## 1 Introduction  
In the main reducer assembly of automobiles, about 80% of the failures occur in the paired gear system [1, 2]. So intelligent real-time monitoring systems can identify the fault gear in a timely fashion, and effectively identify the paired gear fault type. It not only can reduce the scrap rate and decrease rework to reduce production costs, also can reduce the vibration level to improve the working performance of the transmission system and eventually improve the comfort and safety of the car [3, 4]. Due to the complicated assembling process of the main reducer assembly and the complex internal structure, the correlation between each component is very close which results in multi-level failure. So when a function fails inside the main reducer, it may be due to a single fault or multiple faults occurring at the same time [5, 6, 7, 8]. Benefiting from the increasing development and gradual maturity of machine learning and deep learning technologies, intelligent diagnosis of machinery has achieved more breakthroughs, especially in the field of simultaneous fault diagnosis. [PERSON] utilized Bayesian deep learning and between-class learning to create a simultaneous fault diagnosis model [9]. [PERSON] utilized GAN and time-frequency imaging technology for simultaneous fault diagnosis of gearbox [10]. [PERSON] contributed comparative research on several simultaneous fault diagnosis methods based on multi-label classification [11]. [PERSON] used convolutional neural networks to diagnose the simultaneous fault of a wind turbine [12]. Although these researches have achieved satisfactory results, most of the achievements adopted single-label classification to solve simultaneous fault diagnosis. That is, each sample belongs to only one category, and each simultaneous fault type is processed as a special new single-fault type. Furthermore, a diagnostic model based on single-label classification requires a great many simultaneous failure samples to train the diagnostic model. Generally speaking, the principal difficulties of identifying simultaneous fault are arranged as follows:  
1. Simultaneous fault is not a simple combination of multiple single faults, so it is unfeasible to use a simple and traditional model of mechanism to recognize simultaneous faults. 2. In practice, due to the high number of species of all possible simultaneous failure modes, it is impractical to collect ample samples of all simultaneous failure modes to train a fault classification model focused on both single failure modes and simultaneous failure modes recognition. 3. There is no one-to-one correspondence between failure signs and failure cause, and the failure signs have the characteristics of large ambiguity, strong coupling, and uncertainty, which further increase the difficulty of recognition. To tackle the above-mentioned challenges, diagnostic models for single and simultaneous faults must be capable of probabilistic classification, which can clearly reflect the probability of various single fault types. The extreme learning machine (ELM) organizes multiple neurons in a three-layer structure, and the data transmission direction of the neural network is forward [13; 14]. The characteristics of ELM include short training time and simple operation. The classification capability of ELM is superior to that of the BP network and support vector machine (SVM), and its execution time is significantly shorter than that of SVM [15; 16]. [PERSON] et al. used ELM to diagnose failures in real-time for power generation systems in power plants [17]. [PERSON] combined singular value decomposition with ELM in [18]. By combining ELM with an optimized sample entropy algorithm, [PERSON] et al. proposed an automatic recognition system [19]. [PERSON] utilized ELM into the regression and classification problem [20]. At present, there are many derivative algorithms based on ELM. [PERSON] introduced the Bayesian method into ELM to create Bayesian ELM (BELM).