Integrated Framework of LSTM and Physical-Informed Neural Network for Lithium-Ion Battery Degradation Modeling and Prediction

Abstract

Accurate prediction of the State of Health (SOH) of lithium-ion batteries is essential for ensuring their safe and reliable operation. However, traditional deep learning approaches often suffer from challenges such as overfitting, limited generalization capability, and suboptimal prediction accuracy. To address these issues, this paper proposes a novel framework that combines a Long Short-Term Memory (LSTM) network with a Physics-Informed Neural Network (PINN), referred to as LSTM-PINN, for high-precision SOH estimation. The proposed framework models battery degradation using state-space equations and extracts latent temporal features. These features are further integrated into a Deep Hidden Temporal Physical Module (DeepHTPM), which incorporates physical prior knowledge into the learning process. This integration significantly enhances the model’s ability to accurately capture the complex dynamics of battery degradation. The effectiveness of LSTM-PINN is validated using two publicly available datasets based on graphite cathode materials (NASA and CACLE). Extensive experimental results demonstrate the superior predictive performance of the proposed model, achieving Mean Absolute Errors (MAEs) of just 0.594% and 0.746% and Root Mean Square Errors (RMSEs) of 0.791% and 0.897% on the respective datasets. Our proposed LSTM-PINN framework enables accurate battery aging modeling, advancing lithium-ion battery SOH prediction for industrial applications.

1. Introduction

Lithium-ion batteries have become the dominant choice for rechargeable energy storage, widely deployed across portable electronics, aerospace systems, and electric vehicles. With the rapid expansion of the electric vehicle industry, the demand for high-performance lithium-ion batteries continues to rise. However, battery degradation presents significant challenges: it not only incurs high replacement costs but also has a substantial impact on vehicle safety. As a result, Battery Management Systems (BMS) have been extensively developed to monitor and manage battery health [1]. A central function of BMS is the accurate estimation of the battery’s State of Health (SOH) and the prediction of its Remaining Useful Life (RUL) [2]. Reliable SOH and RUL predictions are critical for evaluating battery condition and ensuring the safe operation of electric power systems.

Substantial research efforts [3,4,5,6] have been devoted to improving the prediction of the State of Health (SOH) and Remaining Useful Life (RUL) of lithium-ion batteries, which remains a challenging task despite significant advancements in the field. The recent progress in machine learning across various domains has sparked growing interest in applying data-driven techniques to battery health estimation. These approaches circumvent the complexities of electrochemical modeling by directly learning degradation patterns from historical operational data. This characteristic makes them particularly appealing for practical applications, especially with the emergence of advanced deep learning architectures such as Long Short-Term Memory (LSTM) networks [7], Convolutional Neural Networks (CNNs), and Recurrent Neural Networks (RNNs) [8]. While these models have demonstrated encouraging results, their generalization capability remains limited. Their performance often depends heavily on the specific features extracted from particular datasets or operating conditions, which restricts their transferability and robustness across diverse battery systems.

To bridge the gap between interpretability and flexibility, Physics-Informed Neural Networks (PINNs) have emerged as a promising hybrid approach. By embedding physical laws and constraints into the learning process, PINNs can train effectively on limited data and demonstrate stronger generalization. Recent studies have demonstrated the effectiveness of PINNs in battery state estimation [9]. For PINN-based prediction approaches, the authors in [10] proposed a physics-informed neural network framework combining a predictor and an estimator. The predictor estimates degradation trends, while the estimator corrects the predictions by replacing hard-to-obtain sensor data. The research [11] showed a physics-informed multitask learning approach for battery aging state estimation, aiming to predict the dynamic variation of lithium-ion concentration in solid particles and the electrolyte. Wang et al. [12] further introduced a PINN-based method that first extracted key features from charging data, then used them to accurately estimate SOH. However, these approaches primarily employ linear neural architectures and do not fully exploit the time-series nature of battery data. As a result, they may struggle with unstable performance, especially when training data is limited.

To overcome these limitations, we propose a novel framework that combines a Long Short-Term Memory (LSTM) network with a Physics-Informed Neural Network (PINN), referred to as LSTM-PINN, for high-precision SOH estimation of lithium-ion batteries. In our approach, an LSTM network is first employed to extract latent temporal features from sequential monitoring data. These features are then incorporated as physical priors into the Deep Hidden Temporal Physical Module (DeepHTPM) of the proposed framework, enabling it to more effectively capture the complex degradation dynamics of batteries. The overall training is constrained by a combination of three loss components: SOH prediction loss, implicit PDE loss, and PDE gradient loss. This multi-loss optimization strategy helps improve both the training stability and the generalization of the model. The main contributions of this work can be summarized as follows:

• A novel framework is proposed, which integrates LSTM with PINN in a unified manner. The LSTM is employed to extract latent temporal features from battery monitoring sequences, which are then embedded as physical priors into the Deep Hidden Temporal Physical Module (DeepHTPM). This enables more accurate modeling of the degradation dynamics of lithium-ion batteries.
• We propose a dynamic degradation modeling approach based on the chain rule, which leverages early-stage predictions at each LSTM step to impose additional constraints on the model. This enhances reliability, mitigates overfitting in small datasets, and strengthens generalization performance.
• We assess the performance of our framework using the NASA and CALCE datasets. The experimental results show that LSTM-PINN consistently surpasses all baseline models, achieving Mean Absolute Errors (MAEs) of 0.594% on the NASA dataset and 0.746% on the CALCE dataset, along with Root Mean Square Errors (RMSEs) of 0.791% and 0.897%, respectively.

2. Related Works

Predicting the state of lithium-ion batteries involves constructing dynamic models that simulate the degradation process, offering valuable insights into internal battery changes over time. Existing methods for battery state prediction are typically categorized into three types: physics-based, electrochemical-based, and data-driven approaches.

Physics-based models simulate the underlying physical and chemical mechanisms governing battery behavior. While these models provide detailed insights into degradation patterns and performance under various operating conditions, they require extensive domain knowledge and precise parameterization, which are often difficult to obtain due to the complex internal reactions within lithium-ion batteries. Similarly, electrochemical-based models describe long-term battery degradation as a function of charge-discharge cycles using predefined mathematical relationships. Although these models can estimate future degradation trajectories, they struggle with real-time tracking of capacity variations and often suffer from low predictive accuracy. For example, Yang et al. [13] proposed a semi-empirical model based on Coulombic efficiency to capture degradation trends more effectively, while research in [14] combined experimental aging data with theoretical models to enhance prediction accuracy. Nevertheless, both approaches rely heavily on domain-specific expertise and require complicated modeling procedures with numerous tunable parameters, which limits their scalability and generalizability.

In contrast, data-driven models have shown significant promise in recent years, particularly for systems where physical mechanisms are poorly understood or too complex to model explicitly. These approaches extract latent patterns from historical monitoring data, without requiring explicit knowledge of the underlying physics. Data-driven methods can be broadly divided into linear regression models and time-series forecasting models [15,16], with the latter becoming the mainstream due to their superior ability to capture long-term trends and cyclical behavior. For instance, Ajami et al. [17] employed temporal neural networks to capture long-range dependencies in sequential battery data. Further enhancements were introduced in [16], where CNN and LSTM networks were combined to jointly exploit spatial and temporal features for capacity estimation. Building upon this, ref. [18] incorporated a self-attention mechanism to improve temporal feature extraction. Additionally, Mohamed et al. [19] proposed a data-driven DLinear model for SOH and RUL estimation, emphasizing the growing role of deep learning in Battery Management Systems (BMS). Despite their success, these models often underperform when applied to small, noisy datasets due to their reliance on large-scale training data and high-quality features.

To overcome these limitations, recent studies have explored PINN for battery state estimation. For example, Dev et al. [20] introduced a PINN-based approach that used an adaptive normalized loss function to estimate battery temperature. Cofré-Martel et al. [21] utilized deep neural networks in conjunction with partial differential equations (PDEs) to model complex physical degradation phenomena. However, such models often require extensive experimental measurements and physical parameters that are challenging to obtain. To address this, DeepHPM [22] was proposed to embed prior physical knowledge into neural networks using Automatic Differentiation (AutoDiff), enabling it to learn both discrete and continuous degradation dynamics—linear or nonlinear. Despite these advances, most PINN-based methods still fall short in fully exploiting the rich temporal features embedded in sequential battery data.

In this work, we propose a novel framework that overcomes these limitations by integrating LSTM with PINNs. Unlike existing methods, our approach extracts hidden temporal features from battery monitoring data and uses them as physical priors to guide the dynamic modeling process. These features are fused within a Deep Hidden Temporal Physical Module, which couples the learning capability of time-series models with the interpretability and constraints of physics-based modeling. This fusion enables the proposed framework to accurately capture the degradation behavior of lithium-ion batteries, even under data-limited conditions.

3. Problem Statement

3.1. Battery Degradation Model

SOH is a crucial concept in battery management systems. The SOH of a battery is a metric that reflects battery aging and is defined as the ratio of the battery’s current capacity to its original rated capacity. Once the SOH drops below 80%, the battery is considered to have reached the end of its first life. Since SOH serves as an indicator of the battery’s condition during each charge-discharge cycle, accurately estimating it is essential [23,24]. The SOH can be expressed as follows:

$$u_n={\mathrm{SOH}}_n={\displaystyle \frac{Q_t}{Q_{\mathrm{Nom}}}}\times 100\%$$

(1)

where $Q_{Nom}$ is the rated capacity, and $Q_t$ is the available capacity after the t-th charge-discharge cycle. When a battery operates under consistent conditions, the SOH after each cycle can be denoted as follows:

$$u_n=u\left(t\right)$$

(2)

where $u\left(\right)$ represents the function for predicting the SOH of the battery through t. We can derive the partial differential of the battery’s SOH through $u\left(\right)$. That is, the degradation rate of the battery’s SOH is modeled as follows:

$${\displaystyle \frac{du\left(t\right)}{dt}}=F(t,u;\theta )$$

(3)

Equation (3) can be interpreted as a parameterized PDE, where $F(·)$ is a nonlinear function of t, capacity u, and parameter set $\theta$. This function characterizes the battery’s degradation dynamics. However, due to the difficulty of acquiring electrochemical parameters in practice, this model is challenging to implement. To address this, literature [14] proposes a simplified aging model using regression analysis:

$${\displaystyle \frac{du\left(t\right)}{dt}}=\beta \left[u\left(t\right)\right]$$

(4)

where $\beta$ is a linearized aging rate affected by factors such as charge state, discharge depth, and temperature. While this model provides stable predictions, it may not capture the early-cycle degradation patterns due to variability in battery degradation modes and mechanical heterogeneity [25,26].

3.2. A Dynamic Model Based on Temporal Features

To address these limitations, deep learning approaches directly extract degradation patterns from historical data, eliminating the need for explicit electrochemical modeling. However, their generalizability is often limited by the quality of the extracted features. To improve accuracy, we propose a hybrid model that integrates LSTM with PINN, enabling dynamic and generalizable prediction of battery degradation.

To construct a data-driven battery degradation model, we define SOH as a function of the feature vector $X_N={[x_1,x_2,\dots ,x_{n-1}]}^T$, where $x_k$ is the remaining capacity after the k-th cycle, and $N=\{1,2,\dots ,n-1\}$. We can extend the function of the relationship between $X_N$ and the battery SOH through the Function (2); we have the following:

$$u_n=u\left(X_N\right)$$

(5)

the hidden temporal feature extracted from the input at cycle i using LSTM is given as follows:

$$H_n=N{N}_1(X_N,H_{n-1})$$

(6)

$$u_n=N{N}_2\left(H_n\right)$$

(7)

Formula (6) describes the prediction of the current hidden state $H_n$ using a temporal neural network ($N{N}_1$), based on the input $X_N$ and the previous hidden state $H_{n-1}$. Here, $N{N}_1$ represents the temporal network (e.g., LSTM), which captures the sequence dynamics. Formula (7) illustrates the estimation of the final output un from the current hidden state $H_n$ using a linear network ($N{N}_2$). Combine Formula (5), Formula (6), and Formula (7); we have the following:

$$u_n=u(X_N,H_{n-1})$$

(8)

It can be understood from the composition of the differential equation that the partial derivative of SOH with respect to $X_N$ is as follows:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial u_n}{\partial X_N}}\approx G\left(X_N,u_n,{\displaystyle \frac{\partial u_n}{\partial H_{n-1}}}|\theta \right)\end{array}$$

(9)

$G(·)$ represents the implicit partial differential equation in the deep physical model. where $\theta$ is the parameter of nonlinear function $G(·)$, Since the explicit form of $G(·)$ is difficult to obtain, we propose the DeepHTPM to implicitly model $G(·)$. We have the following:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial u_n}{\partial X_N}}\approx \mathrm{DeepHTPM}\left(X_N,\widehat{u_n},H_{n-1}^{\prime }|\mathrm{\Phi }\right)\end{array}$$

(10)

where $H_{n-1}^{\prime }={\displaystyle \frac{\partial u_n}{\partial H_{n-1}}}$, and $\mathrm{\Phi }$ denotes the trainable parameters of the DeepHTPM.

4. Framework Description

4.1. Framework of LSTM-PINN

LSTM-PINN consists of three main components: a surrogate neural network, the DeepHTPM module, and an automatic differentiator. The surrogate neural network is typically an LSTM in our framework. It takes the input sequence $X_n$ along with an initial hidden state $H_0$, and processes the sequence step by step over time. At each time step, the network receives the current input $x_i$ and the hidden state $H_{n-1}$ from the previous step. It then computes a new hidden state $H_n$, which is passed to the next time step as part of the input. This iterative process continues until the final prediction $\widehat{u_n}$ is produced. In essence, the model updates its hidden state recursively at each step based on the current input and prior hidden state, enabling it to capture temporal dependencies and generate the final output across the entire sequence. $\widehat{u_n}$ is given as follows:

$$\begin{array}{c}\hfill \widehat{u_n}=LSTM\left(X_N,H_{n-1};\gamma \right)\end{array}$$

(11)

To effectively capture the discrepancy between the predicted battery capacity and the actual remaining capacity, the surrogate neural network is trained by minimizing a data-driven loss function. This objective function is formulated as follows:

$$\begin{array}{c}\hfill L_u=\sum _{i=1}^N{\left[LSTM(X_N,H_{n-1})-u_n;\gamma \right]}^2\end{array}$$

(12)

where $\widehat{u_n}=LSTM(X_N,H_{n-1};\gamma )$ denotes the output of the surrogate network parameterized by $\gamma$, and $u_n$ is the ground truth value.

To approximate the unknown dynamic behavior of battery health evolution in the absence of an explicit physical model, the DeepHTPM architecture (illustrated in Figure 1) is introduced. DeepHTPM is designed as a multi-layer fully connected neural network. Each layer applies a non-linear activation function to its outputs, enabling the model to represent complex non-linear relationships. The hyperbolic tangent function is selected due to its smooth differentiability, which is crucial for gradient-based learning.

Inputs to the DeepHTPM module include the original sequence $X_N$, the predicted capacity $\widehat{u_n}$, and the first-order partial derivatives of the hidden states with respect to their corresponding input variables. These derivatives are computed via an automatic differentiation mechanism.

The DeepHTPM parameters $\mathrm{\Phi }$ are optimized using a loss based on a physical-inspired constraint, approximating a PDE model. The PDE consistency loss is given by the following:

$$\begin{array}{c}\hfill L_f=\sum _{i=1}^N{\left[{\displaystyle \frac{\partial u_n}{\partial x_k}}-\mathrm{DeepHTPM}\left(X_N,\widehat{u_n},H_{n-1}^{\prime }|\mathrm{\Phi }\right)\right]}^2\end{array}$$

(13)

where $H_{n-1}={[h_1^{\prime },\dots ,h_{n-1}^{\prime }]}^T$ comprises the required derivatives of hidden states.

To formalize this relation, we define a residual function $f(X_N;\theta ,\mathrm{\Phi })$ as follows:

$$\begin{array}{c}\hfill f(X_N;\theta ,\mathrm{\Phi })=u_x\left(X_N\right)-G(X_N,u_n,H_{n-1}^{\prime }\left|\theta \right)\end{array}$$

(14)

where $u_x={\left[{\displaystyle \frac{\partial u_n}{\partial x_1}},\dots ,{\displaystyle \frac{\partial u_n}{\partial x_{n-1}}}\right]}^T$ represents the vector of partial derivatives of $u_n$ with respect to each input dimension except $x_k$.

To further enforce the smoothness and consistency of the learned residual function, we define a gradient-based PDE residual loss as follows:

$$\begin{array}{c}\hfill L_{f_X}=\sum _{i=1}^N{\left[f_X(X_N;\theta ,\mathrm{\Phi })-0\right]}^2\end{array}$$

(15)

where $f_X(X_N;\theta ,\mathrm{\Phi })={\left[{\displaystyle \frac{\partial f}{\partial x_1}},\dots ,{\displaystyle \frac{\partial f}{\partial x_{n-1}}}\right]}^T$ denotes the gradients of the residual function with respect to each input.

The overall training objective is to minimize a weighted sum of the three loss components:

$$\begin{array}{c}\hfill L={\lambda }_u{L}_u+{\lambda }_f{L}_f+{\lambda }_{f_X}{L}_{f_X}\end{array}$$

(16)

where ${\lambda }_u,{\lambda }_f,{\lambda }_{f_X}$ are weight coefficients determining the relative importance of each loss term and ${\lambda }_u+{\lambda }_f+{\lambda }_{f_X}=1$. Our experimental results indicate that the weighting of individual loss components plays a critical role in model performance. Given the high cost and inefficiency of manual tuning, we explore the use of Bayesian optimization as an automated approach to adjust these loss weight hyperparameters effectively.

4.2. Bayesian Optimization for Hyperparameter Tuning

Hyperparameter tuning is a critical step in training machine learning models, as the choice of hyperparameters can significantly affect performance. Traditional approaches such as grid search and random search require a large number of function evaluations and become intractable when the hyperparameter space is high-dimensional or when each evaluation (i.e., model training and validation) is costly. Bayesian optimization [27,28] offers an efficient alternative by constructing a probabilistic surrogate model of the objective function and using it to guide the search for optimal hyperparameters. The flowcharts of ${\lambda }_u$, ${\lambda }_f$, and ${\lambda }_{f_x}$ using Bayesian optimization are shown in the Figure 2. The flowchart depicts a standard Bayesian optimization process, commonly used for optimizing expensive black-box functions. The procedure begins with an initialization step, where the process starts by selecting random points to evaluate the objective function. Next, a surrogate model, typically a Gaussian process, is fitted to the data collected from these initial points. The acquisition function is then optimized to determine the next point for evaluation, balancing exploration and exploitation. The objective function is evaluated at this selected point, and the surrogate model is updated with the new data. This iterative process continues, checking for convergence criteria after each update. If the convergence criteria are met, the process stops; otherwise, it repeats the cycle of fitting the surrogate model, optimizing the acquisition function, and updating with new data.

4.2.1. Overview of Bayesian Optimization

Bayesian optimization is a sequential model-based optimization framework designed to minimize (or maximize) an expensive-to-evaluate black-box function, which is defined as follows:

$$\begin{array}{c}\hfill f:\mathcal{X}\to \mathbb{R},\end{array}$$

(17)

where $\mathcal{X}\subset {\mathbb{R}}^d$ is the d-dimensional hyperparameter space. At each iteration t, Bayesian optimization maintains:

• A surrogate model (often a Gaussian process) that approximates the unknown objective function $f\left(x\right)$.
• An acquisition function ${\alpha }_t\left(x\right)$ that quantifies the utility of evaluating f at a candidate point $x\in \mathcal{X}$.

The general loop of Bayesian optimization is as follows:

• Initialize: Select an initial set of $n_0$ hyperparameter points ${\left\{x^{\left(i\right)}\right\}}_{i=1}^{n_0}$ (e.g., via Latin hypercube sampling) and observe $y^{\left(i\right)}=f\left(x^{\left(i\right)}\right)$.
• Fit Surrogate: Train the surrogate model (e.g., a Gaussian process) on ${\left\{(x^{\left(i\right)},y^{\left(i\right)})\right\}}_{i=1}^{t-1}$ to obtain a posterior distribution $p\left(f\left(x\right)|{\mathcal{D}}_{t-1}\right)$, where ${\mathcal{D}}_{t-1}={\left\{(x^{\left(i\right)},y^{\left(i\right)})\right\}}_{i=1}^{t-1}$.
• Optimize Acquisition: Find the next query point

  $$\begin{array}{c}\hfill x_t=arg\underset{x\in \mathcal{X}}{max}{\alpha }_{t-1}\left(x\right),\end{array}$$

  (18)

  where ${\alpha }_{t-1}\left(x\right)$ is an acquisition function defined using the posterior predictive mean ${\mu }_{t-1}\left(x\right)$ and variance ${\sigma }_{t-1}^2\left(x\right)$.
• Evaluate Objective: Compute $y_t=f\left(x_t\right)$ (training and validating the model with hyperparameters $x_t$).
• Augment Data: Update ${\mathcal{D}}_t={\mathcal{D}}_{t-1}\cup \left\{(x_t,y_t)\right\}$.
• Repeat: Return to Step 2 until a stopping criterion is met (e.g., budget exhausted or convergence).

4.2.2. Algorithmic Summary

Algorithm 1 summarizes the Bayesian optimization procedure for minimization. In summary, Bayesian optimization provides a principled framework for hyperparameter tuning by iteratively building a probabilistic surrogate of the objective function and using acquisition functions to balance exploration and exploitation. Its ability to find competitive hyperparameter configurations with relatively few evaluations makes it a cornerstone method in Automated Machine Learning (AutoML) pipelines.The training process of LSTM-PINN is shown in Algorithm 2, where $SSE(·)$ denotes computing MSE error by using Equations (12), (13) and (15), respectively, $D_{Trian}=\left\{{X_N,u_n}_{n=1}^N\right\}$ is training data.

Algorithm 1 Bayesian optimization for hyperparameter tuning

Require:  Domain $\mathcal{X}$, initial points ${\left\{x^{\left(i\right)}\right\}}_{i=1}^{n_0}$, acquisition function $\alpha \left(x\right)$, evaluation budget T.   1: Evaluate $y^{\left(i\right)}\leftarrow f\left(x^{\left(i\right)}\right)$ for $i=1,\dots ,n_0$.   2: ${\mathcal{D}}_{n_0}\leftarrow {\left\{(x^{\left(i\right)},y^{\left(i\right)})\right\}}_{i=1}^{n_0}$.   3: for $t=n_0+1$ to T do   4:    Fit Gaussian Process to ${\mathcal{D}}_{t-1}$.   5:    Select next point $$x^{\left(t\right)}\leftarrow arg\underset{x\in \mathcal{X}}{max}\alpha \left(x;{\mathcal{D}}_{t-1}\right).$$   6:    Evaluate $y^{\left(t\right)}\leftarrow f\left(x^{\left(t\right)}\right)$.   7:    Augment data: ${\mathcal{D}}_t\leftarrow {\mathcal{D}}_{t-1}\cup \left\{(x^{\left(t\right)},y^{\left(t\right)})\right\}$.   8: end for   9: Output: ${\displaystyle x^{*}=arg\underset{1\le i\le T}{min}{y}^{\left(i\right)}}$.

In summary, LSTM is a type of RNN capable of capturing long-range dependencies in sequential data. It uses gated mechanisms (input gate, forget gate, output gate) to selectively remember or discard information over time, making it effective for modeling battery SOH time series. PINNs embed physical laws, typically represented by differential equations, as constraints within the neural network’s learning process. This incorporation enhances model robustness, particularly in scenarios with limited or noisy data, thereby improving predictive reliability and accuracy. Hence, we propose a unified framework that integrates LSTM with PINN. This integration facilitates more precise modeling of the degradation dynamics of lithium-ion batteries.

Algorithm 2 The training process of LSTM-PINN

Input:  training data $D_{Trian}=\left\{{X_N,u_n}_{n=1}^N\right\}$, hyper-parameters. Output:  surrogate $LSTM\left(X_N,H_{n-1};\gamma \right)$, dynamic model $G\left(X_N,u_n,H_{n-1}^{\prime }|\theta \right)$.   1: initialize $\theta$, $\gamma$, ${\lambda }_u$, ${\lambda }_f$, ${\lambda }_{f_x}$   2: for each $i\in$Search times do   3:    for each $j\in$epoch do   4:      $\widehat{u}$ ← $u\left(X_{N,}{H}_{n-1};\theta \right)$   5:      ${\widehat{H}}_{n-1}$ ← $AutoDiff(\widehat{u},X_n)$   6:      ${\widehat{X}}_N$ ← $AutoDiff(\widehat{u},H_{n-1})$   7:      $\widehat{G}$ ← $G\left(X_N,\widehat{u_n},{\widehat{H}}_{n-1};\mathrm{\Phi }\right)$   8:      $\widehat{f}$ ← ${\widehat{X}}_N-\widehat{G}$   9:      ${\widehat{f}}_x$ ← $AutoDiff(\widehat{f},X_N)$ 10:      ${\widehat{L}}_u$ ← $SSE(\widehat{u},u)$ 11:      ${\widehat{L}}_f$ ← $SSE(\widehat{f},0)$ 12:      ${\widehat{L}}_{f_x}$ ← $SSE(\widehat{f_x},0)$ 13:      compute loss $\widehat{L}$ by (18) 14:      update $\theta$, $\mathrm{\Phi }$ on loss $\widehat{L}$ 15:    end for 16:    update ${\lambda }_u$, ${\lambda }_f$, ${\lambda }_{f_x}$ on MSE by Bayes 17: end for 18: return models

5. Experiments

5.1. Dataset Description

This study utilizes two publicly available datasets: the NASA lithium-ion battery dataset and the CALCE dataset from the University of Maryland. Key battery characteristics of these two datasets are presented in Figure 3.

The NASA dataset [29] provided by the Prognostics Center of Excellence at NASA Ames involves 18,650 Li-ion cells (2 Ah rated capacity) tested under accelerated aging to simulate hybrid electric vehicle conditions. Batteries were charged using a constant current (1.5 A)–constant voltage (4.2 V) protocol and discharged at 2 A until the voltage limit was reached. End of Life (EOL) was defined as a 30% reduction in capacity.

The CALCE dataset [30,31] developed by the Center for Advanced Life Cycle Engineering supports research in battery health monitoring and fault diagnostics. Similar charging and discharging protocols were used, with discharging down to 2.7 V. EOL was likewise determined when capacity fell below 70% of nominal.

We use a sliding window to extract battery features, dividing each cycle sequence into multiple sub-sequences to generate training samples [32,33]. With a window size of n and battery cycles m, the sequence is divided into $X=[X_1,X_2,\dots ,{X_i\dots ,X}_{m-n}]$, where $X_i=[x_i,x_{i+1},x_{i+2},\dots ,x_{i+n-1}]$ and label $Y_i=x_{i+n}$. This enhances data utilization and modeling of battery degradation. To mitigate variance among battery samples, SOH prediction uses SOH-based normalization for consistency and accuracy. The leave-one-out cross-validation method is used, where one battery is randomly selected as the test set and the remaining batteries are used as the training set until all batteries are used as the overtest set to train the model as a result of cross validation.

5.2. Experiment Results on the NASA and CALCE Datasets

In order to explore the performance of different models in SOH prediction, two sets of experimental cases were designed, labeled as Experiment A and Experiment B, respectively. Experiment A is for predicting the SOH of the NASA battery, and Experiment B is for predicting the SOH of the CACLE battery. In experiment A, we trained LSTM(M1), CNN-LSTM(M2), CNN-LSTM-AM(M3), swin transformer(M4), LSTM-PINN(M5), TPANet(M6),Bayes LSTM-PINN(M7) model to predict the SOH of the NASA battery. In Experiment B, these six models in Experiment A were trained to predict the SOH of the CACLE battery. LSTM [34] simultaneously considers contextual information, with the aim of enhancing the ability to understand the context. CNN-LSTM [16] combines the local feature extraction of a convolutional network and the time-dependent processing of LSTM, which can better extract data features. CNN-LSTM-AM [35] further enhances the focus on important features by adding attention mechanisms. Swin transformer [36,37] leverages a transformer architecture, enabling efficient computation and scalability while maintaining high accuracy. TPANet [38] consists of a triple parallel architecture combining convolutional neural networks and attention units, where each parallel branch extracts capacity degradation features with enhanced sensitivity to regeneration phenomena and long-term dependencies, and is preceded by an FNN-based window selection mechanism to optimize temporal feature input. LSTM-PINN and Bayes LSTM-PINN are the proposed frameworks in this study, both incorporating a surrogate neural network based on LSTM. The Bayes LSTM-PINN further integrates Bayesian optimization for hyperparameter tuning.

(1) Root Mean Square Error (RMSE). RMSE, which is applied to reveal how tightly the observed data clusters around the predicted values, is a commonly used metric to measure the difference between a model’s predicted values and the actual values. The formula of RMSE is given as follows:

$$\begin{array}{c}\hfill RMSE=\sqrt{{\displaystyle \frac{1}{N}}\sum _{i=1}^N{({\widehat{u}}_i-u_i)}^2}\end{array}$$

(19)

where ${\widehat{u}}_i$ denotes the predicted value, $u_i$ is the observed value, and N is the number of data samples.

(2) Mean Absolute Error (MAE). MAE is a primary metric used to measure the accuracy of a model. It is measured as the average absolute difference between the predicted values and the true values. The formula of MAE is the following:

$$\begin{array}{c}\hfill MAE={\displaystyle \frac{1}{N}}\sum _{i=1}^N\left|{\widehat{u}}_i-u_i\right|\end{array}$$

(20)

5.2.1. Experiment A

Experiment A verifies the SOH estimation results of the six models on the NASA batteries, with results presented in

Table 1. The RMSE and MAE of NASA.

		M1	M2	M3	M4	M5	M6	M7
B0005	RMSE	1.696	1.362	1.15	0.906	1.318	0.883	0.866
	MAE	1.098	1.031	0.887	0.805	1.078	0.836	0.746
B0006	RMSE	2.274	1.905	1.721	1.46	1.877	1.403	1.353
	MAE	1.8	1.588	1.404	1.216	1.536	1.22	1.194
B0007	RMSE	1.837	1.681	1.006	0.906	1.497	0.852	0.797
	MAE	1.577	1.318	1.128	1.076	1.345	1.003	0.901
B0018	RMSE	2	1.754	1.447	1.201	1.687	1.124	0.985
	MAE	1.736	1.576	1.247	1.059	1.544	0.921	0.822

. The SOH prediction results and evaluation metrics for each model are depicted in Figure 4 and Figure 5, respectively. From Table 1, it is evident that compared to other models, the prediction errors of LSTM-PINN and Bayes LSTM-PINN are significantly reduced, with both RMSE and MAE showing marked improvement. Notably, LSTM exhibits higher errors, with RMSE and MAE values often exceeding 1.5% and 1.2%, respectively, while CNN-LSTM shows moderate improvements, typically with RMSE around 1.3% and MAE around 1.1%. The swin transformer model performs slightly better, with RMSE and MAE values generally around 1.2% and 1.0%, respectively. The prediction effect of TPANet is second only to ours, reaching the best (RMSE) 0.852% and (MAE) 0.836%. To verify the stability of the framework, a four-fold cross-validation was conducted on each model using this dataset, with experimental results shown in Figure 5. In the cross-validation of the six models, Bayes LSTM-PINN demonstrates the best prediction performance, achieving the lowest RMSE of 0.797% and MAE of 0.746%. As seen in Figure 5, compared to LSTM (with RMSE often above 1.6% and less stable performance), CNN-LSTM, swin transformer, LSTM-PINN, and TPANet, the proposed Bayes LSTM-PINN exhibits higher stability and superior performance in battery SOH prediction, likely due to its enhanced optimization and robust architecture. This performance difference can be attributed to multiple aspects. Firstly, although the traditional LSTM model has the ability of time-series modeling, it has certain limitations in capturing long-range dependencies and nonlinear features, resulting in a relatively high prediction error. CNN-LSTM and CNN-LSTM-AM enhance the ability of local feature extraction by introducing convolutional structures, but it is still difficult to model the complex temporal dynamics during the SOH degradation process. The swin transformer enhances the global modeling ability with the help of the self-attention mechanism and performs relatively better. However, the model has high complexity and strong sensitivity to the amount of data and parameters, which affects its stability. TPANet improves the model performance to a certain extent by introducing the spatio-temporal attention mechanism. However, its architecture still has problems such as parameter redundancy and difficulty in optimization, which limit the improvement of the final accuracy. The Bayes LSTM-PINN we proposed integrates physical priors in the model structure and dynamically adjusts the weights of the multitask loss terms through Bayesian optimization, achieving better generalization ability and lower prediction error. This fully demonstrates the important role of optimization strategies and physical guidance in improving the accuracy of SOH prediction.

5.2.2. Experiment B

In Experiment B, the SOH prediction for CALCE batteries was assessed using seven distinct models: LSTM, CNN-LSTM, CNN-LSTM-AM, swin transformer, LSTM-TPIPNN, TPANet, and the proposed Bayes LSTM-PINN framework. The predictive performance is summarized in

Table 2. The RMSE and MAE of CALCE.

		M1	M2	M3	M4	M5	M6	M7
CS2_35	RMSE	2.957	2.334	1.986	1.669	2.087	1.553	1.438
	MAE	2.024	1.915	1.565	1.453	1.880	1.239	0.965
CS2_36	RMSE	3.016	2.85	1.586	1.49	2.768	1.386	1.324
	MAE	2.085	1.827	1.366	1.24	1.726	1.115	1.097
CS2_37	RMSE	1.951	1.686	1.25	1.015	1.672	0.994	0.897
	MAE	1.455	1.369	0.899	0.791	1.239	0.678	0.594
CS2_38	RMSE	2.303	1.719	1.594	1.282	1.706	1.137	0.964
	MAE	1.497	1.301	0.964	0.869	1.206	0.804	0.7495

and visually represented in Figure 6, with detailed metric distributions provided in Figure 7. The LSTM model, a baseline recurrent neural network, captures temporal dependencies with RMSE and MAE values approximating 2.303% and 1.497%, respectively, across various datasets. The CNN-LSTM model integrates convolutional layers with LSTM to extract spatial features prior to temporal analysis, yielding RMSE and MAE values such as 2.85% and 1.827% for CS2_36. The CNN-LSTM-AM variant enhances this approach with an attention mechanism, yet its RMSE and MAE, e.g., 1.954% and 1.369% for CS2_37, remain inferior to the proposed model. The swin transformer employs a transformer architecture with shifted windows, achieving RMSE and MAE around 1.726% and 1.124% for CS2_35.TPANet employs a fusion model of triple parallel attention and CNN, and uses FNN technology to calculate the window size, which has significantly improved the prediction results. The RMSE has reached 0.994% and the MAE has reached 0.678% for CS2_37. The LSTM-TPIPNN combines LSTM with a physics-informed neural network, improving upon standalone LSTM with RMSE and MAE values like 1.266% and 0.964% for CS2_38. Notably, the Bayes LSTM-PINN framework outperforms all models, exhibiting the lowest RMSE of 0.897% and MAE of 0.594% on CS2_38. This superior performance is attributed to the effective temporal feature extraction via LSTM, augmented by physical model guidance, and optimized through Bayesian techniques for loss weight adjustment, thereby enhancing generalization and robustness in SOH prediction.

5.2.3. Ablation Study on Different Loss Functions

This ablation study evaluates the impact of different loss functions on battery SOH prediction, with experimental results are shown in

Table 3. Ablation results on NASA.

Model	Metrics (%)
$L_u$		✓	✓	✓
$L_f$		✗	✓	✓
$L_{f_x}$		✗	✗	✓
B0005	RMSE	3.461	1.914	1.318
	MAE	2.48	1.534	1.078
B0006	RMSE	3.266	2.648	1.877
	MAE	3.015	2.224	1.536
B0007	RMSE	4.23	3.05	1.497
	MAE	3.84	2.463	1.345
B0018	RMSE	3.436	2.735	1.687
	MAE	2.821	2.23	1.544

and

Table 4. Ablation results on CALCE.

Model	Metrics (%)
$L_u$		✓	✓	✓
$L_f$		✗	✓	✓
$L_{f_x}$		✗	✗	✓
CS2_35	RMSE	3.678	2.744	2.087
	MAE	3.794	2.467	1.880
CS2_36	RMSE	4.627	3.541	2.768
	MAE	4.137	3.171	1.726
CS2_37	RMSE	3.671	2.947	1.672
	MAE	3.4	2.489	1.239
CS2_38	RMSE	3.98	2.764	1.706
	MAE	3.347	2.137	1.206

, and Figure 8 and Figure 9. In the ablation experiment, we can observe that the result of using only $L_u$ on the NASA dataset is the worst, with RMSE and MAE being only 3.266% and 2.48%, respectively. After adding $L_f$, the prediction result significantly improves, with RMSE and MAE reaching 1.917% and 1.534% respectively. Moreover, when $L_{f_x}$ is added along with $L_u$ and $L_f$, this enabled our framework results to reach the optimum, with RMSE and MAE reaching 1.318% and 1.078%, respectively. The results of $L_u$ on the CALCE dataset were the worst, with RMSE and MAE being only 3.671% and 3.4%, respectively. After adding $L_f$, RMSE and MAE reached 2.744% and 2.137%, respectively. And when $L_{f_x}$ was added along with $L_u$ and $L_f$, RMSE and MAE reached 1.672% and 1.206%, respectively. This indicates that the $L_f$ and $L_{f_x}$ proposed by us have a significant impact on the performance improvement of the framework. This experiment also demonstrates the effectiveness and reliability of the deep hidden physical knowledge extracted by DeepHTPM in improving predictive performance. In addition, combining LSTM with PINN significantly contributes to improving prediction accuracy.

5.3. Impact of Loss Function Weights on Prediction Performance

Since the weights of the loss components play a crucial role in balancing the model’s objectives, we employed Bayesian Optimization to tune the coefficients in the composite loss function defined as $L={\lambda }_u{L}_u+{\lambda }_f{L}_f+{\lambda }_{f_X}{L}_{f_X}$. Figure 10 details the influence of the algorithm optimizing the loss weight. The scattered distribution indicates that the parameter space has been widely explored, and the low RMSE area is mainly concentrated within the range of ${\lambda }_u<0.4$ and ${\lambda }_f<0.2$, suggesting that smaller weight parameters may optimize the model performance. The middle area shows a large fluctuation in RMSE, with the color changing from purple to orange, reflecting the sensitivity of the parameter combination to the error. Higher RMSE (0.9–1.1) is more common in the regions where ${\lambda }_u>0.7$ or ${\lambda }_f>0.6$, indicating that excessive weights may lead to overfitting or performance degradation. The optimization algorithm indicates that the parameters are fine-tuned within a lower weight range to achieve the best effect, while avoiding excessive parameters to maintain the stability of the model. As shown in Figure 9, the Bayesian Optimization process quickly converged to configurations that achieved significantly lower RMSE. The final experimental results demonstrate the effectiveness of Bayesian optimization in fine-tuning the loss components in the battery performance prediction task to improve the prediction accuracy.

6. Conclusions

Since PINN provides a flexible way to integrate empirical or physical dynamics models with data-driven neural networks, a fusion framework combining an LSTM network and PINN for SOH prediction of the battery is proposed in this paper. LSTM can effectively handle long-term dependencies in time-series data and mine the hidden temporal features of each time-series state. The DeepHTPM module integrates deep hidden physics models with LSTM to accurately capture the degradation dynamics of the battery. A large number of experiments have verified the effectiveness of the proposed framework. Experimental results demonstrate that the proposed framework significantly improves both prediction performance and stability on the lithium-ion battery datasets. In the future, we plan to conduct a time consumption analysis for different models. Moreover, our next research direction will focus on extracting multiple features for various lithium-ion battery types and improving the accuracy of health state prediction based on these extracted features.