###### Abstract  
As the heart of aircraft, the aero-engine is not only the main power source for aircraft flight but also an essential guarantee for the safe flight of aircraft. Therefore, it is of great significance to find effective methods for remaining useful life (RUL) prediction for aero-engines in order to avoid accidents and reduce maintenance costs. With the development of deep learning, data-driven approaches show great potential in dealing with the above problem. Although many attempts have been made, few works consider the error of the point prediction result caused by uncertainties. In this paper, we propose a novel RUL probability prediction approach for aero-engines with prediction uncertainties fully considered. Before forecasting, a principal component analysis (PCA) is first utilized to cut down the dimension of sensor data and extract the correlation between multivariate data to reduce the network computation. Then, a multi-layer bidirectional gate recurrent unit (BiGRU) is constructed to predict the RUL of the aero-engine, while prediction uncertainties are quantized by the improved variational Bayesian inference (IVBI) with a Gaussian mixture distribution. The proposed method can give not only the point prediction of RUL but also the confidence interval of the prediction result, which is very helpful for real-world applications. Finally, the experimental study illustrates that the proposed method is feasible and superior to several other comparative models. remaining useful life; improved variational Bayesian inference; Gaussian mixture distribution; bidirectional gate recurrent unit; probability prediction +
Footnote †: journal: Medical  
to establish a reliable degradation model for RUL prediction [9; 10]. However, the system structures are complex and surrounding conditions changeable, it is next to impossible to build a universal degradation model with sufficient accuracy in practical applications. The statistical data model-based methods fit historical data with a random relationship model or stochastic process model to obtain the RUL prediction [11; 12]. Though such methods consider the impact of system uncertainty on RUL, they are unable to mine the internal, complex, in-depth features of the monitoring data and, thus, cannot make sufficient utilization of massive monitoring data. The primary advantage of machine learning model-based methods is that they do not require a priori information about system degradation while making use of complex relationships in the monitoring data [13; 14]. The hybrid model-based methods are achieved by combining the advantages of different models [15], but they often require a more complex structure than a single model. Against the backdrop of more and more massive data that we can obtain, machine learning model-based methods have definitely become the mainstream direction. Deep learning-based methods are representative of machine learning model-based methods [16; 17; 18]. In recent years, deep learning has been successfully applied to deal with the RUL prediction problem of aero-engines due to its strong ability in nonlinear modeling. For example, a multi-objective deep belief network ensemble (MODBNE) method was proposed in [19]. It could extract features automatically and estimate the RUL of an aero-engine by building multiple deep belief networks (DBNs). The convolutional neural network (CNN) is also welcomed by many scholars because of its unique advantages in feature extraction. In ref. [20], [PERSON] et al. constructed a new deep convolutional neural network approach, which could predict the RUL of an aero-engine by learning features of the originally collected sensor data without accurate physical or expert knowledge. Considering the high-dimensional characteristics and complexity of aero-engine monitor data, [PERSON] et al.