Correction: A hybrid LSTM–transformer model for accurate remaining useful life prediction of lithium-ion batteries

1. Introduction With the global transition toward cleaner energy and the rapid advancement of electrification technologies, lithium-ion batteries have emerged as essential energy storage components in electric vehicles, renewable energy storage systems, and portable electronic devices[1]. Owing to their high energy density, long cycle life, and low self-discharge rate, lithium-ion batteries have become increasingly important. However, their performance inevitably degrades over time due to repeated charge-discharge cycles, leading to capacity fade and a shortened remaining useful life (RUL)[2]. Accurate prediction of battery state of health (SOH) and RUL is crucial for optimizing battery management, extending service life, reducing maintenance costs, and ensuring system safety[3,4]. In particular, battery failure in electric vehicles or large-scale energy storage systems can result in significant safety hazards and economic losses, making the development of high-accuracy RUL prediction methods a pressing research focus. Precise RUL prediction plays a vital role in optimizing battery management systems. First, it provides a scientific basis for battery replacement and maintenance decisions, thereby lowering operational costs[5]. Second, it enables early identification of potential failures, thus enhancing system safety[6]. Moreover, accurate RUL estimates support battery recycling and second-life applications, contributing to sustainable resource utilization. Nevertheless, several challenges hinder effective RUL prediction. The degradation process is influenced by multiple factors such as temperature, charge/discharge rates, and usage scenarios, exhibiting high nonlinearity and complexity[7]. Real-world operational data often contain noise and missing values, increasing the difficulty of modeling[8]. Furthermore, long-term prediction requires models that can simultaneously capture short-term fluctuations and long-term trends, which is difficult for single-model architectures to achieve. Traditional RUL prediction approaches can be categorized into physics-based and data-driven methods. Physics-based models rely on the electrochemical mechanisms of batteries[9], using complex mathematical formulations to describe the degradation process. However, these methods require detailed knowledge of material properties and operating conditions, involve high computational complexity, and often lack generalizability across different battery types. In contrast, data-driven approaches have gained popularity by learning patterns directly from operational data[10]. With advances in sensor technology and data acquisition capabilities, these methods can effectively model battery behavior using features such as voltage, current, and temperature, showing improved adaptability and predictive accuracy. Data-driven RUL prediction techniques generally fall into three categories: statistical models[11], machine learning methods[12], and deep learning models[13]. Statistical approaches, such as Kalman filtering and particle filtering, model battery degradation probabilistically. For example, Nunes et al.[14] proposed an online RUL estimation method for second-life lithium-ion batteries based on an unscented Kalman filter and degradation curve modeling, validated on six different second-life battery datasets. Despite some success—achieving a worst-case mean absolute percentage error (MAPE) of 5.279% and an R² score of 0.726—statistical models often struggle with nonlinear or complex degradation behaviors. Machine learning approaches such as support vector machines, random forests, and XGBoost have demonstrated promising results in RUL prediction through hand-crafted features. Jafari et al.[15] introduced a hybrid RUL prediction method based on particle filtering and Kalman filtering, where XGBoost was used as the observation model due to its strong nonlinear fitting capabilities. Despite high predictive accuracy based on full-cycle test data, such methods are heavily dependent on the quality of feature engineering and may suffer from overfitting or inefficiency when applied to high-dimensional time-series data. The emergence of deep learning has opened new avenues for RUL prediction. Long Short-Term Memory (LSTM) networks, known for their capability in modeling temporal dependencies, have been widely adopted for battery degradation modeling. LSTM networks utilize gating mechanisms to effectively capture long-term dependencies, making them well-suited for modeling the nonlinear degradation process of batteries. Reza et al.[16] proposed an improved method combining LSTM with the Gravitational Search Algorithm (GSA), using data cleaning to remove noise, replacing anomalies with highly correlated data, and applying normalization. GSA was employed to optimize the LSTM hyperparameters to address key challenges in battery life prediction. However, LSTM models may still face limitations such as vanishing gradients and computational inefficiencies when dealing with long sequences. Recently, Transformer models have demonstrated excellent performance in natural language processing and time-series analysis tasks due to their strong capability in extracting global features[17,18]. The Transformer architecture processes sequences in parallel through multi-head attention mechanisms, effectively capturing long-range dependencies. Nevertheless, its application in battery RUL prediction remains relatively unexplored. Given the above challenges, accurate RUL prediction remains difficult due to the following key factors, (1) The degradation process is highly nonlinear and influenced by external factors such as temperature, cycling rate, and usage scenarios; (2) Real-world data are often noisy and incomplete, complicating the modeling process; (3) Long-term prediction tasks require models to simultaneously capture short-term variations and long-term trends, which is difficult for single models to handle effectively. To address these challenges, this study develops a hybrid deep learning model that captures both local temporal dependencies and global contextual features. As illustrated in Figure 1, we propose a novel LSTM-Transformer hybrid model based on the MIT battery dataset to enhance prediction accuracy and robustness. The LSTM module extracts local temporal dynamics from time-series inputs, while the Transformer module, with its attention mechanism, captures global feature dependencies. The integration of both allows the model to effectively represent complex degradation patterns. The main contributions of this work are as follows, (1) A novel hybrid deep learning architecture combining local temporal modeling and global attention mechanisms is proposed for lithium-ion battery RUL prediction. (2) Temperature-based features are engineered based on battery physical mechanisms to enhance input feature expressiveness. (3) The proposed model is validated on the MIT battery dataset, demonstrating superior prediction accuracy and robustness compared to existing methods, with strong potential for real-world applications. The remainder of this paper is organized as follows: Section 2 introduces the data preprocessing and feature engineering methods; Section 3 details the architecture and training process of the LSTM-Transformer hybrid model; Section 4 presents experimental results and performance evaluation; Section 5 concludes the paper with a summary of findings and future work directions. Through this research, we aim to provide an efficient and accurate solution for lithium-ion battery RUL prediction, offering theoretical and technical support for optimized battery management systems. Figure 1. Schematic diagram of the LSTM-Transformer hybrid model architecture. 2. Data Preprocessing and Feature Engineering Accurate prediction of the remaining useful life of lithium-ion batteries critically depends on high-quality data preprocessing and feature engineering. To construct a time-series modeling– ready prediction dataset, this study systematically performs data cleaning, temperature feature extraction, normalization, target variable transformation, and sample construction based on a sliding window. By incrementally sliding a fixed-length window over the time-series data to extract local segments, representative features or sequential samples are generated. This approach facilitates the capture of local temporal dependencies and dynamic variations[19], thereby ensuring that the input data fed into the model possesses strong representativeness and consistency. The original dataset used in this work is the MIT Battery Dataset, which includes operational data from multiple batteries under various charging and discharging conditions. Key variables include voltage, current, and temperature. Let the raw feature matrix be denoted as n d x   , where n represents the number of samples and d indicates the number of feature dimensions. The target variable is denoted as n y . To explore the linear relationships between different features, we construct a Pearson correlation-based heatmap, as illustrated in Figure 2(a). In this heatmap, red indicates strong positive correlations, while blue denotes strong negative correlations. The heatmap reveals significant correlations among several voltage-related, capacity-related, and temperature-related features. Based on this analysis, redundant features are removed to improve the generalization ability of the model and to mitigate issues related to multicollinearity. (a) (b) Figure 2. (a) Feature correlation heatmap based on Pearson coefficients. (b) Feature space distribution of battery life using Principal Component Analysis. To comprehensively evaluate the structure of the feature space, Principal Component Analysis was applied to the high-dimensional feature data after redundancy removal, as shown in Figure 2(b). The first two principal components were visualized, with a color gradient representing the corresponding battery RUL values. This visualization enables the observation of degradation trends within the feature space. The results indicate that samples with different RUL levels exhibit clear clustering patterns in the two-dimensional PCA space, demonstrating the discriminative capability of the extracted features in characterizing battery degradation states. Temperature, as a critical factor influencing lithium-ion battery aging and performance degradation, plays a key role in modeling degradation behavior. Analysis reveals that the raw temperature features (T1-0 to T3-3) exhibit significant dynamic fluctuations during battery operation and are highly correlated with changes in the RUL curve. To this end, three types of temperature-derived features are designed: temperature mean, temperature range, and temperature fluctuation. The temperature mean reflects the overall thermal load level during battery operation. Elevated operating temperatures accelerate electrolyte decomposition, solid electrolyte interphase (SEI) layer growth, and structural degradation of electrode materials, which are key contributors to capacity fade and internal resistance increase. The temperature range measures the amplitude of temperature variation within each time window, indicating the degree of thermal stress fluctuation. Frequent and intense thermal stress cycles may induce mechanical fatigue or even cracking of electrode particles, exacerbating material degradation and performance decline. Temperature fluctuation, quantified by the standard deviation, characterizes the local instability of temperature over time, typically associated with abnormal conditions such as high-current charge/discharge events and cooling system failures. These factors are prone to cause localized hotspots and accelerate undesirable electrochemical side reactions, ultimately shortening battery life. For each temperature sensor group, the temperature mean is defined as follows: 1 , 0 1 m mean i i j j T T m (1) In this j T denotes the temperature within the group, where 4 m represents the number of in each The corresponding temperature range is defined as follows: , , 0 range i i j i j j m j m T T T (2) The above two features the and of each temperature group, which can such as localized or abnormal In to capture the overall level of the thermal the standard across temperature is and used as an of temperature 2 1 1 T T (3) Let the number of temperature represent the of the temperature and T be the mean temperature across These three temperature-derived features enhance the of the data provide with the battery aging To among features and to improve training is applied to the feature ensuring that each feature has mean and fed into the x Let the mean and standard of each feature and x represent the feature This that features an and which facilitates of gradient during model training and generalization the is applied to the target variable for nonlinear This the data distribution to a thereby enhancing the and accuracy of model The is defined as follows: 2 1, 0 0 1, 2 2 2 y y y y y y y y y denotes the which is using the The variable y a which is for model and error To the of deep time modeling, the dataset is into a sliding window As illustrated in Figure the input window be in and the prediction be For the feature matrix n d x   and the target variable n y each training sample is from the following in d in in in x y y  The number of samples that can be from the dataset is 1 in n The dataset was into training and a of with the training to enhance sample and training robustness. 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