iMOE: prediction of second-life battery degradation trajectory using interpretable mixture of experts

Domain: physics
Published: 2026-02-09
Editorial outcome: minor_revision

Abstract
Retired electric vehicle batteries offer immense potential to support energy infrastructure stability in underdeveloped regions through second-life use, but uncertainties in battery degradation behaviors pose major safety concerns. This work proposes an interpretable mixture of experts (iMOE) network that predicts battery degradation trajectories using partial, field-accessible signals in a single cycling operation. iMOE leverages an adaptive multi-degradation prediction module to classify battery degradation modes using expert weight synthesis learned from battery capacity-voltage and relaxation data. The module produces latent degradation trend embeddings, which are input to a use-dependent recurrent network for long-term degradation trajectory prediction. Validated on three typical use patterns (i.e. consistent operating histories, deeply aged batteries with unknown prior use, and uncertain second-life conditions, including 295 batteries, 93 use conditions, and 84,213 cycles), iMOE achieves an average mean absolute percentage errors (MAPE) of 0.95% with a 0.43 ms inference time for life-long battery degradation trajectory prediction. Compared to state-of-the-art Informer and PatchTST, it reduces computational time and MAPE by 50% and 77%, respectively. Compatible with data sampling in random state of charge regions, iMOE supports a 150-cycle time-horizon degradation trajectory prediction with 1.50% and 6.26% MAPE on average and at maximum, respectively. Notably, iMOE can operate effectively even with pruned 5MB training data while retaining 0.95% MAPE. Broadly, this network offers a deployable, history-free solution for battery degradation trajectory prediction at the time of second-life deployment, redefining how second-life energy storage systems are sensed, evaluated, controlled, and integrated for sustainable energy infrastructures at scale.

Introduction
Lithium-ion batteries have been the core energy storage medium for electric vehicles (EVs) and renewable energy systems1. However, it is projected that by 2030, 120 GWh of in-service batteries will reach end of life (EOL) and be retired from primary use, such as EVs, bringing considerable economic and environmental concerns2,3. The reusing and recycling of these retired EV batteries has been identified emerging solution to improve the affordability and sustainability in battery lifecycle, particularly for the energy infrastructures in underdeveloped regions4–7.

Degradation trajectory is a widely adopted indicator for battery degradation characterization, which can be critical to the safety performance during extended reusing or repurposing process, i.e., second-life of the retired batteries8–11. For example, retired batteries undergo degradation trajectory evaluation before second-life use based on the cell, module or pack-level application requirements12,13. Conventional non-data-driven methods relying on destructive disassembly or full-cycle capacity tests14,15, inflicting extra damage or labor investments. Notably, the economic feasibility of these state measurement methods of retired batteries are impractical due to prohibitive time, momentary, and environmental costs at scale1,16–18.

Data-driven methods have demonstrated promises in predicting degradation trajectories using non-destructive electrical signal data, but they typically require 5–40% of full lifecycle historical data19–22. Moreover, uncertain second-life use conditions can significantly differ from those in first-life, rendering conventional battery degradation trajectory prediction models ineffective, given that the data used for model training were built on constant-condition tests or typical dynamical cycling tests while the degradation trajectory is highly path-dependent23–25. Lu et al. leveraged at least one full cycle of capacity-voltage curves to predict degradation trajectories under uncertain future operating scenarios by learning the relationship between use condition and battery capacity23. Yet, the method encounters limitations due to the stochasticity in state of charge (SOC) of retired batteries and demanding requirement for full discharge-charge data. Moreover, the unavailability of historical cycling data of retired batteries complicates the tracing of initial conditions of degradations, such as impedance rise, loss of lithium inventory, and loss of active material26–29, challenging degradation trajectory prediction effectiveness. At the policy level, global initiatives such as the Battery Data Genome Initiative30 and the European Commission’s July 4, 2025, Delegated Regulation on battery recycling31 aim to enhance data standardization, traceability, and material recovery across the battery lifecycle. However, policy advances still face practical challenges due to frequent ownership change and uncertain second-life use conditions of retired batteries, emphasizing the need for degradation prediction frameworks that rely solely on field-accessible data and adapt to diverse operational environments.

Additionally, due to highly non-linear degradation of second-life batteries, the incorporation of physics knowledge into models has gained increased attention. Recent advances in sensory-based measurements include X-ray imaging32, electrochemical impedance, optical fiber sensing33, acoustic sensing34, and partial charging35, which serve as the data input of the data-driven methods. Nevertheless, most sensing techniques remain at the laboratory stage and are invasive33. Meanwhile, purely data-driven methods struggle to capture internal physical information of retired batteries36. Given the understanding that internal physical state is particularly critical for second-life applications with safety-sensitive considerations, the challenge of the “black-box” nature and lack of interpretability in data-driven methods for battery state prediction should be addressed. Even if progresses have been made in physics-informed neural networks37,38, the promise lies in additional and explicit integration of physical laws into feature engineering process39,40, neural network loss functions36, and transferability metrics41. However, the complex physical constraints involve numerous parameters and prior physical understanding from the modeling of retired batteries, which are still hardly available post-retirement12. Tao et al. employed physics-informed machine learning to achieve full degradation trajectory prediction using early-cycle data, reducing data requirements, while still relying on historical data22. In recent years, the mixture-of-experts (MOE) architecture, a core component in large language models, has demonstrated notable performance in learning heterogeneous data representations. By decomposing complex tasks into sub-tasks handled by “expert” modules with specialized task knowledge, MOE captures multi-level and nonlinear feature representations. The ideal of decomposing tasks into multiple subtasks naturally aligns with coupled degradation mechanisms of batteries. However, in absence of physical interpretability, the learning outcomes of model are confined to purely statistical correlations and fail to reflect true physical mechanisms underlying degradation processes, particularly with diverse cathode chemistries42, historical usages43, and uncertain second-life operating conditions23. Incorporating physical knowledge into the MOE architecture anchors data-driven learning outcomes to typical electrochemical processes, thereby enhancing model interpretability and enabling mechanism-aware degradation trajectory prediction.

To fulfil this gap, this work proposes an interpretable mixture of experts (iMOE) network for predicting degradation trajectories of retired batteries under uncertain second-life use complexities without requiring historical data. As shown in Fig. 1a, the model uses electrical signal data from randomly sampled SOC regions at collection field of retired batteries, interactively incorporating assumed future use conditions to predict degradation trajectories toward extended time horizons in future second-life use. Figure 1b illustrates the designed adaptive multi-mode degradation prediction (AMDP) module, which integrates multiple expert systems’ weights studied from capacity-voltage and relaxation data to achieve classification of degradation modes of retired batteries. AMDP outputs latent degradation trend embedding vectors that are forwarded into a feature-operational recurrent neural network (FORNN), enabling interpretable and extendable prediction of degradation trajectories. In Fig. 1c, model’s applications on battery reusing and recycling are illustrated, highlighting a safety-aware allocation of second-life batteries based on their predicted degradation trajectories toward a sustainable battery circular economy. Without historical data, the proposed approach achieves an average mean absolute percentage errors (MAPE) of 0.88% in predicting degradation trajectories under uncertain and diversified second-life use scenarios, spanning 295 batteries and 93 use conditions and 84,213 cycles. Notably, the required input data can be easily extracted at random SOC region, without charging or discharging retired batteries to anticipated SOC levels. The proposed method can extend degradation trajectory prediction over a 150 cycle of horizon in future second-life use with an MAPE of 1.50%. This work underscores transformative potential of incorporating physics prior principles into predictive health management of retired batteries, enabling broader proactive fault detection and reliability assessment across complex, safety-critical systems, such as large-scale energy infrastructures, with careful consideration of durability, safety, and sustainable operation under data-scarce and data-heterogeneous conditions.

Results
Methodology overview
The objective of this study is to compute battery degradation trajectories under uncertain future use conditions using one cycle data collected at random SOC regions. We use partial charging curves, relaxation voltage, and future conditions (charge/discharge current, temperature) as required model inputs (specific extraction methods are detailed in “Methods”). Input signal selection prioritizes two critical aspects: first, the controllability of charging processes in second-life battery ensures easier signal acquisition; second, relaxation voltage measurements remain unaffected by charge/discharge operations, thus enhancing model applicability. Six physics-informed features extracted from both relaxation voltage and partial charging curves (detailed in Supplementary Figs. 1–6 and Supplementary Notes 1–2) characterize polarization and degradation states. These features drive the model to classify batteries with different aging modes into distinguishable latent subspaces. Notably, physics-informed feature extraction extends beyond the charging and relaxation voltage curves discussed here but serves as the inputs of the AMDP module.

A two-phase collaborative mechanism achieves deep integration of physical insights and data-driven approaches: (1) AMDP, and (2) FORNN. The AMDP module employs “degradation-router” routing to assign expert network weights based on the physical features, adaptively modeling degradation modes under dominant mechanisms for degradation trend representation44, which can be formulated as:

Trend i = \sum_{j=1}^{N_{expert}} G_j(F_i) \cdot Expert_j(Q_i) (1)

where, Trend_i denotes the computed degradation trend vector for the i-th battery sample. it is important to note that “Trend” is a high-dimensional feature vector embedded through a deep neural network, encoding the latent feature representation of the battery’s current health state and short-term degradation trend. Its length is the same as the degradation trajectory we predict. F_i represents the physics-informed features extracted from the field cycle, and Q_i is the partial charging curve (possibly with random initial SOC) for the field cycle. N_{expert} is the total number of expert networks, and G_j(F_i) is the router weight associated with Expert_j, determined by a degradation-router mechanism. Expert_j(Q_i) denotes a specific degradation trend estimator under the j-th dominant degradation mode.

The FORNN module iteratively integrates AMDP predictions regarding latent degradation trends with assumed future load conditions cycle-by-cycle, enabling proactive degradation trajectory computation without historical data, which can be formulated as:

\hat{S}_i = FORNN(Trend_i, Cond_i) (2)

where, \hat{S}_i represents the final predicted degradation trajectory (e.g., capacity fade curve) for the i-th battery sample over the computation time horizon, Trend_i is the short-term trend output from the AMDP module, and Cond_i contains future usage conditions such as charge/discharge current, which can be assumed available. The FORNN leverages the sequential modeling capability to capture how future load profiles affect the long-term degradation starting from the current state.

These two modules exhibit unique but complementary strengths that AMDP computes latent degradation trends from learned degradation modes while FORNN deals with second-life use complexities (see Supplementary Fig. 7 for the detailed machine learning pipeline). The degradation routing simultaneously considers physics-informed features reflecting current existing degradation modes and future load profiles to select optimal experts, forming a closed-loop physics-driven specialization and data-driven fusion architecture. Theoretically, iMOE’s design philosophy benefits from the success of collaborative learning1,45, where the weighted collaboration from multiple networks enhances prediction robustness46–48. In principle, iMOE applies to degradation trajectory computation across the entire lifecycle stage under arbitrary use conditions, including performance computation in other lifetime stages despite considerably different internal degradation modes.

Dataset
This work addresses the computation challenge of degradation trajectories for retired batteries under uncertain future use conditions using only partial cycling data at test field, i.e., no historical data is assumed. To validate computation performance when historical and future operating conditions are consistent but historical data is unavailable, we select the Uniform-Life (UL) dataset49 for demonstration experiment. To validate computation performance when the historical operating conditions are unknown and the batteries are in a deeply aged state, we select the late-stage degradation (LSD) dataset43 for demonstration experiment. To validate computation performance when the second-life use conditions change significantly as compared to their first-life counterpart, we select the two-phase second-life (TPSL) dataset23 for demonstration experiment.

The UL dataset comprises batteries tested under different use conditions, with each batch cycled under consistent conditions throughout their entire lifespan until EOL, comprising three batches totaling 130 commercial 18650 cells. Batch 1 consists of LiNi_{0.86}Co_{0.11}Al_{0.03}O_2 positive electrode (NCA battery) with 3500 mAh nominal capacity and cutoff voltages of 2.65–4.2 V. Batch 2 contains LiNi_{0.83}Co_{0.11}Mn_{0.07}O_2 positive electrode (NCM battery) with 3500 mAh nominal capacity and cutoff voltages of 2.5–4.2 V. Batch 3 includes 42(3) wt% Li(NiCoMn)O_2 blended with 58(3) wt% Li(NiCoAl)O_2 positive electrode (NCM+NCA battery) with 2500 mAh nominal capacity and cutoff voltages of 2.5–4.2 V. All cells underwent cycling in thermal chambers under three temperatures (25°C, 35°C, 45°C) with variable charge rates (from 0.25C to 4C) and fixed 1C discharge rate. Each battery experienced identical operational profiles throughout its full lifecycle until reaching EOL (see Supplementary Table 1 for details). Degradation trajectories in Fig. 2a demonstrate cycling variability.

The LSD dataset contains 86 commercial batteries with LiNi_{0.5}Co_{0.2}Mn_{0.3}O_2 cathodes and graphite anodes. These 2.4 Ah batteries (3.7 V nominal) underwent two-phase testing, In Phase 1, the batteries were grouped into 16 distinct charge-discharge protocols to induce diverse degradation behaviors as the state of health (SOH) decreased from 100% to 80% (details in Supplementary Table 2). In Phase 2, all 86 cells were re-cycled from 80% to 50% SOH under a unified low-rate protocol (0.5C charge/0.2C discharge) to simulate second-life operation. Degradation trajectories in Fig. 2b demonstrate the ability to predict the degradation trajectories of second-phase batteries under unknown and varying historical operating conditions.

The TPSL dataset contains batteries subjected to diverse second-life scenarios, where they undergo both randomized and standardized operating conditions following their primary usage phase, containing 77 pieces of 18650 batteries with LiCoO_2 and LiNi_{0.5}Co_{0.2}Mn_{0.3}O_2 cathodes and graphite anodes. These 2.4 Ah batteries (3.7 V nominal, from 3.0 to 4.2 V cutoffs) underwent two-phase testing: Phase 1 involved 20 cycles of 0.5C constant-current constant-voltage (CCCV) charging and 2C discharging. Phase 2 divided batteries into two subgroups: (1) TPSL-Random (55 cells) subjected to stochastic charging profiles with current rates (1C/2C/3C) changed randomly every 5 cycles that follow a uniform statistical distribution, combined with fixed 3C discharge; (2) TPSL-Fixed (22 cells) cycled under pre-determined charge/discharge current combinations (details in Supplementary Table 3). Figure 2c shows degradation trajectories under uncertain use conditions. In the TPSL-fixed dataset, different magnitudes of charge and discharge currents lead to vastly different degradation trajectories. Meanwhile, in the TPSL-Random dataset, due to variations in the magnitude of the charging current, the difference in the maximum discharge capacity between adjacent cycles can even exceed 0.3 Ah, disproving the assumption of identical operational history for prediction reliability. Figure 2d reveals substantial capacity dispersion at cycles 21, 60, and 100 post-load variation, demonstrating how uncertain loads interact with battery degradations. Figure 2e details the input data construction process, where each sample combines raw charging curve sequences with physics-informed features, see “Methods” (data processing) and Supplementary Note 2.

Model performance and generalization capability
To validate the robustness of the proposed method under unknown historical data and uncertain second-life use conditions, we conduct evaluations in four real-world application scenarios (UL, LSD, TPSL-Random, TPSL-Fixed). iMOE adaptively computes nonlinear degradation modes under uncertain use conditions through the AMDP module, see Supplementary Table 4 for the associated model parameter. Performance metrics (RMSE, MAPE, R²) are shown in “Methods” (Evaluation metric). It is emphasized that all experiments use current-cycle data with random initial SOC regions extracted from field conditions to compute degradation trajectories spanning dozens to hundreds of cycles, rather than EOL points (see the prediction horizon selection in Supplementary Note 3). As a time-series computation task, we benchmark the model performance against state-of-the-art (SOTA) sequence computation models50,51 PatchTST and Informer (see Supplementary Note 4 and Supplementary Table 4 for detailed information). In Fig. 3a, existing research demonstrates that battery degradation can be divided into three typical while distinct degradation phases: SEI formation, SEI thickening, and lithium plating8,52. During the battery’s lifecycle, the formation and thickening of the SEI layer impacts the battery’s capacity degradation in early stages, while lithium-ion deposition typically occurs in the later stages of the battery’s use and is one of the main causes of sharp capacity decline. However, it is important to note that the capacity knee at the EOL is not always directly related to lithium plating. In some cases, the capacity decline at the EOL may also be influenced by other degradation mechanisms, such as SEI thickening, loss of active material, or electrolyte degradation, which can contribute to the sharp capacity drop observed in later stages. The degradation patterns across these phases exhibit significant differences in both available capacity and subsequent aging rates, especially in the slope of the degradation trajectory, with each phase reflecting distinct degradation behaviors53,54. Supplementary Fig. 8 illustrates that the definition of degradation stages and mechanisms can be regarded as typical and reasonable. We randomly select representative samples from each degradation stage to analyze performance, where each sample reflects distinct dominant degradation modes with varying capacity retention and aging rates.

While PatchTST and Informer show errors across early, middle, and late degradation phases, iMOE demonstrates superior stability, delivering accurate current-cycle capacity estimation while generating degradation trajectory predictions consistent with phase-specific degradation rates. Supplementary Figs. 9–11 validate iMOE’s effective prediction capability across full lifecycle samples.

Figure 3b reveals that in TPSL data, baseline methods display limited capability in capturing long-term trends after future load changes and completely fail during initial load transitions. In contrast, iMOE effectively identified transitional characteristics between phases through future usage condition integration. Supplementary Figs. 12-15 further demonstrate iMOE’s successful learning of relationships between future operating conditions and capacity performance. In LSD dataset, the baseline models exhibit larger errors in predicting the later stages of degradation trajectories. This is mainly due to the unknown and varying operating conditions in the first phase, which result in highly heterogeneous degradation behaviors during deep aging. In contrast, our proposed method, benefiting from the AMDP module, can achieve more stable and reliable prediction performance.

Figure 3c demonstrates that iMOE exhibits significant and stable advantages in both MAPE and R² metrics with the cross-dataset performance comparison setting. For UL dataset, iMOE achieved its worst prediction result on the NCA-25-1-1 dataset with an average MAPE of 0.98%, while maintaining MAPE below 1.00% across all UL conditions, showcasing robust prediction performance. PatchTST delivers the second-best results with an average MAPE of 0.85% and a maximum MAPE of 1.55% in the UL dataset, indicating that its patch-based tokenization strategy captures aging variations more effectively than single-point sampling. Informer performs the worst in this scenario, with an average MAPE of 1.48% and a maximum MAPE of 2.13%, suggesting that its sparse attention mechanism may lose critical information and fail to detect subtle temporal differences in voltage-capacity curves during such extreme prediction tasks. In LSD dataset, iMOE achieves an average MAPE of 1.82% for batteries in the deep degradation stage, whereas the Informer reaches an average MAPE of 2.68%. This result indicates that purely data-driven models struggle to make relatively accurate predictions under the unknown and varying operating conditions of the first-phase usage. In the TPSL dataset, iMOE shows even greater performance advantages over baseline methods, achieving an average RMSE of 0.05 compared to 0.38 for PatchTST and 0.39 for Informer, further confirming the necessity of incorporating future operating conditions in second-life applications.

In Fig. 3d, the scenario analysis confirms that iMOE achieves an average MAPE of 0.52% with a standard deviation of 0.28% for the UL dataset, 1.81% with a standard deviation of 0.13% for the LSD dataset, 2.96% with a standard deviation of 0.44% for the TPSL-Fixed dataset, and 2.81% with a standard deviation of 0.15% for the TPSL-Random dataset. iMOE performed better on the UL dataset with continuous operating history than on the LSD dataset with unknown historical conditions and deep degradation, as well as on the TPSL dataset with highly variable operating conditions (see Supplementary Table 5 for detailed information). This is primarily because continuous operation provides more stable degradation patterns, allowing the model to more accurately capture battery aging behaviors. In contrast, the deep degradation in the LSD dataset exhibits stronger nonlinearity due to differences in aging mechanisms, while the random load switching in the TPSL dataset introduces additional cycle-level nonlinear complexities, increasing the difficulty of degradation trajectory computation. Nevertheless, all MAPE values remained below 3.00%, demonstrating that the proposed model can effectively compute degradation trajectories under uncertain future conditions using only current-cycle data without relying on historical records.

In Fig. 3e, we analyze the relationship between model computational efficiency and performance to meet the real-world deployment requirements. After training, iMOE requires 0.43 ms to infer the entire degradation trajectory in the coming hundreds of cycles incorporating future operating conditions for a single retired battery, while achieving an MAPE of 0.93%. In comparison, baseline methods PatchTST and Informer required 0.58 ms and 0.87 ms per battery, with MAPEs of 4.15% and 4.73%, respectively. iMOE thus achieves the most robust prediction accuracy while emphasizing lightweight real-world deployment.

Unlike existing approaches that rely on fixed voltage or SOC regions and require time-consuming SOC recalibration, we validate iMOE’s robustness across five random initial SOC regions. In Fig. 3f, model prediction performance was relatively poorer at an initial SOC of 50%, with MAPEs of 1.44%, 3.35%, and 3.43% for the UL, TPSL-Random, and TPSL-Fixed datasets respectively, accompanied by standard deviations of 0.84%, 0.14%, and 0.50%. Reducing the initial SOC to 20% improved average performance by 63.8%, 16.2%, and 13.7%, respectively.

The observed performance disparity can be attributed not only to the reduced volume of training data available at higher initial SOC from a machine learning perspective but also, more fundamentally, to the inherent lack of adequate Open Circuit Voltage feature information in high-SOC data segments from a physicochemical standpoint, which hinders the model’s ability to differentiate degradation modes such as lithium inventory loss and active material loss. Furthermore, as the computations for the UL dataset do not involve cycle-level condition changes, the model outcomes rely more heavily on the data volume at the initial SOC, resulting in more pronounced performance degradation as the initial SOC increases.

Rationalization of statistical model performance
Here we explore the fundamental mechanisms underlying performance improvements. In Fig. 4a, we first randomly select NCA battery test samples from the UL dataset to visualize the weight assignments of the degradation router throughout their full lifecycle. Without requiring historical data, the degradation router partitions battery lifecycle degradations into three distinct phases using partial cycling data, which is evidenced by existing literature8,52. It is observed that early-retirement samples are predominantly governed by Expert Networks 1 and 4, mid-life retired batteries exhibit higher weights for Expert Networks 2 and 3, while late-life retired batteries are primarily regulated by Expert Network 5. These results demonstrate iMOE’s capability to identify evolving degradation patterns, with clear functional specialization among expert networks for different degradation stages. Mechanistically, this suggests Expert Networks 1, 3, and 5, respectively, dominate initial SEI layer formation, subsequent thickening, and lithium plating processes, consistent with prior research on primary battery degradation modes52. Expert Network 4 maintains high weights during both early and late degradation phases, potentially reflecting its statistical correlation with high degradation rates, i.e., slope of the degradation trajectories, rather than specific mechanistic contributions.

Figure 4b takes Battery 1 as an example to deeply analyze the dynamic correlation between individual cycling samples and expert network weight assignments during capacity degradation. Single-cycle samples at similar degradation stages exhibit clustering characteristics in expert weight allocation, demonstrating highly consistent expert category selection. Meanwhile, the degradation router adaptively adjusts the weights of different expert networks based on the features of individual cycling samples, thereby achieving differentiated prediction of degradation variations both between cycles and across degradation stages. Supplementary Fig. 16 further confirms the observed expert weight dependencies across different battery material systems and second-life use conditions.

We perform T-SNE dimensionality reduction experiments on expert weights across the entire test battery cohort (details in Supplementary Note 5) to visualize the separability of decision boundaries among expert networks. In Fig. 4c, despite significant inter-cell variability, samples from early and late degradation stages form distinct decision boundaries, as demonstrated in Supplementary Figs. 17–19 that iMOE has generalizability across battery material systems and operating conditions. These findings raise a critical question: can the trained iMOE model directly classify retired batteries based on degradation router-generated weight patterns? In essence, this question is an interpretability analysis showing why AMDP should work well.

In Fig. 4d, for the 130 UL dataset batteries retired at different SOH levels (classify solely based on expert weights, see Supplementary Note 6 for criteria), those retired at 95% SOH achieve an “Excellent” classification confidence of 91.5%, while batteries retired at 75% SOH receive a “Degraded” rating of 96%. This further demonstrates that the model utilizes different primary experts for prediction based on the dominant degradation mechanisms at various stages, thereby enabling the reverse classification of retired batteries at different degradation stages according to expert selection. Supplementary Figs. 20–22 and Supplementary Table 6 visualize the expert weight distributions across different SOH, with marked differences further demonstrating our proposed method’s novelty and simplicity of AMDP module that can classify the degradation mode using a few electric measurements.

Figure 4e examines post-classification computation accuracies of degradation trajectories. iMOE achieves MAPEs of 0.47%, 0.52%, and 0.47% for batteries retired at 95%, 85%, and 75% SOH, respectively. While predictions remain robust for 95%-85% SOH, samples at 75% SOH show maximum outliers, likely due to coupling of accumulated and other unobserved degradation mechanisms.

In Fig. 4f, controlled experiments with noise addition validate feature importance and expert network specialization. By individually applying Gaussian noise (σ=1) to each physical feature while holding other features at their original value, we observe elevated errors from baseline MAPE of 1.13% when features are perturbed with noise. Supplementary Figs. 23–24 present the performance analysis under conditions of missing values and reduced feature inputs. With the same amount of noise injections, the feature whose perturbation causes the largest decrease in degradation trajectory computation accuracy is regarded as the most important. It reveals that feature extracted from relaxation voltage that is reflective of internal resistance demonstrate greater noise resilience than those from capacity-voltage curves (particularly their mean values, after adding noise, the MAPE increased from 1.13% to 1.38%, representing a 22% relative increase, as these means directly reflect charge storage capacity per voltage increment).

The observed performance decrease caused by misrouting samples to expert networks not specialized for their specific degradation phases (e.g., late-stage samples erroneously routed to Expert Network 1 instead of 5) confirms the functional divergence among expert networks, enabling history-free computation of degradation mode. Supplementary Figs. 25–26 show reduced specialization when degradation trajectories are geometrically similar, while demonstrating that iMOE’s superior performance does not strictly rely on any single physical feature. Supplementary Fig. 27 and Supplementary Note 7 presents the SHAP and Spearman importance analysis for different features. We further analyze the model results after removing features with lower SHAP importance in Supplementary Fig. 28. The results indicate that, in certain scenarios, redundant features can be removed to maintain a balance between computational efficiency and accuracy.

In Fig. 4g, we investigate the contribution of the AMDP module to predict accuracy using an ablation experiment. When replacing AMDP with a simple linear layer while keeping other components unchanged, comparative analysis of the embedded representations (see Supplementary Note 8 for details) shows that while the ablated model can roughly distinguish degradation stages, it fails to capture subtle SOH variations within similar stages. This results in discontinuous embeddings and unreasonable trajectory predictions, highlighting AMDP’s critical role in enabling robust long-term predictions under history-free settings, Supplementary Fig. 29 directly shows the magnitudes of different cycling trend metrics. By decoding implicit semantics of expert weights, iMOE transforms from a prognostic tool into an interpretable decision engine, bypassing traditional sequential assessment workflows that require extensive data curation and model calibration in real-world deployment.

Uncertainty analysis
Under the assumption of available historical data and consistent second-life operational conditions, we extend our analysis to scenarios where partial historical data is accessible. While this significantly simplifies the prediction task, it does not align with the complex data composition of real-world applications, using polynomial fitting and MLP-based approaches to map historical trajectories to future degradation (see implementation details in Supplementary Note 9). In Fig. 5a, without requiring battery-specific mechanistic analysis, a simple polynomial regression and single-layer MLP achieved a 0.79% and a 0.93% MAPE, respectively. We demonstrate the iMOE’s capability in computing degradation trajectories using historical data regarding maximum capacity to validate its adaptability to data-accessible scenarios (see Supplementary Fig. 30 and Supplementary Note 10 for implementation details), With 10 historical maximum capacity points accessible, iMOE achieves an average MAPE of 0.53% when predicting the next 50 cycles. We stress that these historically-dependent methods become infeasible when historical data is unavailable in real-world settings, underscoring the practical limitations of such assumptions. To quantify component contributions of AMDP and FORNN, we designed two ablation studies: (1) replacing AMDP with a linear layer (iMOE-linear); and (2) substituting FORNN with a standard RNN (iMOE-woFO). After replacing the AMDP architecture with a linear layer directly, the overall MAPE increased from 0.95% to 1.19%, which aligns with the observation in Fig. 4e that the embedding vectors became indistinguishable between different degraded samples after removing the AMDP architecture. This demonstrates that the adaptive integration of predictions from different degradation modes contributes to robust and high-precision estimation. When replacing FORNN with a standard RNN, the model exhibited a significant decline in prediction performance for TPSL dataset. This further underscore the necessity of incorporating future load conditions in second-life scenarios, as future operating conditions often differ from those in their first-life service patterns.

Figure 5b evaluate the iMOE performance under limited training data access (data partitioning details are provided in Supplementary Note 11). When training data was reduced from 100% to 40%, the model performance maintained stable on UL dataset, achieving an average MAPE of 0.75% with a 0.56% standard deviation. However, a linear performance decrease was observed on TPSL dataset, with an average MAPE increasing to 3.86% with a 0.88% standard deviation. When training data was reduced to 20% (32 batteries in total, pruned training data size 5 MB), both datasets exhibited noticeable error and uncertainty increase, with MAPE reaching 1.07% for UL and 5.30% for TPSL. This training data access experiment indicates that learning future operating conditions’ impact on battery degradation trajectories requires higher data curation costs, whereas the UL dataset demonstrates relatively lower data access dependency. Despite this observed data access challenge, Supplementary Tables 7–9 compare baseline model performance under scarce data access, validating iMOE’s superior generalization capability in few-shot learning scenarios.

The decomposition of degradation subspaces and dominant mechanism prediction depend on the number of experts and TopK selection (see “Methods”). Figure 5c examines the impact of number of experts on the iMOE performance. The iMOE demonstrates relative insensitivity to hyperparameter modifications, across experiments with five distinct expert network quantities, the average MAPE was 0.98%. when employing 4 experts, the model showed comparatively poorer performance with an average MAPE of 1.03%, while the 6-expert configuration exhibited inferior stability with a standard deviation of 1.04%. After carefully balancing practical deployment requirements against performance metrics, we ultimately adopted a configuration with 5 experts and TopK = 2.

Different application scenarios have varying requirements for prediction horizons. In practical applications, it is desirable to predict degradation trajectories as far as possible while maintaining acceptable performance. As shown in Fig. 5d, we present results across different prediction lengths. When the prediction length is reduced, model performance improves, achieving an average MAPE of 0.80%. In most cases, model performance degrades with increasing prediction length. Nevertheless, even when predicting 150 future cycles from current-cycle data, the average MAPE remains below 1.50%, demonstrating iMOE’s robustness in long-term forecasting. Supplementary Tables 10–12 compare our method’s performance with baseline approaches under extended prediction horizons. The results show that all models exhibit performance degradation as prediction length increases. When predicting future 100-cycle degradation trajectories, PatchTST and Informer achieve average MAPEs of 8% and 7.98%, respectively, while iMOE maintains a lower average MAPE of 1.47%, enabling more accurate long-term predictions. We further analyzed the model’s predictive performance under unknown and varying historical operating conditions at different levels of deep degradation. As shown in Fig. 5e, across six SOH intervals from 80% to 50%, the model achieved consistently good prediction accuracy, with an average MAPE of 1.81% and a standard deviation of 0.16%. However, when SOH fell within the 55–50% range, the prediction accuracy decreased, with an average MAPE of 2.00%, reflecting the increased difficulty of modeling degradation trajectories when multiple degradation mechanisms interact at advanced aging stages.

Methods
Datapreparation
Inputsignals.Weconsidertwosignalsineachcycle:partialchargingprofilethatstartsatarandomvoltage(i.e.,arandominitialSOC)andendsatthecutoffvoltage.Relaxationvoltagecurverecordedimmediatelyaftercharging.LetI(t),V(t),andt denotetheinstantaneouscurrent,voltage,andtime,respectively.WetrackthebatteryvoltagefromV_{min},whichvariesacrosssamplesduetorandomSOC,uptothecutoffvoltageV_{end}.

Partialchargingcurve.Wedividetherange[V_{min},V_{end}]intoM equalintervals.Thepartialchargingcurveisrepresentedasthecumulativeinputchargeasafunctionofvoltage:

q = [q_0(V),q_1(V),...,q_M(V)], q(V) = \int_{t(V_{min})}^{t(V_{min}+i\Delta V)} I(t) dt (3)

In experiments, we record 50 segments at equal intervals (i.e., M=50), yielding a 50-dimensional sequence.

Relaxation voltage curve. To capture the relaxation behavior, we measure the open-circuit voltage v_k(t) for k=0,1,…,M at 30-minute intervals after charging concludes:

v = [V_0(t) V_1(t) ... V_M(t)] (4)

This sequence of length M+1 reflects how the polarized voltage recovers over time.

Feature engineering. From partial charging and relaxation voltage signals, we extract 12 features derived from physical considerations, including polarization over charge, relaxation slope, and voltage plateau transitions. All 12 features are normalized to [0,1] before entering the model. The final input to the iMOE consists of the partial charging curve q \in R^{50} and the feature vector F_i \in R^{12} derived from both the charging and relaxation profiles.

Model architecture
The proposed interpretable mixture of deep expertized learning network consists of two modules, Adaptive Multi-degradation Prediction (AMDP) module and a Future-Operation Recurrent Neural Network (FORNN), to include current-cycle degradation phenomena and prospective operating conditions that may arise in second-life applications.

Adaptive multi-degradation prediction (AMDP)
Formally, let F_i \in R^{12} be physics-informed features for the i-th sample, and q = [q_0(V),q_1(V),...,q_M(V)] be its partial charging curve. The AMDP model defines the expert networks, each denoted Expert_j, j=1,2,...,E. A degradation mode routing F_i \to g produces a probability weight g(F_i) across these experts:

g(F_i) = Softmax(TopK(Softmax(H(F_i), k))) (5)

where, H(F) \in R^E is a feed-forward transformation with a noise term:

H(F_i) = F_i W_g + \psi Softplus(F_i W_{noise}) (6)

where, W_g \in R^{12 \times E} and W_{noise} \in R^{12 \times E} are learned parameters, and \psi \in N(0,1) is drawn from a standard Gaussian. The top-k experts are selected and normalized by a Softmax function.

Retired batteries may undergo multiple degradation mechanisms, e.g., SEI formation, SEI thickening, loss of active material, and lithium plating, that can coexist or interact. To flexibly and specifically model these different pathways, AMDP module enables a MOE structure by fusing predictions from multiple expert submodels, each “specialized” in a certain degradation mode. Within the AMDP enabled MOE framework, an Expert is not confined to a single degradation mode; instead, it is represented as a weighted superposition of countably many implicitly defined “typical degradation mode”. Let the j-th Expert be denoted as:

Expert_j(q_i) = \sum_{\theta \in \Omega} w_j(\theta) \phi(q_i; \theta) (7)

with the latent degradation mode indexed by \theta, and w_j(\theta) is the weight that Expert_j assigns to corresponding degradation mode. The output is a degradation trend estimate under the j-th dominant. The final AMDP output for sample i is a weighted sum:

Trend_i = \sum_{j=1}^{E} g_j(F_i) Expert_j(q_i) : R^N \to R^L (8)

Where Trend_i \in R^{1 \times L} represents the preliminary degradation trend output by AMDP module, and L indicates length of the predicted future degradation trajectory. Expert_j denotes the predictor for a specific degradation mode, where g_j(F_i) is the weight for expert j. This step produces a latent degradation trend vector Trend_i for subsequent cycles, reflecting the dominant mechanism(s) underpinned by F_i. Hence, the model adaptively fuses multiple degradation mode subspaces into a short-term trend embedding for subsequent long-horizon computation.

Future-operation recurrent neural network (FORNN)
Second-life applications involve future load conditions that can alter degradation pace as well as the maximum available capacity in the current cycle. To include these influences, we employ a recurrent neural network that takes as input AMDP, as output Trend_i and a set of future load parameters:

C_i = [I_{charge}^{(1)}, I_{discharge}^{(1)}, T^{(1)}, ..., I_{charge}^{(L)}, I_{discharge}^{(L)}, T^{(L)}] (9)

where C_i \in R^{L \times 3}, I_{charge}^{(‘)}, I_{discharge}^{(‘)}, and T^{(‘)} represent the charge current, discharge current, and temperature in future cycle ‘. For each future load condition, its length is equal to L, and the length of the degradation trajectory we need to predict from AMDP: Trend_i. We concatenate these load parameters with the short-term trend:

X_i = Concat(Trend_i, C_i) (10)

The LSTM processes X_i \in R^{L \times 4} sequentially to predict the capacity trajectory \hat{S}_i over L future cycles:

\hat{S}_i = LSTM(X_i) (11)

By iterating cycle by cycle, FORNN accommodates a prediction horizon that the user chooses, merging the degradation modes signature from AMDP with the load sequence the battery will experience under the second-life complexities and uncertainties.

Model training
We train the AMDP and FORNN modules jointly by minimizing a loss function that balances trajectory accuracy with degradation router diversity. Let S_i be the ground-truth capacity trajectory of length L for sample i, the true capacity label represents the maximum discharge capacity of that cycle. Let \hat{S}_i be the corresponding model output. The trajectory fidelity objective is

‘_{traj} = \frac{1}{N} \sum_{i=1}^{N} ||\hat{S}_i - S_i||^2 (12)

where the summation is over the training set of size N.

Because MOE can exhibit “expert collapse” (where one expert is used for all samples), we include a regularization term to encourage usage across experts. For a mini-batch, the router weight assigned to expert j is averaged as

A_j = \sum_{i \in Batch} g_j(F_i) (13)

We encourage these averages to avoid large variation by penalizing their coefficient of variation:

‘_{router} = Var(A_1, A_2, ..., A_E) / Mean(A_1, A_2, ..., A_E)^2 + \epsilon (14)

where \epsilon is a constant set to 10, Var and Mean are the variance and mean operator, respectively. The total loss is:

‘ = \alpha ‘_{traj} + \beta ‘_{router} (15)

with \alpha and \beta controlling the trade-off between trajectory accuracy and expert diversity. We set \alpha = 0.75 and \beta = 0.25 in our experiments.

Evaluation metrics
Root mean square error (RMSE)

RMSE = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (\hat{S}_i - S_i)^2} (16)

Mean absolute percentage error (MAPE)

MAPE = \frac{1}{N} \sum_{i=1}^{N} |\frac{\hat{S}_i - S_i}{S_i}| \times 100\% (17)

Coefficient of determination (R^2)

R^2 = 1 - \frac{\sum_{i=1}^{N} (\hat{S}_i - S_i)^2}{\sum_{i=1}^{N} (S_i - \bar{S})^2} (18)

where, N is the total number of samples (i.e., cycles), \hat{S}_i and S_i are the computed and true capacity, respectively. \bar{S} is the sample mean of the ground-truth capacities in the computation horizon.

Data availability
Raw data can be found here23,43,49. Source data are provided as a Source Data file. Source data are provided with this paper.

Code availability
The code used in this work has been deposited at the Zenodo repository here59.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Dear Author,\nThanks for sharing your research work. This reviewer has carefully considered the contents, quality, and extent, providing\nthe following feedback.\nThe state-of-the-art analysis is weak. The use of historical data and some examples of existing works are discussed, but not\nstrong enough to formulate the research questions.\nSimilarly, the proposed PIMOE, AMDP, and FORNNs are not justified, not enough existing works are discussed, and no\nmotivation is available behind choosing this path.\nWhat is this 3rd type of battery mentioned as NMC+NCA? What is the cathode composition?\nIn Supplementary Figure 8a/b, there are some red lines referring to low SoH that existed since the beginning. How is it\npossible?\nWhy does Supplementary Figure 10 show spikes appear after every 100 cycles, probably referring to rest periods, while\nothers do not exhibit such?\nSupplementary figures 11-14 show time-to-time capacity recovery, which is strange. Could the authors explain this?\nThere are many figures presented in the supplementary document, but this reviewer does not find the significance. Rather,\nthey could be grouped/tabled or less sampled/categorized. It is more important to state the findings from these curves. Could\nthe authors motivate the choice behind?\nThere is confusion about the availability of the selected parameters from noisy, missing, or partial curves. It is unclear in the\nmanuscript if not absent.\nMany acronyms are not explained at their first use. Please consider the documents separately.\nSupplementary tables should be placed at the beginning of the supplementary file.\nThe UL and TPSL datasets are referred to literature, meaning the PIMOE framework is the only novel contribution in this\nresearch. The work seems to have weaker novelty. Could the authors comment on this or guide to the novel contributions?\nWhy PatchTST and Informer have bad predictions with TPSL datasets?\nIn the main manuscript, in the early section of page 23, the authors tried to link the aging mechanisms to the expert networks.\nWhat is the validity of such a hypothesis?\nThe research is regarding PI-physics informed parameters, but there is no discussion of these parameters on the impact of\naging and how they are relevant to be called physical parameters. They are driven by electrical tests and more statistical\ndata.\nWhat is the justification for Figure 4d, where the SEI and Li-plating are plotted separately, while these events could happen\nin the same cell, depending on the activated mechanisms (vs conditions).", "reviewer_2": "This is an interesting paper addressing the challenge of predicting battery degradation trajectories in second-life\napplications using a novel physics-informed mixture of experts (PIMOE) architecture. The manuscript is generally well-\nwritten, and the proposed approach is innovative, combining domain knowledge with machine learning for interpretable\npredictions. The results are compelling and relevant to both academia and industry.\nHowever, while the paper shows promise, there are several important issues that must be addressed before it can be\nconsidered for publication.\n1) The approach assumes that future load conditions are known in advance, which is a strong assumption in real-world\nsecond-life scenarios. Although this is briefly acknowledged in the discussion, a more thorough examination of the\nlimitations and implications of this assumption is warranted. Furthermore, scenarios involving uncertain or stochastic future\nuse should be considered or at least simulated.\n2) While the model is claimed to be \"physics-informed,\" its interpretability largely relies on correlations extracted from\ncapacity-voltage and relaxation curves. These features, while derived from empirical electrochemical knowledge, do not\ntruly integrate physics-based modeling such as differential equations or parameter estimation from equivalent circuit models.\nA clearer distinction between physics-informed and statistically correlated is needed.\n3) The paper compares its approach to two baselines (Informer and PatchTST), which are relevant time-series models.\nHowever, additional baselines from battery-specific modeling domains—such as hybrid physics-informed neural networks or\ndirect state-of-health estimation models—would strengthen the empirical evidence. Also, it is unclear whether the baselines\nwere retrained under the same conditions, which affects reproducibility.\n4) The following two recent and highly relevant works in the field of imitation learning-based degradation modeling are not\ncited and must be referenced:\n• Pozzi, A., Incremona, A. & Toti, D. \"Imitation learning-driven approximation of stochastic control models\". Applied\nIntelligence, 55, 838 (2025). https://doi.org/10.1007/s10489-025-06704-x\n• Pozzi, A., Incremona, A., Toti, D. \"Neural Network-Based Imitation Learning for Approximating Stochastic Battery\nManagement Systems,\" IEEE Access, vol. 13, pp. 71041-71052, 2025. https://doi.org/10.1109/ACCESS.2025.3563300\nThese works are directly relevant to the idea of modeling stochastic system dynamics using data-driven expert systems and\nshould be included for context and completeness.\n5) The current model provides deterministic predictions. In safety-critical applications like battery reuse, it is essential to\nassess the uncertainty or confidence bounds of the predicted degradation trajectories. The paper mentions this briefly in the\ndiscussion but does not quantify it in the results. A demonstration or discussion of uncertainty-aware predictions (e.g., via\nensembles or Bayesian methods) is encouraged.", "reviewer_3": "This a well written paper on a general topic of interest to the battery community. I can recommend publication if the authors\naddress the following comments.\n1) Spearman coefficient & SHAP values\nThe authors rely on their physics-informed router + experts structure, but it would strengthen the paper to validate feature\nimportance explicitly:\n• A simple Spearman correlation analysis between input features and outputs (SOH, degradation modes) could confirm\nmonotonic relationships and highlight redundancies among the 12 features.\n• A SHAP (Shapley Additive Explanations) analysis on the trained neural model could provide insight into which features\nthe network relies on most. This would help connect the “physics-informed” explanation with the actual feature usage inside\nthe NN.\n• Suggestion: Add a supplementary analysis of feature importance (Spearman + SHAP) to improve interpretability and\nconfidence in the model.\n2) Reduction in Number of features\nThe manuscript states that the input consists of 12 features derived from partial charge + relaxation voltage profiles. While 12\nis not excessive for modern ML, some may be correlated or redundant.\n• Suggestion: Discuss whether feature selection or dimensionality reduction (e.g., PCA, correlation filtering, or SHAP\npruning) could simplify the model without significant loss in accuracy. This also makes the physics-informed routing more\ninterpretable.\n3) Effect of random SOC starting point\nThe work does not discuss how the initial SOC (random starting point of partial charging) influences the extracted features.\nIn practice, this can strongly affect the relaxation trajectory and hence the prediction accuracy.\n• Ignoring this factor may lead to optimistic performance compared to real-world deployment, where SOC is not always\ncontrolled.\n• Suggestion: Even if beyond the current study’s scope, add a short discussion of this limitation and how future work might\naddress it (e.g., SOC normalization, SOC-aware routers). This would show awareness of practical challenges.\n4) Aging trajectory\nOne of the key assumptions in the developed model is that the aging trajectory can be divided into three distinct zones: SEI\nformation, SEI thickening, and lithium plating. However, this assumption lacks sufficient rationale or supporting evidence for\nthe aging datasets analyzed. The authors should elaborate on this degradation classification, ideally providing experimental\nor model-based justification. Furthermore, it would be helpful if the authors discuss the sensitivity of their methodology/result\nto the choice of degradation mechanism. For instance, how would the approach change if loss of active material were\nconsidered as one of the dominant degradation pathways?\n5) Dataset\nThe authors should make it clearer that the aging datasets used in this work are obtained from the literature and provide the\ncorresponding references.\n6) Typos and grammatical mistakes\n• Line 56: relavent → relevant\n• Line 65: Larbor → labor\n• Line 65: awared → aware\n• Line 79: the demanding → demanding\n• Line 95: implication → implication\n• Line 123: \"uncertain secondary use condition\" → \"uncertain secondary use conditions\"\n• Line 279-280: \"a interpretability\" → \"an interpretability\"\n• Line 389: exhitbit → exhibit\n• Line 399: extenstive → extensive", "reviewer_4": "Authors describe a method for estimation of Li-ion cell capacity and combine it with predictive model that allows forecasting\nof future capacity degradation trends. Authors claim that the proposed method allows history-free estimation of State-of\nHealth (SOH) of the battery hence eliminating a need for historical usage records or extensive lab tests. Elimination of data\nrequirements is a very valuable technical goal since unavailability of historical usage or test data is a biggest obstacle for\naccurate estimation of SOH.\nWhile authors claim history-free method, the training process used for training of the estimation and prediction components\nof the proposed model relies on extensive test data collected offline. I think that authors need to revise some of their claims\nrelated to this discrepancy. That training data is the missing ‘history’. It is very well known that with sufficiently rich\ndegradation data collected offline one can train a data-driven model to achieve pretty much any desired accuracy.\nI recommend the authors to put a bit more effort in connecting the proposed modeling method to physics of Li cell\ndegradation. Without this connection between selected ML features and vaguely mentioned degradation modes the paper\ndoes not stand out from a long list of data-driven modeling papers for SOH estimation.\nSome more technical comments:\n- Line 130: Please, explain what the Trend vector is exactly. Is it a vector of capacity or SOH values over one cycle vs time?\nSeveral cycles?\n- It would be very valuable to a reader to see an explanation of why/how selected AMDP features contain information about\nSOH of the cell. Without this info a reader stays uneducated about why the proposed method should work and paper reads\njust like yet another ‘we stacked several MLMs to achieve good KPIs’ paper.\n- Figure 2c and subsequent figures show fluctuations in measured capacity by double-digit percent. How is this possible?\nHow was exactly capacity measured in experiments?\n- There is a large drop in capacity from First to Second life on Figure 2c. Why is there such a drop?\n- Figure 3b, and 2c show not only drops but also jumps up to 20% in measured capacity. Can you, please provide\nexplanation of what is going on there? This is not physical behavior for NCM/NCA cells.\n- Line 199-200. Capacity knee at EOL of a cell, ‘third phase’, is not always related to Li-plating.\n- Line 207. Again, not clear why measured capacity jumps around and why the model follows this trend.\n- Line 243. I think the conclusion about use of higher SOC range is a bit ML model development centric. A physical reason\ncould be that the snippet of data starting at SOC > 50% simply does not contain enough information about OCV features that\ncan help the model to distinguish between components of degradation such Li loss vs AML.\n- Figure 3: What is the prediction horizon? Are we predicting x-number of cycles ahead or the entire second life until certain\nSOH?\n- Line 375, Discussion: PIMOE needs reach experimental offline test data to train on according to the described\nmethodology. Please state that explicitly.\n- Line 461, equation 5: What is exactly the ‘basis response’ in equation 5? What are exactly the degradation modes?\n- I think if degradation modes are described that may help to explain ‘physics-informed’ part of the title. Without this\nexplanation I do not see any ‘physical’ components in the proposed model.\n- Equation 6: what are the units of ‘Trend’? Is it in Ah or normalized SOH value? Is it a vector of values vs time or delta V? L\nis the number of samples in a cycle or many cycles?\n- Equation 7: Future load parameters, I(t), T(t). Are these time vectors of dimension N?\n- Equation 10: How are true capacity labels generated?\nNoticed typos:\n- Line 89, ‘interatively’\n- Line 100, ‘extends’\n- Line 170, ‘pecices’\n- Line 389, ‘valuse’"}, "author_response": "Response to Reviewers\nResponse to Reviewer #1\nDear Reviewer,\nThe authors respond to your suggestions point by point and carefully revise the manuscript and\nsupplementary information, where you can find corresponding revisions using “track changes”\nmode. Changes are colored in blue in revised clean version. The authors truly hope that the responses\nappropriately address your justified concerns, and that revised manuscript meets your expectations.\n\nComment 1\nThe state-of-the-art analysis is weak. The use of historical data and some examples of existing works\nare discussed, but not strong enough to formulate the research questions.\nResponse to Comment 1\nThank you for the comment. The authors have revised the paper in the following aspects:\n The authors elaborated on existing methods and literature, providing a detailed discussion of\nthe limitations of current studies that rely on historical data and fixed future operating\nconditions for predicting degradation trajectories of retired batteries.\n The authors provided a detailed outlook on future work in the Discussion section.\nThe authors have made modifications in the introduction.\n“Degradation trajectory is a widely adopted indicator for battery degradation characterization,\nwhich can be critical to the safety performance during extended reusing or repurposing process, i..e,\nsecond-life of the retired batteries1–4. For example, retired batteries undergo degradation trajectory\nevaluation before second-life use based on the cell, module or pack-level application requirements5,6.\nConventional non-data-driven methods relying on destructive disassembly or full-cycle capacity\ntests7,8, inflicting extra damage or labor investments. Notably, the economic feasibility of these state\nmeasurement methods of retired batteries are impractical due to prohibitive time, momentary, and\nenvironmental costs at scale 9–12. Data-driven methods have demonstrate promises in predicting\ndegradation trajectories using non-destructive electrical signal data, but they typically require 5%\n-40% of full lifecycle historical data13–15.Moreover, uncertain second-life use conditions can\nsignificantly differ from those in first-life, rendering conventional battery degradation trajectory\nprediction models ineffective, given that the data used for model training were built on constant-\ncondition tests or typical dynamical cycling tests while the degradation trajectory is highly path-\ndependent16–18. Lu et al. leveraged at least one full cycle of capacity-voltage curves to predict\ndegradation trajectories under uncertain future operating scenarios by learning the relationship\nbetween use condition and battery capacity16. Yet, the method encounters limitations due to the\nstochasticity in state of charge (SOC) of retired batteries and the demanding requirement for full\ndischarge-charge data. Moreover, the unavailability of historical cycling data of retired batteries\ncomplicates the tracing of initial conditions of degradations, such as impedance rise, loss of lithium\ninventory, and loss of active material 19–22, challenging degradation trajectory prediction\neffectiveness. At the policy level, global initiatives such as the Battery Data Genome Initiative23 and\nthe European Commission’s July 4, 2025 Delegated Regulation on battery recycling24 aim to\nenhance data standardization, traceability, and material recovery across the battery lifecycle.\nHowever, the policy advances still face practical challenges due to frequent ownership change and\nuncertain second-life use conditions retired batteries, emphasizing the need for degradation\nprediction frameworks that rely solely on field-accessible data and adapt to diverse operational\nenvironments.\nAdditionally, due to highly non-linear degradation of second-life batteries, incorporation of physics\nknowledge into models has gained increased attention. Recent advances in sensory-based\nmeasurements include X-ray imaging25, electrochemical impedance, optical fiber sensing26,\nacoustic sensing27, partial charging28, which serves as the data-input of the data-driven methods.\nNevertheless, most sensing techniques remain at laboratory stage and are invasive26. Meanwhile,\npurely data-driven methods struggle to capture internal physical information of retired batteries29.\nGiven the understanding of internal physical state is particularly critical for second-life\napplications with safety-sensitive considerations, the challenge of the \"black-box\" nature and lack\nof interpretability in data-driven methods for battery state prediction should be addressed. Even if\nprogresses have been made in physics-informed neural networks, the promise lies in additional and\nexplicit integration of physical laws into feature engineering process, neural network loss functions,\nand transferability metrics. However, the complex physical constraints involve numerous\nparameters and prior physical understanding from the modeling of retired batteries, which is still\nhardly available post-retirement5. Tao et al. employed physics-informed machine learning to achieve\nfull degradation trajectory prediction using early-cycle data, reducing data requirements, while still\nrelying on historical data30 . In recent years, mixture-of-experts (MOE) architecture, a core\ncomponent in large language models, has demonstrated notable performance in learning\nheterogeneous data representations. By decomposing complex tasks into sub-tasks handled by\n“expert” modules with specialized task knowledge, MOE captures multi-level and nonlinear feature\nrepresentations. The ideal of decomposing tasks into multiple subtasks naturally aligns with coupled\ndegradation mechanisms of batteries. However, in absence of physical interpretability, the learning\noutcomes of model are confined to purely statistical correlations and fail to reflect true physical\nmechanisms underlying degradation processes, particularly with diverse cathode chemistries31,\nhistorical usages32, and uncertain second-life operating conditions16. Incorporating physical\nknowledge into the MOE architecture anchors data-driven learning outcomes to typical\nelectrochemical processes, thereby enhancing model interpretability and enabling mechanism-\naware degradation trajectory prediction.”\nThe authors have made modifications in the discussion.\n“ PIMOE model shows great potential to transform the battery reusing and recycling industry\nlandscape by reducing the need of manual-assisted testing approach to an automated data-driven\ndecision support system. However, it must be acknowledged that PIMOE approach primarily relies\non electrochemical characteristics (such as voltage-capacity relationships and relaxation voltage)\nfor macroscopic diagnostics. While extensive validation confirms the physical understanding of\nthese features13,33, they essentially remain statistical correlations instead of physical casualty. It is\nrecommended that future work should focus on non-invasive in-situ sensing signals 26,27,34, such as\nvibration sensing, strain sensing, ultrasonic signals, fiber optic sensing to enhance internal state\nobservation for a physics-informed modeling, instead of physically interpretable presented in this\nwork. Building a \"physics-statistics\" fusion framework has the potential to improve the\ninterpretability of PIMOE 29, for example, parameters from battery electrochemical or equivalent\ncircuit models could be updated online as an intermediate physical interpretation layer for\ndegradation trajectory prediction. Integrating primary governing equations of battery modeling\nwith data-driven methods would also provide physical interpretability35.\nWith acknowledging the assumption that future use conditions can be assumed, it is reasonable for\nverifying the effectiveness of PIMOE under predefined uncertain scenarios with considerable\nuncertainties, specifically, consistent operating histories, deeply aged batteries with unknown prior\nuse and varying second-life conditions with 95 use conditions being validated. However, despite\npractical applications such as home energy storage and integrated photovoltaic-storage systems\noften have pre-defined operational plans, the actual operating conditions of battery energy storage\nsystems remain even more uncertain and are difficult to predict 36. Although this work has discussed\nthe importance of introducing future use conditions in the FORNN structure, the analysis of the\nuncertainty of future use conditions needs to be extended to highly random and time-varying use\nscenarios. Thus, future work could focus on introducing load conditions at finer temporal\nresolutions to ensure the validity of predicted use conditions of retired batteries. Another aspect is\nthat random initial SOC is an important factor in the data availability of retired batteries. Although\nthe effectiveness of PIMOE in predicting degradation trajectory under different initial SOC\nconditions has been validated, the initial SOC of retired batteries is more uncontrollable due to the\ntime-consuming nature of acquisition such information. Future work suggests considering SOC\nnormalization during the feature extraction phase or introducing an SOC-aware routing mechanism\nin the model, to better handle battery degradation difference under different SOC states. Although\nthe paper has considered three very different application conditions, namely UL and TPSL-fixed,\ngiven that the second-life application of the battery is safety-critical, more extreme conditions need\nto be investigated. This is because the robustness of the dataset to non-steady-state use conditions,\nsuch as the battery with thermal runaway or internal short circuits, has not been included.\nConsidering the bias of the training data is crucial, and it is necessary to assess the uncertainty and\nconfidence intervals of the predicted degradation trajectories, as UL and TPSL data hardly cover\nall environmental fluctuations under actual service conditions, such as thermal non-uniformity at\nthe battery pack level. Future work is suggested to collect real-world variables that include different\nmanufacturers, degradation mechanisms, and cell-to-cell variations.\n\nComment 2\nSimilarly, the proposed PIMOE, AMDP, and FORNNs are not justified, not enough existing works\nare discussed, and no motivation is available behind choosing this path.\nResponse to Comment 2\nThank you for the comment. The authors have revised the paper in the following aspects:\n The PIMOE framework includes two core modules, AMDP and FORNN. The authors have\nrevised the Introduction section to emphasize the importance of physical interpretability and\nnecessity of incorporating future operating conditions, providing a more comprehensive\nexplanation of our motivation.\n“ Degradation trajectory is a widely adopted indicator for battery degradation characterization,\nwhich can be critical to the safety performance during extended reusing or repurposing process, i..e,\nsecond-life of the retired batteries1–4. For example, retired batteries undergo degradation trajectory\nevaluation before second-life use based on the cell, module or pack-level application requirements5,6.\nConventional non-data-driven methods relying on destructive disassembly or full-cycle capacity\ntests7,8, inflicting extra damage or labor investments. Notably, the economic feasibility of these state\nmeasurement methods of retired batteries are impractical due to prohibitive time, momentary, and\nenvironmental costs at scale 9–12. Data-driven methods have demonstrate promises in predicting\ndegradation trajectories using non-destructive electrical signal data, but they typically require 5%\n-40% of full lifecycle historical data13–15.Moreover, uncertain second-life use conditions can\nsignificantly differ from those in first-life, rendering conventional battery degradation trajectory\nprediction models ineffective, given that the data used for model training were built on constant-\ncondition tests or typical dynamical cycling tests while the degradation trajectory is highly path-\ndependent16–18. Lu et al. leveraged at least one full cycle of capacity-voltage curves to predict\ndegradation trajectories under uncertain future operating scenarios by learning the relationship\nbetween use condition and battery capacity16. Yet, the method encounters limitations due to the\nstochasticity in state of charge (SOC) of retired batteries and the demanding requirement for full\ndischarge-charge data. Moreover, the unavailability of historical cycling data of retired batteries\ncomplicates the tracing of initial conditions of degradations, such as impedance rise, loss of lithium\ninventory, and loss of active material 19–22, challenging degradation trajectory prediction\neffectiveness. At the policy level, global initiatives such as the Battery Data Genome Initiative23 and\nthe European Commission’s July 4, 2025 Delegated Regulation on battery recycling24 aim to\nenhance data standardization, traceability, and material recovery across the battery lifecycle.\nHowever, the policy advances still face practical challenges due to frequent ownership change and\nuncertain second-life use conditions retired batteries, emphasizing the need for degradation\nprediction frameworks that rely solely on field-accessible data and adapt to diverse operational\nenvironments.\nAdditionally, due to highly non-linear degradation of second-life batteries, incorporation of physics\nknowledge into models has gained increased attention. Recent advances in sensory-based\nmeasurements include X-ray imaging25, electrochemical impedance, optical fiber sensing26,\nacoustic sensing27, partial charging28, which serves as the data-input of the data-driven methods.\nNevertheless, most sensing techniques remain at laboratory stage and are invasive26. Meanwhile,\npurely data-driven methods struggle to capture internal physical information of retired batteries29.\nGiven the understanding of internal physical state is particularly critical for second-life\napplications with safety-sensitive considerations, the challenge of the \"black-box\" nature and lack\nof interpretability in data-driven methods for battery state prediction should be addressed. Even if\nprogresses have been made in physics-informed neural networks, the promise lies in additional and\nexplicit integration of physical laws into feature engineering process, neural network loss functions,\nand transferability metrics. However, the complex physical constraints involve numerous\nparameters and prior physical understanding from the modeling of retired batteries, which is still\nhardly available post-retirement5. Tao et al. employed physics-informed machine learning to achieve\nfull degradation trajectory prediction using early-cycle data, reducing data requirements, while still\nrelying on historical data30 . In recent years, mixture-of-experts (MOE) architecture, a core\ncomponent in large language models, has demonstrated notable performance in learning\nheterogeneous data representations. By decomposing complex tasks into sub-tasks handled by\n“expert” modules with specialized task knowledge, MOE captures multi-level and nonlinear feature\nrepresentations. The ideal of decomposing tasks into multiple subtasks naturally aligns with coupled\ndegradation mechanisms of batteries. However, in absence of physical interpretability, the learning\noutcomes of model are confined to purely statistical correlations and fail to reflect true physical\nmechanisms underlying degradation processes, particularly with diverse cathode chemistries31,\nhistorical usages32, and uncertain second-life operating conditions16. Incorporating physical\nknowledge into the MOE architecture anchors data-driven learning outcomes to typical\nelectrochemical processes, thereby enhancing model interpretability and enabling mechanism-\naware degradation trajectory prediction.”\n\nComment 3\nWhat is this 3rd type of battery mentioned as NMC+NCA? What is the cathode composition?\nResponse to Comment 3\nThank you for the comment. In this paper, \"NCM+NCA\" refers to a hybrid cathode material system\ncomposed of 42 (±3) wt.% Li(NiCoMn)O₂ blended with 58 (±3) wt.% Li(NiCoAl)O₂, with graphite\nused as the anode material. The authors have revised the paper in the following aspects:\n The authors have provided a more detailed explanation of the battery material composition and\ncharacteristics in the main text\n The authors have revised Supplementary Tables 1 and 2 avoid potential ambiguity.\nWe have made the following modifications to the article content:\n“The UL dataset comprises batteries tested under different use conditions, with each batch cycled\nunder consistent conditions throughout their entire lifespan until EOL, comprising three batches\ntotaling 130 commercial 18650 cells. Batch 1 consists of LiNi Co Al O positive electrode\n0.86 0.11 0.03 2\n(NCA battery) with 3500mAh nominal capacity and cutoff voltages of 2.65–4.2 V. Batch 2 contains\nLiNi Co Mn O positive electrode (NCM battery) with 3500mAh nominal capacity and cutoff\n0.83 0.11 0.07 2\nvoltages of 2.5–4.2 V. Batch 3 includes 42 (3) wt.% Li(NiCoMn)O blended with 58 (3) wt.%\n2\nvoltages of 2.5–4.2 V. All cells underwent cycling in thermal chambers under three temperatures\n(25°C, 35°C, 45°C) with variable charge rates (from 0.25C to 4C) and fixed 1C discharge rate.\nEach battery experienced identical operational profiles throughout its full lifecycle until reaching\nEOL (see Supplementary Table 1 for details). Degradation trajectories in Fig. 2a demonstrate\ncycling variability, while Supplementary Fig. 28 illustrates significant remaining useful life (RUL)\ndifferences among batteries.\n”\nModifications have been made in the Supplementary Table 1 and 2:\nWorking conditions Nomina\nCut-off\nMaterial l Numbe\nCharg Discharg Temperatur voltag\nCell types capacit r of\ne e e e\ny cells\nC rate C rate (°C) (V)\n(Ah)\nNCA Li0.86Ni0.86Co0.11Al0.0 0.5 1 25 3.5 2.65- 66\n0.5 1 45\n1 1 25\n0.25 1 25\n42 (3) wt.% 0.5 1 25\nNCM+NC Li(NiCoMn)O2 blended 0.5 2 25 2.5–\n2.5 9\nA with 58 (3) wt.% 4.2\n0.5 4 25\nLi(NiCoAl)O2/ Graphit\n0.5 1 25\nLi0.86Ni0.86Co0.11Al0.0\nNCM 0.5 1 35 3.5 2.5-4.2 55\n0.5 1 45\n\nComment 4\nIn Supplementary Figure 8a/b, there are some red lines referring to low SoH that existed since the\nbeginning. How is it possible?\nResponse to Comment 4\nThank you for the comment. Some of the red lines indicate discrepancies in the early stage of battery\ndegradation, reflecting relatively large deviations between predicted and true values during the\ninitial phase—rather than implying low initial state of health in the battery's label. The authors have\nrevised the paper in the following aspects:\n The authors have revised all the figures to ensure clear and unambiguous presentation.\nModifications have been made in the Supplementary Figure 8-10:\n“Predicting the degradation trajectory of arbitrary NCM material battery samples throughout their\nfull lifecycle. Using a sliding window approach for data extraction and prediction (see\nSupplementary Note 1 for details), the method predicts the next 50 cycles for individual batteries\nbased solely on partial data from the current cycle. (a)-(c) present prediction results under different\noperating conditions. The solid black line represents the true SOH values, while the color scheme\ndenotes the predicted values for the health states of different samples used as input. (Naming\nconventions are specified in Supplementary Table 1).”\n\nComment 5\nWhy does Supplementary Figure 10 show spikes appear after every 100 cycles, probably referring\nto rest periods, while others do not exhibit such?\nResponse to Comment 5\nThank you for the comment. Battery with NCM+NCA materials underwent additional\nelectrochemical impedance spectroscopy tests during the capacity degradation experiment. The tests\nwere conducted at full charge state within a frequency range of 10 kHz to 0.01 Hz with a sinusoidal\namplitude of 250 mA. Additionally, a 60-minute open-circuit voltage rest period was implemented\nprior to each impedance test. The rest of the period and testing procedures resulted in a slight\ncapacity recovery of the battery. The authors retained this cycle in the experiment as such transient\ncapacity recovery phenomena are commonly observed in real-world scenarios, even though\nexcluding it might have further improved prediction accuracy.\nThe authors have revised the paper in the following aspects:\n The authors made the following modifications to the Supplementary Table 1:\nThe authors have made the following modifications to the Supplementary Table 1:\n“The table presents the material types, operating conditions, nominal capacities, cutoff voltages,\nand number of batteries used in our experimental dataset. All batteries in this study underwent\nidentical load cycling throughout their full lifecycle until aging. The dataset is designated as the UL\nDataset33, and we adhere to the naming convention established by the data providers. For instance,\nthe notation \"NCA-45-05-1\" refers to an NCA battery tested at 45°C with 0.5C CCCV charging and\n1C discharging conditions, with similar naming logic applying to other cases. For the NCM + NCA\nbattery, the electrochemical impedance is conducted every 50 cycles at full charge in a range of 10\nkHz to 0.01 Hz (6 data points per decade of frequency) with a sinusoidal amplitude of 250 mA. 60\nmin are set at the open circuit voltage before the electrochemical impedance tests. The UL Dataset\nencompasses 3 material types, 11 operating conditions, and 130 batteries in total. Notably, for each\nsample, we utilize only the current cycle's data without requiring any historical cycle information.”\n\nComment 6\nSupplementary figures 11-14 show time-to-time capacity recovery, which is strange. Could the\nauthors explain this?\nResponse to Comment 6\nThank you for the comment. The TPSL dataset incorporates 56 types of randomized charge-\ndischarge protocols to simulate diverse secondary-use conditions with high complexity and\nstochasticity. These varying protocols result in significant differences in the maximum available\ncapacity even for the same battery across consecutive cycles. This inherent variability invalidates\nmethods that assume consistent future operating conditions (e.g., classical time-series models). In\nreal-world secondary applications of retired batteries, future load profiles are often unpredictable,\nfurther highlighting the complexity of our task. The authors acknowledge that without further\nclarification, this aspect may cause confusion among readers.\nThe authors have revised the paper in the following aspects:\n The authors made the following modifications to the article content:\n The authors have made the following modifications to the Supplementary Table 2:\nThe authors have made the following modifications to the article content:\n“Fig. 2c shows degradation trajectories under uncertain use conditions. In the TPSL-fixed dataset,\ndifferent magnitudes of charge and discharge currents lead to vastly different degradation\ntrajectories. Meanwhile, in the TPSL-Random dataset, due to variations in the magnitude of the\ncharging current, the difference in the maximum discharge capacity between adjacent cycles can\neven exceed 0.3 Ah, disproving the assumption of identical operational history for prediction\nreliability.”\nThe authors have made the following modifications to the Supplementary Table 2:\n“The table presents the materials, operating conditions, nominal capacities, cutoff voltages, and\nnumber of batteries used in our experimental dataset. After the initial 20 cycles, the load conditions\nwere modified for these batteries, with 22 batteries undergoing different constant operating\ncondition cycles and 55 batteries subjected to random operating condition cycles. The dataset is\ndesignated as the TPSL Dataset16.\nIn this study, \"TPSL-Random\" refers to batteries operating under random conditions during their\nsecond-life usage, while \"TPSL-Fixed\" indicates batteries operating under fixed conditions during\ntheir second-life usage. The UL Dataset encompasses a total of 66 second-life operating conditions\n(55 random and 11 fixed) across 77 batteries. In the TPSL-Fixed dataset, batteries under different\nfixed conditions exhibited significantly divergent degradation trajectories. In the TPSL-Random\ndataset, the random operating conditions resulted in notable maximum capacity differences even\nbetween adjacent cycles. This further underscores the necessity of incorporating future\noperating conditions as conditional input. Notably, for each sample, we utilize only the current cycle's data\nwithout requiring any historical cycle information. “\n\nComment 7\nThere are many figures presented in the supplementary document, but this reviewer does not find\nthe significance. Rather, they could be grouped/tabled or less sampled/categorized. It is more\nimportant to state the findings from these curves. Could the authors motivate the choice behind?\nResponse to Comment 7\nThank you for the comment. The authors have revised the paper in the following aspects:\n some figures have been grouped and presented in table format.\n\nComment 8\nThere is confusion about the availability of the selected parameters from noisy, missing, or partial\ncurves. It is unclear in the manuscript if not absent.\nResponse to Comment 8\nThank you for the comment. The authors acknowledge that in reality, noisy, incomplete, or partially\nmissing curves are very common and represent an urgent issue that needs to be considered.\nTherefore, in Supplementary Figure 27, The authors have already evaluated the robustness of the\nproposed method by adding noise to the extracted physical features. However, your suggestion is\ncrucial evaluating only the physical feature with added noise is insufficient.\nThe authors have revised the paper in the following aspects:\n The authors have made the following modifications to Supplementary Figure 23 to evaluate\nthe model performance under conditions of incomplete intervals and noisy data.\nThe authors have made the following modifications to the Supplementary Note 2:\n“ Supplementary Figure 23. Analysis of the Impact of Sensor Noise and Missing Values on Results\nIn real-world applications, sensor sampling errors are inevitable. This section aims to investigate\nthe impact of sensor sampling errors on estimation results. Specifically, we applied three levels of\nrandom Gaussian noise to partial charging curves and relaxation voltages extracted from the field.\nAs shown in Figure (a), different colors represent varying magnitudes of Gaussian noise. On the\nselected dataset, as the sampling error increases, the model's error also increases linearly.\nIn extreme cases, incomplete data sampling or missing values may occur. We also considered the\nimpact of such scenarios on the results. Specifically, we randomly removed 20%, 40% of the\nsampling points, with missing values handled using simple linear interpolation. As shown in Figure\n(b), different colors represent different levels of missing data. On the selected dataset, as the amount\nof missing data increases, the model's error also rises. “\n\nComment 9\nMany acronyms are not explained at their first use. Please consider the documents separately.\nResponse to Comment 9\nThank you for your comment. The authors have revised the paper in the following aspects:\n The authors have double-checked and revised the explanations of acronyms upon their first\nappearance.\n\nComment 10\nSupplementary tables should be placed at the beginning of the supplementary file.\nResponse to Comment 10\nThank you for the comment. The authors have placed all supplementary tables at the beginning of\nthe supplementary file to ensure a more intuitive presentation of the supplementary file.\n\nComment 11\nThe UL and TPSL datasets are referred to literature, meaning the PIMOE framework is the only\nnovel contribution in this research. The work seems to have weaker novelty. Could the authors\ncomment on this or guide to the novel contributions?\nResponse to Comment 11\nThank you for the comment. Although the UL and TPSL datasets are indeed sourced from existing\nliterature, we would like to emphasize that the core innovation of the PIMOE framework lies in its\nphysics-interpretable MOE architecture for “history-free” degradation trajectory prediction of\nsecond-life batteries. This method enables the interpretable calculation of degradation trajectories\nfor retired batteries in deployment scenarios using only partially available field signals, without\nrequiring complete historical data.\nThe authors have revised the paper in the following aspects:\n The authors further analyzed a newly released retired battery dataset to more comprehensively\nevaluate the superiority of the proposed method under different second-life scenarios.\n The authors have added descriptions of the newly introduced dataset and revised Figure 2 to\nprovide more comprehensive coverage of retired battery scenarios.\n The authors have included performance analyses of the new dataset and modified Figure 3 to\nevaluate the prediction performance of deeply degraded batteries\n The authors have added performance evaluations across different SOH intervals in the\nuncertainty analysis section.\n The authors have added detailed information of the new dataset in Supplementary Table 3.\nThe authors further analyzed a newly released retired battery dataset to more comprehensively\nevaluate the superiority of the proposed method under different second-life scenarios. The authors\nhave added the following description to the main text:\n“ The LSD dataset contains 86 commercial batteries with LiNi Co Mn O cathodes and\n0.5 0.2 0.3 2\ngraphite anodes, These 2.4 Ah batteries (3.7 V nominal) underwent two-phase testing, In Phase 1,\nthe batteries were grouped into 16 distinct charge–discharge protocols to induce diverse\ndegradation behaviors as the SOH decreased from 100% to 80% (details in Supplementary Table\n3). In Phase 2, all 86 cells were re-cycled from 80% to 50% SOH under a unified low-rate protocol\n(0.5 C charge / 0.2 C discharge) to simulate stationary energy-storage operation. Degradation\ntrajectories in Fig. 2b demonstrate Phase 2 cycling variability, we predict the deep degradation\ntrajectories of second-phase batteries under unknown and varying historical operating conditions.”\n\nComment 12\nWhy PatchTST and Informer have bad predictions with TPSL datasets?\nResponse to Comment 12\nThank you for the comment. The TPSL dataset incorporates 66 types of randomized charge-\ndischarge protocols to simulate diverse secondary-use conditions characterized by high complexity\nand stochasticity. Under such variable operational profiles, existing methods including classical\ntime series models like PatchTST and Informer—often assume future operating conditions remain\nconsistent with historical data. Although capable of modeling battery temporal degradation, these\napproaches fail to adapt to changing future operational patterns, leading to suboptimal prediction\nperformance. The fundamental limitation lies in their inability to effectively capture the correlation\nbetween the future maximum available capacity of second-life batteries and the uncertainty of future\nusage conditions.\nThe authors have made the following modifications to the article content:\n“Fig. 3b reveals that in the TPSL data, baseline methods exhibited limited capability in capturing\nlong-term trends following future load changes and failed completely during the initial load\ntransition phase. This is because models like PatchTST and Informer, as classical time-series models,\npredict the future by mining temporal patterns from historical data. However, in the TSPL dataset,\nthe uncertain charge and discharge currents significantly impacts the maximum available capacity\nof the current cycle. In contrast, PIMOE effectively identified transitional characteristics between\nphases by integrating future usage conditions. Supplementary Figs. 11-14 further demonstrate\nPIMOE's successful learning of the relationships between future operating conditions and capacity\nperformance.\n\nComment 13\nIn the main manuscript, in the early section of page 23, the authors tried to link the aging\nmechanisms to the expert networks. What is the validity of such a hypothesis?\nResponse to Comment 13\nThank you for the comment. In Figure 4a, a clear correlation can be observed between battery\nsamples at different degradation stages and the corresponding expert weights. In Figure 4b and\nSupplementary Figures 19–21, the authors further demonstrate the relationship between aging\nmechanisms and expert networks more directly through t-SNE clustering. In Supplementary Figures\n16–18, samples with different SOH values show distinct expert network selections. However, a\ndeeper connection between the underlying degradation mechanisms and the expert networks still\nneeds to be established.\nThe authors have revised the paper in the following aspects:\n The authors further illustrate the correlation between degradation mechanisms at different\nstages of degradation and the expert network by analyzing the IC curves of the dataset.\n The authors further validated the rationality of these physical features through SHAP analysis\nand Spearman correlation coefficients.\nThe authors further illustrate the correlation between degradation mechanisms at different stages of\ndegradation and the expert network by analyzing the IC curves of the dataset.\n“ Supplementary Figure 28. IC Aging Curve Analysis of the Dataset\nThis section aims to demonstrate the rationality and universality of dividing battery aging into\nmultiple stages using IC curves, and to explain how these parameters reveal the battery degradation\nmechanisms and reflect their physical significance. This assumption is based on numerous studies\non lithium battery degradation patterns. For instance, existing research suggests that during the\nbattery's lifecycle, the formation and thickening of the SEI layer affect the capacity degradation in\nthe early stages, while lithium-ion deposition typically occurs in the later stages of battery use and\nis one of the main causes of the sharp capacity decline.\nIC curves reveal internal changes in the battery by showing the subtle variations between battery\ncapacity and voltage, especially during the aging process, where the SEI layer growth, lithium-ion\nloss, and internal electrochemical reactions significantly impact the IC curve. As shown in the figure,\ndifferent colors represent different SOH samples. In the early charging and discharging cycles, the\ncurve displays sharp peaks, indicating the battery is in a healthy state with significant and rapid\ncapacity changes, mainly reflecting the formation of the SEI layer and surface chemical reactions.\nAs the cycle count increases, the peaks of the IC curve become more rounded and shift to the right,\nand longer flat sections appear, indicating that the battery's reversible capacity is gradually\ndecreasing. When the battery enters a more severe degradation phase, the peaks of the IC curve\ngradually decrease. At this point, degradation mechanisms such as lithium-ion deposition and loss\nof active materials dominate the capacity decline. The widening of the peaks in the curve and the\nsharp capacity drop indicate that the battery's energy storage capacity is severely compromised.\n\nComment 14\nThe research is regarding PI-physics informed parameters, but there is no discussion of these\nparameters on the impact of aging and how they are relevant to be called physical parameters. They\nare driven by electrical tests and more statistical data.\nResponse to Comment 14\nThank you for the comment. The PI-physics informed parameters adopted in this study have clear\nphysical meanings and are selected based on battery electrochemical principles. In Supplementary\nFigures 1-6, The authors directly display the correlation between these physical features and battery\naging. The authors have revised the paper in the following aspects:\n The authors have revised the title of the manuscript to avoid potential ambiguity.\n The authors have revised the introduction to further discuss physical modeling methods\n The authors have revised the discussion section to clearly outline the current limitations of the\nmethod and the direction for future improvements:\nTo avoid any potential confusion for readers, the authors have revised the title of the paper.\n“PIMOE: Physically interpretable mixture of experts network for battery degradation trajectory\nprediction amid second-life complexities.”\n\nComment 15\nWhat is the justification for Figure 4d, where the SEI and Li-plating are plotted separately, while\nthese events could happen in the same cell, depending on the activated mechanisms.\nResponse to Comment 15\nThank you for your comment. As you rightly pointed out, SEI growth and lithium plating may occur\nsequentially within the same battery through different activation mechanisms. The authors\nacknowledge that categorizing battery conditions based solely on dominant degradation\nmechanisms is oversimplified and overly broad. The original separation of the three mechanisms in\nFigure 4d did causes ambiguity. The authors have revised Figure 4d to eliminate this\nmisunderstanding and have further clarified this point in the main text.\n“ In Fig 4d, for the 130 UL dataset batteries retired at different SOH levels (classified solely based\non expert weights, see Supplementary Note 7 for criteria), those retired at 95% SOH achieved an\n\"excellent\" classification confidence of 91.5%, while batteries retired at 75% SOH received a\n\"scrap\" rating of 96%. This further demonstrates that the model utilizes different primary experts\nfor prediction based on the dominant degradation mechanisms at various stages, thereby enabling\nthe reverse classification of retired batteries at different degradation stages according to expert\nselection.”\n\nResponse to Reviewer #2\nThis is an interesting paper addressing the challenge of predicting battery degradation trajectories\nin second-life applications using a novel physics-informed mixture of experts (PIMOE) architecture.\nThe manuscript is generally well-written, and the proposed approach is innovative, combining\ndomain knowledge with machine learning for interpretable predictions. The results are compelling\nand relevant to both academia and industry.\nHowever, while the paper shows promise, there are several important issues that must be addressed\nbefore it can be considered for publication.\nDear Respected Reviewer,\nDear Reviewer,\nThe authors respond to your suggestions point by point and carefully revise the manuscript and\nsupplementary information, where you can find corresponding revisions using “track changes”\nmode. Changes are colored in blue in revised clean version. The authors truly hope that the responses\nappropriately address your justified concerns, and that revised manuscript meets your expectations.\n\nComment 1\nThe approach assumes that future load conditions are known in advance, which is a strong\nassumption in real-world second-life scenarios. Although this is briefly acknowledged in the\ndiscussion, a more thorough examination of the limitations and implications of this assumption is\nwarranted. Furthermore, scenarios involving uncertain or stochastic future use should be considered\nor at least simulated.\nResponse to Comment 1\nThank you for the comment. You pointed out the limitation of the assumption that “future load\nconditions are fully known” in real-world second-life battery applications. However, it is important\nto note that the degradation trajectory of a battery is inherently path-dependent, as its future\ndegradation rate and maximum usable capacity are closely influenced by operating conditions.\nAlthough the proposed method can be extended to long-term degradation trajectory prediction and\nhas been extensively validated across different prediction horizons, applying short-term predictions\nin deployment scenarios can greatly mitigate the impact of uncertainty or randomness in future\noperating conditions on prediction accuracy. The authors have comprehensively analyzed\nrepresentative application scenarios of retired batteries to verify the effectiveness of the proposed\nmethod and further supplemented evaluations on deeply degraded retired battery datasets to\nminimize the influence of random variability.\n\nResponse to Reviewer #3\nThis a well written paper on a general topic of interest to the battery community. I can recommend\npublication if the authors address the following comments.\nDear Reviewer,\nThe authors respond to your suggestions point by point and carefully revise the manuscript and\nsupplementary information, where you can find corresponding revisions using “track changes”\nmode. Changes are colored in blue in revised clean version. The authors truly hope that the responses\nappropriately address your justified concerns, and that revised manuscript meets your expectations.\n\nComment 1\nSpearman coefficient & SHAP values. The authors rely on their physics-informed router + experts\nstructure, but it would strengthen the paper to validate feature importance explicitly:\n• A simple Spearman correlation analysis between input features and outputs (SOH, degradation\nmodes) could confirm monotonic relationships and highlight redundancies among the 12 features.\n• A SHAP (Shapley Additive Explanations) analysis on the trained neural model could provide\ninsight into which features the network relies on most. This would help connect the “physics-\ninformed” explanation with the actual feature usage inside the NN.\n• Suggestion: Add a supplementary analysis of feature importance (Spearman + SHAP) to improve\ninterpretability and confidence in the model.\nResponse to Comment 1\nThank you for the comment. This suggestion has significantly enhanced the rigor and interpretability\nof our research. Following your advice. The authors have revised the paper in the following aspects:\n The authors have conducted a comprehensive supplementary analysis involving both\nSpearman's coefficient and SHAP values.\n The authors have made the following modifications to the Supplementary Figure 22:\n\nResponse to Reviewer #4\nAuthors describe a method for estimation of Li-ion cell capacity and combine it with predictive\nmodel that allows forecasting of future capacity degradation trends. Authors claim that the proposed\nmethod allows history-free estimation of State-of-Health (SOH) of the battery hence eliminating a\nneed for historical usage records or extensive lab tests. Elimination of data requirements is a very\nvaluable technical goal since unavailability of historical usage or test data is a biggest obstacle for\naccurate estimation of SOH.\nDear Reviewer,\nThe authors respond to your suggestions point by point and carefully revise the manuscript and\nsupplementary information, where you can find corresponding revisions using “track changes”\nmode. Changes are colored in blue in revised clean version. The authors truly hope that the responses\nappropriately address your justified concerns, and that revised manuscript meets your expectations.\n\nComment 1\nWhile authors claim history-free method, the training process used for training of the estimation and\nprediction components of the proposed model relies on extensive test data collected offline. I think\nthat authors need to revise some of their claims related to this discrepancy. That training data is the\nmissing ‘history’. It is very well known that with sufficiently rich degradation data collected offline\none can train a data-driven model to achieve pretty much any desired accuracy.\nResponse to Comment 1\nThank you for the comment. The PIMOE indeed relies on offline testing data from laboratory\nexperiments for initial training. This characteristic stems from the fundamental need for high-quality\nlabeled data in deep learning methods. The training data comes from battery full lifecycle testing\nconducted in laboratory environments, and the statement about not requiring historical data can\nindeed cause confusion for readers. The authors have revised the paper in the following aspects:\n The authors have revised the abstract to emphasize that historical data is not required in\ndeployment scenarios, thereby avoiding potential confusion for readers.\n The authors made modifications in the Broader Context to emphasize that our approach has\nlower data requirements and higher applicability in practical applications.\n The authors have revised the introduction to explain the challenges of predicting degradation\ntrajectories in scenarios where historical data are unavailable.\n The authors clarified in the main text the different data requirements for the two phases: offline\ntraining and online application.\n The authors have added a deeply degraded battery dataset to the experimental design to\ndemonstrate that the proposed method does not rely on historical usage conditions.\n\nComment 2\nI recommend the authors to put a bit more effort in connecting the proposed modeling method to\nphysics of Li cell degradation. Without this connection between selected ML features and vaguely\nmentioned degradation modes the paper does not stand out from a long list of data-driven modeling\npapers for SOH estimation.\nResponse to Comment 2\nThank you for the comment. Although different degradation stages exhibit distinct characteristics,\ntheir corresponding electrochemical behaviors follow relatively unique yet universally observable\npatterns. These features can be extracted through physics-informed feature engineering, while the\nMOE network decomposes the complex task into sub-tasks handled by specialized expert modules\ncapable of capturing multi-level and nonlinear feature representations. The concept of decomposing\na complex problem into multiple sub-tasks inherently aligns with the coupled nature of degradation\nmechanisms in batteries. The authors have revised the paper in the following aspects:\n The authors have revised the title of the manuscript to avoid potential ambiguity.\n The authors have revised the introduction to further discuss physical modeling methods\n The authors further illustrate the correlation between degradation mechanisms at different\nstages of degradation and the expert network by analyzing the IC curves of the dataset.\n The authors further validated the rationality of these physical features through SHAP analysis\nand Spearman correlation coefficients.\n The authors have revised the discussion section to clearly outline the current limitations of the\nmethod and the direction for future improvements:\n\nComment 3\nLine 130: Please, explain what the Trend vector is exactly. Is it a vector of capacity or SOH values\nover one cycle vs time? Several cycles?\nResponse to Comment 3\nThank you for the comment. The trend vector is a latent feature representation generated from field-\nacquired current cycle data, designed for predicting capacity degradation over multiple future cycles.\nIn most conventional methods, the output of this vector is directly used as the prediction value.\nHowever, the authors argue that incorporating future load conditions is essential for meaningful\nreal-world lifespan prediction, as future operating conditions significantly influence capacity\ndegradation. In the original design, this vector does not account for the impact of future load\nconditions on degradation trajectories; rather, it reflects a preliminary prediction of future\ndegradation based solely on the current health state. The length L of the trend vector strictly\ncorresponds to the number of prediction cycles."}, {"round": 2, "reviewer_comments": {"reviewer_1": "In Supplementary Table 1, please check the material composition typos. The NMC composition is just copied and pasted\nfrom NCA.\nThanks for adding the LSD dataset to the analysis.\nRegarding the linked aging mechanisms, the identified/mentioned mechanisms (i.e. SEI thickness growth) should be\ndescribed using words like 'may be' or 'probably' as they could not be validated within the scope, although they are proven in\nother works. The batteries are always different from one to another and trigger different levels of mechanisms under known\ncircumstances.\n(Remarks on code availability)\nThe code seems clean but did not run it to verify.", "reviewer_2": "I appreciate the thorough revision. All my previous concerns have been fully addressed, and I have no further comments.\n(Remarks on code availability)\nThe code package appears to be well organized and adequately documented. It includes a clear README. I was able to\nrun the code without particular difficulties, and the results align with those reported in the manuscript.", "reviewer_3": "The authors have delicately revised the paper based on my comments and I am satisfied with the revised version. As such, I\nrecommend the publication of this work.\n(Remarks on code availability)", "reviewer_4": "I would like to thank the authors for putting the effort in addressing reviewer’s questions and suggestions. The manuscript\nhas improved, and most important technical aspects of the proposed method are now more readable and easier to\nunderstand. I think the results are valuable and should be published.\nMy main concern is still about the claim of physical interpretability claimed in the title and abstract. From my point of view the\nproposed method does not really provide any physical interpretability of the results. Weight assignments by PIMOE experts\nare a result of ML model training. The model simply learns to differentiate between different capacity degradation trends\nprovided in the training data. The model uses its available flexibility to put more weight on one part of sub-model closer to\nBOL and more weight on another part of sub-model later in cells’ life. There is nothing physical about it, it is simply a result of\na properly setup optimization problem. Fig 30 in supplemental material and its notes does not really tell a reader about\nphysical changes in the cells: it does not show how much Li or active materials have been lost or how kinetic parameters of\nthe cells have changed. The way it is presented, it is just a collection of curves that ML model learned to map to\ntarget values.\nI suggest keeping the references to physical interpretations in the main text and supplemental materials as is, but to modify\nthe title and remove the corresponding claim from the abstract and the introduction. It will be fairer to a reader and avoid any\nconfusion.\nSupplementary note 5 refers to figure 4e. Could not find anything related to Trend on Fig 4e of the article.\nTables 1,2,3 in supplementary material pdf are not properly formatted, difficult to read.\n(Remarks on code availability)"}, "author_response": "Response to Reviewers\nResponse to Reviewer #1\nDear Reviewer,\nThe authors respond to your suggestions point by point and carefully revise the manuscript and\nsupplementary information, where you can find corresponding revisions using “track changes”\nmode. Changes are colored in blue in revised clean version. The authors truly hope that the responses\nappropriately address your justified concerns, and that revised manuscript meets your expectations.\n\nComment 1\nIn Supplementary Table 1, please check the material composition typos. The NMC composition is\njust copied and pasted from NCA.\nResponse to Comment 1\nThank you for the comment. The authors have corrected the typographical errors regarding the NCM\nbattery material composition and have performed a thorough review of the entire manuscript.\nComment 2\nThanks for adding the LSD dataset to the analysis.\nResponse to Comment 2\nThank you for your valuable comments, which have made a significant contribution to this research.\nThe authors consider your perspectives to be crucial to the proposed methods in this study. We hope\nthat the findings of this research can draw adequate attention from the academic community to the\nemerging field of retired battery echelon utilization.\nComment 3\nRegarding the linked aging mechanisms, the identified/mentioned mechanisms (i.e. SEI thickness\ngrowth) should be described using words like 'may be' or 'probably' as they could not be validated\nwithin the scope, although they are proven in other works. The batteries are always different from\none to another and trigger different levels of mechanisms under known circumstances.\nResponse to Comment 3\nThank you for the comment. The authors acknowledge the complexity of various degradation\nmechanisms across different batteries. Accordingly, the descriptions regarding definitive\ndegradation mechanisms have been revised in the main text.\nComment 4\nThe code seems clean but did not run it to verify.\nResponse to Comment 4\nThank you for the comment. The authors have revised the README file instructions to make it\neasier for readers to run and reproduce the experimental results.\nResponse to Reviewer #2\nI appreciate the thorough revision. All my previous concerns have been fully addressed, and I have\nno further comments. The code package appears to be well organized and adequately documented.\nIt includes a clear README. I was able to run the code without particular difficulties, and the results\nalign with those reported in the manuscript.\nDear Reviewer,\nThank you for your valuable comments, which have made a significant contribution to this\nresearch. The authors consider your perspectives to be crucial to the proposed methods in this study.\nWe hope that the findings of this research can draw adequate attention from the academic\ncommunity to the emerging field of retired battery echelon utilization.\nResponse to Reviewer #3\nThe authors have delicately revised the paper based on my comments and I am satisfied with the\nrevised version. As such, I recommend the publication of this work.\nDear Reviewer,\nThank you for your valuable comments, which have made a significant contribution to this\nresearch. The authors consider your perspectives to be crucial to the proposed methods in this study.\nWe hope that the findings of this research can draw adequate attention from the academic\ncommunity to the emerging field of retired battery echelon utilization.\nResponse to Reviewer #4\nI would like to thank the authors for putting the effort in addressing reviewer’s questions and\nsuggestions. The manuscript has improved, and most important technical aspects of the proposed\nmethod are now more readable and easier to understand. I think the results are valuable and should\nbe published.\nDear Reviewer,\nThank you for your valuable comments, which have made a significant contribution to this\nresearch. The authors consider your perspectives to be crucial to the proposed methods in this study.\nWe hope that the findings of this research can draw adequate attention from the academic\ncommunity to the emerging field of retired battery echelon utilization.\nThe authors respond to your suggestions point by point and carefully revise the manuscript and\nsupplementary information, where you can find corresponding revisions using “track changes”\nmode. Changes are colored in blue in revised clean version. The authors truly hope that the responses\nappropriately address your justified concerns, and that revised manuscript meets your expectations.\n\nComment 1\nMy main concern is still about the claim of physical interpretability claimed in the title and abstract.\nFrom my point of view the proposed method does not really provide any physical interpretability of\nthe results. Weight assignments by PIMOE experts are a result of ML model training. The model\nsimply learns to differentiate between different capacity degradation trends provided in the training\ndata. The model uses its available flexibility to put more weight on one part of sub-model closer to\nBOL and more weight on another part of sub-model later in cells’ life. There is nothing physical\nabout it, it is simply a result of a properly setup optimization problem. Fig 30 in supplemental\nmaterial and its notes does not really tell a reader about physical changes in the cells: it does not\nshow how much Li or active materials have been lost or how kinetic parameters of the cells have\nchanged. The way it is presented, it is just a collection of curves that ML model learned to map to\ntarget values.\nI suggest keeping the references to physical interpretations in the main text and supplemental\nmaterials as is, but to modify the title and remove the corresponding claim from the abstract and the\nintroduction. It will be fairer to a reader and avoid any confusion.\nResponse to Comment 1\nThank you for the comment. The authors have made the following revisions:\n The authors have revised the title of the manuscript.\n The authors have updated the relevant expressions in the Abstract.\n The authors have refined the descriptions and phrasing in the Introduction.\nThe authors have revised the title of the manuscript.\n“ iMODEL: Prediction of second-life battery degradation trajectory using interpretable mixture\nof deep expertized learning”\nThe authors have updated the relevant expressions in the Abstract.\n“Retired electric vehicle batteries offer immense potential to support energy infrastructure stability\nin underdeveloped regions through second-life use, but uncertainties in battery degradation\nbehaviors pose major safety concerns. This work proposes an interpretable mixture of deep\nexpertized learning (iMODEL) network that predicts battery degradation trajectories using partial,\nfield-accessible signals in a single cycling operation. iMODEL leverages an adaptive multi-\ndegradation prediction module to classify battery degradation modes using expert weight synthesis\nlearned from battery capacity-voltage and relaxation data. The module produces latent degradation\ntrend embeddings, which are input to a use-dependent recurrent network for long-term degradation\ntrajectory prediction. Validated on three typical use patterns (i.e. consistent operating histories,\ndeeply aged batteries with unknown prior use, and uncertain second-life conditions, including 295\nbatteries, 93 use conditions, and 84,213 cycles), iMODEL achieves an average mean absolute\npercentage errors (MAPE) of 0.95% with a 0.43 ms inference time for life-long battery degradation\ntrajectory prediction. Compared to state-of-the-art Informer and PatchTST, it reduces\ncomputational time and MAPE by 50% and 77%, respectively. Compatible with data sampling in\nrandom state of charge regions, iMODEL supports a 150-cycle time-horizon degradation trajectory\nprediction with 1.50% and 6.26% MAPE on average and at maximum, respectively. Notably,\niMODEL can operate effectively even with pruned 5MB training data while retaining 0.95% MAPE.\nBroadly, this network offers a deployable, history-free solution for battery degradation trajectory\nprediction at the time of second-life deployment, redefining how second-life energy storage systems\nare sensed, evaluated, controlled, and integrated for sustainable energy infrastructures at scale.”\nThe authors have refined the descriptions and phrasing in the Introduction.\n“To fulfil this gap, this work proposes a interpretable mixture of deep expertized learning\n(iMODEL) network for predicting degradation trajectories of retired batteries under uncertain\nsecond-life use complexities without requiring historical data. As shown in Fig. 1a, the model uses\nelectrical signal data from randomly sampled SOC regions at collection field of retired batteries,\ninteractively incorporating assumed future use conditions to predict degradation trajectories\ntoward extend time horizons in future second-life use. Fig. 1b illustrates the designed adaptive multi-\nmode degradation prediction (AMDP) module, which integrates multiple expert systems' weights\nstudied from capacity-voltage and relaxation data to achieve classification of degradation modes of\nretired batteries. AMDP outputs latent degradation trend embedding vectors that are forwarded into\na feature-operational recurrent neural network (FORNN), enabling interpretable and extendable\nprediction of degradation trajectories. In Fig. 1c, model’s applications on battery reusing and\nrecycling are illustrated, highlighting a safety-aware allocation of second-life batteries based on\ntheir predicted degradation trajectories toward a sustainable battery circular economy. Without\nhistorical data, the proposed approach achieves an average MAPE of 0.88% in predicting\ndegradation trajectories under uncertain and diversified second-life use scenarios, spanning 295\nbatteries and 93 use conditions and 84,213 cycles. Notably, the required input data can be easily\nextracted at random SOC region, without charging or discharging retired batteries to anticipated\nSOC levels. The proposed method can extend degradation trajectory prediction over a 150 cycle of\nhorizon in future second-life use with an MAPE of 1.50%. This work underscores transformative\npotential of incorporating physics prior principles into predictive health management of retired\nbatteries, enabling broader proactive fault detection and reliability assessment across complex,\nsafety-critical systems, such as large-scale energy infrastructures, with careful consideration of\ndurability, safety, and sustainable operation under data-scarce and data-heterogeneous conditions.”\nComment 2\nSupplementary note 5 refers to figure 4e. Could not find anything related to Trend on Fig 4e of the\narticle.\nResponse to Comment 2\nThank you for the comment. The authors have corrected the figure citation in Supplementary Note\n5, changing Fig. 4e to Fig. 4g.\n“We define a vector Trend to represent the output of the AMDP module and visualize the hidden\noutputs of input samples across the full lifecycle to explain the performance improvement of\niMODEL. In Fig.4g, using the NCA-35-05-1 battery, we normalize each value of the Trend vector\nacross different samples for clearer interpretation, thereby demonstrating how the model internally\nprocesses samples at different aging stages and clarifying its contribution to degradation trajectory\nprediction. For the input X of the iMODEL model, the Trend is obtained through the AMDP module.”\nComment 3\nTables 1,2,3 in supplementary material pdf are not properly formatted, difficult to read.\nResponse to Comment 3\nThank you for the comment. The authors have double-checked all PDF documents and implemented\nthe required changes."}]

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