## 1 Introduction  
The reliability of equipment is crucial for smooth production processes in modern industry and particularly vital in smart manufacturing where mechanical systems are highly automated and integrated. Rolling-element bearings (REB) are critical structural components in a great variety of mechanical systems such as electric motors, gas turbines, aircraft engines, etc. Bearing faults are one of the root causes of secondary defects developed on connected structures [1, 2], leading to potential safety hazards, unscheduled system downtime, significant maintenance cost and downtime loss. In the field of predictive maintenance, the estimation of remaining useful life (RUL) for bearings has attracted growing attention given its fundamental role in reducing systematic overhaul and preventing catastrophic failures through timely replacement. The RUL of a piece of equipment is defined as the duration from the current time to the end of the service life [3]. The RUL of bearings can be regarded as the duration from the current inspection until functional failures occur. Different from the mean time to failure (MTTF) [4], which is a reliability measure representing the average time to replacement due to unrepairable failure, the bearing RUL is dynamically estimated (or a random variable) based on the current health condition and the predefined end-of-life(EOL) threshold. Indeed, even under the same operating conditions, variations in bearing lifespan can be prominent due to manufacturing errors, mounting mistakes and differences in working conditions, as evidenced by a few public bearing lifetime experiments, e.g., the IMS data [5], the PROGNOSTIA data [6] and the XJTU-SY data [7]. Thereby, RUL prediction for individual bearings is significant for a system to operate at peak performance through the optimization of the maintenance schedule. To date, researchers have made considerable efforts in the development of RUL prediction models for the predictive maintenance of REBs with promising performance in preliminary applications. In general, these models can be categorized into physics-based models and data-driven models. The physics-based models rely on the carefully crafted full bearing life models based on the Hertz contact stress theory and the fatigue life limit, such as the Weibull fatigue life model, the Lundberg-Palmgren model and the Ioannides-Harris model, which associate the bearing fatigue life with the critical shear stress and the number of stress cycles to failure [8]. The 10-percent life or L10 life model widely used in the industry provides a reference for the basic rating life of different types of rolling bearings using the equivalent dynamic loads, as specified in the international standard ISO 281:2007 [9]. Specifically, researchers from the US National Aeronautics and Space Administration presented a technical tutorial outlining the theoretical variations and applications of various bearing life prediction models [10]. These models provide important references to determine bearing life and reliability, while the complexity of failure dynamics and individual differences may still undermine the practicability of the RUL estimations in a dynamical fashion. For example, as pointed out by [PERSON], the use of inconsistent fatigue limit could lead to the over-prediction of bearing life [10]. On the other hand, data-driven models attempt to establish an RUL predictor using statistical (or) machine learning (ML) approaches based on historical condition monitoring (CM) measurements, such as temperature data, oil debris data, vibration data, acoustic emission data, etc. In ML regression, the bearing RUL is regarded as the response variable of a linear or nonlinear system based on the CM data input, where dominant features are recognized to efficiently map the input data to the target RUL. Examples of recent applications of ML models to bearing RUL predictions include support vector regression [11], random forest regression [12] and deep learning networks with sophisticated architectures [13].