Enhanced Convolutional Neural Network (CNN)-Long Short-Term Memory (LSTM) attention model with adaptive loss for lithium-ion battery state of health estimation

Materials

The proposed model was implemented using Python 3.11.5 with TensorFlow as the primary deep learning framework. All experiments were conducted in an Anaconda-managed environment. The experiments were executed on a high-performance cloud computing platform (AutoDL), equipped with a 16-core Intel Xeon(R) Gold 6430 CPU, 120 GB of RAM, and an NVIDIA RTX 4090 GPU with 24 GB memory. This setup ensured efficient training and evaluation of deep learning models under large-scale battery data. All dependencies are documented in the provided GitHub repository at https://github.com/Benmoshangsang/SOH-predict-2025. The repository includes modular code for each experimental task, including baseline training, loss function comparison, hyperparameter tuning, ablation studies, and data augmentation. Utility scripts for preprocessing, parameter configuration, and model definition are also included. The raw battery dataset used in this study was sourced from NASA, comprising voltage, current, and temperature measurements under various operating conditions. To improve model robustness and generalization, three augmentation strategies—slicing, flipping, and linear interpolation—were applied to generate synthetic data. All raw and augmented datasets, along with model outputs and results, are accessible via Zenodo at https://doi.org/10.5281/zenodo.15239011.

Model performance was rigorously evaluated using a comprehensive suite of regression-based metrics, including: MAE, MSE, mean absolute percentage error (MAPE), root mean square error (RMSE), R-squared (R²), centered root mean square difference (CRMSD), median absolute difference (MAD), and normalized RMSE (nRMSE). These metrics were carefully selected to capture both absolute and relative prediction accuracy, and were uniformly applied across all experimental modules to ensure consistency and comparability.

The following sections detail the core methodological components employed in this study, including the data augmentation pipeline, the CNN-LSTM architecture enhanced with attention mechanisms, the ablation strategy, and the hyperparameter optimization process.

Data augmentation

Linear interpolation is a widely used technique for estimating missing values in lithium-ion battery datasets by generating intermediate data points between existing observations (Xiong et al., 2024). For SOH prediction, given two known data points ( $t_0$, $V_0$) and ( $t_1$, $V_1$), representing voltage at times $t_0$ and $t_1$, the linear interpolation formula is:

(1) $$V_{\left(t\right)}=V_0+{\displaystyle \frac{\left(V_1-V_0\right)}{\left(t_1-t_0\right)}\cdot \left(t-t_0\right)}$$where $t$ is a time point between $t_0$ and $t_1$, and $V_{\left(t\right)}$ is the predicted voltage value. $V_0$ represents the voltage at the starting time $t_0$, and $V_1$ represents the voltage at the ending time $t_1$. This technique is applied to various parameters including voltage, current, capacity, and temperature to produce more continuous and smooth data. This enhanced dataset contributes to improving the model’s SOH estimation accuracy by providing a more thorough depiction of battery behaviour over time.

Slicing is a data preprocessing technique particularly effective for time series analysis. It involves segmenting the original long-sequence data into multiple fixed-length subsequences, thereby enabling downstream machine learning models to more efficiently process and analyse the temporal patterns within these segments (Xie et al., 2024).

Given the original time series data $X=\left[x_1,x_2,\dots ,x_T\right]$, where $T$ indicates the length of the sequence, the subsequence $X_i$ after slicing is defined as:

(2) $$X_i=\left[x_{iS+1},x_{is+2},\dots ,x_{is+L}\right]$$where $L$ represents the slicing length, which indicates the length of each subsequence after slicing. S indicates slicing step, representing the number of steps to move forward in the original sequence each time a new subsequence is created. $X_i$ is the i-th subsequence obtained after slicing. $x_{iS+1},x_{is+2},\dots ,x_{is+L}$ represent elements of the i-th subsequence, extracted from the original sequence $X$ according to the slicing step S and slicing length $L$.

This method facilitates the identification of localized patterns and trends within specific time windows, enabling the model to more effectively learn temporal dynamics. It is particularly valuable for detecting short-term variations in battery performance that may signal longer-term degradation trajectories.

Flipping is a data augmentation technique that involves reversing the order of a time series sequence. This technique helps to create additional training samples from the original data, enhancing the diversity and robustness of the training set (Chou & Nguyen, 2024).

Given the original time series data, for a time series data $X=\left[x_1,x_2,\dots ,x_T\right]$, the flipped sequence $X_{flip}$ is:

(3) $$X_{flip}=\left[x_T,x_{t-1},\dots ,x_1\right]$$where $X$ represents the original time series data, and $T$ is the length of the sequence, indicating the total number of elements in the sequence. $X_{flip}$ is the flipped or reversed sequence of $X$, starting from the last element and going to the first.

This method applies to all parameters (voltage, current, temperature) to mimic various charging and discharging cycles. Flipping increases the dataset’s diversity, providing the model with a wider range of scenarios to learn from, which enhances its generalization capability and robustness in real-world applications.

Cross-validation is a technique for assessing model performance by partitioning the dataset into several subsets. The model is methodically trained and validated on various combinations of these subsets. This approach helps evaluate the model’s effectiveness and its ability to generalize to new, unseen data (Ahmadzadeh, Zahrai & Bitaraf, 2024).

Given a dataset D with N samples, and K folds, cross-validation involves:

(1) Split the D into K equally sized subsets $D_1,D_2,\dots ,D_k$.

(2) For each fold k (where k = $1,2,\dots ,K$), designate $D_k$ as the validation set and utilize the remaining K − 1 subsets as the training set.

(3) Compute the evaluation metric, such as MAE or MSE, for each fold, and then average the results across all folds.

This method is applied across all relevant parameters—voltage, current, and temperature—to construct a composite dataset that captures a diverse range of operational conditions. The crossing strategy enhances the model’s ability to generalize by exposing it to mixed behavioural patterns, enabling better adaptation to different battery states and environmental scenarios. By integrating characteristics from multiple data sequences, this technique introduces greater variability and complexity, which in turn improves the deep learning model’s capacity to manage heterogeneous real-world conditions with enhanced robustness and predictive accuracy.

Uncertainty-guided adaptive Bayesian optimization (UGABO)

UGABO is a strategy that combines uncertainty estimation with adaptive exploration-exploitation trade-offs in Bayesian optimization. The core idea is to dynamically adjust the balance between exploration and exploitation by guiding the process based on uncertainty estimates, thereby achieving more efficient optimization (Zangirolami & Borrotti, 2024). UGABO begins with initial hyperparameter settings and iteratively adjusts them using optimization algorithms. This approach gradually converges to an optimal configuration, balancing accuracy, robustness, and computational efficiency.

Gaussian process regression (GPR) is commonly used to model the objective function in Bayesian optimization. It provides a probabilistic framework to estimate the function and its uncertainty (Li et al., 2024).

Given a set of observed data points $D={\left\{\left(x_i,y_i\right)\right\}}_{i=1}^n$, where each pair $\left(x_i,y_i\right)$ represents an input $x_i$ and its corresponding observed output $y_i$. $x_i$ represents the i-th input in the observed dataset, and $y_i$ is the corresponding function value or observed output for input $x_i$. The corresponding function value, a gaussian process assumes that the function $f\left(x\right)$ follows a multivariate normal distribution over any finite set of points:

(4) $$f\left(x\right)\sim GP\left(\mu \left(x\right),k\left(x,x^{\mathrm{\prime }}\right)\right)$$where $f\left(x\right)$ represents the target function, representing the function value at the input x. $GP\left(\mu \left(x\right),k\left(x,x^{\mathrm{\prime }}\right)\right)$ denotes that $f\left(x\right)$ is modeled as a Gaussian process with a mean function $\mu \left(x\right)$ and a covariance function $k\left(x,x^{\mathrm{\prime }}\right)$. $\mu \left(x\right)$ represents Mean function (often assumed to be zero) and $k\left(x,x^{\mathrm{\prime }}\right)$ indicates Covariance function (kernel function), such as the Radial Basis Function (RBF) kernel.

Acquisition functions guide the selection of the next evaluation point by balancing exploration and exploitation (Duankhan et al., 2024). The Upper Confidence Bound (UCB) acquisition function incorporates both the mean and the uncertainty of the prediction, controlled by a parameter $\kappa$:

(5) $${\alpha }_{UCB}\left(x\right)={\mu }_n\left(x\right)+\kappa {\sigma }_n\left(x\right)$$where ${\alpha }_{UCB}\left(x\right)$ represents the UCB acquisition function value, which evaluates the potential of selecting point x. ${\mu }_n\left(x\right)$ denotes the predicted mean at point x. ${\sigma }_n\left(x\right)$ represents the predicted uncertainty (standard deviation) at point x. $\kappa$ is a parameter that controls the trade-off between exploration and exploitation, higher values of $\kappa$ encourage exploration by giving more weight to the uncertainty term ${\sigma }_n\left(x\right)$, and lower values of $\kappa$ favor exploitation by focusing more on areas with high predicted mean ${\mu }_n\left(x\right)$.

UGABO dynamically adjusts the exploration-exploitation balance based on the uncertainty estimates. This ensures that the algorithm focuses more on exploration in regions of high uncertainty and on exploitation (Daliri et al., 2024).

The trade-off parameter $\kappa$ can be dynamically adjusted using the maximum uncertainty ${\sigma }_{max}$:

(6) $${\kappa }_{\left(t\right)}={\kappa }_0\cdot {\displaystyle \frac{{\sigma }_{max}}{{\sigma }_n\left(x\right)}}$$where ${\kappa }_{\left(t\right)}$ represents the trade-off parameter at iteration step t, controlling the balance between exploration and exploitation during the optimization process. ${\kappa }_0$ indicates initial trade-off parameter, ${\sigma }_{max}$ denotes the maximum uncertainty observed in the current iteration, guiding the adjustment of the trade-off parameter to favor exploration when uncertainty is high. ${\sigma }_n\left(x\right)$ represents the uncertainty associated with the prediction at point x during the n-th iteration, used to normalize the trade-off adjustment.

The complete UGABO process is illustrated in Fig. 1.

Multi-scale adaptive Wasserstein-Huber loss (MSAWH loss)

MSAWH is a method that dynamically adjusts the $\delta$ parameter of the Huber loss and incorporates multi-scale Wasserstein distances, aiming to build a more robust and flexible loss function by combining the strengths of both components, the specific workflow is shown in Fig. 2.

The Huber loss function is defined as:

(7) $$L_{\delta }\left(a\right)=\{\begin{array}{ccc}{\displaystyle \frac{1}{2}{a}^2}& if\hfill & \left|a\right|\le \delta \hfill \\ \delta \left(\left|a\right|-{\displaystyle \frac{1}{2}\delta }\right)\hfill & if\hfill & \left|a\right|>\delta \end{array}$$where $L_{\delta }\left(a\right)$ denotes the Huber loss function, $a$ is the error term, the difference between the predicted value and the true value, and $\delta$ is the threshold that determines whether the loss behaves quadratically (for small errors) or linearly (for larger errors).

During training, the parameter $\delta$ is dynamically adjusted based on the statistical properties of the errors, allowing it to adaptively change to provide more appropriate penalties across different error ranges. By dynamically adjusting the $\delta$ parameter of the Huber loss, the model can handle outliers more robustly.

Calculation of the Wasserstein distance at different scales to capture distributional differences at various levels is given as follows (You, Shung & Giuffrè, 2024).

The Wasserstein distance is defined as:

(8) $$W\left(p,q\right)=in{f}_{\gamma \in \mathrm{\Pi }\left(p,q\right)}{E}_{\left(x,y\right)\sim \gamma }\left[d\left(x,y\right)\right]$$where $W\left(p,q\right)$ represents the Wasserstein distance between the probability distributions p and q, $inf$ denotes the infimum (greatest lower bound). It is used here to find the minimum value of the expected distance over all possible joint distributions. $\gamma$ represents a joint distribution over the product space of x and y, where the marginals are p and q respectively. $\mathrm{\Pi }\left(p,q\right)$ denotes the set of all joint distributions with marginals $p$ and $q$. $E_{\left(x,y\right)\sim \gamma }$ represents the expected value (expectation) with respect to the joint distribution γ. It is the average value of the function $d\left(x,y\right)$ when pairs $\left(x,y\right)$ are sampled according to γ, and $d\left(x,y\right)$ is the distance metric between points $x$ and $y$.

Multi-scale Wasserstein distances capture distributional differences at various scales, enhancing the model’s sensitivity to information across different levels.

Combine the multi-scale Wasserstein distance with the adaptively adjusted Huber loss to form a new loss function:

(9) $$L_{MSAWH}=\alpha L_{\delta }\left(a\right)+\beta {\sum }_i{w}_i{W}_i\left(p,q\right)$$where $L_{MSAWH}$ represents the combined loss function, specifically designed to integrate the MSAWH loss. This function combines the Wasserstein distance and Huber loss components to adapt to various data distributions. $\alpha$ and $\beta$ are weighting parameters that control the influence of the Huber loss and Wasserstein distance components in the combined loss function. ${\sum }_i{w}_i{W}_i\left(p,q\right)$ represents the sum of Wasserstein distances calculated at different scales, with each term in the sum being weighted by $\omega i$, and $\omega i$ is the weight for the Wasserstein distance at the i-th scale, allowing the model to adjust the importance of each scale’s contribution to the total loss. $W_i\left(p,q\right)$denotes the Wasserstein distance between distributions p and q at the i-th scale. This term measures the distance between distributions at multiple resolutions or granularities.

Combining Wasserstein distance with Huber loss provides a flexible framework that can adapt to different types of data distributions and error characteristics.

Model predictive control (MPC) error correction

The MPC method is primarily used for prediction and control issues in dynamic systems (Zhao et al., 2024b). In this article, MPC is used to adjust the predicted time series data to more accurately match the actual values. Specifically, it dynamically adjusts predicted values to bring them closer to observed actual values. The optimization of the cost function reduces the error between predicted and actual values. During this process, it also smooths out anomalies in the predicted values (Gu, Wang & Liu, 2024).

In the MPC control approach, the cost function is mainly utilized to assess the discrepancy between the predicted values and the actual values (Han, Park & Lee, 2024). The formula is:

(10) $$J\left(x\right)=\sum _{t=0}^{N-1}(y_{true}\left[t\right]-\left(y_{pred}\left[t\right]+x\left[t\right]\right){)}^2$$where $J\left(x\right)$ represents the cost function, which is used to quantify the discrepancy between the predicted values and the actual values over a specified prediction horizon. $t$ represents the time step within the predicted horizon. N is the prediction horizon (the total number of future steps over which the cost function is evaluated). $y_{true}$ represents the actual value or ground truth for the target variable at time step t, $y_{pred}$ represents the predicted value at time step t, $x\left[t\right]$ represents the adjustment value applied at time step t, which is used to fine-tune or correct the prediction.

CNN-LSTM-Attention

The CNN is a deep learning architecture specifically designed to process data with a grid-like topology, such as images and time series (Zeghina et al., 2024). By leveraging local connections and weight sharing, CNNs effectively extract spatial or temporal features from input data, reducing the number of parameters and computational complexity (Zhang et al., 2024a). Their primary function is to automatically extract and learn high-level features from data, enabling the model to achieve high accuracy and efficiency in tasks such as classification, detection, and prediction (Kheddar et al., 2024).

A typical 1D-CNN consists of an input layer, a convolutional layer, a pooling layer, a fully connected layer, and an output layer. The input layer acts as the gateway for data. In the convolutional layer, features are extracted using a particular convolutional kernel, often referred to as a feature detector. The feature extraction process follows this formula:

(11) $$S\left(i\right)=\left(X\ast W\right)\left(i\right)=\sum _{m=0}^{M-1}X\left(i+m\right)\cdot W\left(m\right)$$where $S\left(i\right)$ represents the output of the convolution operation, $X$ denotes the input feature map, $W$ refers to the convolutional kernel, $M$ corresponds to the height of the convolutional kernel, respectively, and $i$ denotes the position of the output, $m$ is the index variable used in the summation, representing the positions within the convolutional kernel as it slides over the input feature map.

Ultimately, the CNN architecture combines outputs from various convolutional layers, utilizing Dropout and Batch Normalization to boost the model’s generalization ability and stability. This structure ensures effective learning from both temporal data and local features, while enhancing prediction performance through the fusion of multi-level features (Man et al., 2024).

LSTM is a variant of recurrent neural networks (RNNs) specifically designed to capture long-term dependencies within sequential data (Ehteram et al., 2024). LSTMs overcome the problems of vanishing and exploding gradients found in traditional RNNs by incorporating memory cells and gating mechanisms, which are specialized units designed for this purpose (Al-Selwi et al., 2024). In the CNN-LSTM-Attention model, LSTM is used to process time series data, such as voltage, current, and temperature over time. LSTM can capture the temporal dependencies in these sequences, providing more accurate predictions (He et al., 2024). An LSTM network consists of four key components: the input gate, forget gate, output gate, and cell state (Ehteram et al., 2024). Each component is vital for the LSTM’s operation, but the cell state stands out as the most critical. The cell state functions like a conveyor belt, enabling information to pass through multiple time steps without alteration, which is essential for preserving long-term dependencies (Zhao et al., 2024a). LSTM is governed by the cell state formula:

(12) $$C_t=f_t\cdot C_{t-1}+i_t\cdot {\tilde{C}}_t$$where $C_t$ represents the cell state at the current time step t, t is the time step in the sequence, and $f_t$ denotes the forget gate’s activation vector at time step t, which determines how much of the previous cell state $C_{t-1}$ should be retained, $C_{t-1}$ is the cell state at the previous time step (t − 1), $i_t$ is the input gate’s activation vector at time step t and ${\tilde{C}}_t$ is the candidate cell state at time step t.

In the CNN-LSTM-Attention model, LSTM plays a crucial role in capturing long-term dependencies in time series data. By combining CNN and Attention mechanisms, the model can more accurately predict the SOH of lithium batteries. The final outputs of the model include predicted SOH (make sure consistent) values and relevant evaluation metrics to assess the model’s performance.

The Attention mechanism is an advanced technique employed to improve the efficiency of models when processing sequential datasets. This approach allows the model to focus more on the relevant parts of the data by assigning varying levels of importance to different segments of the input sequence (Hassanin et al., 2024). By dynamically assigning different weights to various segments of the input sequence, this approach allows the model to focus on the most relevant information for the current task (Guo et al., 2024). For predicting the SOH in lithium batteries, the Attention mechanism can boost the performance of the CNN-GRU model by highlighting the time steps most relevant to SOH variations, thereby increasing the accuracy of the predictions (Du et al., 2024).

The core of the Attention mechanism lies in its ability to dynamically adjust the importance of input information, making the model more accurate in time series prediction. In lithium battery SOH prediction, the Attention mechanism helps the model better identify and utilize important time step data, thereby improving prediction performance and accuracy.

Evaluation metrics for prediction performance

To accurately quantify the deviation between forecasted outcomes and actual values, it is essential to implement appropriate performance assessment metrics. Let y = $\left[y_1,y_2,\dots ,y_n\right]$ represents the vector of actual values, with the average value denoted by $\overline{y}$. The vector representing the forecasted values is denoted by $\hat{y}=\left[\widehat{y1},\widehat{y2},\dots ,\widehat{yn}\right]$, and the relationship between them is captured by relevant assessment metrics. The main assessment metrics are listed below:

MAE:

(13) $$MAE\left(y,\hat{y}\right)={\displaystyle \frac{1}{n}\sum _{i=1}^n\left|yi-\widehat{yi}\right|.}$$

MSE:

(14) $$MSE\left(y,\hat{y}\right)={\displaystyle \frac{1}{n}\sum _{i=1}^n(yi-\widehat{yi}{)}^2.}$$

MAPE:

(15) $$MAPE={\displaystyle \frac{1}{n}\sum _{i=1}^n{\displaystyle \frac{yi-\widehat{yi}}{yi}.}}$$

RMSE:

(16) $$RMSE\left(y,\hat{y}\right)=\sqrt{{\displaystyle \frac{1}{n}\sum _{i=1}^n{(y_i-\widehat{yi})}^2}}.$$

R²:

(17) $$Rsquare\left(y,\hat{y}\right)=\left|1-{\displaystyle \frac{\sum {(y_i-\hat{y}i)}^2}{\sum {\left[y_i-mean\left(\hat{y}i\right))\right]}^2}}\right|.$$

CRMSD:

(18) $$CRMSD=\sqrt{{\displaystyle \frac{1}{N}\sum _{i=1}^N{\left[\left(\hat{y}i-\overline{\hat{y}}\right)-\left(yi-\overline{y}\right)\right]}^2}}.$$

MAD:

(19) $$MAD\left(y,\hat{y}\right)=median\left|yi-\hat{y}i\right|.$$

nRMSE:

(20) $$nRMSE\left(y,\hat{y}\right)={\displaystyle \frac{RMSE\left(y,\hat{y}\right)}{\overline{y}}.}$$

Process flow of UGABO-CNN LSTM Attention-MSAWH Loss-MPC model

The detailed workflow of the UGABO-CNN-LSTM-Attention-MSAWH Loss-MPC model is depicted in Fig. 3, outlining data preparation, model training, prediction, and evaluation stages. The process ensures a structured and reproducible approach for LIB SOH evaluation.

Raw data description

To analyse the degradation performance of LIBs in electric vehicles and support the development of deep learning prediction models, the NASA lithium battery dataset serves as the foundation data source (Saha & Goebel, 2007). This dataset includes four batteries: B0005, B0006, B0007, and B00018. The basic testing parameters of these four batteries are summarized in Table 2. These batteries are initially charged in a constant current (CC) mode at 1.5 A until they reach 4.2 V, then charged in a constant voltage (CV) mode until the current decreases to 20 mA. The discharge process occurs in CC mode at 2 A until the voltages of batteries 5, 6, 7, and 18 drop to 2.7, 2.5, 2.2, and 2.5 V, respectively.

The battery is subjected to continuous charge and discharge cycles until its capacity meets the predetermined end-of-life criteria. Figure 4 illustrates the degradation trends of four lithium-ion batteries (B0005, B0006, B0007, and B0018) in terms of capacity (A), current (B), and voltage (C) over time. Overall, the three subplots collectively demonstrate how different cells degrade under the same testing protocol and highlight the importance of modelling multi-dimensional inputs for accurate SOH estimation.

Evaluation of CNN LSTM attention framework

This section validates the proposed deep learning-based SOH prediction system for lithium-ion batteries using the NASA dataset. As outlined in the workflow (Results), data preprocessing, augmentation, and partitioning were key steps to ensure robust evaluation. To enhance dataset diversity, data augmentation methods—including linear interpolation, cross-enhancement, flipping, and slicing—were applied. The dataset was partitioned into training, validation, and test sets, with flexible parameter control to enable different experimental setups.

Validation sets were dynamically generated through K-fold cross-validation (k = 3), and the independent test set was reserved exclusively for final performance evaluation. These rigorous methodologies ensured unbiased and reliable assessment of the model’s predictive capabilities, as presented in the following results.

It is important to note that, unless explicitly stated otherwise (e.g., the baseline model), all experimental configurations integrate UGABO, MSAWH Loss, and MPC. These core components are fundamental to the system’s design and will not be reiterated in subsequent sections.

As illustrated in Fig. 5, the proposed CNN-LSTM-Attention model was trained and tested on datasets generated by four distinct data augmentation techniques: linear interpolation, cross-validation sampling, flipping, and slicing. Across all four scenarios, the predicted capacity trajectories (in red) closely match the actual measured values (in blue), indicating high consistency in capturing the underlying degradation patterns.

The results demonstrate that the integration of multiple data augmentation techniques significantly enhances the model’s adaptability to diverse input patterns and strengthens its resilience against noise and distributional shifts.

Analysis and comparison of different LIB capacity prediction models

This section examines the strengths and weaknesses of different capacity prediction methods. The accuracy of models such as CNN-LSTM-Attention, CNN-GRU-Attention, CNN-GRU, CNN, CNN-LSTM, LSTM, and LSTM-Attention is evaluated. The method resembles that described in ‘Analysis and Comparison of different Hyperparameter Optimization Method’, with the forecast results illustrated in Fig. 6.

As illustrated in Fig. 6 (capacity trend comparison), multiple deep learning architectures such as CNN-GRU-Attention and CNN-LSTM-Attention demonstrate satisfactory capacity prediction performance. However, a comparative analysis across all seven architectures indicates that CNN performs relatively poorly. Although CNNs are well-suited for capturing spatial patterns, their ability to model temporal dependencies in sequential battery data is limited, making them less effective in SOH estimation tasks. Similarly, while LSTM captures temporal dependencies, its standalone performance is slightly lower, likely due to limitations in feature extraction from raw multivariate inputs. Although certain parameters of CNN-GRU-Attention occasionally outperform those of CNN-LSTM-Attention, the latter exhibits overall smaller prediction errors and superior robustness across different evaluation scenarios. Therefore, we ultimately selected the CNN-LSTM-Attention method for subsequent model development and validation.

Analysis and comparison of different hyperparameter optimization method

In the hyperparameter optimization comparison experiments, the proposed UGABO optimization algorithm was compared with Random Search, Tree-structured Parzen Estimator (TPE), Particle Swarm Optimization (PSO), and Bayesian Optimization under the same model structure. As shown in Fig. 7, the proposed UGABO algorithm consistently outperforms other mainstream hyperparameter optimization methods—including Bayesian Optimization, PSO, Random Search, and TPE—across various data augmentation strategies (interpolation, flipping, slicing, crossover). The box plot in Fig. 7 illustrates that UGABO achieves the narrowest error distribution and the lowest median MAE and RMSE values, reflecting both accuracy and robustness.

Overall, these results validate that UGABO not only improves model generalization but also offers stable convergence and stronger resistance to noise in high-dimensional search spaces. The consistent dominance of UGABO across multiple metrics and datasets reinforces its practical utility in optimizing deep learning-based SOH prediction models.

Comparison with various loss functions

In this section, the proposed MSAWH Loss, based on the Wasserstein distance, is compared with Huber Loss, Log-Cosh Loss, Mean Squared Logarithmic Error (MSLE), and Smooth L1 Loss. The evaluation metrics used for comparison include R², MSE, and RMSE. The results are presented in Fig. 8. The testing conditions are similar to those described in ‘Analysis and comparison of different hyperparameter optimization methods’, with the only difference being the type of loss function used in the same model architecture.

As shown in the box plot of Fig. 8, the MSAWH loss function results in a narrower interquartile range and fewer outliers compared to other methods, indicating more stable and reliable predictions.

These trends confirm that MSAWH is not only robust to noisy data but also maintains generalization across different data distributions.

The prediction errors are concentrated within a smaller range, demonstrating that the model exhibits strong robustness in most cases.

Therefore, the integration of multi-scale Wasserstein distance with Huber loss proves particularly effective in handling battery degradation variability and mitigating the influence of abnormal values or distribution shifts.

Evaluation of ablation experiments

This section primarily employs ablation experiments to validate the importance of each component, ensuring the model’s robustness. The experiments are divided into five groups:

(1) Baseline Model: A CNN-LSTM-Attention model that does not utilize UGABO optimization, MSAWH Loss, or MPC error correction. It employs fixed hyperparameters without any hyperparameter optimization methods and uses the Huber Loss function with no error correction algorithm.

(2) Complete model.

(3) Removal of UGABO Hyperparameter Optimization.

(4) Removal of MSAWH Loss Function.

(5) Removal of MPC Error Correction Algorithm.

From Tables 3 to 6, the results indicate that removing any of the three proposed components—UGABO, MSAWH, or MPC—leads to a noticeable degradation in model performance, as reflected by metrics such as MAE and RMSE. Specifically, excluding UGABO results in the highest increase in prediction errors, underscoring its critical role in guiding efficient hyperparameter optimization. Omitting MSAWH widens the error distribution and increases the number of outliers, demonstrating its importance in enhancing error tolerance and mitigating the impact of anomalous data. Likewise, eliminating the MPC module introduces significant fluctuations in the predicted sequences, revealing its effectiveness in smoothing temporal predictions and reducing cumulative error propagation.

Compared to the baseline model, the improved model achieves a reduction in MAE from 0.01653 to 0.01313 and RMSE from 0.02619 to 0.02132, representing decreases of 20.6% and 18.6%, respectively. Additionally, the average R² improves from 0.97855 to 0.9798, marking a relative increase of 0.1%, which further validates the effectiveness of the proposed enhancements.