# A generic physics-informed machine learning framework for battery remaining useful life prediction using small early-stage lifecycle data

Weikun Deng, Hung Le, Khanh T.P. Nguyen, Christian Gogu, Kamal Medjaher, Jérôme Morio, Dazhong Wu

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## ► To cite this version:

Weikun Deng, Hung Le, Khanh T.P. Nguyen, Christian Gogu, Kamal Medjaher, et al. A generic physics-informed machine learning framework for battery remaining useful life prediction using small early-stage lifecycle data. Applied Energy, 2025, 384, pp.125314. 10.1016/j.apenergy.2025.125314. hal-05482347

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https://ut3-toulouseinp.hal.science/hal-05482347v1

Submitted on 28 Jan 2026

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# A generic physics-informed machine learning framework for battery remaining useful life prediction using small early-stage lifecycle data

Weikun Deng<sup>a ID</sup>,*, Hung Le<sup>d</sup>, Khanh T.P. Nguyen<sup>a</sup>, Christian Gogu<sup>b ID</sup>, Kamal Medjaher<sup>a ID</sup>, Jérôme Morio<sup>c ID</sup>, Dazhong Wu<sup>d ID</sup>

## ARTICLE INFO

### Keywords:

- Physics informed machine learning

- Dual branches parallel framework

- Three steps training strategy

- Solid electrolyte interphase growth

- Lithium-ion batteries

- Remaining discharging cycles prediction

## ABSTRACT

Predicting the remaining useful life (RUL) of fast-charging lithium-ion batteries using early-stage lifecycle data is remains challenging due to limited run-to-failure data and lack of knowledge on battery degradation mechanisms. To address this issue, a generic Physics-Informed Machine Learning (PIML) framework is developed. The PIML framework consists of two parallel branches: a physics-informed (PI) branch and a data-driven branch. The PI branch is a neural network stacked by the linear projection layers with embedded physics knowledge, while the data-driven branch is a task-specific machine-learning model. In addition, a three-step training strategy is introduced, including (1) Training the data-driven branch, (2) Training the PI branch for aligning physical consistency without updating the hyperparameters in the data-driven branch, and (3) Fine-tuning both branches simultaneously to achieve optimal performance. To validate this framework, a physics-based model that represents the growth of solid electrolyte interphase (SEI) and a dilated convolutional neural network are implemented in the PI and data-driven branches, respectively. The solid electrolyte interphase-informed dilated convolutional neural network (SEI-DCN) model is demonstrated on the Stanford-MIT-Toyota-battery dataset. Using only four lifecycle data, the SEI-DCN model achieves very high prediction accuracy compared to standard dilated CNNs and other state-of-the-art models under various testing conditions and lifetime ranges. Moreover, the framework is generalizable to different physics-based battery degradation models.

## 1. Introduction

Due to high performance in terms of capacity, rechargeability, maintenance requirements, and environmental friendliness, lithium-ion batteries play a vital role in energy storage systems for a wide range of industrial and daily life applications such as automotive, aerospace, telecommunication and industrial production systems. Precise prognostics in lithium battery health are crucial to ensure safe, reliable, and cost-effective utilization of their full potential [1]. Particularly with fast-charging batteries, as few as 25 rapid charging cycles can initiate irreversible damage to batteries [2]. Predicting the Remaining Useful Life (RUL) is a significant focus in battery health prognostics. The RUL of a battery is typically represented as the number of operational cycles left before the battery reaches the end of its lifespan. This endpoint is commonly defined as when the battery's capacity diminishes to either 80% or 70% of its nominal capacity [3].

Currently, two primary paradigms exist for Lithium-ion battery prognostics. The first model's battery degradation mechanisms aim to identify common patterns that cause and permeate the degradation process, especially usage patterns related factors [4,5]. Initially, efforts focused on developing empirical formulas [6], semi-empirical models [7], or building physical knowledge related to real degradation [8]. These empirical or semi-empirical models are designed to fit degradation curves using predefined parametric functions [9,10]. Although they have been proven effective when applied to laboratory data under controlled conditions, extending these models to complex real-world battery systems reduces predictive accuracy and potential bias [11]. Therefore, many studies attribute the primary cause of battery aging to chemical changes at the negative electrode's electrode/electrolyte interface [12]. Specifically, the solid electrolyte interface (SEI) which happens at the beginning of usage is responsible for capacity degradation and increased resistance, resulting in power loss [13]. Other

<footnote><sup>a</sup> Laboratoire Génie de Production, LGP, Université de Technologie Tarbes Occitanie Pyrénées (UTTOP), 47 Avenue d'Azereix, 65000, Tarbes, France</footnote>

<footnote><sup>b</sup> Institut Clément Ader (ICA), Université de Toulouse, ISAE-SUPAERO, UPS, CNRS, INSA, Mines Albi, 3 rue Caroline Aigle, 31400 Toulouse, France</footnote>

<footnote><sup>c</sup> ONERA/DTIS, Université de Toulouse, F-31055 Toulouse, France</footnote>

<footnote><sup>d</sup> Additive Manufacturing and Intelligent Systems Laboratory, University of Central Florida Orlando, United States</footnote>

<footnote>* Corresponding author.
E-mail address: weikun.deng@enit.fr (W. Deng).</footnote>
factors, such as alterations in active material and electrode composition, also contribute to aging, though to a lesser extent [14]. Identifying the mechanism factors is difficult because battery degradation arises from various interconnected nonlinear processes [15], making it impossible to accurately describe all these degradation processes with a single quantitative model. Secondly, observing the electro-chemical state's evolution is nearly impossible with current sensing techniques because it involves a range of state-dependent parameters that are extremely challenging to calibrate [16]. Thus, fully understanding battery degradation physics remains a challenge due to ambiguous analytical relationships and unclear parameter estimates.

The second paradigm is data-driven-based RUL prediction, where statistical and machine learning (ML) techniques use extensive data to automatically establish the connection between cycles data and RUL, as summarized in research [17]. In particular, neural networks (NN) that consist of a series of linear and nonlinear transformations facilitate the extraction of highly representative features across multiple levels. The typical NN models applied in battery prognostics, such as auto-encoder (AE), deep neural network (DNN), deep belief network (DBN), convolution neural network (CNN), recurrent neural network (RNN), and generative adversarial network (GAN) are reviewed in [18]. The ML accuracy of RUL prediction depends greatly on the available charge/discharge data, the lithium-ion batteries system's complexity, and the appropriate selection of NN structure and hyperparameters [19]. However, the large battery monitoring datasets are often huge in data volume rather than degradation information. Complete run-to-failure data on battery charge/discharge cycles is rarely accessible, and usage history is often unknown, which challenges the data-intensive nature of ML. Furthermore, these ML models do not cover the system's underlying physics, making their performance enhancements more difficult [20]. Therefore, data-driven methods are facing a sparse data and physics-agnosticism dilemma.

Especially, accurate early-cycle based lifetime prediction is vital for applications such as fast-charging optimization, production evaluation, and battery design. This task is challenging due to limited data, but recent studies have developed innovative deep-learning methods to address this issue. One method integrates early-cycle voltage, current, and temperature data into a high-dimensional tensor, employing a convolutional sparse autoencoder and regularized logistic regression for classification, achieving high accuracy even with minimal data [21]. Another approach, the attention-assisted temporal convolutional memory-augmented network (ATCMN), predicts remaining useful life (RUL) using only 10 cycles of incomplete data, utilizing attention modules and temporal convolution to learn latent features, demonstrating superior performance and generalizability across different battery chemistries and conditions [22]. Additionally, a three-stage deep learning framework combines convolutional neural network-based latent feature extraction with handcrafted features and wrapper feature selection, improving early-stage battery lifetime predictions and generalizing well to various operational conditions [23].

On these basis, predicting the RUL of batteries falls into a regime where limited data and partial understanding of the degradation mechanism intersect [24]. In this context, “Physics-informed Machine Learning (PIML)” emerged as a promising solution to improve RUL prediction performance in recent years. Therefore, our paper first offers a detailed review of PIML in battery prognostics(see Section 2). Furthermore, we identify the challenges in applying PIML from the perspective of the architecture and learning strategies. Accordingly, we propose a new PIML paradigm for battery RUL prognostic. It is a novel parallel dual-branch framework (See Section 3) with the effective branch-independently multi-step training methodology outlined in Section 4, marking the major contribution of this paper. This paradigm provides a flexible and generalizing solution for knowledge embedding from the perspective of viewing PI parts as ML extensions. We validate the effectiveness of our proposed method on a degraded dataset of fast-charging lithium batteries in Section 5. Meanwhile, its flexibility and generalization performance are tested in Section 6. Finally, the conclusion and future work are presented in Section 7.

## 2. Related works

PIML in battery seeks to seamlessly incorporate physics knowledge into the cycle data processing pipeline in ML. As summarized in the surveys [25,26], there are three main PIML categories: “Physics-informed features (PIF)”, “Physics-guided algorithm structures (PGAS)”, and “Physics-informed training constraints (PITC)”. The potential for different physics knowledge that can be utilized to integrate battery degradation physics into ML in different ways is discussed in Guo et al.'s review [27]. The general summary of different categories is presented in Table 1.

In detail, PIF extends traditional “Feature engineering” and “Surrogate model-based data generation” approaches by utilizing physics knowledge to guide data processing and augmentation [28,29]. Regarding the knowledge type, the Equivalent circuit model (ECM) [28] and pseudo-2D model [30] are the most commonly used analytical relations to describe the internal electro-chemical processes of battery degradation. These analytical relationships also include numerical modeling of thermogenesis and heat transfer to mimic various battery operating conditions, which are solved by finite elements or volume methods [31]. The simulated results are the supplementary input information. In addition, it is worth noting that the geometrical characteristics of the data distribution and its feature shape are also regarded as physical knowledge, such as indicating the battery aging process by a left shift of the discharge curve and a decrease in the area under the energy curve [32] or identify Loss of Lithium Inventory (LLI) and Loss of Active Material (LAM) based on IC curve peak shape [33]. These geometric characteristics are the empirical awareness of monitoring phenomena, often pointing to specific battery degradation processes. Training the PIF model is generally purely data-driven, with the complexity of algorithms and PIML models increasing in line with the complexity of the input data. For simple data features, models like Gaussian regression [28,32] and one-dimensional LSTM [30] suffice for mapping features to degradation. For inputs with structural characteristics, such as 2D images or inputs integrating multiple physical models like the five-equivalent-stress model [34] or the SOC estimation model [35], more complex structures including CNN-LSTM, Two-Dimensional LSTM [33], dimensional grid LSTM [31], and Stacked LSTM [34] are utilized. PIF is straightforward to implement, but it is crucial to notice that PIF modifies the input space without altering and considering the inputs' physical relationships, which are also important for the final prognostic decision. In addition, possible biases in feature extraction guided by incomplete physics are difficult to address by the following model learning process.

“Physics-guided algorithm structures (PGAS)” improve traditional ML's physics consistency by integrating analytical relations from battery degradation models into the design and optimization of ML algorithms, such as the Nernst, Butler-Volmer, Fick's laws, and Arrhenius equations [34,36-40]. It replaces ML's local operators with physical modules and links them through logical relations between equations like decoupling voltage and current sequences or differential-algebraic relations [35,41]. PGAS encodes unknown physics equation parameters as ML trainable parameters, with MLPs modeling corrections for physics-based computations. For example, internal resistance and maximum charge are incorporated as trainable parameters that merge physics with learning [42]. PGAS employs deep structures like DNNs and Autoencoders [41,42] alongside RNNs and LSTMs. What is important to note is that the deeper the embedded position of physical knowledge, the stronger the physics constraints on the final output. [43] PGAS adjusts the local input-output relationships of ML models. Existing PGAS designs use physical operators as intermediate bilaterals in the ML data flow. This paradigm could negatively impact results if the embedded knowledge and its integration are not aptly considered. Moreover, embedding at a superficial layer minimally affects the outcomes regarding physical consistency, while overly deep embedding rigidifies network outputs, necessitating strict adherence
and experience with ML for controlling the embedding method, positioning, and form [44,45]. Accordingly, developing an effective PGAS is challenging.

Compared to PGAS and PIF, “Physics-informed training constraints (PITC)” integrates physical constraints directly into the objective function, affecting the optimization of parameters on a global scale, thereby facilitating machine learning models to approximate physical systems with reasonable accuracy [46]. PITC incorporates physical knowledge in two primary forms: (1) Derived from parameter relationships or phenomena, such as charge conservation, electrode averaging equations, and the assumption of discharge capacity’s monotonic decrease with a concave downtrend [41,47]. (2) Enforcing outputs to satisfy derived relationships, often through Partial Differential Equations (PDEs) or Ordinary Differential Equations (ODEs), such as linking capacity loss rate to health state, cycle number, initial Solid-Electrolyte Interphase (SEI) formation, and capacity loss, and solid-phase diffusion equations [42,48]. A novel approach distinct from traditional fixed models involves calibrating physics-based simulation models to showcase a bidirectional synergy between informed ML and physics. For example, to adapt to data-driven Long Short-Term Memory (LSTM) networks. This integration allows for real-time predictions via LSTM, complemented by precise but delayed health indicator evaluations from simulation models. An adaptive mechanism triggers model updates with new simulation data for LSTM retraining if prediction errors surpass a defined threshold [49]. The problem of solving the governing equations directly is converted to an optimization problem with a loss function to find partial differential equation solutions. However, higher-order or inappropriate PDEs can contaminate the back-propagation gradient and prevent optimal convergence [50]. Forcing physical target constraints on the output can lead to computationally expensive and difficult-to-converge model training results.

We assert that the key to addressing the above-mentioned problems lies in rethinking the relationship between informed knowledge and the ML module from a novel perspective. Therefore, we advocate for adopting a parallel (or dual branch) architecture, wherein a portion of the PIML is incorporated into the ML model as external plug-in modules for ML. Sections 3 and 4 will detail our proposed approach.

## 3. Dual-branch physics-informed machine learning framework

The novel physics-informed machine learning model is based on a dual parallel branch end-to-end framework shown in Fig. 1. The hybrid model comprising a Physics-Informed (PI) branch and a data-driven branch. The data-driven branch can operate independently as a standalone data-driven model. The PI branch significantly enhances the functionality of the data-driven branch by contributing augmented features $p$ to the latent space of the data-driven model. PI branch also outputs the features $p_{out}$ for final decision. Without integrating the data-driven branch’s output layer, the standalone PI branch falls short of generating predictions that align with expectations. Therefore, the PI branch is a complementary extension to the data-driven branch within this model.

In detail, the physics-informed (PI) and data-driven branches share some of their inputs but process them differently. The data-driven branch processes the integrated input space formed by all measured samples through an encoding module. Therefore, the dimension of each training batch is [batchsize, $m$, $len$, $n$]. This encoding module aims to uncover latent trends within the same measurement type and hidden features among different measurements. The architecture of the “Encoding module” and the following “Final output module” are tailored to specific data processing needs.

Conversely, the PI branch leverages finite physics measurements selected by the embedded formula as inputs. These inputs are processed individually or jointly by a finite number of each embedding input-related formula $f_i(\cdot)$ in the shallow layer of the PI branch as shown in Fig. 1. For example, $f_2$ considers the physical relationship between

the two input variables simultaneously. There are many similar connections between cross-layers that are customized based on the logical relationships between different equations and parameters encapsulated in the physical knowledge. The output of each custom layer with $f_i(\cdot)$ as the activation function will be merged to generate two-dimensional (2D) feature tensors. These 2D features will, on the one hand, be incorporated into the hidden layer space of the data-driven branch to provide physical information gain. On the other hand, they will be fed into the subsequent custom layers which embedded output-related physical formulas $F$ in the deeper position of the PI branch. The resulting features $p_{out}$ that satisfy the physical output relationship will be incorporated into the final output of the data-driven branch as physics consistency constraints. The undetermined parameters within $f_i(\cdot)$ and $F$ serve as learnable parameters within the PI branch, where the NN weights $w$ in custom layers are used to represent the coefficients of the formula, and the bias $b$ is the constant term.

This paper uses a battery solid electrolyte interphase healthy indicator mode for building the formula-embedded layers in the PI branch as shown in Fig. 2. More details regarding the models are presented in Sections 3.2 and 3.3.

### 3.1. End to end input

Raw sensor data are initially truncated to construct the inputs $x_{in}$. The sliding window technique splits continuous time series data into overlapping training samples from the start to the end of the monitoring sequence, as shown in Fig. 3. For instance, when truncating temperature measurements, the sliding window length parameter $m$ determines the number of samples in each window, with each sample having a fixed dimension of [$len$, 1]. Another parameter, $\delta$, consecutive windows to ensure the continuity of the truncated samples. Each training sample gathers data from the period starting at $t_i + m \times len$, retaining its 2D-dimensional structure for subsequent channel merging and splitting processes. As a result, each training sample $x_j$ (where $i = 1, 2, ..., n$) has a dimension of [$m$, $len$, 1]. The number of discharge cycles remaining $c_i$ at the time point $t_i + m \times len$ is recorded as the sample’s RUL label. The parameters $len$ and $m$ determine the extent of cyclic data covered in a training sample. Smaller values of $len$ and $m$ can enhance the proposed model’s practical value by increasing the data’s granularity and thereby improving the model’s prediction accuracy.

It is worth noting that within each truncated data segment, we have used a fixed length of time rather than a fixed number of cycles, as pinpointing the exact timing of the 3-4 charge/discharge cycles is difficult and results in a variable length of the input sequence segments. The window length $len$ roughly contains the information of 3-4 complete charge/discharge cycles. Since this is an end-to-end model for online prediction in real-world applications, “sample $i$” in Fig. 3 may start or end at any point within a discharge or charge cycle. Although the specific start and end points within a cycle can vary, the fixed time length ensures that the data spans approximately 3-4 cycles. This model’s predictive ability does not rely on complete cycle granularity, making it more practical for real-world applications.

### 3.2. End to end data-driven branch

The data-driven branch employs an end-to-end (E2E) deep learning (DL) paradigm to acquire degradation-related features for the battery RUL prediction from raw data directly. This approach eliminates the need for intricate signal processing or expert domain intervention.

The input comprises three parallel channels: temperature, voltage, and current. The samples within each channel are partitioned into multiple batches. In each learning batch, these independent physics measurements are combined through “Concatenate” layers to construct the inputs $X_{in}$ with the dimension [$m$, $len$, 3].

A Dilated convolution Neural Network (D-CNN) is shown in Fig. 4 to demonstrate the application of the E2E paradigm above. D-CNN has
<figure><img src="image_5.png" /><figcaption>Fig. 1. Embedding physical knowledge into NN models in the form of parallel architectures.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 2. SEI informed DCNN model diagram.</figcaption></figure>

a series of dilated convolution layers with increasing dilation factors (*dilated*<sub>rate</sub>) to encapsulate the interdependences between temperature, voltage, and current concerning the battery's discharge cycles. By

leveraging dilated convolutions, DCN can encompass larger receptive fields of each layer by incorporating gaps between weight connections to build positional jump connections.
<figure><img src="image_7.png" /><figcaption>Fig. 3. Truncate the original sequence in a fixed-length time window to construct an end-to-end input.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 4. Schematic diagram of D-CNN with a data flow.</figcaption></figure>

Subsequently, the final feature representation is obtained through a dense layer with $d_1$ neurons, yielding the data-driven feature $h_i$ (dimension: $[batch \ size, d_1]$).

### 3.3. Physics informed branch

The physics-informed branch utilizes the “physics-embedded algorithm structure” paradigm, as detailed in [45], to obtain the physics-consistency features and insights relevant to degradation.

The PI branch utilizes the SEI growth formula proposed by Attia et al. [51] as shown in Eqs. (1) and (2). This approach incorporates temperature-dependent kinetic models to effectively correlate SEI growth with cycle life, providing a robust and analyzable framework for predicting degradation under varying charging conditions. In Eq. (1), $D$ is the SEI degradation parameter. $D_0$ represents the initial or baseline SEI degradation parameter. $E_a$ is the effective activation energy for SEI growth, determined by the Arrhenius relation. $k_B$ denotes Boltzmann's constant, a fundamental physical constant. $T$ is the monitoring temperature of the battery. This expression captures the exponential dependence of SEI degradation, aligning with the Arrhenius equation's

insights into temperature-dependent chemical processes.

$$D = D_0 \exp \left( - \frac{E_a}{k_B T} \right) \qquad (1)$$

The $p_{out}$ of is estimated using the equation:

$$p_{out} = \frac{C}{D} + b \qquad (2)$$

In this context, at the data-flow level, the calculation of $D$ relies solely on the temperature input ($\hat{x}_i$), which is processed through a stacked dense layer to produce the hidden layer output $\hat{T}$. This intermediate representation ($\hat{T}$) is then transformed using a custom-designed physics-informed (PI) layer, where the activation function is defined by Eq. (1). The resulting output, $D$, serves as a critical input to the 2D fully connected layer, where it is used to compute $p$, a PI-derived hidden feature. This feature ($p$) is subsequently integrated into the data-driven branch for feature fusion, as well as the PI branch, to generate the physically informed insights $p_{out}$. $p_{out}$ refers to an approximate lifespan indicator, as it is not the final RUL output of the proposed PIML model. This distinction arises due to the simplified nature of the models, often incorporating empirical parameters and assumptions that do not fully capture the complexities of actual battery operation and degradation phenomena. This formulation emulates the Paris laws in material fatigue and posits that a battery's cycle life is inversely proportional to the rate of SEI degradation, adjusted by the empirical factor $C$.

Considering the input-output relationships and the logical relationship between Eqs. (1) and (2), only the temperature data are taken as the input of the PI branch, and the cross-layer connections are used in Fig. 2. Eqs. (1) and (2) are the activation functions of these custom design layers.

### 3.4. Merge the two branches' information

In the output part, the final “Full connectivity neural network (FCNN)” considers the output $p_{out}$ of the PI branch in the decision-making process and the data insights from feature $p$. FCNN is tailored for the many-to-one process to generate the output of the remaining discharging cycles.

The data flow processing of the entire PIML framework can be represented by the following equation. In data-driven branch, the input tensors $x_1, \dots, x_n$ undergo processing through $L$ hidden layers, where each layer progressively extracts higher-level features. This process can be modeled in Eq. (3):

$$h^{[l]} = g^{[l]} \left( \mathbf{W}_h^{[l]} g^{[l-1]} \left( \mathbf{W}_h^{[l-1]} \dots g^{[2]} \left( \mathbf{W}_h^{[2]} g^{[1]} \left( \mathbf{W}_h^{[1]} \mathbf{x} + \mathbf{b}_h^{[1]} \right) + \mathbf{b}_h^{[2]} \right) \dots + \mathbf{b}_h^{[l-1]} \right) + \mathbf{b}_h^{[l]} \right) \qquad (3)$$
$$y = h_f \left\{ w_1 s^{[r]} \left( \mathbf{W}_d^{[r]} s^{[r-1]} \left( \mathbf{W}_d^{[r-1]} \cdots s^{[2]} \left( \mathbf{W}_d^{[2]} s^{[1]} \left( \underbrace{\mathbf{W}_f^{[p]} + \mathbf{W}_f^{[h_l]} h^{[l]} + \mathbf{b}_f^{[1]}}_{\text{Feature fusion}} + \mathbf{b}_d^{[2]} \right) \cdots + \mathbf{b}_d^{[r-2]} \right) \right) + w_2 p_{\text{out}} + \mathbf{b}_d^{[r]} \right\} \quad (5)$$

Box I.

Each layer applies a linear transformation (via the weights $W^{[l]}$ and biases $b^{[l]}$) followed by a non-linear activation function $g^{[l]}$. The final network output is a continuous transformation product from the inputs with dimensions $[m, len, n]$ to the last hidden layer output features $h_l$.

Simultaneously, in K layers PI-branch, the entire process can be represented as a data flow in the NN through Eq. (4):

$$p = g^{[K]} \left( \mathbf{W}_p^{[K]} F^{[K-1]} \left( \mathbf{W}_p^{[K-1]} \cdots g^{[2]} \left( \mathbf{W}_p^{[2]} f^{[1]} \left( \mathbf{W}_p^{[1]} \mathbf{x} + \mathbf{b}_p^{[1]} \right) + \mathbf{b}_p^{[2]} \right) \cdots \right. \right. \\ \left. \left. + \mathbf{b}_p^{[K-1]} \right) + \mathbf{b}_p^{[K]} \right) \quad (4)$$

Comparing Eq. (4) with Eq. (3), it is worth noting that the custom architecture in the PIML branch replaces the conventional intermediate hidden layer to represent the physics input-output relationship.

In summary, the complete proposed PIML model can be represented by Eq. (5) which is given in Box I.

In this context, $p$ is defined and elucidated in Eq. (4), while $h_l$ is formulated and expounded upon in Eq. (3). The weights $w_1$ and $w_2$ pertain to the final output layer of the proposed PIML model, while $b^{[r]}$ represents the bias associated with this final output layer.

## 4. A novel PIML training method

The difficulty of successfully combining a small-parameter physics-informed branch with a large-parameter data-driven branch in parallel architectures becomes clear when we follow the traditional approach of directly training the suggested PIML model. This challenge, underscored by Eq. (5), involves balancing physical insights and data-driven complexities.

In detail, the PI and data-driven branches exhibit notable disparities in the size and complexity of their parameter spaces. Consequently, these differences can lead to inconsistent optimization rates during training, thereby presenting a formidable obstacle to achieving simultaneous convergence of the two branches. Furthermore, while large-parameter branches are susceptible to overfitting, their small-parameter counterparts may suffer from underfitting. Hence, numerous approaches in the realm of PIML may fall short of achieving the anticipated synergy between these models, even impeding to preserve the original data-driven model's fitting capabilities.

This paper proposes that a novel training method for the proposed dual branch model. We believe that the starting point for training a PIML model after embedding physical knowledge should be the original pre-trained ML model, rather than training it as a completely new model. Such a PIML model can be guaranteed to perform at the lower bound of the ML model's performance by a suitable training method.

Therefore, we develop this branch-independent learning paradigm which consists of three steps: “Data-driven pre-training”, “Physics-informed alignment”, and “Joint training” as shown in Fig. 5.

In “Step 1: data-driven pretraining”, only the data-driven branch is trained, including the “Feature extractor” and the “Decision module”. The parameters of the PI-branch are set to untrainable. In addition, the PI branch is initialized with a 0-mean to ensure that its output $p$ and $p_{\text{out}}$ are 0. At this point, it learns useful weights and biases only for the data-driven branch, i.e., the error feedback optimizes a data-driven pre-trained branch.

In “Step 2: physics-informed alignment”, the pre-trained “Feature extractor” in data-driven branch is loaded and frozen with the trainable “Decision module”. Meanwhile, the parameters of the PI branch are initialized randomly and are made trainable. The parameters in the final FCNN are relearned. In this configuration, the data-driven branch functions act as a fixed feature extractor for PI branch. The PI branch will be adapted and optimized concerning the knowledge already available in the data-driven branch, which improves the fit between the two. On this basis, the whole model is physically aligned.

Given that the pre-trained data-driven branch has already reached convergence, coupled with the PI branch with a smaller parameter space, the overall proposed PIML model exhibits enhanced ease of training. The independent training of the PI and data-driven branches eliminates the hampering interference of weight coupling during random initialization learning. In particular, compared to existing PIMLs that require training the entire model from scratch, “Step 2” only requires training a small number of PI branches with a small number of parameters, which saves training resources, especially given that existing public pre-trained models can be used for the data-driven branch.

After completing “Step 2”, it can evaluate the appropriateness, relevance, and integration of embedded knowledge and the informed ways after the “physics-informed alignment” training step. It compares the model’s performance with that of the original pre-trained model. If there is an improvement, the method deems the knowledge integration acceptable.

Building upon the preceding steps, this paper proposes “Step 3: joint training”. The fundamental objective of this approach extends beyond rectifying the inconsistency between the physical insights of the data-driven approach and the embedded physical knowledge. It also addresses enhancing the incomplete physical perspectives through data mining within the data-driven framework. Consequently, once the data-driven branch has been fine-tuned to a local optimum, demonstrating strong performance, and the PI branch has been optimized to a local optimum, considering its alignment with the data-driven branch’s performance, the pre-training weights of the PI branch are loaded. Subsequently, the weights of the data-driven branch and PI branch are configured as trainable, enabling the entire model to undergo retraining. This iterative process aims to bring the model closer to achieving a global optimal solution for the problem.

As shown in Fig. 6, this approach preserves the original architecture's ability to fit the data, a feature of considerable importance for enhancing the generalization and broader applicability of PIML. Although the traditional PIML learning can add physical consistency to the ML model, but the physical consistency is there and ML itself in the data-driven conditions of the optimization objectives are inconsistent or even want to conflict with the situation, want to let a single model at the same time to meet the existence of potentially conflicting multiple optimization objectives, the model will appear to end up in the Pareto surface to take the global sub-optimal solution of the case, for this reason, we are from the structural design of the model, will be satisfied with the For this reason, we design the structure to make the subjects that satisfy different optimization objectives independent of each other, and at the same time, in the learning strategy, we make
<figure><img src="image_9.png" /><figcaption>Fig. 5. Novel PIML learning mechanism: a multi-step learning process based on pre-training and physical alignment in a parallel architecture.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 6. Schematic illustration of the mechanism of multi-step learning strategies.</figcaption></figure>

the potentially conflicting learning objectives separate from each other, which guarantees that the final model has the ability to approximate the better optimum under the condition that the two branches reach their respective localities and the PI-branch is locally optimal with the incorporation of the insights of data-driven insights.

This learning paradigm shift is the key to bringing PIML to a wider world of applications, as it directly allows PIML to use a large number of mature data-driven pre-trained models to form a wide variety of PIML models by simply mounting PI-branches loaded with different knowledge.

# 5. Case study

## 5.1. Data description

The fast-charging lithium-ion battery dataset used as case study is developed by the MIT-Stanford team and described in the article [51]. This dataset comprises three experiments. The battery sets from “2017-05” and “2017-06” are set up for model training and testing. Initially, a random data splitting method equally divides the dataset into training and testing subsets. This involves shuffling the dataset indices before splitting them into two groups to ensure a balanced partition. As clarified in [51], the battery sets recorded in “2018-04” are different from the sets in “2017-05” and “2017-06”. It can be used as the second test set to evaluate the model’s generalization ability on entirely new data.

## 5.2. The configuration of the PIML model

Table 2 presents the benchmark model’s data processing and optimized learning hyperparameters for reference.

Combining the optimized DCNN architecture as the data-driven branch in the proposed PIML framework, the hyperparameters of the “PI branch” and the “decision network” in Fig. 2 are detailed in Table 3. Additionally, the coefficients utilized in the physics model equations, Eqs. (1) and (2), are also provided in Table 3 for reference.

## 5.3. Comparative study

A standard DCNN model developed by Hong et al. [52] serves as a benchmark model for investigating the superior performance of the PIML paradigm. The DCNN model has been validated on the MIT-Stanford dataset [52,53]. In addition to directly referencing the existing models in the literature that has been validated on the MIT-Stanford dataset, we further performed structural parameter optimization on the DCNN in developed by Hong et al. [52] to construct a DCNN that outperforms the original one. This new DCNN is also used as the model in the data-driven branch of the SEI-DCN proposed in this paper. The key distinction between DCNN in SEI-DCN and benchmark model lies in the presence of a PI branch and the incorporation of “Physics-informed alignment” and “Joint training” throughout the model. Consequently, we set the control variables as detailed in Table 4.
<table><caption>**Table 2**
Data processing and learning hyperparameters for benchmark experiments.</caption><thead><tr><th>Parameters</th><th>Physics meaning/Description</th><th>Setting</th></tr></thead><tbody><tr><td>len</td><td>The length of time represented by the sliding window</td><td>2500 (s), approximately 3 4 cycle length</td></tr><tr><td>δ</td><td>The overlap area of two windows</td><td>2250 (s)</td></tr><tr><td colspan="3"><strong>Hyperparameters</strong></td></tr><tr><td>Input channels</td><td>Number of types of input monitoring quantities</td><td>3</td></tr><tr><td>Batch size</td><td>The number of processed samples in one forward and backward pass</td><td>256</td></tr><tr><td>Input shape</td><td>Input data shape</td><td>[Batch size, 1, 2500]</td></tr><tr><td>Concatenate<sub>in</sub></td><td>Inputs' concatenation layer output shape</td><td>[Batch size, 2500, 3]</td></tr><tr><td>Conv1D-Bat<sub>n</sub></td><td>The number of 1D convolution layers with batch normalization after inputs concatenation</td><td>3</td></tr><tr><td>Kernel size</td><td>The 1D convolution layer kernel size</td><td>3</td></tr><tr><td>Conv1D-Bat<sub>d</sub></td><td>1D convolution layer output shape</td><td>[Batch size, 2500, 64]</td></tr><tr><td>Dilated<sub>rate</sub></td><td>Jump interval of dilated convolution</td><td>[1, 2, 4, 8, 16]</td></tr><tr><td>K<sub>num</sub></td><td>The number of kernels used in 1D convolution layer</td><td>64</td></tr><tr><td>Padding mode</td><td>The specific way in NN in which extra values (usually zeros) are added to the edges of input data</td><td>Casual</td></tr><tr><td>Flatten</td><td>Flattened output shape</td><td>(None, 160 000)</td></tr><tr><td>Stack-Dense 1</td><td>3 stacked dense layers as the decision NN in the data-driven branch</td><td>K<sub>num</sub> in each dense layer: [128, 64, 1]</td></tr><tr><td>Activation 1</td><td>Activation function for all dense and convolution layers</td><td>ReLU</td></tr><tr><td>Optimizer</td><td>Type of optimizer used</td><td>Adam (learning rate = 0.001)</td></tr><tr><td>Validation ratio</td><td>How much data is randomly divided in the training set as the validation set</td><td>0.2</td></tr><tr><td>Loss function</td><td>Loss function used for model training</td><td>Mean squared error</td></tr></tbody></table>

<table><caption>**Table 3**
Hyperparameters in physics informed branch.</caption><thead><tr><th>Parameters</th><th>Physics meaning/Description</th><th>Setting</th></tr></thead><tbody><tr><td>Units<sub>1</sub></td><td>Number of neurons corresponding to SEI growth's Arrhenius equation terms</td><td>64</td></tr><tr><td>E<sub>α</sub></td><td>Activation energy parameter in the Arrhenius equation, indicative of SEI growth rate</td><td>0.122 eV</td></tr><tr><td>k<sub>B</sub></td><td>Boltzmann constant, relates temperature to energy</td><td>1.380649 × 10<sup>-23</sup> J/K</td></tr><tr><td>D<sub>0</sub></td><td>Pre-exponential factor in the Arrhenius equation, a rate constant</td><td>1.0 s<sup>-1</sup></td></tr><tr><td>Extractor</td><td>Frozen data-driven branch serves as the feature extractor for the feature fusion and decision fusion</td><td>Inputs: 3 × [Batch size, 1, 2500]<br>Output 1, data-driven branch feature: [Batch size, 64]<br>Output 2, data-driven branch RUL: [Batch size, 1]</td></tr><tr><td>Fusion<sub>d</sub></td><td>Number of features from different branches in concatenate layer</td><td>Concatenate layer dimension: [Batch size, 128]<br>Feature from data-driven branch: [Batch size, channel 1–64]<br>Feature from PI branch: [Batch size, channel 64–128]</td></tr><tr><td>Dense-PI</td><td>Dense layer preceding the physics-informed (PI) layer</td><td>Number of neurons: 64</td></tr><tr><td>PI-output</td><td>Output layer of the PI branch, predicting the physical informed metrics</td><td>Output dimension: [Batch size, 1]</td></tr><tr><td>Stack-Dense 1</td><td>Three successive dense layers in the decision-making neural network branch</td><td>Neurons in each layer: [128, 64, 1]</td></tr><tr><td>Activation 2</td><td>Activation function for all dense layers in the decision network</td><td>ReLU</td></tr></tbody></table>

<table><caption>**Table 4**
Schematic representation of the differences in SEI-DCN compared to the original DCNN model.</caption><thead><tr><th>Control variables</th><th>Benchmark DCNN</th><th>SEI-DCN</th></tr></thead><tbody><tr><td>Learning strategy</td><td>One-time supervised training</td><td>The proposed 3 steps training strategy</td></tr><tr><td>Network architecture</td><td>Multi-Channel Input - Deep Null Convolution - Stacked Fully Connected Layers Output</td><td>Adding a parallel PI branch to the DCNN</td></tr><tr><td>Parameters scale</td><td>20,559,681</td><td>20,749,976</td></tr></tbody></table>

In evaluating the performance of ML models, two key metrics are commonly employed: the number of monitoring data cycles required for prediction and the calculation accuracy of the predicted RUL, which are indicated by the predicted mean absolute error (MAE) of the remaining discharging cycles. The number of data cycles required is a critical measure [20] for predicting remaining discharging cycles, making timely decision-making and achieving efficient resource allocation. In the time span of the sample used to pre-RUL, the less lifecycle involved, the higher model capacity to make accurate predictions with limited data. Concurrently, the mean absolute error (MAE) quantifies the agreement between the model's predictions and the actual cycle life, indicating its reliability.

The encoder structure we selected has already been validated on the benchmark dataset as a published paper in *Applied energy* [52], maintaining the same structure, hyperparameters, and sample truncation lengths for consistent comparison. The key difference of our approach compared to the state-of-the-art DCNN is the addition of a PI branch and the application of our proposed multi-step training

method. This controlled variable experiment aims to highlight the specific advantages of our method.

Moreover, our framework is flexible and can accommodate other data-driven models beyond DCNN. Notable alternatives that effectively utilize partial cycle information include the bilateral- branched visual transformer with dilated self-attention or RNN-LSTM. These options are also worth exploring for their potential benefits, as their ability in this “small-early cycles” scenarios are already detailed in Refs. [54], and [55].

### 5.3.1. Comparison with benchmark model in SOTA

Table 5 compares various ML models' performance. This includes several SOTA models and the DCNN model reproduced and optimized specifically in this paper, serving as an ablation study. Table 5 highlights significant advancements in predicting cycle errors for battery life estimation, with the proposed parallel model standing out by achieving the lowest predicted MAE of 15 cycles. This outcome is achieved using a data sample truncated to 2500 s, typically spanning 3 to 4 cycles. It
<table><caption>**Table 5**
Performance of different models.</caption><thead><tr><td>Model</td><td>Required cycles length</td><td>Predicted cycle error (MAE)</td></tr></thead><tbody><tr><td>Shallow MLP [52]</td><td>4</td><td>150</td></tr><tr><td>MLP [52]</td><td>4</td><td>174</td></tr><tr><td>Full model [53]</td><td>100</td><td>99</td></tr><tr><td>Variance [53]</td><td>100</td><td>112</td></tr><tr><td>SVR [52]</td><td>100</td><td>245</td></tr><tr><td>CNN [52]</td><td>4</td><td>82</td></tr><tr><td>CNN + LSTM [52]</td><td>4</td><td>72</td></tr><tr><td>DCNN in research [52]</td><td>4</td><td>65</td></tr><tr><td>DCNN in our research</td><td>4</td><td>55</td></tr><tr><td>The proposed PIML model</td><td>4</td><td>15</td></tr></tbody></table>

demonstrates that our SEI-DCN can deliver more precise usage insights with short-term data, enhancing the adaptability of prediction algorithms to real-time user habits. In addition, using a platform equipped with a single 16 GB NVIDIA T4 Tensor Core GPU, the prediction time for one test sample is approximately $3.5 \times 10^{-5}$ s.

### 5.3.2. Comparison with predictions solely based on the data-driven branch

The qualitative and quantitative comparison is made between the DCNN model alone and the SEI-DCN model, whose data-driven branch is identical to the DCNN model in terms of structure, parameters, inputs, and outputs. The qualitative prediction results are detailed in Fig. 7, with a detailed quantitative comparison of predictions for each test cell presented in Fig. 8. Fig. 7's "Identity line" represents perfect predictions, where predicted values match true values. The "Regression line" indicates the general trend of predictions. Ideally, the "Regression line" should align with the "Identity line". The blue scatter points show the actual predictions. Results in Fig. 7 reveal that both models effectively capture the linear decay trend of remaining discharging cycles. However, our SEI-DCN model outperforms traditional DCNNs by providing more accurate predictions, aligning better with trends, reducing prediction deviation, with fewer outliers. This improvement underscores the value of incorporating physics and tailored learning strategies, providing the enhanced system security prognostic service. To further compare the two models, we utilized box plots to visualize the predictions for each group. Fig. 8 presents the results of this comparison. The range of Remaining Discharge Cycles (RDC) values for each test battery is categorized as follows: Test Batteries 1 and 2 have RDC values spanning [0, 1200], while Test Batteries 3, 14, and 15 span [0, 1000]. Batteries 4, 7, 8, 9, 16, 17, and 18 exhibit ranges of [0, 800], and Test Batteries 6, 19, and 20 span [0, 600]. In contrast, Test Batteries 21 through 35 primarily fall within [0, 450], except for Test Battery 30, which uniquely covers [0, 500]. This distribution highlights the comprehensive evaluation of the model across both extended and shorter lifecycle ranges, demonstrating its robustness in predicting battery degradation across varying RDC values.

Both methods accurately predicted the RUL decreasing trend for batteries with varying lifespans (45 to 1200 remaining charge cycles) despite the measurements missing gaps (batteries 19, 27). However, early in the battery life, the DCNN's predictions are significantly higher than actual values, in contrast to the more accurate SEI-DCN (our approach) predictions for batteries 4, 11, 12, and 29. This accuracy makes SEI-DCN more suitable for early-life task planning in fast-charging batteries. Towards the end of the battery life, with RUL nearing zero, DCNN's predictions remained overly optimistic compared to actual values and SEI-DCN predictions for batteries 1, 2, 23, and 28. This suggests that SEI-DCN provides more reliable information for safely operating fast-charging batteries in their late degradation stages. Moreover, a comparison of overall prediction errors revealed that SEI-DCN had a narrower error distribution with fewer outliers, indicating its greater robustness and reliability over the purely data-driven DCNN.

It is worth noted that the gap appears in Battery 17 and 29, given that our model is designed as an end-to-end solution ready for deployment, we did not perform extensive data cleaning, which includes

retaining instances of sparse data. The dataset authors assert [53] that raw data for each cell is available in CSV format. It is important to note that the CSV files occasionally exhibit discrepancies in test time and step time, with test time sometimes resetting to zero mid-cycle. Additionally, temperature measurements were taken by securing T-type thermocouples to the exposed battery can with thermal epoxy (OMEGATHERM 201) and Kapton tape after removing a small portion of plastic insulation. These temperature measurements are unreliable due to potential variations in thermal contact between the thermocouples and the battery can. Sometimes, the thermocouples lost contact during cycling, resulting in data gaps. These errors were considered while directly reading the raw data, leading to noticeable gaps in the data for cells 17 and 29. However, the prediction results demonstrate that the model maintained robust end-to-end predictive capability even under these conditions. This underscores the reliability of the method, particularly its ability to make accurate predictions using small-cycle data without relying on long historical sequences.

In test cell 5, SEI-DCN slightly the underperformed compared to DCNN, with a predicted MAE of 28 versus 22, even though the upper and lower quartiles of SEI-DCN's box plots are closer. This demonstrates that SEI-DCN has more predicted outliers smaller than the true value for cell 5, whereas the predicted outliers for the DCNN are all larger than the true value. Combined with the overall prediction results, we argue that the predictions of the SEI-DCN are biased towards conservatism by the constraints of knowledge of physics. Moreover, in cells 6 to 9 and 17, SEI-DCN has worse performance with an MAE higher than 15 compared to its performance in other cells. This suggests the potential for further model improvements and adjustments to enhance its generalization across different cells by finding more representative physics and applying long-term learning.

### 5.3.3. Comparison of performance robustness test results on new data without relearn conditions

The performance of the DCNN and SEI-informed DCNN models are compared on the data collected in experiment "2018-04" without retraining (see Table 6).

The battery sets recorded in "2018-04" differ from those in "2017-05" and "2017-06" due to distinct charging policies and resulting variations in cycle life distributions. The overview of the cycle life distribution is in Table 6. Specifically, the "2018-04" set exhibits more batteries in the higher cycle life ranges ([1000–1500] and [1500–2000]), whereas the earlier sets have more batteries in the lower cycle life ranges ([0–500] and [500–1000]). For instance, the "2018-04" set includes charging policies like 5C(67%)-4C and 5.9C(60%)-3.1C, which are different from those in "2017-05" and "2017-06" such as 3.6C(80%)-3.6C and 4C(80%)-4C. These differences highlight the impact of varied charging protocols on battery degradation patterns, making the "2018-04" set an ideal second test set to evaluate the model's generalization ability on entirely new data. The robustness testing results are shown in Fig. 9.

In this generalization ability test, several observations are made that are illustrated in Fig. 9.:
<figure><img src="image_11.png" /><figcaption>Fig. 7. Demonstration of predicted trajectory results for remaining discharging cycles of test battery packs.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 8. Box plots the prediction errors of remaining discharging cycles (RDCs) for DCNN and SEI-DCN (our approach). (The x-axis of each subplot represents the true RUL, while the y-axis represents the predicted RUL. The embedded boxplot of each subplot has boxes on the left representing the prediction error of DCNN as a benchmark model and boxes on the right representing the prediction error of SEI-DCN as a PIML model. The number above the boxplot is the predicted MAE).</figcaption></figure>
<table><caption>**Table 6**
Cycle life distribution.</caption><thead><tr><th>Frequency of cycle life distribution</th><th>2017-05-12</th><th>2017-06-30</th><th>2018-04-12</th></tr></thead><tbody><tr><td>[0–500]</td><td>0</td><td>35</td><td>0</td></tr><tr><td>[500–1000]</td><td>36</td><td>12</td><td>21</td></tr><tr><td>[1000–1500]</td><td>10</td><td>1</td><td>18</td></tr><tr><td>[1500–2000]</td><td>0</td><td>0</td><td>5</td></tr></tbody></table>

<figure><img src="image_13.png" /><figcaption>**Fig. 9.** Predicted trajectory results for remaining discharging cycles of test battery packs on the new dataset.</figcaption></figure>

1. It is clear that although the purely DCNN model captures the overall trend, there is a significant difference between the predicted and actual values, indicating that the accuracy of the model could be significantly improved.

2. Considering that the RUL span in the training set does not include the case where the remaining discharging cycle life is greater than 2000, it can be seen that DCNN does not have the prediction ability in that case. Still, SEI-DCN does match the real data significantly better, which proves that SEI-DCN has a superior generalization ability by embedding physical knowledge.

3. The predictions of the SEI-DCN model (our approach) have higher variability on this new dataset, along with outliers with relatively large deviations from the true values in the deep degradation stage. However, its performance declined much less than that of the DCNN.

4. The SEI-DCN model's MAE is 81. Whereas the pure DCNN is 311, which quantifies the significant generalization advantage our the proposed physics informed approach.

5. The sudden increases in deviation observed in Fig. 9 are primarily attributable to the introduction of a new dataset characterized by distinct cycle life ranges and charging policies, coupled with the limited representation of late-stage battery life samples in the training data. Specifically, some samples may be nearing the end of their life cycle, while under similar conditions, other batteries remain in the midst of their degradation phase. This overlap results in pronounced deviations, particularly near the terminal values of the life cycle.

6. Although our PIML model mitigates some of these challenges by incorporating embedded physical knowledge, further enhancements are necessary to fully capture the complex degradation behaviors present in the new dataset.

## 6. Investigating the learning mechanism of the proposed PIML

In this section, our main focus lies in examining the learning mechanism underlying the proposed approaches. This analysis investigates how the PIML model effectively amalgamates physical and data-driven

knowledge into the prediction process, as detailed in Section 6.1. Furthermore, we explore the adaptability of the framework and training methodology for screening embedded physical knowledge in more detail in Section 6.2.

### 6.1. Mechanisms for improving data-driven model performance

In a hybrid model that combines data-driven and physics-informed branches, efficiently exploiting useful features from both branches is essential. We focus on the proposed PIML model's treatment of fused features $[h_t, p]$ within the concatenate layer with dimensions [Batch size, 128]. This layer effectively merges data-driven features spanning channels 1 to 64 with PI features spanning channels 64 to 128. We plotted a 3D schematic of the channel weights to illustrate the integration process, as shown in Fig. 10.

We transformed it from a cartesian coordinate system to a cylindrical coordinate system which unveils the relative significance assigned to the features from each branch. This visualization is a valuable tool for understanding the rationale behind the prioritization of certain features and the consequential impact of their weights on prediction outcomes. The weights distribution map generated after the two-step training process reveals a dense matrix of weights across numerous units in the dense layer. This indicates a comprehensive consideration of features across various channels during the model's decision-making process.

When the model is trained in three steps, Fig. 10(b) shows a sparser distribution of weights with more pronounced extremes. This suggests that the model learns to assign more pronounced weights to specific hidden layer features after completing the three-step joint training process. This is desirable in many machine learning tasks, as it is often associated with a more refined and potentially more generalized model. At this point, the weights for PI branch have decreased in magnitude. We believe that the knowledge from both branches has been unified and integrated into a more expressive data-driven branch. Weight distribution analysis reveals that the model selectively emphasizes features after the joint training step, indicating a more refined and robust model. The PI branch is a regularization mechanism, guiding the data-driven branch towards a better optimum and enhancing the model's generalization ability.
<figure><img src="image_14.png" /><figcaption>(a) Channel weights distribution in the “Physical alignment” pre-trained data-driven model for processing the integrated features [h<sub>l,p</sub>].</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(b) Channel weights distribution in the “Joint training” pre-trained PIML model for processing the integrated features [h<sub>l,p</sub>].</figcaption></figure>

**Fig. 10.** Schematic depiction of the modulation in channel weights subsequent to integrating features across various branches during the multi-step learning phases. (The model above has been trained in the second step, and the model below has been trained in the full three steps.)

## 6.2. Evaluating the robustness of the PI branch for knowledge switching

To evaluate the flexibility of PI branching in knowledge switching, this study changes the knowledge model embedded in the PI-branch while keeping the data-driven branch intact, using a new empirical SEI model [56] to constrain the output relationship of the PI branch. The new empirical formula elucidates the relationship between the thickness (L(t)) of the SEI layer and time (t). It provides the following Eq. (6):

$$L(t) = \sqrt{\frac{2D_s C_s^0 \epsilon t}{c_p}} \qquad (6)$$

The entire SEI-informed DCNN model is shown in Fig. 11.

The model in Fig. 2 is denoted as “PIML model 1”, the model in Fig. 11 is denoted as “PIML model 2” Their comparison results are shown in Table 7.

Figs. 11 and 2 illustrate specific application cases of the proposed framework shown in Fig. 1, demonstrating its generalized performance in both PIML establishment and application performance assurance. The primary distinction between these cases lies in the type of knowledge incorporated into the PI branch: the knowledge in Fig. 2 is

derived directly from the dataset, while that in Fig. 11 is sourced from another SOTA research. These validation cases highlight two key characteristics of the framework: (1) standardized integration of diverse physical knowledge, ensuring adaptability across different scenarios and datasets; and (2) robustness and consistency in model performance, maintaining reliable outcomes regardless of variations in the incorporated physical knowledge. Together, these attributes showcase the flexibility and reliability of the framework in accommodating diverse physical insights while preserving overall model integrity. Table 7 summarizes the performance metrics for two PIML models, each designed with distinct SEI growth knowledge. Within our proposed parallel framework, it becomes evident that models presented in Eqs. (1) and (2) outperform the model described in Eq. (6) after two steps. This suggests that PIML model 1 and its incorporation of physics exhibit superiority within the same data-driven branch and informed framework.

Another conclusion that can be drawn is that the three-step training process outperforms the two-step training process for both of the proposed PIML models. Specifically, the MAPE of “PIML model 1” improves from 10.9% to 8.4%, and the cycle error is reduced from
<figure><img src="image_16.png" /><figcaption>Fig. 11. SEI layer thickness growth model informed DCNN functional module integration diagram.</figcaption></figure>

<table><caption>**Table 7**
Performance metrics of different PIML models.</caption><thead><tr><th rowspan="2">Model</th><th colspan="2">Mean absolute percentage error (%)</th><th colspan="2">Cycle error (MAE)</th></tr><tr><th>After “Physics-informed alignment” training</th><th>After “Joint training” training</th><th>After “Physics-informed alignment” training</th><th>After “Joint training” training</th></tr></thead><tbody><tr><td>PIML model 1</td><td>10.9</td><td>8.4</td><td>15</td><td>11</td></tr><tr><td>PIML model 2</td><td>12.7</td><td>8.3</td><td>17</td><td>9</td></tr></tbody></table>

15 to 11. Similarly, the MAPE of PIML Model 2 improved from 12.7% to 8.3%, and the cycle error was reduced from 17 to 9. Even though the overall MAPE declined by only a limited number of percentage points, the actual forecast cyclic error was reduced by a very large amount. These improvements suggest that the additional training step effectively enhances the model's accuracy and ensures the model's flexibility in embedding different physics knowledge. This optimization enables improved generalization and more accurate predictions. This highlights that even if the initial knowledge is sub-optimal in that it is not directly generating certain RUL-related metrics, but rather an empirical understanding of changes in system behavior (e.g., SEI thickness growth), it can be refined through this complementary training step to achieve superior performance.

## 7. Conclusion and future works

A generic physics-informed machine learning (PIML) framework for battery RUL prediction was developed. The PIML framework consists of two parallel branches: a data-driven model tailored to the requirements of the task, and a hybrid model that incorporates machine learning components designed around the input-output dynamics of physical models. To better coordinate the training of different parts of this framework, a three-step training strategy was introduced to build an optimal joint model between two branches. A solid electrolyte interphase healthy indicator model informed dilated convolutional neural network (SEI-DCN) was developed according to this framework. DCNN was used to build the data-driven branches of SEI-DCN. The physics-based model estimates the indicator about solid electrolyte interphase (SEI) growth is used as the hidden layer of the hybrid model in the

PI branch of SEI-DCN. SEI-DCN was validated on the MIT–Stanford fast-charging lithium-ion battery dataset and compared to other models in SOTA. The results demonstrate that our method outperforms SOTA models. The SEI-DCN achieved a mean absolute error (MAE) of 15 using only 4 lifecycle data, whereas the conventional DCNN, despite having the same structure, hyperparameters, and pre-learned weights as the SEI-DCN's data-driven branches, had an MAE of 65. Furthermore, absent re-learning, data-driven DCNN models lost their predictive ability for completely new data that is different from the training dataset. Compared to the 311 MAE of DCNN, the SEI-DCN model has an MAE of 81, which maintained the prediction accuracy and achieved the average level in SOTA on the new data testing.

This study also assessed how well the PI branch could accommodate knowledge exchange across different SEI models. It was observed that replacing the SEI degradation model with a model estimating SEI layer thickness resulted in poorer performance (MAE of 17) compared to the original SEI-DCN model (MAE of 15) when the SEI-DCN underwent only the initial two-step training, halting at the “Physics alignment” stage. However, completing the “Joint training” third step improved the performance of the SEI layer thickness estimation model informed DCNN, yielding an MAE of 9, comparable to the original SEI-DCN's MAE of 11. These findings suggest that evaluating a model's performance after two-step training can gauge the efficacy of the integrated knowledge and the adopted paradigm. The co-training third step has the potential to notably enhance the model's performance, especially when incorporating sub-optimal knowledge or embedding knowledge in sub-optimal methods.

Currently, PI and data-driven branches integrate at the level of hidden layer features and outputs. Future work should concentrate on
more advanced integration techniques, specifically the dynamic position adjustment of the PI output's embedding within the data-driven branch. This would involve varying the embedding across different layers in response to changes in inputs and targets. Additionally, the process through which embedded physics knowledge influences both branches warrants more research. In the end, more cross-disciplinary and cross-application research using the framework or similar PIML architectural approaches is desired to promote PIML development. While the current study focuses on SEI growth as the dominant degradation mechanism under the experimental conditions of the MIT-Stanford dataset, future work should consider the roles of LLI (Loss of Lithium Inventory) and LAM (Loss of Active Material) to better capture the complex, intertwined degradation processes in real-world scenarios. Integrating models for LLI and LAM into the framework will enable a more comprehensive assessment of their combined effects, improving the prediction and understanding of battery degradation across diverse operating conditions. Moreover, Capacity diving — characterized by a nonlinear and rapid decline in battery capacity, arises from factors such as lithium metal deposition, electrode saturation, abnormal resistance growth, and mechanical deformations. While our dataset does not explicitly represent these mechanisms, our approach provides systematic pathways for addressing this challenge through staged and intelligent charging protocols, material and thermal management enhancements, and advanced battery management systems (BMS). By integrating physicochemical optimizations, such as improved electrode materials and thermal regulation, with dynamic model learning in the building advanced BMS. Extending our approach to the BMS involves real-time monitoring of battery status and developing dynamic model parameter learning mechanisms linked to charging parameters. Such advancements would protect the battery from overcharging and other adverse conditions, thereby actively preventing Capacity diving.

### CRediT authorship contribution statement

**Weikun Deng:** Writing – original draft, Visualization, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Hung Le:** Investigation, Formal analysis, Data curation. **Khanh T.P. Nguyen:** Writing – review & editing, Supervision, Conceptualization. **Christian Gogu:** Writing – review & editing, Supervision, Methodology, Funding acquisition, Conceptualization. **Kamal Medjaher:** Writing – review & editing, Supervision, Resources, Project administration. **Jérôme Morio:** Writing – review & editing, Supervision, Formal analysis. **Dazhong Wu:** Writing – review & editing, Supervision, Resources, Formal analysis.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

This research was in part supported by the Thomas Jefferson Fund of the Embassy of France in the United States and the FACE Foundation, United States. This research also extends its gratitude to Tarbes and the X-IMS ANR project for their valuable support.

### Data availability

Data will be made available on request.

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