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Degradation-Aligned Self-Supervised Learning for State of Health Estimation of Lithium-Ion Batteries under Label Sparsity
Jiaqi Yao, Julia Kowal
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 97%
Last extracted: 8/23/2026, 2:25:05 AM
Summary
This paper proposes a degradation-aligned self-supervised learning (SSL) framework for estimating the State of Health (SOH) of lithium-ion batteries under conditions of label sparsity. The method utilizes a CNN-GRU model to learn aging-consistent representations from unlabeled charging data via a cycle-order ranking pretext task. The model is then fine-tuned on a small fraction (1%) of labeled data, achieving high accuracy (MAE 1.718%, RMSE 2.329%) despite limited and unevenly distributed labels.
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Jiaqi Yao → affiliatedwith → Technische Universität Berlin
confidence 99% · Jiaqi Yao a,∗ , Julia Kowal a a Department of Electrical Energy Storage Technology (EET), Technische Universität Berlin
CNN-GRU → estimates → State of Health
confidence 99% · enabling robust SOH estimation after fine-tuning on sparsely labeled data
Degradation-Aligned Self-Supervised Learning → uses → CNN-GRU
confidence 98% · we propose a degradation-aligned self-supervised learning (SSL) framework based on a convolutional neural network-gated recurrent unit (CNN-GRU) model
Degradation-Aligned Self-Supervised Learning → solves → Label Sparsity
confidence 97% · enabling robust SOH estimation after fine-tuning on sparsely labeled data
CNN-GRU → achieves → MAE 1.718%
confidence 95% · where the MAE of 1.718% and RMSE of 2.329% can be achieved on the test cell
Cycle-Order Ranking → ispretextfor → Degradation-Aligned Self-Supervised Learning
confidence 95% · through a cycle-order ranking objective as the pretext task for pretraining
CNN-GRU → learnsfrom → Unlabeled Data
confidence 95% · learns aging-consistent representations from unlabeled data through a cycle-order ranking objective
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Abstract
Abstract:An accurate estimation of the state of health (SOH) underpins a safe and optimized use of the battery system. Although compelling, data-driven SOH estimation models typically require large amounts of high-quality labeled cycling data, while in practice such labels are often sparse in both quantity and coverage. Therefore, in this work, we propose a degradation-aligned self-supervised learning (SSL) framework based on a convolutional neural network-gated recurrent unit (CNN-GRU) model, which learns aging-consistent representations from unlabeled data through a cycle-order ranking objective as the pretext task for pretraining, thereby enabling robust SOH estimation after fine-tuning on sparsely labeled data. Test results showcase that the proposed ranking-based SSL approach proves to endow the pretrained model with degradation-aligned information from unlabeled data, and after fine-tuning the model can carry out accurate, robust SOH estimation, even when only an extremely limited amount of 1% of unevenly distributed labeled training data is available, where the MAE of 1.718% and RMSE of 2.329% can be achieved on the test cell. In addition, in-depth analyses are presented regarding the influences of label distribution of battery degradation data. We believe this work could shed new light on SOH estimation of lithium-ion batteries under label sparsity in real-world applications.
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- Source: https://arxiv.org/abs/2608.16612v1
- Canonical: https://arxiv.org/abs/2608.16612v1
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Graphical Abstract Degradation-Aligned Self-Supervised Learning for State of Health Esti- mation of Lithium-Ion Batteries under Label Sparsity Jiaqi Yao, Julia Kowal Self-SupervisedPretraining Downstream Adaptation Fine-Tuning Encoder ParametersFrozen Data Preparation DataCollection ... Cell1Cell 2 Cell k Data Preprocessing Cleaning Formatting Partial Labeling DataSplitting Model InitializationPairwiseRankingAging Score Validation Save Model LoadEncoder SOHEstimation Evaluation arXiv:2608.16612v1 [eess.SP] 17 Aug 2026 Highlights Degradation-Aligned Self-Supervised Learning for State of Health Esti- mation of Lithium-Ion Batteries under Label Sparsity Jiaqi Yao, Julia Kowal • A ranking-based SSL framework is proposed for SOH estimation under label sparsity. • A CNN-GRU model is developed for SOH estimation with local feature fusion. • Insights are presented on the influences of label distribution of degradation data. Degradation-Aligned Self-Supervised Learning for State of Health Estimation of Lithium-Ion Batteries under Label Sparsity Jiaqi Yao a,∗ , Julia Kowal a a Department of Electrical Energy Storage Technology (EET), Technische Universität Berlin, Einsteinufer 11, 10587, Berlin, Germany Abstract An accurate estimation of the state of health (SOH) underpins a safe and optimized use of the battery system. Although compelling, data-driven SOH estimation models typically require large amounts of high-quality labeled cycling data, while in practice such labels are often sparse in both quantity and coverage. Therefore, in this work, we propose a degradation-aligned self-supervised learning (SSL) framework based on a convolutional neural network-gated recurrent unit (CNN-GRU) model, which learns aging-consistent representations from unlabeled data through a cycle-order ranking objective as the pretext task for pretraining, thereby enabling robust SOH estimation after fine-tuning on sparsely labeled data. Test results showcase that the proposed ranking-based SSL approach proves to endow the pretrained model with degradation- aligned information from unlabeled data, and after fine-tuning the model can carry out accurate, robust SOH estimation, even when only an extremely limited amount of 1% of unevenly distributed labeled training data is available, where the MAE of 1.718% and RMSE of 2.329% can be achieved on the test cell. In addition, in- depth analyses are presented regarding the influences of label distribution of battery degradation data. We believe this work could shed new light on SOH estimation of lithium-ion batteries under label sparsity in real-world applications. Keywords: Lithium-ion batteries, Battery management systems, State of health estimation, Deep learning, Self-supervised learning ∗ Corresponding author. Department of Electrical Energy Storage Technology (EET), Technische Universität Berlin, Einsteinufer 11, 10587, Berlin, Germany Email address: jiaqi.yao@tu-berlin.de (Jiaqi Yao) 1. Introduction The growing urgency of global climate change has significantly accelerated the transition to clean, sustainable energy systems [1, 2, 3, 4]. Renewable energy sources such as wind [5] and solar power [6, 7] have been rapidly developed and deployed worldwide. However, their inherent intermittency and volatility pose significant chal- lenges to the stability and reliability as an energy supply, thereby creating a strong demand for dependable, efficient energy storage systems [8]. Benefiting from many advantages, including high energy and power densities [9, 10], low self-discharge [11], and longevity [12], lithium-ion batteries have been widely utilized across numerous fields, from portable consumer electronics [13, 14] to electric transportation [15, 16]. Regardless of the specific application, an accurate determination of the state of health (SOH) is indispensable for lithium-ion batteries and is a core functionality of battery management systems (BMSs). At some time t, the SOH of a battery can be defined either based on the capacity or the internal resistance [17, 18]: SOH C (t) = C actual (t) C rated (1) SOH R (t) = R actual (t)− R EOL R rated − R EOL (2) where C actual (t) and C rated respectively denote the actual battery capacity at time t and the rated capacity, while R actual (t), R EOL , and R rated respectively denote the actual battery internal resistance at time t, the predefined end-of-life (EOL) internal resistance, and the rated internal resistance. Usually, the capacity-based definition of SOH C is utilized without further clarification, as in this work. SOH is a defined metric for quantifying battery degradation. The determination of SOH plays a vital role in battery systems [19, 20, 21], as it not only provides essential information for the assessment of remaining useful life (RUL) and the early scheduling of nec- essary maintenance, but also underpins a safe and optimized use of the battery by supporting the prevention of unexpected failures and improving the overall lifespan utilization of the battery systems. However, SOH is closely related to the complex underlying electrochemical pro- cesses and can hardly be directly measured; thus, it has to be estimated from the mea- surable information during operation [22]. In general, SOH estimation approaches can be categorized into three types [23, 24]: direct measurement approaches, model- based approaches, and data-driven approaches. Direct measurement approaches esti- mate the SOH based on the knowledge acquired from measurements, such as deriving the actual usable capacity by applying coulomb counting on a standardized discharge 2 capacity test [25]. Although straightforward and easy to implement, such approaches can hardly be applied in real applications, as the fully controlled operating condi- tions and complete discharge cycles required are generally unavailable in practice. Model-based SOH estimation approaches can be further categorized into two groups, namely degradation modeling approaches and parameter identification approaches [24]. Degradation modeling approaches exploit various models that describe battery degradation, such as empirical models [26, 27] and electrochemical models [28, 29], to infer the current usable capacity, while parameter identification approaches cast the problem of SOH estimation as a parameter identification problem utilizing battery models like equivalent circuit models (ECMs) [30] and electrochemical models [31], and nonlinear state observers like extended Kalman filters (EKFs) [32], unscented Kalman filters (UKFs) [33], and particle filters [34]. Despite the fact that model- based approaches are generally considered accurate and robust, in-depth domain knowledge is required for the modeling of the electrical, aging, and even thermal be- havior of the battery, and specifically designed characterization tests are compulsory to parameterize these models. In recent years, data-driven solutions have come into the spotlight for various applications. Especially in the field of deep learning, prosperous progresses are being made. In the context of battery SOH estimation, data-driven approaches aim to capture the underlying degradation patterns directly from operational data without requiring prior knowledge of battery electrochemical dynamics. However, most ex- isting works choose to manually extract input features for data-driven models due to the belief that handcrafted health indicators can more explicitly characterize bat- tery degradation and thus reduce the difficulty of model learning with increased interpretability [35]. In Ref. [36], the authors proposed a lightweight local health indicator extraction approach for multi-stage fast charging protocols, where the seg- mented charging data of certain state of charge (SOC) windows was used as the input features, which were then fed into a hybrid deep learning model for the map- ping of SOH estimates. The proposed approach achieved precise SOH estimation, with mean absolute errors (MAEs) and root mean square errors (RMSEs) below 1%. In Ref. [37], the authors proposed an improved gated recurrent unit (GRU) network combined with the whale optimization algorithm for SOH estimation, where the time duration in the voltage ranges from 3.55 V to 3.75 V and from 3.80 V to 4.15 V during constant current (C) charging was extracted as the two input features. Test results showed that the proposed method is able to achieve an average error of less than 1% and presents good generalization capability. In Ref. [38], the authors conducted a comprehensive characterization of the aging behavior patterns of lithium-ion bat- teries, where five aging patterns were identified, covering the rate of voltage change 3 during discharging and C charging, the duration of C charging, the rate of current change during constant voltage (CV) charging, and the rate of temperature change during CV charging. These aging patterns were then used as health indicators for the input of deep neural networks (DNNs). Test results showed that the proposed health indicators consistently improved estimation performance across different DNN-based SOH estimators. In Ref. [39], incremental capacity analysis (ICA) was utilized for feature extraction, where the incremental capacity differences of five voltage intervals were selected as the input feature candidates of a multi-layer perceptron (MLP) for SOH and RUL estimation. Test results showed that the proposed approach could achieve relative error rates below 3% for SOH estimation. In Ref. [40], the authors focused on real-life electric vehicle (EV) driving scenarios and proposed a snapshot- based approach with a long short-term memory (LSTM) network for flexible SOH estimation, where the voltage and current sequences of partial charging segments with a fixed window size were used as the model input. In addition, the authors fused the snapshot-based and conventional history-based approaches to develop a noise-robust approach. Test results showed that the proposed approaches were able to achieve an average error of less than 2.46% over all presented experiments. In Ref. [41], on the other hand, the authors directly took the full charging curves of voltage, current, and temperature as the input features of the proposed GRU-convolutional neural network (CNN) for SOH estimation. Test results on the National Aeronautics and Space Administration (NASA) dataset and the Oxford battery degradation dataset showcased that the proposed approach was able to achieve accurate SOH estimation with the maximum estimation error of less than 4.3%, demonstrating the fact that deep learning models are capable of automatically extracting useful features from the measured cycling data and mapping them into accurate SOH estimates. However, data-driven SOH estimation approaches rely on large amounts of high- quality labeled battery cycling data, where the SOH labels are typically obtained through standardized checkup tests that cover the entire lifespan of the batteries un- der controlled conditions. Such calibration tests are time-consuming and expensive for laboratories, and can seldom be conducted in real-world deployment. Further- more, late-life SOH labels are inherently more difficult to collect, as reaching the deep-degradation regime requires extensive cycling over a long period of time with considerable experimental costs, resulting in a pronounced imbalance in label dis- tribution across the battery lifespan. As a result, in practical application scenarios, battery aging data often face the problem of label sparsity, both in quantity and coverage, while large volumes of unlabeled operational cycling data are available. Self-supervised learning (SSL), a branch of machine learning, is specifically aimed at such challenges. SSL methods enforce the model to learn meaningful data rep- 4 resentations without relying on manual labels by exploiting underlying data struc- tures or automatically constructed pretext tasks derived from the unlabeled data itself [42, 43, 44]. In general, SSL techniques can be categorized into four types [44]: generative approaches, which learn representations by reconstructing the input data or predicting the missing parts of it [45, 46]; contrastive approaches, which learn representations by forcing similar samples to be closer and dissimilar samples to be more distant in the latent space [47, 48]; contrastive generative approaches, which combine both generative and contrastive objectives and exploit the advan- tages of both [49, 50]; context-based approaches, which learn representations by leveraging contextual information of the samples, such as temporal order and spa- tial structures [51, 52]. In fact, SSL techniques have been sparsely applied in some previous works in the field of battery SOH estimation to address the problem of label sparsity. In Ref. [53], the authors proposed an SSL framework utilizing a reconstruction-based generative approach with an auto-encoder-decoder, aiming to address the problems of limited labeled data and underutilized measurements dur- ing degradation. In their work, an auto-encoder-decoder was trained to reconstruct the partial capacity-voltage curve as the pretext task. After the SSL pretraining, the trained encoder was transferred to the downstream network for SOH estima- tion, which was then fine-tuned using the sparse labeled data. Test results showed that the proposed framework was able to achieve a robust, accurate SOH estimation using a very limited amount of labeled data. In Ref. [54], the authors proposed a self-supervised framework incorporating weak labels to reduce the demand for large amounts of annotated battery aging data for deep learning-based SOH estimation approaches, where the raw data was first processed into three-dimensional feature maps with enriched information. Afterwards, the authors exploited the generative pretext tasks of masked image reconstruction and charging capacity estimation for pretraining. Similar to the aforementioned work, a small amount of labeled data was then used for fine-tuning on the downstream task of SOH estimation. Test results showed that the proposed model demonstrated strong generalization even with only a few labeled data points. On the other hand, in Ref. [55], the authors proposed a multi-level contrastive SSL approach with dynamic embedding, aiming to leverage the multi-level physical temporal dependence of the data to solve the problem of label scarcity. Frequency domain information was extracted through the discrete Fourier transform, after which the embedding was obtained using their proposed multi-scale dynamic embedding method. Contrastive learning combined with mask reconstruc- tion was utilized for SSL pretraining. Finally, the pretrained model was fine-tuned using limited labeled data for SOH estimation. As a matter of fact, current SSL approaches in the field of SOH estimation tend 5 to focus on reconstruction-based pretext tasks, where the model is trained to recover the input or its masked parts, as such strategies help the model capture general underlying patterns of the data. However, the learned representations are not neces- sarily aligned with the battery degradation process that is most relevant to the task of SOH estimation. In contrast, the relative aging order between cycles provides a much more direct and task-relevant self-supervised signal, since battery aging is inherently an ordered process. Therefore, in this work, we propose a degradation- aligned SSL framework that learns aging-consistent representations from unlabeled C charging curves through a cycle-order ranking objective as the pretext task for pretraining, thereby enabling robust SOH estimation after fine-tuning on only a lim- ited amount of labeled data for practical scenarios where data labels are sparse in both quantity and coverage. In addition, we develop a CNN-GRU model that ex- tracts local patterns from charging curves via convolutional layers and then integrates these features sequentially through recurrent units for accurate SOH estimation as well as self-supervised pretraining. To the best of our knowledge, this work is the first to exploit the intrinsic information behind the cycle order on battery aging for self-supervised SOH estimation. The main contribution of this work is as follows: 1. A ranking-based SSL framework is proposed to learn degradation-consistent representations from unlabeled C charging curves via cycle-order ranking as pretraining for robust SOH estimation in scenarios where only sparsely labeled data are available for fine-tuning. 2. A CNN-GRU model is developed to extract and sequentially integrate the local patterns from charging curves for accurate SOH estimation as well as self-supervised pretraining. 3. In-depth analyses are presented regarding the influences of label distribution of battery degradation data. The rest of this paper is structured as follows: Section 2 introduces the proposed SSL framework for SOH estimation of lithium-ion batteries under label sparsity, with a detailed explanation of the developed model, the ranking-based SSL algorithm, and the overall workflow. Section 3 describes the experimental setups, including data preparation, settings of the presented models and experiments, and the utilized evaluation metrics. In Section 4, comprehensive results of a variety of experiments are presented, together with in-depth analyses. The key takeaways of this work are summarized in Section 5. 6 2. Methodology 2.1. CNN-GRU Model for Self-Supervised Pretraining and SOH Estimation 2.1.1. Convolutional Neural Networks 풙 ퟎ 풙 푻 ... 풙 ퟏ 풙 ퟐ 풙 푻−ퟏ Padding Data Points Convoluted Data Points Outputs Input Layer Hidden Layer OutputLayer 풚 ퟎ 풚 ퟏ 풚 푻 ... 풚 ퟐ 풚 푻−ퟏ Figure 1: Working scheme of an example two-layer 1D CNN. CNNs [56] are a class of deep learning models that apply trainable convolution kernels with local receptive fields and shared weight parameters to extract local patterns from structured data. Fig. 1 shows the working scheme of an example two- layer 1D CNN with a kernel size of k = 7. The dilation factor d and the stride s are assumed to be 1. In order to preserve the input length after each convolution operation, the same padding technique is adopted in the network. Specifically, when the kernel size k is an odd number, the padding size on both sides of the 1D sequence is given by: p = k− 1 2 (3) With the same padding technique, the output sequence will have the same length as the input sequence without being shortened. In this case, the output vector at time step t, namely y t , is dependent on the inputs x t−p , x t−p+1 ,..., x t ,..., x t+p−1 , x t+p , where the inputs with indices outside the original index range of the input sequence are the appended paddings. More generally, at some time step t, the 1D convolution operation under same padding G(t) can be formulated as: 7 G(t) = (X ∗ F )(t) = k−1 X i=0 F (i)· x t+i−p (4) where X is the input sequence and F denotes the convolution kernel with kernel size k. CNNs effectively capture local patterns in input data with relatively few parame- ters, which is why they are widely used across tasks such as image recognition, object detection, and signal processing. In our context of battery SOH estimation, the cy- cling curves of batteries can also be viewed as structured one-dimensional signals, where adjacent measurements often exhibit strong local correlations. Since battery degradation gradually alters the shape and patterns of the cycling measurement tra- jectories, CNNs are a natural choice for the extraction of local patterns correlated with aging from such data, which is why they are utilized as part of the network backbone in this work for SOH estimation as well as the self-supervised pretraining. 2.1.2. Gated Recurrent Units 풉 풕−ퟏ σ σ 푡푎푛ℎ 풙 풕 풉 풕 풓 풕 풛 풕 ෩ 풉 t + 1− Figure 2: Internal architecture of GRU. GRUs [57] are an advanced variant of recurrent neural networks (RNNs) that are designed for sequential dependency modeling while mitigating the problem of van- ishing gradient in vanilla RNNs. With the introduction of gating mechanisms, GRUs can regulate the information flow actively and update the hidden state selectively based on the importance of historical and current information. In addition, the lean 8 internal architecture of GRUs facilitates high computational efficiency and lower de- pendencies on the amount of required training data, especially when compared with the other frequently applied variant of RNN with gating mechanisms, namely LSTM [58], making them a perfect choice for our case of SOH estimation under label spar- sity. The internal architecture of the GRU is demonstrated in Fig. 2. At time step t, given the input vector x t , the update gate z t and reset gate r t can be calculated as: z t = σ(W xz x t + W hz h t−1 + b z )(5) r t = σ(W xr x t + W hr h t−1 + b r )(6) where σ(·) denotes the sigmoid activation function, h t−1 denotes the hidden state of the previous time step, and W and b denote the respective weight matrix and bias vector. The candidate hidden state ̃ h t can thereby be computed by: ̃ h t = tanh(W xh x t + W h (r t ⊙ h t−1 ) + b h )(7) where⊙ denotes the Hadamard product, namely element-wise multiplication. In the end, the final output of the hidden state h t is updated as: h t = (1− z t )⊙ h t−1 + z t ⊙ ̃ h t (8) The reset gate controls the forgetting and retention of the historical information for the calculation of the candidate hidden state. The candidate hidden state com- bines the current input and the selected historical information. The new hidden state is the final output at the current time step from the fusion of the previous hidden state and candidate hidden state controlled by the update gate. In this work, the GRU is utilized to sequentially integrate the local patterns extracted by the previous CNN from C charging curves, thereby comprehensively capturing the underlying patterns correlated with battery degradation for self-supervised representation learn- ing and downstream SOH estimation. 2.1.3. CNN-GRU Model Fig. 3 shows the architecture of the proposed CNN-GRU model. The model con- sists of two parts, namely the encoder and head. The encoder aims to learn the underlying patterns from the input charging curves and is composed of two stacked CNN blocks and one GRU layer, where each CNN block further consists of one 1D convolutional layer and one rectified linear unit (ReLU) as the activation function. The CNN is responsible for extracting local patterns from the input charging curves, 9 Linear ReLU Linear Head Ranking Reconstruction Regression ReLU 1D Convolution CNN Block CNN Block GRU Encoder 푼 풄 / 푼 풄 / 푆푂퐻 푠 푎푔푒 Figure 3: The proposed CNN-GRU model for self-supervised pretraining and SOH estimation. and the GRU is used to sequentially integrate the extracted features so as to capture their temporal dependencies. Based on the learned latent representation of the en- coder, the head is further used to accomplish the target task, either self-supervised pretraining or SOH regression, which is composed of two fully connected (FC) lay- ers and ReLU in between as the activation function. The same head architecture 10 is used for both self-supervised pretraining and SOH estimation, but of course with different loss functions and sizes for the outputs. For the proposed ranking-based self-supervised pretraining, the head is used to generate an aging score s age based on the learned representations. For reconstruction-based self-supervised pretraining, the head is used to reconstruct the respective input charging curve ˆ U c based on the embedded vector generated by the encoder. For the downstream SOH estimation task, the head performs regression for ˆ SOH. We use the C segment of charging as input, since, in contrast to the highly dynamic discharge process with substan- tial fluctuations in the operating conditions, the charging process generally follows a clearer protocol and is acquired under more stable conditions for EVs and elec- tronics [36, 40]. As a result, it is more feasible to charge data to build high-quality, large-scale datasets that are continuously accumulated across the entire usage his- tory, with strong comparability across different cycles. In addition, during the C charging stage, the curve shape often exhibits evident pattern shifts with battery aging, making it a strong indicator of battery health. Since previous works have showcased that deep learning models are able to extract the aging-related features from the charging curve automatically [41], we use the voltage measurement of the C charging segment U c directly as the input of the model. 2.2. Cycle-Order Ranking for Degradation-Aligned SSL Battery aging is an inherently progressive process, in which later cycles generally correspond to deeper degradation states than earlier ones. Such an ordinal rela- tionship provides a natural, intuitive source of self-supervision even in the absence of explicit SOH labels. Motivated by this observation, we introduce an intra-cell cycle-order ranking objective for self-supervised pretraining to encourage the model to learn degradation-aligned representations from unlabeled charging data. Specif- ically, for each sampled mini-batch during training, the samples are first grouped according to their cell IDs, and ranking is only performed within each cell group. This design avoids unreliable comparisons across different cells, whose degradation trajectories may differ due to inter-cell variability. Within each cell group, random sample pairs are constructed for efficient optimization. In order to improve the relia- bility of the ranking signal, pairs with identical cycle numbers or with cycle-number differences smaller than a predefined threshold d min are excluded, since their degrada- tion order is either undefined or too weak to provide clear supervision due to possible capacity recovery phenomena. For a valid pair of samples (i,j) from the same cell, the ranking label is defined as: 11 Algorithm 1 Computation of intra-cell cycle-order ranking loss for a sampled mini- batch. Require: Mini-batch(c i ,n i ,s i ) B i=1 , where c i is the cell ID, n i is the cycle number, and s i is the predicted aging score of sample i; minimum cycle gap d min 1: G ← GroupByCell(1,...,B;c i B i=1 )▷ group sample indices by cell ID 2: S ←∅▷ store group-wise ranking losses 3: for all G m ∈G do 4: if |G m | < 2 then 5:continue▷ at least two samples are needed 6: end if 7: ̃ G m ← Permute(G m ) ▷ generate a shuffled index order for subsequent paring 8: P m ←(i,j)| i∈ G m , j ∈ ̃ G m , n i ̸= n j , |n i − n j |≥ d min ▷ construct sample pairs with ambiguous pairs removed 9: if |P m | = 0 then 10:continue▷ skip if no valid pair remains 11: end if 12: y ij ← ( +1, n i > n j −1, n i < n j , ∀(i,j)∈P m ▷ later cycles should receive larger aging scores 13: L m ← 1 |P m | P (i,j)∈P m ln(1 + exp(−y ij (s i − s j ))) ▷ group-wise logistic ranking loss 14: S ←S∪L m ▷ collect valid group losses 15: end for 16: if |S| = 0 then 17: return None▷ no valid ranking supervision in this mini-batch 18: else 19: L rank ← 1 |S| P L m ∈S L m ▷ average over valid cell groups 20: return L rank 21: end if y ij = ( +1, if n i > n j −1, if n i < n j (9) where n i and n j denote the corresponding cycle numbers. Here, y ij = +1 indicates that sample i comes from a later cycle and is expected to have a larger aging score than sample j, as the pairwise logistic ranking loss is designed to be: 12 ℓ ij = ln(1 + exp(−y ij (s i − s j )))(10) where s i and s j are the aging scores predicted by the model under training. It is referred to as a logistic ranking loss because it is derived from the negative log- likelihood of a logistic model, in which the probability of a correct pairwise order p ij is modeled as a sigmoid function of the margin y ij (s i − s j ), namely: p ij = 1 1 + exp(−y ij (s i − s j )) (11) where the margin indicates both whether the predicted order is correct and how confidently the model makes this prediction. Minimizing the negative log-likelihood of this probability p ij yields the loss ℓ ij , which encourages the score difference s i −s j to be consistent with the relative cycle order. For the m-th valid cell group, withP m denoting the set of valid sample pairs constructed within that group, the group-wise ranking loss is then computed as: L m = 1 |P m | X (i,j)∈P m ℓ ij (12) Finally, the overall ranking loss for the mini-batch is obtained by averaging over all valid cell groups: L rank = 1 M M X m=1 L m (13) where M is the number of valid cell groups in the mini-batch. Algorithm 1 summa- rizes the intra-cell cycle-order ranking objective for a sampled mini-batch. During ranking-based self-supervised pretraining, the model, consisting of the encoder and the head, is expected to predict a dimensionless aging score from the C charging curve input. By explicitly enforcing that the predicted aging scores follow the relative degradation order across cycles, the proposed ranking objective encourages the encoder to capture the underlying patterns correlated to the degrada- tion process from unlabeled charging curves. In this way, the learned representations become better aligned with the evolution of battery degradation, which is closely related to the downstream SOH estimation task, thereby benefiting cases with label sparsity in both quantity and coverage. 13 Self-SupervisedPretraining Downstream Adaptation Fine-Tuning Encoder ParametersFrozen Data Preparation DataCollection ... Cell1Cell 2 Cell k Data Preprocessing Cleaning Formatting Partial Labeling DataSplitting Model InitializationPairwiseRankingAging Score Validation Save Model LoadEncoder SOHEstimation Evaluation Figure 4: Workflow of the proposed SSL framework for SOH estimation under label sparsity. 2.3. Degradation-Aligned SSL Framework Fig. 4 shows the overall workflow of the proposed SSL framework for SOH esti- mation under label sparsity. First, as data preparation, the collected battery cycling aging data undergo a series of preprocessing procedures, including cleaning of dirty samples and outliers, formatting for a unified structure, and labeling the data par- tially based on the experiment settings to simulate practical scenarios where the battery aging data are under label sparsity, both in quantity and coverage. The preprocessed data are then split into training, validation, and test sets for parame- 14 ter optimization, hyperparameter fine-tuning, and objective evaluation, respectively. The unlabeled data will be used for self-supervised pretraining, while the partially labeled subset will be used for fine-tuning the pretrained model during downstream adaptation. During self-supervised pretraining, the CNN-GRU model introduced in Section 2.1, composed of an encoder and a head, is first initialized. Afterwards, the model is trained on the pretext task with unlabeled charging data using the ranking- based SSL algorithm proposed in Section 2.2. The outcome of the pretraining can be validated by inspecting the correlation between the pretrained model’s predicted aging score and the ground-truth SOH values, if available. The pretrained model will be saved and transferred to downstream adaptation on the task of SOH esti- mation, where only the pretrained encoder will be loaded, and a new head will be instantiated for SOH regression. Subsequently, the labeled subset of data will be used for fine-tuning the model, during which only the new head is updated, with the parameters of the pretrained encoder frozen. The fine-tuned model is ready for accurate, robust SOH estimation on new cells despite label sparsity in the training data. 3. Experimental Setup 3.1. Data Preparation Table 1: Specifications of the utlized CX2 battery. ParameterData Nominal Capacity1.35 Ah Cell ChemistryLCO Cell FormatPrismatic End-of-Charge Voltage4.20 V End-of-Discharge Voltage2.70 V In this work, the experiments and analyses are based on the public battery aging dataset, which is widely used in the research community, from the Center for Ad- vanced Life Cycle Engineering (CALCE) at the University of Maryland [59]. Specif- ically, the cells CX2-34, CX2-36, CX2-37, and CX2-38 are utilized in this work, of which the specifications are shown in Table 1. These prismatic cells have lithium cobalt oxide (LCO) cathodes and a rated capacity of 1.35 Ah. The four cells un- derwent cyclic aging from the beginning of life until different depths of degradation. For charging, they went through a C-CV protocol, where the cells were charged 15 under the constant current rate of 0.5 C until the end-of-charge voltage of 4.2 V was reached, after which CV charging carried on until the charging current dropped be- low 0.05 A. For discharging, the cells went through a C protocol with a current rate of 1 C [18] until the end-of-discharge voltage of 2.7 V was reached. 0.6725 0.6750 0.6775 Current [A] 0100020003000400050006000 Time [s] 3.5000 3.8500 4.2000 Voltage [V] 020040060080010001200 Cycle Number 0.5000 0.7500 1.0000 SOH 0 250 500 750 1000 1250 Cycle Number (a) 0.6740 0.6760 Current [A] 0100020003000400050006000 Time [s] 3.5000 3.8500 4.2000 Voltage [V] 0200400600800100012001400 Cycle Number 0.5000 0.7500 1.0000 SOH 0 250 500 750 1000 1250 Cycle Number (b) 0.6740 0.6750 Current [A] 0100020003000400050006000 Time [s] 3.5000 3.8500 4.2000 Voltage [V] 02004006008001000 Cycle Number 0.6000 0.8000 1.0000 SOH 0 200 400 600 800 1000 Cycle Number (c) 0.6750 0.6755 Current [A] 0100020003000400050006000 Time [s] 3.5000 3.8500 4.2000 Voltage [V] 02004006008001000120014001600 Cycle Number 0.5000 1.0000 SOH 0 500 1000 1500 Cycle Number (d) Figure 5: Aging profiles of the cells, including the current and voltage measurement during C charging and the evolution of SOH. (a) Cell CX2-34. (b) Cell CX2-36. (c) Cell CX2-37. (d) Cell CX2-38. BatteryML [60], an open-source platform for machine learning-based battery degradation diagnostics, is used for preliminary preprocessing of the raw cycling data, including the initial integration and ordering of the measurement files, com- pilation of basic information, and rough cleaning of dirty entries. Afterward, the voltage measurements of the C charging fragment are extracted for each cycling curve and resampled to a fixed length of 300 points to facilitate later training and 16 to enforce the network to learn the underlying patterns from the partial charging curve rather than simply using the sequence length as the indicator. The corre- sponding SOH labels are calculated based on Equation 1 using the full discharge capacity. Consequently, more thorough data cleaning is performed, removing out- liers based on anomalies in current and voltage readings, as well as on the rolling median absolute deviation (MAD) of the lengths of C charging segments at the cell level. The preprocessed aging profiles of the cells are shown in Fig. 5. As can be observed, the overall shape and pattern of the voltage measurement during C charging demonstrate an obvious shift through the aging process: the initial voltage reading becomes higher due to impedance rise, and the duration of the C fraction becomes shorter due to the combination of loss of lithium inventory (LLI), loss of active material (LAM), and impedance rise, which intuitively underpins the reliabil- ity of using voltage measurement during C charging as the input feature for SOH estimation. The capacity degradation trends of the cells appear similar, where the four cells are aged to different depths of degradation. Based on this observation, CX2-37 and CX2-38 are selected for training because together they cover relatively early and late degradation stages, enabling the model to learn from a wider range of aging conditions. CX2-36 and CX2-34, which fall between these two extremes in terms of degradation depth, are used for validation and testing, respectively. In this way, the validation and test cells remain unseen during training, while their degra- dation states are still enclosed by the training distribution rather than lying outside it. This split is therefore intended to enable a more stable and meaningful evaluation of cross-cell generalization. 3.2. Experiment Settings Z-score normalization is utilized on the training set to accelerate convergence in this work using the following formula: z = x− μ σ (14) where z is the normalized value, x is the raw value, μ is the mean, and σ is the standard deviation. The utilized CNN-GRU models all have the same architectural hyperparameters regardless of whether they are for self-supervised pretraining or SOH estimation. For the encoder part, the two-layer CNN uses a slightly larger kernel size of 7 to capture local patterns across more adjacent points and increases the number of channels from 1 to 32 and then to 64. Replicate values are used for the same padding on both sides of the input sequence. After the CNN, a one-layer GRU with an input size of 64 and an output size of 128 is used to fuse the extracted local patterns. For the head part, the fully connected layers have different numbers 17 of neurons for different tasks. For ranking-based self-supervised pretraining and for SOH regression, the two fully connected layers decrease the size of the latent vector gradually from 128 to 64, and then to 1, yielding a scaler of aging score or SOH estimate. For the reconstruction pretext task, which is used as a comparison in the experiments, the two fully connected layers gradually increase the size of the latent vector from 128 to 256, and then to 300. The batch size of 256 is used in this work. Automatic hyperparameter fine-tuning of the learning rate is conducted for all models, including both self-supervised pretraining and fine-tuning, using the hyperparameter optimization framework Optuna [61], utilizing the tree-structured Parzen estimator (TPE) algorithm, based on their performance on the validation set for a fair comparison. The ranking-based self-supervised pretraining uses a minimum cycle gap of d min = 50 to mitigate disturbances to the supervision signal caused by the capacity recovery effect during the aging test. Besides the proposed ranking-based SSL (SSL-Rank) approach, we further apply two other SSL approaches for pretraining as comparison, including the most frequently used reconstruction-based SSL (SSL-Recon) approach and a multi-objective SSL (SSL-MO) approach. As the pretext task, SSL-Recon aims to reconstruct the voltage curves during C charging through the encoder and the head, thereby encouraging the model to learn informative representations that preserve structural characteristics of the input charging signals. And as the name indicates, SSL-MO combines reconstruction and ranking as its pretext task in order to test the possibility that these two self-supervised objectives can complement each other and thereby yield more informative representations for downstream SOH esti- mation. The SSL models are first pretrained using the respective SSL strategy on the unlabeled data, after which a new head will be instantiated on top of the pretrained encoder for adaptation using the limited labeled data. During the adaptation, the pretrained encoder will be frozen, meaning only the freshly instantiated head will be updated. Certainly, at the same time, we train ordinary supervised learning (SL) models using the limited labeled data for SOH estimation as a baseline. 3.3. Evaluation Metrics The regression performance of the SOH estimation task is evaluated using the fol- lowing four evaluation metrics in this work, namely MAE, RMSE, maximum absolute error (MAX), and the coefficient of determination R 2 : MAE(y, ˆy) = 1 n n X i=1 |y i − ˆy i |(15) 18 RMSE(y, ˆy) = v u u t 1 n n X i=1 (y i − ˆy i ) 2 (16) R 2 (y, ˆy) = 1− n P i=1 (y i − ˆy i ) 2 n P i=1 (y i − ̄y) 2 (17) MAX( ˆy) = max|y i − ˆy i | n i=1 (18) where y and ˆy denote the labels and the estimates, respectively. n is the number of samples, while y i and ˆy i denote the label and the estimate of the i-th sample, respectively. ̄y is the mean of the labels y. The four evaluation metrics for regression have different emphases. MAE is an intuitive measure for the average magnitude of estimation errors, while RMSE gives greater weight to large errors due to the squaring operation. R 2 is used to evaluate how well the estimations capture the overall variation trend of the labels during fitting, and MAX reflects the worst-case estimation deviation across all samples. In addition, Spearman’s rank correlation coefficient ρ is employed to evaluate the monotonic relationship between the predicted aging scores and the ground-truth SOH values after the self-supervised pretraining, thereby assessing whether the learned representations can consistently reflect the real degradation process. It is defined as: ρ = cov(R s , R y ) σ R s σ R y (19) where R s and R y denote the ranks of the predicted aging scores and the ground- truth SOH labels, respectively. cov denotes covariance, and σ R s and σ R y denote the standard deviations of the corresponding rank variables. ρ is defined in the range from -1 to 1, where ρ = 1 indicates a perfect positive monotonic relationship, and ρ =−1 indicates a perfect negative monotonic relationship. 4. Results and Discussion In this section, we present the results and analyses of a variety of experiments conducted to evaluate the pretrained model via degradation alignment, compare different SSL and SL approaches on sparsely labeled data, and examine the influence of label distribution. 19 4.1. Evaluation of the Pretrained Model via Degradation Alignment In this experiment, we aim to evaluate the pretrained model using the proposed ranking-based SSL approach before downstream fine-tuning. To this end, the pre- trained model is used to generate an aging score for each input charging curve, and the resulting scores are compared with the corresponding ground-truth degradation process in order to verify their alignment. Since the proposed ranking objective is designed to encode the relative degradation order of battery cycles into the learned representations and then map it into an unscaled aging score prediction, effective pretraining is expected to produce aging scores that vary monotonically with SOH. Therefore, both the overall relationship between the predicted aging scores and the degradation process, and their Spearman’s rank correlation coefficient are analyzed to assess whether the pretrained model has successfully captured degradation-aligned information from unlabeled charging data. Fig. 6 shows the correlation between the aging score predicted by the pretrained model under the proposed ranking-based SSL and the ground-truth degradation processes of the four cells, namely CX2-37, CX2-38, CX2-36, and CX2-34, across the training, validation, and testing sets. Since the learned aging score of the pretrained model is an unscaled value without a unit, we first normalize it into the range from 0 to 1 for better visualization, as well as the ground-truth capacity fade, using the following formula of min-max normalization: x ′ = x− x min x max − x min (20) where x ′ is the normalized value, x is the original value, and x max and x min denote the maximum and minimum of the variable, respectively. In the left column of Fig. 6, namely Fig. 6a, Fig. 6c, Fig. 6e, and Fig. 6g, we compare the aforementioned normalized aging scores with the ground-truth capacity fade over the entire lifespan of each cell. As can be observed, despite the minor local fluctuations, the aging scores predicted by the pretrained model generally demonstrate a clear monotonic overall trend that is well aligned with the capacity degradation trajectory of each cell. In particular, as the battery ages and the capacity gradually fades, the predicted aging scores consistently evolve in the corresponding direction, indicating that the pretrained model has successfully captured degradation-relevant information from unlabeled charging data. This is particularly evident for cells CX2-37 and CX2-38, namely the two training cells, as the ranking objective was directly imposed on them during the training process. Especially for the training cell CX2-38, the aging scores predicted by the pretrained model can even reconstruct the different degradation paces of the ground-truth degradation process, showing clearly different slopes before 20 02004006008001000 Cycle Number 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Value CX2_37 Normalized Ground-Truth Capacity Fade Normalized Learned Aging Score (a) 0.600.650.700.750.800.850.900.951.00 SOH 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Value =0.997 CX2_37 Normalized Learned Aging Score (b) 02004006008001000120014001600 Cycle Number 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Value CX2_38 Normalized Ground-Truth Capacity Fade Normalized Learned Aging Score (c) 0.20.30.40.50.60.70.80.91.0 SOH 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Value =0.998 CX2_38 Normalized Learned Aging Score (d) 0200400600800100012001400 Cycle Number 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Value CX2_36 Normalized Ground-Truth Capacity Fade Normalized Learned Aging Score (e) 0.50.60.70.80.91.0 SOH 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Value =0.996 CX2_36 Normalized Learned Aging Score (f) 0200400600800100012001400 Cycle Number 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Value CX2_34 Normalized Ground-Truth Capacity Fade Normalized Learned Aging Score (g) 0.50.60.70.80.91.0 SOH 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Value =0.997 CX2_34 Normalized Learned Aging Score (h) Figure 6: Correlation between the aging score predicted by the pretrained model under ranking- based SSL and the respective ground-truth degradation process. (a) Cell CX2-37, normalized aging score and capacity fade. (b) Cell CX2-37, normalized aging score and SOH. (c) Cell CX2-38, normalized aging score and capacity fade. (d) Cell CX2-38, normalized aging score and SOH. (e) Cell CX2-36, normalized aging score and capacity fade. (f) Cell CX2-36, normalized aging score and SOH. (g) Cell CX2-34, normalized aging score and capacity fade. (h) Cell CX2-34, normalized aging score and SOH. and after the 1350th cycle. For the validation cell CX2-36, from around the 600th cycle, the predicted aging score begins to show slight divergence from the ground- 21 truth capacity fade, although the monotonicity and alignment are not compromised, and the dip indicating capacity recovery at around the 1300th cycle is well captured by the pretrained model. Similarly, on the test cell CX2-34, the predicted aging score generally aligns quite well with the ground-truth capacity fade, despite local fluctuations at the beginning of life (BOL) and a slight divergence in the middle of life. The capacity recovery phenomenon at around the 1150th cycle is well captured by the pretrained model based on the unlabeled C charging data as well. In the right column of Fig. 6, namely Fig. 6b, Fig. 6d, Fig. 6f, and Fig. 6h, the correlation between the normalized aging score and the ground-truth SOH is showcased for each cell, with Spearman’s rank correlation coefficient calculated and marked in each case. The normalized aging scores generally exhibit a clear monotonic decreasing trend as the ground-truth SOH increases, or equivalently, a monotonic increasing trend as the battery degrades. This indicates that the predicted aging scores are strongly consistent with the underlying aging progression of each cell. Although a few outliers can still be observed locally, the overall relationship remains highly correlated. The calculated Spearman’s rank correlation coefficients ρ, namely - 0.997 for cell CX2-37, -0.998 for cell CX2-38, -0.996 for cell CX2-36, and -0.997 for cell CX2-34, are all extremely close to -1, indicating an almost perfect monotonic inverse relationship between the predicted aging score and the ground-truth SOH for all cells, which further supports the fact that our proposed ranking-based SSL approach is able to enforce the pretrained model to extract degradation-aligned information from unlabeled charging data, such that the learned aging scores consistently reflect the underlying battery aging progression across the entire lifespan. 4.2. Model Performance Comparison under Label Sparsity In this experiment, we aim to compare the performance of the models pretrained using different SSL strategies after fine-tuning, as well as under complete SL with no pretraining on the unlabeled data. More specifically, we consider three different SSL approaches: SSL-Rank, our proposed SSL approach using the cycle-order objec- tive as the pretext task; SSL-Recon, the most frequently used SSL approach using reconstruction of the input C charging curves as the pretext task; SSL-MO, which combines reconstruction and ranking as its pretext task in order to test the possibil- ity that these two self-supervised objectives can complement each other and thereby yield more informative representations for downstream SOH estimation. All models are first pretrained on the unlabeled C charging data using their respective SSL strategies and then fine-tuned using sparsely labeled C charging data. In addition, we train a pure SL model from scratch without pretraining on the same sparsely la- beled C charging data. Here, we consider a sparse-label setting in which only data 22 above the 80% SOH threshold are labeled, and the proportion of labeled samples varies from 1% to 100%, where 100% of labeled data corresponds to approximately 1200 training samples. This setup is consistent with practical scenarios, since 80% SOH is commonly adopted as the EOL criterion for lithium-ion batteries, and thus data from this earlier degradation region are more likely to be available than la- beled samples from deep-degradation stages, whose acquisition requires much longer cycling time and higher experimental cost. The labeled data points for fine-tuning or complete SL are sampled uniformly from all data points within the defined SOH range for each training cell. Fig. 7 shows the SOH estimation results on the test cell CX2-34 of different models after fine-tuning or complete training on different ratios of labeled data. At first glance, we can see that all models pretrained using SSL strategies are able to carry out satisfactory SOH estimation under different ratios of labeled data from 1% to 100% on the entire lifespan of the test cell, while the pure SL model without pretraining shows great fluctuations in its estimation under ratios of labeled data above 50% and fails the task of SOH estimation under ratios of labeled data below 40%, because the number of samples available for SL from scratch becomes too small for the model to capture the underlying battery degradation dynamics. Compared with models pretrained with the other two SSL approaches, the proposed SSL-Rank model consistently outperforms the rest of the models and is able to track the ground- truth degradation progression accurately under all scenarios with different ratios of labeled data, even when the labeled data is extremely sparse, as low as 1%. In general, since the labeled data are considered to be in the earlier degradation region, the estimations of the models fit better to the early-life ground-truth SOHs, while slight divergences can be observed towards the EOL. Nevertheless, the proposed SSL-Rank model is able to carry out an SOH estimation that not only is steady and smooth in early life, but also converges well to the ground-truth in late life, as shown in the zoomed-in sections. In comparison, the SSL-Recon model exhibits significantly higher oscillations in its estimation, and the estimation errors of the SSL-Recon and SSL-MO models become more pronounced in the deep-degradation regime. Fig. 8 is the box plot of the absolute error of different models for SOH estimation under different ratios of labeled data. The aforementioned observations can be better confirmed in this visualization. The three SSL models achieve consistently good performance across all testing scenarios with varying proportions of labeled data. The SSL-Recon model and the SSL-MO model achieve comparable overall performance in most cases, though the latter typically produces more outliers. The proposed SSL- Rank model is able to outperform the rest of the models in every testing scenario, regardless of whether the labeled proportion is abundant or scarce, with fewer outliers 23 0200400600800100012001400 Cycle Number 0.4 0.5 0.6 0.7 0.8 0.9 1.0 SOH Labeled Ratio 100% Ground Truth SL SSL-Recon SSL-MO SSL-Rank (a) 0200400600800100012001400 Cycle Number 0.4 0.5 0.6 0.7 0.8 0.9 1.0 SOH Labeled Ratio 50% Ground Truth SL SSL-Recon SSL-MO SSL-Rank (b) 0200400600800100012001400 Cycle Number 0.4 0.5 0.6 0.7 0.8 0.9 1.0 SOH Labeled Ratio 40% Ground Truth SL SSL-Recon SSL-MO SSL-Rank (c) 0200400600800100012001400 Cycle Number 0.4 0.5 0.6 0.7 0.8 0.9 1.0 SOH Labeled Ratio 30% Ground Truth SL SSL-Recon SSL-MO SSL-Rank (d) 0200400600800100012001400 Cycle Number 0.4 0.5 0.6 0.7 0.8 0.9 1.0 SOH Labeled Ratio 20% Ground Truth SL SSL-Recon SSL-MO SSL-Rank (e) 0200400600800100012001400 Cycle Number 0.4 0.5 0.6 0.7 0.8 0.9 1.0 SOH Labeled Ratio 10% Ground Truth SL SSL-Recon SSL-MO SSL-Rank (f) 0200400600800100012001400 Cycle Number 0.4 0.5 0.6 0.7 0.8 0.9 1.0 SOH Labeled Ratio 5% Ground Truth SL SSL-Recon SSL-MO SSL-Rank (g) 0200400600800100012001400 Cycle Number 0.4 0.5 0.6 0.7 0.8 0.9 1.0 SOH Labeled Ratio 1% Ground Truth SL SSL-Recon SSL-MO SSL-Rank (h) Figure 7: SOH estimation results on the test cell CX2-34 of different models after fine-tuning or complete training on different ratios of labeled data. (a) 100% of labeled data available. (b) 50% of labeled data available. (c) 40% of labeled data available. (d) 30% of labeled data available. (e) 20% of labeled data available. (f) 10% of labeled data available. (g) 5% of labeled data available. (h)1% of labeled data available. in addition. The baseline SL model can still carry out relatively good SOH estimation when the amount of labeled data is sufficient, namely 50% or more, but is not able to learn the underlying degradation dynamics and thus fails to carry out SOH 24 100504030201051 Labeled Ratio [%] 0 5 10 15 20 25 30 Absolute Error [%] SL SSL-Recon SSL-MO SSL-Rank Figure 8: Box plot of the absolute error for SOH estimation under different ratios of labeled data. estimation when fewer labeled data are available. It is intriguing that even in cases where 100% or 50% labeled data is available, the SL model is still outperformed by all the SSL models, indicating that leveraging unlabeled charging data through SSL pretext tasks not only compensates for label scarcity, but also improves the model initialization and guides it toward a more favorable parameter space for downstream optimization, so that the model can make better use of the labeled data during fine-tuning and achieve superior SOH estimation performance even when labels are relatively abundant. Table 2 summarizes the experiment results on different models’ performance un- der varying ratios of labeled data above 80% SOH. The proposed SSL-Rank model leads across all four evaluation metrics in all testing scenarios with different labeled data ratios, achieving MAEs of 1.715%, 1.722%, 1.692%, and 1.718% for labeled data ratios of 100%, 50%, 10%, and 1%, respectively. The single-objective SSL-Recon model is outperformed by the dual-objective SSL-MO model in most test cases, ex- cept at 20% and 10% labeled data ratios, in terms of MAE. In the case when 30% of labeled data is available, the SSL-MO model achieves an MAE of 2.283%, which is lower than the MAE of 2.449% of the SSL-Recon model, while the RMSE of 3.311% of the SSL-MO model is higher than the RMSE of 3.169% of the SSL-Recon model, indicating that the estimation of the SSL-MO model contains more outliers. The SL model achieves MAEs of 4.017% and 4.031% in test scenarios with labeled data ratios 25 Table 2: Comparison of different models’ performance under varying ratios of labeled data above 80% SOH. Labeled Ratio [%] Model Evaluation Metrics MAE [%]RMSE [%]R 2 MAX [%] 100 SSL-Rank1.7152.2550.97210.458 SSL-Recon2.7683.7890.92112.874 SSL-MO2.4093.4070.93614.556 SL4.0175.3840.84119.625 50 SSL-Rank1.7222.3020.97110.937 SSL-Recon2.5573.3660.93811.380 SSL-MO2.3703.3030.94013.889 SL4.3015.3810.84118.038 40 SSL-Rank1.6992.2590.97210.724 SSL-Recon2.5903.4010.93611.168 SSL-MO2.3623.3700.93814.576 SL12.20314.317-0.12728.848 30 SSL-Rank1.7072.2790.97110.815 SSL-Recon2.4493.1690.94511.838 SSL-MO2.2833.3110.94014.869 SL12.44214.593-0.17129.566 20 SSL-Rank1.6922.2410.97210.624 SSL-Recon2.5643.3500.93811.878 SSL-MO2.5973.5880.92915.300 SL12.18414.290-0.12328.659 10 SSL-Rank1.6922.2490.97210.686 SSL-Recon2.6553.5280.93211.839 SSL-MO2.8033.8730.91816.210 SL11.96614.039-0.08427.791 5 SSL-Rank1.6932.2580.97210.749 SSL-Recon2.8063.7890.92112.592 SSL-MO2.7213.7560.92215.753 SL11.96914.043-0.08527.803 1 SSL-Rank1.7182.3290.97011.157 SSL-Recon3.0754.2930.89914.228 SSL-MO2.6893.7070.92415.689 SL11.95714.029-0.08227.749 of 100% and 50%, respectively, but collapses when fewer labeled data are available, with MAEs around 12%. In all test scenarios, the proposed SSL-Rank model is able 26 to maintain an R 2 of over 0.970, indicating its strong performance in SOH regres- sion. The MAXs of around 10% typically characterize the final estimation error in the deepest-degradation regime due to significantly accelerated aging in this stage. The results of this experiment showcase the effectiveness and robustness of the proposed SSL framework across different labeled data proportions under the practical earlier-life labeling setting. Even with an extremely limited amount of labeled data, the proposed method is still able to maintain a high estimation accuracy, further demonstrating that the self-supervised pretraining stage has successfully endowed the model with informative and degradation-aligned prior knowledge that can be effectively transferred to downstream SOH estimation under label sparsity in both quantity and coverage. 4.3. Analysis on the Influence of Label Distribution In this experiment, we aim to evaluate the SSL-Rank model’s performance under different distributions of sparsely labeled data and thereby analyze the influence of label distribution on the task of SOH estimation. Since the test results in Section 4.2 already show that the model pretrained utilizing our proposed ranking-based SSL approach is able to carry out accurate and robust SOH estimation after fine-tuning, even only on an extremely small number of labeled data, we explore the different distributions on rather small labeled ratios here, namely 10%, 5%, 2%, and 1% of the whole training set, where 10% corresponds to approximately 260 samples. The four studied types of distribution are uniform, random, early-only, and late-only. For the uniform setting, the labeled samples are selected to cover the entire available SOH range as evenly as possible. For the random setting, the labeled samples are randomly drawn from all available data. In the early-only setting, only samples from the extreme early degradation stage, namely above 85% SOH, are retained as labeled data, whereas in the late-only setting, only samples from the late degradation stage, namely below 60% SOH, are labeled. This experiment is designed as a supplementary analysis to the main experiment in Section 4.2, with the focus shifted from the overall effectiveness of the proposed SSL framework to the specific influence of label distribution of the training set under very limited labeling budgets. Fig. 9 shows the regression performance of the proposed SSL-Rank model fine- tuned with 10% or 5% of labeled data sampled from different distributions. At first glance, we can see that the type of distribution of the labeled data has some minor influence on the overall regression performance. The R 2 decreases from 0.979 to 0.977 when 10% of the labeled data are sampled randomly from the whole lifespan instead of uniformly, and decreases from 0.981 to 0.977 when the labeled ratio is 5%, indicating that a more even coverage of the degradation trajectory is still beneficial. 27 0.50.60.70.80.91.0 True SOH 0.5 0.6 0.7 0.8 0.9 1.0 Estimated SOH R 2 = 0.979 Labeled Ratio 10% Uniform Estimation Ground Truth (a) 0.50.60.70.80.91.0 True SOH 0.5 0.6 0.7 0.8 0.9 1.0 Estimated SOH R 2 = 0.977 Labeled Ratio 10% Random Estimation Ground Truth (b) 0.50.60.70.80.91.0 True SOH 0.5 0.6 0.7 0.8 0.9 1.0 Estimated SOH R 2 = 0.974 Labeled Ratio 10% Early-Only Estimation Ground Truth (c) 0.50.60.70.80.91.0 True SOH 0.5 0.6 0.7 0.8 0.9 1.0 Estimated SOH R 2 = 0.983 Labeled Ratio 10% Late-Only Estimation Ground Truth (d) 0.50.60.70.80.91.0 True SOH 0.5 0.6 0.7 0.8 0.9 1.0 Estimated SOH R 2 = 0.981 Labeled Ratio 5% Uniform Estimation Ground Truth (e) 0.50.60.70.80.91.0 True SOH 0.5 0.6 0.7 0.8 0.9 1.0 Estimated SOH R 2 = 0.977 Labeled Ratio 5% Random Estimation Ground Truth (f) 0.50.60.70.80.91.0 True SOH 0.5 0.6 0.7 0.8 0.9 1.0 Estimated SOH R 2 =0.037 Labeled Ratio 5% Early-Only Estimation Ground Truth (g) 0.50.60.70.80.91.0 True SOH 0.5 0.6 0.7 0.8 0.9 1.0 Estimated SOH R 2 = 0.981 Labeled Ratio 5% Late-Only Estimation Ground Truth (h) Figure 9: Regression performance of the proposed SSL-Rank model fine-tuned with different ratios and distributions of labeled data. (a) 10% labeled data, uniform distribution. (b) 10% labeled data, random distribution. (c) 10% labeled data, early-only distribution. (d) 10% labeled data, late-only distribution. (f) 5% labeled data, uniform distribution. (f) 5% labeled data, random distribution. (g) 5% labeled data, early-only distribution. (h) 5% labeled data, late-only distribution. The early-only distribution, on the other hand, seems to yield worse training out- comes: the R 2 is 0.974 when the labeled ratio is 10%, whereas the model fails the SOH estimation task when the ratio decreases to 5%. In fact, we have also tried to 28 sample 10% of the labeled data only from the range above 90%, where the model fails as well. The deterioration in model performance when labeled data are only available for the extremely early life stage is due to the limited diversity of charging curves in this stage. As shown in Fig. 5, the pattern shift in the voltage curve during C charging is minor in the early-degradation stage, but it accelerates significantly toward the late-degradation stage, meaning that labeled samples drawn from the early-only distribution are not diverse enough and thus the model can hardly gener- alize on such data. At the same time, we observe that the lack of data diversity can be compensated to some degree by larger data quantity, as showcased by the regres- sion performance under 10% and 5% of early-only labeled data. On the contrary, the late-only distribution leads to the overall performance even better than the uniform distribution. A plausible reason is that the voltage curves in this stage exhibit much more pronounced aging-related variations and pattern shifts, so that the labeled sam- ples provide stronger supervisory signals for calibrating the pretrained model to the SOH regression task. In addition, a major source of error for uniform and random distributions is the divergence between the estimation and the ground truth toward the EOL, where significantly accelerated aging happens. This is also compensated for when more labeled samples in the late-degradation region are available for model fine-tuning. Fig. 10 visualizes the MAEs of the SSL-Rank model’s estimation with different ratios and distributions of labeled data for fine-tuning across different SOH windows. In general, the estimation error in the SOH range below 60% is significantly higher than in other SOH windows for each case due to the accelerated late-life degradation. The model fails the SOH estimation task when fine-tuned on less than 5% of labeled data under early-only distribution, causing the different scales of the bar graphs. In Fig. 10a, we can clearly observe an increasing trend in the SOH estimation error in the late-life SOH window below 60% from uniform distribution to random distribu- tion and then to early-only distribution. Sampling labeled data from the late-only distribution increases the estimation error in the early-life SOH window above 90%, but at the same time significantly decreases the estimation error in the late-life SOH window below 60% as well as in the middle-life SOH windows from 60% to 90%, resulting in an improvement in the overall performance. Table 3 summarizes the estimation results of the proposed SSL-Rank model fine- tuned under different distributions of sparsely labeled data. Many of the aforemen- tioned observations can be confirmed. The model fine-tuned with late-only labeled data outperforms the rest of the label distributions consistently in terms of MAE, with the MAEs of 1.320%, 1.369%, 1.369%, and 1.350% for cases where 10%, 5%, 2%, and 1% of the data are labeled, respectively. This suggests that, under the same 29 UniformRandomEarly-OnlyLate-Only Label Distribution 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 MAE [%] Labeled Ratio 10% (90%, 100%] (75%, 90%] (60%, 75%] 60% (a) UniformRandomEarly-OnlyLate-Only Label Distribution 0.0 5.0 10.0 15.0 20.0 MAE [%] Labeled Ratio 5% (90%, 100%] (75%, 90%] (60%, 75%] 60% (b) UniformRandomEarly-OnlyLate-Only Label Distribution 0.0 5.0 10.0 15.0 20.0 MAE [%] Labeled Ratio 2% (90%, 100%] (75%, 90%] (60%, 75%] 60% (c) UniformRandomEarly-OnlyLate-Only Label Distribution 0.0 5.0 10.0 15.0 20.0 MAE [%] Labeled Ratio 1% (90%, 100%] (75%, 90%] (60%, 75%] 60% (d) Figure 10: Bar graph of the MAEs of the SSL-Rank model’s estimation with different ratios and distributions of labeled data for fine-tuning across SOH windows. (a) 10% of labeled data available. (b) 5% of labeled data available. (c) 2% of labeled data available. (d) 1% of labeled data available. labeling budget, label coverage over the more informative late-degradation region is particularly beneficial for downstream SOH estimation, which indicates that, when only a very limited number of labels can be acquired, prioritizing samples from the later degradation stage may be more advantageous than distributing labels evenly or concentrating them in the early-life stage. The failure of the model fine-tuned with early-only labeled data in cases of fewer labeled data shows that the lack of data di- versity and data quantity jointly affect the effectiveness of downstream fine-tuning. However, to some extent, these two factors can compensate for each other: limited diversity may be partially alleviated by a larger number of labeled samples, while lim- 30 Table 3: Evaluation of the SSL-Rank model’s performance under different distributions of sparsely labeled data. Labeled Ratio [%] Distribution Evaluation Metrics MAE [%] RMSE [%]R 2 MAX [%] 10 Uniform1.5691.9460.9799.032 Random1.5662.0370.9779.515 Early-Only1.6152.1600.97410.058 Late-Only1.3201.7440.9837.605 5 Uniform1.5191.8580.9817.724 Random1.5862.0320.9779.199 Early-Only11.68213.728-0.03727.632 Late-Only1.3691.8620.9817.565 2 Uniform1.5081.8040.9827.480 Random1.5161.9350.9798.791 Early-Only11.65413.694-0.03127.572 Late-Only1.3691.8630.9817.659 1 Uniform1.5891.9040.9808.054 Random1.6352.0130.9788.753 Early-Only11.26613.2400.03627.212 Late-Only1.3501.8390.9817.420 ited quantity may be mitigated if the labeled data cover more informative and diverse degradation stages. These findings further confirm the effectiveness of the proposed ranking-based SSL strategy, which enables the model to capture degradation-relevant information from unlabeled data and thus alleviates the dependence of downstream fine-tuning on both label quantity and label diversity. 5. Conclusion An accurate estimation of SOH underpins a safe and optimized use of the bat- tery system. Although compelling, data-driven SOH estimation models rely on large amounts of high-quality labeled cycling data, where the SOH labels are typically obtained through standardized checkup tests that cover the entire lifespan of the batteries under controlled conditions. However, in practical application scenarios, battery aging data often face the problem of label sparsity in both quantity and 31 coverage due to the time-consuming and costly nature of such lifespan-wide calibra- tion tests, while large volumes of unlabeled operational cycling data are available. Therefore, in this work, we propose a degradation-aligned SSL framework that learns aging-consistent representations from unlabeled C charging curves through a cycle- order ranking objective as the pretext task for pretraining, thereby enabling robust SOH estimation after fine-tuning on only a limited amount of labeled data for practi- cal scenarios where data labels are sparse in both quantity and coverage. In addition, we develop a CNN-GRU model that extracts local patterns from charging curves via convolutional layers and then integrates these features sequentially through recurrent units for accurate SOH estimation as well as self-supervised pretraining. Comprehen- sive experiments are conducted on the CALCE battery aging dataset to evaluate the model performance after self-supervised pretraining and after fine-tuning on sparsely labeled data. Test results not only demonstrate that the proposed ranking-based SSL approach proves to endow the pretrained model with degradation-aligned infor- mation from unlabeled data, where Spearman’s correlation coefficients between the learned aging scores and the ground-truth SOHs are extremely close to -1, but also showcase that the proposed SSL-Rank model can carry out accurate, robust SOH estimation after fine-tuning, even when only an extremely limited amount of 1% of unevenly distributed labeled training data is available, where an MAE of 1.718% and an RMSE of 2.329% can be achieved on the test cell. In addition, in-depth analyses are presented regarding the influences of label distribution of battery degradation data. So far, only one cell model under one cycling condition has been studied, so it would be meaningful for future work to extend the proposed SSL framework to cross-cell and cross-condition settings that better reflect the variability encountered in real-world applications. Considering the practical motivation and the encouraging experimental results, we believe this work could shed new light on SOH estimation of lithium-ion batteries under label sparsity in real-world applications. 32 Data Availability The public dataset utilized in this work can be accessed through Ref. [59]. CRediT Authorship Contribution Statement Jiaqi Yao: Conceptualization, Methodology, Software, Validation, Formal Anal- ysis, Investigation, Resources, Data Curation, Writing - Original Draft, Writing - Review & Editing, Visualization. Julia Kowal: Writing - Review & Editing, Su- pervision. Acknowledgement We thank the High-Performance Computing Cluster of TU Berlin ZECM for the GPU resources and the Open Access Publication Fund of TU Berlin for the support. 33 Abbreviations The following abbreviations are used in this manuscript: BMSBattery Management System BOLBeginning of Life CALCECenter for Advanced Life Cycle Engineering CCConstant Current CVConstant Voltage CNNConvolutional Neural Network DNNDeep Neural Network ECMEquivalent Circuit Model EKFExtended Kalman Filter EOLEnd of Life EVElectric Vehicle FCFully Connected GRUGated Recurrent Unit ICAIncremental Capacity Analysis LAMLoss of Active Material LCOLithium Cobalt Oxide LLILoss of Lithium Inventory LSTMLong Short-Term Memory MADMedian Absolute Deviation MAEMean Absolute Error MAXMaximum Absolute Error MLPMulti-Layer Perceptron NASANational Aeronautics and Space Administration ReLURectified Linear Unit RMSERoot Mean Square Error RNNRecurrent Neural Network RULRemaining Useful Life SLSupervised Learning SOCState of Charge SOHState of Health SSLSelf-Supervised Learning SSL-MOMulti-Objective SSL SSL-Rank Ranking-Based SSL SSL-Recon Reconstruction-Based SSL TPETree-Structured Parzen Estimator UKFUnscented Kalman Filter 34 References [1] Q. 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