There are two problems when the Shapley value is employed to interpret deep neural networks. The first issue is that the computational complexity increases exponentially with the number of players. The other issue is that the contribution evaluation index cannot effectively reflect the nonlinearity of the classification function (i.e., SoftMax), which is often neglected in previous studies. To address these challenges, a method for fast estimating the Shapley value based on the stratified sampling and the Mann–Whitney test (SSMW-Shap) is proposed in this work. In SSMW-Shap, a new contribution index is designed to accurately measure the contribution of each player by leveraging the distance between the outputs of two specific neurons, accounting for the nonlinearity of SoftMax and the efficiency of the Shapley value. Based on the proposed index, a simplified two-player coalition evaluation method is built to select important affiliates for each player, significantly reducing the computational complexity of the Shapley value. Then, the Shapley value is fast estimated by combining the stratified sampling and the Mann–Whitney test. In this process, the Mann–Whitney test is employed to estimate the difference between the samples and the population, and sample expansion is executed for the failed test, improving the estimation accuracy. Finally, a simple but reasonable method based on the proposed index is designed to quantitatively evaluate the explanation accuracy of each method. The proposed method is verified using two classic classification networks trained on two bearing datasets.

The simulation signals generated by the bearing dynamics model have a big gap with the actual signals, which limits their efficacy in bearing fault diagnosis. Therefore, it is valuable to build an accurate digital twin model of faulty rolling bearing . Firstly, a multi-degree-of-freedom bearing fault dynamics model is constructed in the virtual space for generating the vibration responses of bearing parts. Then considering that the frequency spectrum contains more characteristic information than the time-domain signal, a frequency-domain bi-directional long short-term memory (Bi-LSTM) cycle generative adversarial network (CycleGAN) named FBC-GAN is proposed to construct the frequency-domain coupling mapping relationship between the multipart vibration responses and the measured signals. In the proposed network, Bi-LSTM is used for enhancing the feature extraction ability. Meantime, a new spectrum-constraint loss is proposed to ensure the frequency-domain mapping. Next, the simulated fault bearing signals close to the actual signals are generated by FBC-GAN and Fourier transform . Finally, the results of two experiments show the superiority of the proposed method over other advanced data augmentation methods in bearing fault diagnosis with the imbalanced samples.

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.