paper proposes a novel feature selection method based on bearing fault diagnosis called Feature-to-Feature and Feature-to-Category- Maximum Information Coefficient (FF-FC-MIC), which considers the relevance among features and relevance between features and fault categories by exploiting the nonlinearity capturing capability of maximum information coefficient. In this method, a weak correlation feature subset obtained by a Feature-to-Feature-Maximum Information Coefficient (FF-MIC) matrix and a strong correlation feature subset obtained by a Feature-to-Category-Maximum Information Coefficient (FC-MIC) matrix are merged into a final diagnostic feature set by an intersection operation. To evaluate the proposed FF-FC-MIC method, vibration data collected from two bearing fault experiment platforms (CWRU dataset and CUT-2 dataset) were employed. Experimental results showed that accuracy of FF-FC-MIC can achieve 97.50%, and 98.75% on the CWRU dataset at the motor speeds of 1750 rpm, and 1772 rpm, respectively, and reach 91.75%, 94.69%, and 99.07% on CUT-2 dataset at the motor speeds of 2000 rpm, 2500 rpm, 3000 rpm, respectively. A significant improvement of FF-FC-MIC has been confirmed, since the p-values between FF-FC-MIC and the other methods are 1.166 × ${10}^{-3}$ , 2.509 × ${10}^{-5}$ , and 3.576 × ${10}^{-2}$ , respectively. Through comparison with other methods, FF-FC-MIC not only exceeds each of the baseline feature selection method in diagnosis accuracy, but also reduces the number of features.

In order to realize single fault detection (SFD) from the multi-fault coupling bearing data and further research on the multi-fault situation of bearings, this paper proposes a method based on features self-extraction of a Sparse Auto-Encoder (SAE) and results fusion of improved Dempster–Shafer evidence theory (D–S). Multi-fault signal compression features of bearings were extracted by SAE on multiple vibration sensors’ data. Data sets were constructed by the extracted compression features to train the Support Vector Machine (SVM) according to the rule of single fault detection (R-SFD) this paper proposed. Fault detection results were obtained by the improved D–S evidence theory, which was implemented via correcting the 0 factor in the Basic Probability Assignment (BPA) and modifying the evidence weight by Pearson Correlation Coefficient (PCC). Extensive evaluations of the proposed method on the experiment platform datasets showed that the proposed method could realize single fault detection from multi-fault bearings. Fault detection accuracy increases as the output feature dimension of SAE increases; when the feature dimension reached 200, the average detection accuracy of the three sensors for bearing inner, outer, and ball faults achieved 87.36%, 87.86% and 84.46%, respectively. The three types’ fault detection accuracy—reached to 99.12%, 99.33% and 98.46% by the improved Dempster–Shafer evidence theory (IDS) to fuse the sensors’ results—is respectively 0.38%, 2.06% and 0.76% higher than the traditional D–S evidence theory. That indicated the effectiveness of improving the D–S evidence theory by evidence weight calculation of PCC.

As is well known, differential algebraic equations (DAEs), which are able to describe dynamic changes and underlying constraints, have been widely applied in engineering fields such as fluid dynamics, multi-body dynamics, mechanical systems, and control theory. In practical physical modeling within these domains, the systems often generate high-index DAEs. Classical implicit numerical methods typically result in varying order reduction of numerical accuracy when solving high-index systems. Recently, the physics-informed neural networks (PINNs) have gained attention for solving DAE systems. However, it faces challenges like the inability to directly solve high-index systems, lower predictive accuracy, and weaker generalization capabilities. In this paper, we propose a PINN computational framework, combined Radau IIA numerical method with an improved fully connected neural network structure, to directly solve high-index DAEs. Furthermore, we employ a domain decomposition strategy to enhance solution accuracy. We conduct numerical experiments with two classical high-index systems as illustrative examples, investigating how different orders and time-step sizes of the Radau IIA method affect the accuracy of neural network solutions. For different time-step sizes, the experimental results indicate that utilizing a 5th-order Radau IIA method in the PINN achieves a high level of system accuracy and stability. Specifically, the absolute errors for all differential variables remain as low as 10 − 6 , and the absolute errors for algebraic variables are maintained at 10 − 5 . Therefore, our method exhibits excellent computational accuracy and strong generalization capabilities, providing a feasible approach for the high-precision solution of larger-scale DAEs with higher indices or challenging high-dimensional partial differential algebraic equation systems.