## 1 Introduction  
Renewable energy is gradually becoming an important option to replace fossil energy [1]. As one of the most popular renewable energy storage solutions, lithium-ion batteries have the advantages of high energy density, long cycle life, and low pollution [2]. However, the lifespan of lithium-ion batteries will be reduced through recycling, and improper usage will expedite battery aging [3]. When the aging reaches a certain level, the lithium-ion battery will not work properly [4]. Therefore, lithium-ion batteries are often equipped with a battery management system (BMS) to accurately predict the remaining useful life (RUL) of the battery [5]. The RUL is one of the important indicators to measure the aging degree of lithium-ion batteries, and it is closely correlated with the state of health (SOH) of lithium-ion batteries [6]. When the SOH of lithium-ion batteries falls below a certain threshold, their operational performance will inevitably deteriorate [7]. This threshold is called the end-of-life (EOL), usually when SOH drops to 80% of the rated value [8,9]. The moment at which the SOH exceeds this threshold can be referred to as the end-of-monitoring (EOM). Therefore, the RUL is commonly defined as the total number of charge-discharge cycles a battery can undergo from the EOM to the EOL [10], expressed as follows  
\[RUL=n_{EOL}-n_{EOM} \tag{1}\]\(n_{EOL}\) is the number of cycles when the new battery cycles to the threshold, and \(n_{EOM}\) represents the current cycle number. Recently, many methods have been proposed to solve the RUL prediction problem. These methods are primarily categorized as model-based methods and data-driven methods [11]. The model-based method primarily establishes the correlation between measurable physical parameters and the RUL by leveraging the electrochemical mechanism or circuit characteristics of lithium-ion batteries [12; 13]. The literature [14] investigates an empirical exponential growth model for the resistance degradation of lithium-ion batteries, using particle filtering (PF) to evaluate aging data at different time intervals indicative of system health. The literature [15] explores the performance of different circuit topologies for diffusion processes. Although the aforementioned research has contributed to the study of remaining life prediction for lithium-ion batteries, developing an accurate battery model requires a significant amount of expertise [16]. Additionally, the RUL of lithium-ion batteries is not only limited by their aging mechanism but also closely linked to complex operating conditions and battery types [17]. When the battery's working environment is disturbed, the stability of the model will face challenges [18]. Therefore, model-based methods are often insufficiently flexible. Based on the data-driven method to extract the aging law of lithium-ion batteries from battery aging data, it can be more flexibly adapted to online prediction [19]. It mainly includes the signal processing method, neural network method, and machine learning method [20]. The signal processing method usually summarizes the law from the time domain and frequency domain characteristics of the signal. In the literature [21], the empirical mode decomposition (EMD) method is adopted to decompose the capacity curve into linear and nonlinear trends, which is convenient for research separately. Unlike signal processing methods, neural network methods are often utilized in the form of enhancement or combination to predict the RUL of lithium-ion batteries. Literature [22] combined the convolutional neural network (CNN), the bidirectional long short-term memory (Bi-LSTM), and the Bayesian network to develop a Bayesian neural network (BNN). This method utilizes only a small amount of data to accurately predict the RUL of batteries. Literature [23] constructed the deep learning model based on bi-directional long short-term memory(Bi-LSTM) with the addition of an attention mechanism(AM) to focus on the important parts of the batteries' features.