Battery PHM spans degradation-mechanism understanding, state-of-health (SoH) estimation, RUL prediction, and decision-making for maintenance and operation across the battery life cycle. Comprehensive reviews summarize degradation modes, operational factors, and modeling paradigms and emphasize that interpretability and reliability are central to real-world PHM adoption [1].

Early-life prediction targets rapid life estimation using only initial cycles to enable screening and early decisions [3–6]. Severson et al. [3] established a widely used benchmark by showing that early-cycle voltage-curve-derived features (notably ΔQ−V) can be predictive of cycle life before substantial capacity fade. Subsequent studies explored alternative representations and learning models, including image-based encodings with CNNs [4,6], probabilistic regressors such as GPR with degradation-pattern recognition [5], and graphical feature constructions based on ΔQ−V IC/ΔQ curves [22]. Broader-condition work further examined early-life prediction under varying usage conditions and proposed degradation-informed or hierarchical approaches to improve extrapolation [23], while cross-condition deep learning frameworks such as BatLiNet aim to improve reliability across diverse ageing conditions via inter-cell learning mechanisms [24].

These efforts demonstrate the potential of early-cycle data while also highlighting practical trade-offs in small datasets: higher-capacity end-to-end models can be less transparent and more sensitive to modeling choices. In addition, recent perspective work emphasizes that feature selection should match the deployment objective, because protocol-encoding features can bias apparent performance and complicate transferability [16]. Our work adopts a pragmatic stance by combining physics-informed ΔQ indicators with generic statistics and an explicit correlation-structure representation to study whether a lightweight tabular pipeline can remain competitive without introducing a complex end-to-end graph learner.

Recent adjacent literature has also explored graph-based battery health modeling and alternative health metrics under related but non-identical tasks, including graph-based SOH estimation from partial discharge segments [25], alternative battery-health metrics such as Health EquiMetrics [26], and matrix-profile-based online knee-onset identification for lithium-ion battery SOH estimation [27]. Because many recent SOH-oriented and alternative battery-health studies use different prediction targets, sensing windows, and evaluation protocols, we discuss them as methodological context rather than as direct numerical baselines for the early-life cycle-life regression task considered here.

Physics-informed machine learning (PIML) integrates physical constraints, electrochemical knowledge, or equivalent-circuit insights into learning pipelines, for example, by embedding governing equations into model structures or training objectives [7–10]. PIML can offer improved physical consistency and may enhance generalization when adequate domain information and data support are available. However, PIML often introduces additional modeling layers and expanded hyperparameter spaces, which can be burdensome in small-sample regimes and less accessible for practitioners without deep modeling expertise [1].

Graph-based learning has also received substantial attention for capturing structured dependencies among sensors, cells, or features. Recent approaches include spatio-temporal models with dynamic graphs [11], correlation- or attention-based GNN variants [12–14], and graph-attention–recurrent hybrids supported by explainable feature fragments [15]. These models can be powerful, but many treat the graph as an internal latent structure whose influence is not always straightforward to interpret directly; moreover, implicit graph learning can raise concerns about tuning effort and reproducibility in small datasets.

In contrast, our study should be understood as a graph-assisted feature-engineering case study. We use graph theory to construct a structurally explicit descriptor at the feature-correlation level. Specifically, we build a feature-by-feature correlation graph using Pearson coefficients [19] and compute low-dimensional Laplacian Eigenmaps embeddings [20] as compact representations of correlation structure. The contribution, therefore, lies in explicit spectral feature engineering rather than in a novel graph-learning architecture.

Most battery life-prediction studies report static regression metrics such as RMSE, MAE, or R². While these metrics quantify pointwise accuracy, they do not describe whether the RUL trajectory implied by a predicted cycle life remains inside practically relevant error bands over time. In PHM, RUL is often evaluated as a trajectory, motivating measures that summarize error-band agreement and temporal consistency rather than final pointwise error alone [17,18].

Saxena et al. [17] formalized offline prognostic metrics including RA, PH, and α-λ. RA measures the fraction of predictions within a specified tolerance band; PH summarizes how early predictions enter a tighter band; and α-λ captures agreement within a chosen tolerance window. Sharp [18] further discussed how such measures can be communicated to users with varied backgrounds. Despite their relevance, these trajectory-oriented metrics are still not routinely used in early-life battery benchmarks, where RMSE/R² remain the dominant reporting metrics [3–6]. In this work, we therefore report both regression metrics and PHM-style error-band metrics computed on the implied RUL trajectories induced by each model’s early-life cycle-life estimate.

Accordingly, our work evaluates models using both regression metrics and PHM-style threshold-based accuracy measures, providing a complementary view of pointwise prediction quality and error-band agreement under batch shift.

We use the open Severson benchmark [3], which contains 124 commercial A123 APR18650M1A LFP/graphite cells cycled at 30°C under 72 one-step and two-step fast-charging policies. Cycle life is defined as the number of cycles until the cell reaches 80% of nominal capacity [3]. For each cell, the dataset provides cycle-resolved current, terminal voltage, charge/discharge capacity, time, temperature, and internal-resistance-related measurements.

Following Severson et al. [3], we formulate early-life cycle-life prediction as a supervised regression problem. For each cell, we use early-life cycles 1–100 as inputs and predict the total cycle life as the target.

For our evaluation, the dataset is organized into three batches (Batch 1–3). Batches 1–2 form the main development domain and are split into Train (41 cells; odd-indexed cells) and Primary Test (43 cells; even-indexed cells), whereas Batch 3 (40 cells) is retained as the Secondary Test set. Across all three batches, the common protocol backbone is fast charging from 0% to 80% SOC under one of the benchmark charging policies, an internal-resistance measurement at 80% SOC, a uniform 1C CC-CV top-off from 80% to 100% SOC to 3.6 V, and identical 4C CC-CV discharge to 2.0 V [3].

The batches nevertheless differ in their exact rest schedules and in the placement of short pauses around the 80% SOC/internal-resistance/discharge steps. Severson et al. report 1 min and 1 s rests after reaching 80% SOC and after discharge, respectively, for the 12 May 2017 batch; 5 min rests at both locations for the 30 June 2017 batch; and 5 s rests after reaching 80% SOC, after the internal-resistance test, and both before and after discharge for the 12 April 2018 batch [3]. Together, these protocol and distributional differences motivate treating Batch 3 as a protocol-shifted secondary evaluation domain rather than as an in-domain test split. The batch-specific rest/pause schedules and their roles in the evaluation are summarized in Table 1.

Because several point features are computed from current, time, and capacity trajectories over cycles 1–100, differences in rest timing and pause placement can directly alter the empirical distribution of the statistical-feature channel even when the underlying chemistry is unchanged.

After feature extraction, each cell is represented as a single tabular feature vector. Early-life time series are summarized into physics-informed indicators, generic statistics, and correlation-structure descriptors, yielding a regression task suitable for tree-based gradient boosting.

Our feature engineering comprises three groups: (i) physics-informed ΔQ−V health indicators, (ii) data-driven statistical features, and (iii) Laplacian features that encode correlation structure among the statistical features.

Using the three feature groups above, we define five feature configurations to isolate the incremental value of adding statistics and correlation structure on top of the ΔQ−V baseline.

ΔQ Only. ΔQ−V HI (7) only. This is closest to the Severson-style ΔQ−V baseline and serves as a physics-informed reference.

Regression model. We use XGBoost [21] for all feature sets. XGBoost is well-suited to tabular data, captures non-linear interactions, and is relatively parameter-efficient in small-sample regimes. Because tree-based models tolerate heterogeneous feature scales, we combine ΔQ−V HI, Stats, and Laplacian features without requiring additional normalization for the regressor.

Training and hyperparameter tuning. To enable fair comparison across feature sets, we use a common training protocol. We perform 5-fold cross-validation on the 41 training cells. Key hyperparameters (e.g., tree depth, learning rate, number of estimators, min_child_weight, subsample, colsample_bytree, and λ regularization) are optimized using Optuna-based hyperparameter optimization, with cross-validation RMSE as the objective. The final model is retrained on the full training set using the selected hyperparameters and evaluated on the Primary and Secondary test sets. All scalers, correlation graphs, Laplacian bases, and PCA/RP/TSVD transforms were fit on the training cells only and then applied unchanged to the Primary and Secondary test splits.

Table 4 summarizes the feature-set configurations.

We evaluate each feature configuration using both standard regression metrics and PHM-oriented error-band metrics.

Regression metrics. We report RMSE and R² for pointwise cycle-life prediction accuracy.

PHM-style error-band metrics. In addition to RMSE and R², we assess PHM-style behavior using trajectory-oriented metrics [17,18]. For each cell, the model outputs a single early-life estimate of total cycle life. From this estimate we construct an implied RUL trajectory by subtracting the cycle index t from the predicted cycle life, while the true RUL is obtained by subtracting t from the observed cycle life. Because the model does not generate sequentially updated predictions, these metrics are applied to the implied trajectory induced by the fixed early-life estimate. We then compute relative error along the trajectory and report:

RA@0.2: the fraction of time points where the relative error is within ±20%;

We report RMSE, R², and the three PHM metrics for both the Primary and Secondary test sets. This dual evaluation complements pointwise regression accuracy by adding an error-band view of performance under domain shift.

Cross-validation on the 41-cell training set was used for model selection and hyperparameter tuning. In this small-sample setting, richer feature configurations often appeared promising in CV, but the held-out ordering changed across the Primary and Secondary domains.

Accordingly, the main scientific interpretation in this section is anchored in the held-out test results and the accompanying bootstrap intervals. The additional matched-dimensionality comparisons and the k-sensitivity analysis then provide comparative and robustness evidence for the graph-derived descriptor.

Tables 5 and 6 report the held-out test results for the five feature configurations. Taken together, they highlight the contrast between in-domain performance on Batches 1–2 and shifted-domain generalization on Batch 3. The split difference is substantial not only protocol-wise (Section 3.1) but also distributionally: the Primary split has median cycle life 520 cycles, whereas the Secondary split has median 964.5 cycles. The bootstrap intervals reported in Tables 5 and 6 quantify uncertainty for RMSE and R² only; RA, PH, and α-λ are reported as point estimates.

To examine whether the Hybrid Append behavior could be reproduced by generic dimensionality reduction alone, we conducted an additional comparison against three non-graph alternatives at a matched 79-feature scale. These alternatives were obtained by appending 20 features from PCA, Gaussian Random Projection, and Truncated SVD to the 59-feature ΔQ + Statistics baseline, as shown in Fig. 4. This supplementary analysis helps interpret the appended structural descriptor alongside the main five-configuration held-out results.

All compared variants were evaluated under a common train/Primary/Secondary split and XGBoost protocol, and every learned transform was fit on the training cells only before being applied unchanged to the Primary and Secondary test splits.

In this comparison, with all learned transforms estimated from the training cells and then applied unchanged to the test splits, Hybrid Append achieved CV R² 0.7497, Primary R² 0.8185, and Secondary R² −0.3205. The best matched-dimensionality non-graph baseline reached Primary R² 0.8107 and Secondary R² −0.1955, while +RandomProjection20 achieved the highest CV R². The comparison is therefore mixed rather than uniform: the graph-derived embedding remained competitive, but its margin over simpler reductions at the same feature dimensionality depended on the evaluation split. These results instead position Hybrid Append as the study’s principal hybrid configuration, whose added value is best understood as complementary and split-dependent.

Fig. 5 compares RA@0.2, PH@0.1, and α-λ (0.15) across feature sets on both test domains. These PHM-style error-band metrics highlight differences that are not fully captured by RMSE and R² alone. For compact visualization, Fig. 5 displays the scores on a 0–1 scale, whereas Tables 5 and 6 report the corresponding values in percentage form. Because each model produces a single early-life cycle-life estimate per cell, the scores are computed on the implied RUL trajectory defined in Section 3.4.

On the Primary split, Hybrid Append and Integrated Laplacian tie for the highest RA@0.2 (88.4%), Hybrid Append and VIF + Laplacian tie for the highest PH@0.1 (60.5%), and VIF + Laplacian achieves the highest α-λ(0.15) (76.7%). On the Secondary split, VIF + Laplacian reaches the highest RA@0.2 (70.0), Hybrid Append reaches the highest PH@0.1 (45.0), and ΔQ + Statistics, Hybrid Append, and VIF + Laplacian tie on α-λ(0.15) (60.0).

Viewed together with the held-out RMSE/R² results, the PHM-style metrics suggest that the contribution of the appended structural channel is not expressed identically across evaluation criteria. Thus, no single configuration dominates across all criteria; instead, the preferred feature set depends on whether the emphasis is on pointwise regression error or stricter error-band agreement. In particular, the Hybrid Append configuration is most naturally interpreted as improving selected error-band behavior while maintaining competitive pointwise accuracy on the in-domain split and improving PH@0.1 relative to ΔQ + Statistics on both test domains (60.5 vs. 55.8 on Primary; 45.0 vs. 42.5 on Secondary).

To complement the aggregate metrics, Fig. 6 plots observed vs. predicted cycle life for the Primary and Secondary test cells across the five feature configurations. The Primary-domain points cluster much closer to the identity line than the Secondary points, consistent with the protocol differences summarized in Table 1 and the shifted-domain performance reported in Table 6.

Among the augmented feature sets, the Primary scatter for ΔQ + Statistics, Hybrid Append, and Integrated Laplacian is comparatively tight, consistent with the stronger in-domain RMSE/R² values in Table 5. The broader Secondary dispersion of the integrated variant mirrors its weaker shifted-domain behavior, while Hybrid Append remains visually close to ΔQ + Statistics in pointwise scatter but achieves slightly stronger PH@0.1. Viewed together with Tables 5 and 6 and Fig. 5, these plots indicate that added feature channels help in-domain, whereas shifted-domain behavior remains more metric-dependent.