## 1 Introduction  
Rolling bearings are important components of rotating machinery in the automotive, aviation and marine fields. The fault diagnosis of bearings is very helpful for rotating machinery to maintain safe operation with high performance. On the one hand, in practical engineering applications, the bearing vibration characteristics are vulnerable to many factors, such as internal and external noise, which results in significant challenges for the fault diagnosis of bearings' inner and outer rings [1]. On the other hand, real-time monitoring is required to perform real-time control for engine rolling bearing components in the fault diagnosis of rolling bearings. Therefore, both the prediction accuracy and computation time should be taken into consideration in the actual application. From the perspective of the input signal format, bearing fault diagnosis methods can be divided into raw-vibration-based methods and feature-extraction-based methods. The raw vibration signal can be directly used as input for a deep learning (DL) method because of its powerful intelligent learning and data mining ability, such as convolutional neural network (CNN) [2, 3], generative adversarial network [4] and capsule neural network [5]. Of course, there are also studies that combine DL methods with feature extraction methods, such as FFP-DL [6], deep feature transfer learning [7] and reinforcement ensemble deep transfer learning network [8]. Although fault diagnosis based on DL usually has a higher prediction accuracy, it needs a great quantity of data, which results in long computational time. Therefore, many researchers try to use machine learning (ML) methods for fault diagnosis with high computational efficiency, such as RF [9], SVM [10], artificial neural network (ANN) [11] and k-nearest neighbor (KNN) [12]. In general, the fault diagnosis accuracy of ML methods is not as good as that based on a DL method. However, the advantages of ML methods are obvious, namely, simple implementation, fast computation and low demand for training data. In addition, if we can design appropriate feature extraction methods based on actual needs, the performance of ML methods would be comparable to that of DL methods. As a classical ML framework, an ANN is still an effective mean to solve complex problems. With outstanding performance, ANNs have resulted in great successes and set new records in handling different tasks, such as the identification of handwritten numeral and target classification. It is until today that ANNs are still superior to other methods in terms of classification tasks. Compared with the DL method, the structure of an ANN is much simpler, which indicates high computational efficiency. Although the prediction accuracy of an ANN is usually not as good as that of DL methods, it can be improved significantly if the network structure is designed reasonably. Therefore, how to reasonably design a network structure to achieve high performance is the key point in solving complex problems with ANNs. However, theoretical guidance on the selection of a network structure is not available at present, and trial-and-error methods are still used to adjust the structure of ANNs in practical applications. In addition, the feature extraction of the input data is another key issue to improve an ANN's performance. In recent years, many studies have successfully adopted an ANN for bearing fault diagnosis. [PERSON] applied a multilayer perceptron ANN combined with the genetic algorithm for bearing fault detection, where the genetic algorithm is used to selected the characteristic parameters of the classifiers and the input features [13]. Based on the pattern recognition technique, [PERSON] proposed a novel ANN method for the fault diagnosis in induction motors [14]. [PERSON] applied an ANN to develop a vibration-based machine learning model for fault diagnosis in rotating machines, and the ANN parameters, such as the number of layers, number of neurons and types of functions, were optimized [15].