## 2 Fault Diagnosis of Rolling Bearing Based on VMD and ANN
### Architecture of the Artificial Neural Network  
Because of their strong capacity for adaptation and self-organization and nonlinear mapping ability, ANNs have been widely used in the fault diagnosis of bearings. There are many ANN structures, such as a linear ANN, backpropagation ANN, radial basis function ANN and probabilistic neural network. A linear ANN uses the least mean square training function to adjust the weight and bias of the network. Although the computational speed of a linear ANN is very fast, it is only suitable for tackling linear separable problems. A radial basis function ANN and probabilistic neural network have fast convergence speed, but their prediction accuracy is not ideal for the prediction of bearing fault diagnosis [13]. A BP ANN is a kind of multilayer perceptron, which is the most widely used neural network structure at present. Therefore, this paper applied a BP ANN to perform the fault diagnosis of rolling bearings. A brief flowchart of the ANN is shown in Figure 5. The input for the ANN was the normalized energy of each IMF's amplitude frequency. The shape of the total input feature data was 8000 \(\times\) 8. The target input data were labels of the bearing fault types represented by a one-hot code with the shape of 8000 \(\times\) 10. The neuron nodes of the input and output layers were 8 and 10, respectively, which correspond to the 8 normalized energy features of the input data and 10 fault types of rolling bearings. The number of hidden layer was three, and each hidden layer contained 13 neurons. The error was used for the fault diagnosis. The soft-max activation function was applied for the fault type classification of rolling bearings in the output layer. The output of the ANN was fed into the soft-max activation function to judge the bearing fault type according to the 10 probability values. Figure 4: The amplitude frequency of each IMF based on discrete Fourier transform. AF, amplitude frequency. The implementation and training process of the proposed method was conducted with MATLAB on a computer with a i7-6700 CPU and 32 GB memory. The ANN architecture built using MATLAB is shown in Figure 6. The input data were divided into a training dataset, validation dataset and testing dataset. The parameters of the ANN, such as the number of hidden layer, number of neuron nodes and training function, were optimized using a trial-and-error method for the different fault types of rolling bearings. The performance of an ANN is mainly affected by the number of hidden layers, number of neuron nodes in each hidden layer and the training function. In order to simplify the parameter optimization of the ANN, each hidden layer had the same number of neuron nodes. The prediction accuracy for the number of hidden layers, ranging from 1 to 5, and the number of neurons in each layer, ranging from 5 to 15, were studied in this paper. The training functions, including Levenberg-Marquardt, Bayesian regularization, BFGS quasi-Newton, resilient backpropagation and scaled conjugate gradient, were used to build the ANN. Through our studies, we found that the Bayesian regularization training function performs the best. In addition, the commonly used activation functions included sigmoid, tanh, ReLU, leaky ReLU and ELU. Through research, it was found that these activation functions have little impact on the prediction accuracy of fault diagnosis. Therefore, the ReLU activation function with high computational efficiency was used in the input and the hidden layers of the ANN. The optimal parameters for an ANN used for rolling bearing fault diagnosis are shown in Table 2. \begin{table}
\begin{tabular}{l c} \hline Number of Samples & 8000 \\ Number of Neuron Nodes in the Input Layer & 8 \\ Number of Hidden Layers & 3 \\ Number of Neuron Nodes in the Hidden Layer & 13 \\ Number of Neuron Nodes in the Output Layer & 10 \\ Activation Function of the Input/Hidden Layers & ReLU \\ Activation Function of the Output Layer & Soft-max \\ Training Function & Bayesian Regularization \\ \hline \end{tabular}
\end{table}