Mechanistically guided residual learning for battery state monitoring throughout life

Domain: physics
Published: 2026-01-15
Editorial outcome: major_revision

Abstract
Battery monitoring requires high accuracy and robustness throughout the entire lifespan to ensure safe and optimal operations. Here we introduce mechanistic leading residual learners to enhance the monitoring of battery charge and health states, as well as guide safety warnings, targeting large-scale applications. Leveraging prior knowledge from real-time filtering as primary guidance, complemented by mechanistic and statistical features, our approach significantly improves accuracy and robustness. We propose and validate two general residual learning pipelines, namely the correction model and compensation model, across various scenarios, encompassing different battery types, loading profiles, aging conditions, and environmental conditions, using three aging datasets under dynamic cycling. Our method achieves a relative root mean square error reduction of over 50% from the results observed in prior estimations. The extrapolation capability under unseen conditions, along with interpretability, enhances both accuracy and trustworthiness. The models remain effective even with reduced training data and sampling frequency, maintaining the potential for practical electric vehicle applications. Application demonstrations confirm the efficacy in providing continuous monitoring across lifespan without the need for offline testing and model calibration during operation.

Introduction
Lithium-ion batteries are extensively utilized to support sustainable development, yet they present significant challenges in accurate state monitoring due to their diverse chemistry and types, usage conditions, and the complex aging processes encountered in electrified transportation1–4. To ensure safe operations and guide optimal management, state of charge (SOC) and state of health (SOH) are two of the most concerned indices to indicate the battery potential in real-time and in a long-time horizon5–7. Developing a closed-loop framework for battery monitoring is crucial for enhancing battery management and avoiding safety hazards, particularly as batteries degrade8–10. Traditionally, specific tests are conducted to verify the estimations of SOC and SOH independently or under several sparse conditions. However, real-world applications demand effective battery state monitoring that accounts for continuous aging under dynamic loads and varying temperatures.

Widely adopted methods for battery state monitoring include model-based and data-driven approaches, each offering distinct advantages but also having certain limitations. Model-based methods are known for their robustness, particularly when incorporating high-fidelity battery models, though they tend to be more complex. Model-based methods are usually constructed with filtering methods such as the Kalman filter for internal state estimation. In contrast, data-driven methods, which map the relationships between the features and battery states via machine learning (ML) models, offer greater flexibility but often suffer from limited generalization capabilities11. Consequently, hybrid monitoring structures that combine model-based filtering approaches with ML have gained popularity, aiming to enhance both accuracy and robustness12–15. One straightforward method involves fusing the outputs of different sub-models, e.g., filtering-based estimation and ML estimation, to mitigate uncertainties associated with any single model11,16,17. Due to the large variances of the estimation from data-driven approaches, especially for SOC estimation, one popular approach employs filtering techniques afterward to enhance the estimation from ML models16,18–20. However, the implementation in SOC estimation usually adopts the real reference, i.e., coulomb counting, as the observation, which makes it hard to obtain accurate calculations in real applications. On the other hand, ML excels at modeling uncertain relationships, and the estimated errors from model-based methods vary and are hard to describe21,22. Therefore, the sequential framework of the model-based method followed by ML was proposed, where ML is employed after the filtering method to improve the initial estimation from model-based methods23–25. The main idea is that ML models the uncertainties and the errors of the mechanistic model-based methods, which have been employed for both battery modeling and temperature estimation. Nevertheless, the specific pipeline is designed for a specific purpose and has only been verified under several specific tests, limiting its practical large-scale application. Systematic design for general hybrid pipelines integrating model-based filtering and ML methods with applications using dynamic cycling for the entire lifespan is a practical yet unsolved challenge. Additionally, physics-informed neural network models have shown the potential to enhance physical interpretability and reduce data requirements in battery modeling26,27. However, the sensitivity of these models to the tolerance of physical laws can greatly impact their predictive accuracy. Conversely, when reliable prior knowledge is incorporated, data-driven models demonstrate more stable improvements in predictive performance.

Feature extraction is one key factor that hugely influences the accuracy of ML models28–32. Methods for statistical calculation can extract highly effective features28,29. However, these features are often sensitive to varying operating conditions, such as different current rates and temperatures, which can reduce model performance across diverse scenarios. While domain adaptation techniques can reduce the discrepancies between the feature distributions, thus enhancing the generalization of monitoring models under different conditions, they also increase model complexity and computational demands, hindering the application for real-time state estimation33,34. Therefore, improving the suitability and coherence of features across different conditions is crucial for enhancing model accuracy and generalizability. Additionally, statistical features typically lack physical interpretability, complicating the explanation of ML models. Thus, incorporating features with mechanistic meaning can significantly contribute to more interpretable monitoring systems35,36. Recent research has increasingly focused on the interpretability of ML models, which provides deeper insights into their internal workings and facilitates better optimization37–40.

Most of the existing work on SOC estimation has been limited to specific operational scenarios, often neglecting performance under real-world conditions where the model must function continuously across a battery’s entire lifespan. Health predictions are similarly conducted under idealized conditions, assuming consistent availability of data from uniform charging and discharging cycles, an assumption that is difficult to meet in practical applications. Studies on health monitoring frequently consider ideal scenarios with stable working conditions, while in practice, the varying conditions pose significant challenges to maintaining the effectiveness of monitoring models. The continuous verifications of the SOC and SOH joint estimation model are expected to be conducted during the entire lifespan, which is the practical demand of the battery management system, instead of under several specific tests. Moreover, battery safety conditions are crucial for monitoring, which is key to avoiding hazards like thermal runaway. The potential of continuous state monitoring during dynamic aging to guide safety warnings worthy deeper studies.

Weproposemechanisticleadingresiduallearningsforcontinuousmonitoringofbatterystates,specificallySOCandSOH,undervaryingdynamicworkingconditionsthroughouttheentirelifespan.Thegeneralpipelineisadaptableforbatterymodelingandmonitoringwithdifferentpracticalrequirements.Usingthreedatasetsincorporatingdifferentbatterytypesagingunderdifferentloadingandtemperatureconditionsforverifications,wedemonstratetheeffectivenessofourmodelworkingunderdifferentdegradationpatternstoemulatereal-worldusagescenarios.Theproposedmechanisticleadingresidualmodelsintegrateamodel-basedapproachwithML,wherepriorestimationsandmechanisticfeaturesderivedfromthebatterymodelareusedasinputsforresiduallearners,includingacorrectionmodelandacompensationmodel,forenhancedstatemonitoring.ThisgeneralpipelineworkseffectivelyforbothSOCandSOHandconstructsahybridclosed-loopframeworktosupportthemonitoringofthebatterystatesfortheentirelifespan,regardlessofthevaryingloadingprofiles,temperatureconditions,andworkingranges(SOCwindows)forlarge-scalerepresentationsofuncontrollable,randomlyvarying,andunseenworkingscenarios.Themodelisalsobeneficialforearlythermalsafetywarningsduringaging.ThroughinterpretableMLandpracticaldemonstrations,theproposedmodelenhancesaccuracy,robustness,andtrustworthiness,therebyincreasingitspotentialforreal-worldapplications.TheoverallframeworkisillustratedinFig.1,wherethepathwayoftheapplicationandtheflowchartofthepipelineforresiduallearningaredemonstrated.AdditionalmodeldetailsareprovidedinFig.S1.

Results
We employed three datasets to evaluate the proposed mechanistic leading residual learning models under dynamic loadings for the full lifespan. Three types of batteries, including pouch, prismatic, and cylindrical cells, which were cycled with urban, highway, and real-world dynamic profiles, are used for verifications under different scenarios.

To emulate the dynamic working and the varying environmental conditions encountered in real-world applications, the first dataset subjected 13 pouch batteries (nominal capacity of 8 Ah) to aging under dynamic discharge across the entire lifespan at different temperatures34. Urban, highway, and hybrid profiles were employed and are illustrated in Fig. S2, while Fig. S3 presents the variations in voltage curves observed during the aging process. Capacity degradation curves relative to equivalent full cycles are shown in Fig. S4, demonstrating that urban dynamic loading profiles generally result in longer battery life compared to highway-based loading, with varying temperature conditions significantly affecting the degradation patterns. There are in total 27840 discharge samples (represented as the dynamic discharge cycles) generated in dataset 1.

Besides, 3 large-format prismatic cells with a nominal capacity of 50 Ah are used in the second aging test until they die at around 20% SOH with sudden death, i.e., rapid degradation in a few cycles. We employed WLTP (Worldwide Harmonized Light Vehicles Test Procedure) to discharge the cells and used both CC and pulse-constant current (PCC) for charging, as shown in Fig. S5. The pulse was set with an average rate the same as the CC (2C), while with twice the amplitude. The frequency and duty cycle are 1 Hz and 50%, respectively. The capacity, current, voltage, and temperature curves during aging are shown in Fig. S6. The open circuit voltage (OCV) test was conducted using C/20 constant charging and discharging. The sampling frequency is 10 Hz, and the current and voltage variations for the pulses are tracked precisely. This dataset benefits wide applications, including state monitoring, life prediction, and thermal safety warning. The detailed information of the tested pouch and prismatic cell is listed in Table S1. The sample size, i.e., dynamic discharge curves until capacity drops below 50% of the fresh capacity, is 8590.

Finally, datasets using 25 cylindrical cells from Ref. 41., which were aged with synthetic highway and urban, as well as real city driving profiles, are employed to verify our model for use under different unseen working conditions. Typical current and voltage curves during aging under different loading profiles are shown in Fig. S7, indicating varying discharge patterns observed among different cycling, which therefore cause diverse working scenarios and aging trajectories. We used the first dynamic test after the diagnostic test as the dynamic cycling data during aging, the C/2 discharge capacity before the dynamic cycle as the referenced capacity, and the C/40 discharge curve of a new cell as the OCV curve. There are 483 dynamic discharge curves generated for the model evaluation.

To promote the real-time application of the pipelines, the mechanistic model adopts an equivalent circuit model in this work. For real-time battery SOC estimation, online parameter identification and adaptive filtering methods are employed (see Note S1 for a detailed procedure), where the SOC prior estimations and mechanistic model parameters (e.g., resistance and polarization features) are obtained. Coupled with SOC estimation and coulomb counting for released capacity calculation, the battery capacity can be derived to calculate the prior estimation of battery SOH. The capacity here represents the releasable discharge capacity under the corresponding discharging condition with the specific loading profile, temperature, and aging conditions. Meanwhile, measured parameters (e.g., voltage, current, and released capacity) and identified mechanistic parameters (e.g., identified resistance, OCV, and polarization resistance/capacitance) during dynamic discharging are used to calculate health features, which are composed of several statistical values including mean value, standard deviation, and min and max values of these sequential parameters from a random discharge phase. Detailed feature calculations are described in Methods and Notes S2–S3.

Here, we introduce mechanistic leading residual learners, incorporating a correction model and a compensation model to enhance the state monitoring results from prior estimations. As shown in Fig. 1b and S8, the prior estimations and extracted features serve as input for the ML to output an enhanced estimation and a residual for the prior estimation compensation. Therefore, the function of the correction model is to enhance the prior estimation by direct state predictions, while the compensation model enhances the estimations by adding the predicted residuals to the prior estimations. The prior estimation from the mechanistic model provides stable guidance, and those feature matrices help capture the expected battery states, condition indicators from the real battery responses, as well as the mechanistic features describing the internal conditions. The ML model is built to compensate for the error caused by model uncertainties and other influences, such as OCV variations under varying aging and temperature conditions. The effectiveness of this method is demonstrated through two key states: real-time SOC and the more slowly varying SOH. Note that the compared prior estimations represent the conventional model-based filtering methods. The standard coulomb counting-based calculation is employed to obtain the referenced real SOC and SOH for the evaluations of the conventional model-based filtering methods and the proposed residual learning models, while it is not used in any part of the modeling pipelines. Additionally, our model can be easily extended to monitor other relevant states as required, for example, the sensorless temperature estimations based on the mechanistic features and the lamped-mass thermal model-based prior estimations24.

Results for online parameter identification-based voltage and SOC estimation under varying aging and temperature conditions for pouch cells are shown in Fig. S9. Examples of the variations of the estimated model parameters during aging under three different dynamic profiles are shown in Fig. S10, indicating stable mechanistic feature extractions under different working and aging conditions. To emulate uncertain initializations, error noises (ranging within 5%) are randomly added to the initial SOCs and battery capacities. The findings indicate that accurate and robust estimations are achieved despite these initial errors, maintaining stability across different loading, aging, and temperature conditions. However, errors may still become significant during prolonged usage. By incorporating the estimated SOC from the filtering method as prior knowledge, along with mechanistic and measurable features, our mechanistic leading residual learner further enhances estimation accuracy and robustness. Demonstrations of the monitoring through the correction model and compensation model are presented in Fig. 2a, where the estimation errors underscore the significant potential for enhanced accuracy. Additional estimation results are provided in Fig. S11. Both pipelines exhibit similar performance, as evidenced by the symmetrical estimation distributions shown in Fig. 2b, and the densities match the regular usage conditions. Results when only prior estimation is used as input are provided in Fig. S12 and Table S2, which indicate the importance of mechanistic features to enable more accurate and robust estimations. The root mean square error (RMSE) for the two residual learners is reduced to 0.57% and 0.56%, respectively, a significant improvement from the initial 2.20% (see Table S2 for detailed RMSE and mean absolute error (MAE)).

Evaluating battery health under practical dynamic working conditions and varying depths of discharge (DOD) is challenging due to the diverse operational requirements. To meet real-world dynamic working applications, we predict battery health based on random DOD in partial dynamic discharges. As shown in Fig. S13, the SOC ranges indicate the adaptability of the residual learners to monitor battery health status under uncertain working DODs with varying loading profiles. The corresponding average temperature distribution highlights the model’s generalization across a broad temperature range (20–50°C). The varying working conditions are reflected in the density of capacity distributions relative to running cycles, which underscores the irregularity of degradation patterns and the necessity for timely health estimation during usage. By considering random DODs in real-world applications alongside the estimated SOC, the prior capacity is calculated to guide the residual learners. As depicted in Fig. 2c, the correlation coefficient between the prior estimation and real capacity is 0.959, indicating stable estimations, although large residuals exist. The estimations obtained from the model-based methods are stable despite the moderate accuracy, indicating their great effectiveness in guiding ML models.

Inspired by prior estimations and the mechanistic states, which are derived through sequential information (i.e., the measured data and mechanistic states) during discharging, the residual learners are trained to enhance the accuracy and reliability of health monitoring, thereby guiding safer and more effective operations. The predictions of the two residual learners, i.e., corrected SOH and compensations (ΔSOH) for the prior SOH respectively, shown in Fig. 2d–e demonstrate the coefficient of determination (R2) larger than 0.98 and 0.84 respectively. Here, the SOH is calculated by dividing the present capacity (Cap_p) by the nominal capacity (Cap_n). The MAE for the SOH predictions (as listed in Table S3), based on these residual learners, are as low as 0.779% and 0.783%, respectively. Error distributions for the prior SOH and the enhanced estimations, presented in Fig. 2f, highlight significant improvements in accuracy and reliability, with smaller average errors and narrower distribution ranges.

Maintaining model performance across different SOC ranges during dynamic discharging is challenging but crucial for practical applications. We evaluated the error distributions of two residual learning pipelines across varying SOC ranges under different working conditions. The tightly concentrated error distributions with low values, as shown in Fig. 3a, b, demonstrate the robustness and high accuracy of the models under diverse driving conditions. The absence of significant bias across error sizes and SOC intervals further ensures reliable model performance under random usage scenarios. Error distributions relative to running cycles, as depicted in Fig. S14, also confirm robustness across different aging stages. Our residual learners are adaptable to various machine-learning algorithms, making them versatile for different practical requirements and computational resources. We demonstrate the performance of six different machine-learning algorithms using the two mechanistic learning residual learning pipelines, as shown in Fig. 3c. All results reflect average performance over ten iterations, each with a distinct training-testing set, with estimation errors consistently below 2.2% across the different models. Notably, significant improvements were achieved even with more lightweight machine-learning models like KNN (K-nearest neighbors) and DT (decision tree), highlighting their potential for real-world implementations where computational resources may be limited. The pipelines for residual learning are also suitable for other ML algorithms, indicating great generalizability over various application requirements. Detailed descriptions of the ML algorithms are provided in Note S4.

We then evaluate the model performance with different ratios of data splitting for training and testing, since different working conditions cause varying aging patterns. Fig. 3e, f show the results of the MAE for the two residual learners under 10 times validations with different split datasets, and the results of RMSE and R2 are demonstrated in Fig. S15. With the increased training ratio from 0.1 to 0.9, a clear trend of accuracy increment was seen by the reduction trends on MAE and RMSE. There are no obvious differences for the model-based prior estimations, witnessed by the error differences varying within 0.1% under different testing cases in Fig. 3d. Obvious improvements are seen before the training ratio surpasses 50%, and improvement after that becomes smaller. As shown in Fig. S16, where the training data and testing data are demonstrated, after the training ratio surpasses 50%, the testing data distributions are almost covered by the training data distributions. However, even when the training data only takes up 10% of the whole data, i.e., the testing data covers many unseen conditions from the training data, the accuracy is still high (less than 1.2% for MAE and 1.7% for RMSE). Having R2 greater than 0.96 under all testing scenarios indicates high extrapolation capability. One major reason is that the model-based estimations provide stable and reliable prior knowledge for the residual learning pipeline, while the mechanistic features also help improve the accuracy under different scenarios.

To further understand the impactful features for the residual learners, SHAP (SHapley Additive exPlanations) analysis is employed42–46. Additionally, a correlation coefficient map is provided in Fig. S17. As shown in Fig. 3g, the prior knowledge derived from battery model-based estimations exerts the dominant influence on the predictions of the residual learner, effectively ensuring robust data-driven estimations, which is also proven by the stable estimations in Fig. 3d. These model-based filtering estimations demonstrate greater resilience to unseen conditions due to the mechanistic constraints, offering reliable information that helps mitigate the primary limitation of ML models, namely poor generalization under unseen conditions. Mechanistic features also significantly impact SOH predictions for the two residual learners, underscoring their importance in enhancing accuracy. The detailed SHAP values for all the features are shown in Figs. S18a–b. Not all features with high correlation coefficients shown in Fig. S17 have had a high impact on the output. We found that some of those features are redundant to other highly impacted features from the high mutual correlation coefficients. For this case, the random forest model helps remove redundant features and improves monitoring accuracy, eliminating the tedious manual feature selection process47. Comparisons with models using only the mechanistic features (excluding prior estimations) or only the prior estimations (excluding physical features) are presented in Figs. S12 and S19 for SOC and SOH monitoring, demonstrating that our mechanistic leading residual learners achieve better accuracy and robustness. Interestingly, the model relying solely on prior knowledge performs the worst, likely because similar prior estimations under different conditions can mask real differences, which mechanistic features can reveal to improve accuracy across scenarios. The mean absolute SHAP values for SOC residual learners, shown in Fig. 3h, further indicate that prior estimations from the filter provide the most critical guidance for the ML model, while mechanistic information, such as OCV and resistance, aids the estimations, and direct measurements have less impact.

We further employed our model for the state monitoring of different battery types with different aging patterns caused by different cycling conditions. The dynamic aging of large-format prismatic cells is verified. As shown in Fig. 4a, three large-format prismatic cells are aged with CC/PCC charging and WLTP dynamic discharging, which experience accelerated aging after SOH drops below the knee points. The average current of the pulses remains the same as the constant current, as shown in Fig. 4b. While different battery impedances cause the voltage to vary differently, and according to the heat generation, the cells have higher temperatures using pulse current compared to the constant current. Around 10°C difference for the maximum temperature and 3.5°C difference for the average temperature during cycling are seen for the two different charging protocols. These large thermal condition differences during cycling become the main cause of the different lifetimes for the cells under constant or pulse currents. Previous related works have revealed that pulse currents have an influence on battery lifetime. The battery aging is coupled and influenced by current profiles, temperature, current rates, etc., thus, we advocate coupling mechanisms and battery types during evaluations for the lifetime extension strategy designs.

We continuously age the batteries until they die, where accelerated aging is seen. As shown in Fig. S20, the battery is heavily swelled after aging, and the thermal safety issue (largely and rapidly increased temperature) occurred as demonstrated in Fig. 5a. Interestingly, the battery capacity degradation accelerated earlier (around 100 cycles) than the thermal safety issue happened, and the identified mechanistic resistances also show earlier detection. Therefore, accurate estimations of the battery capacity and mechanistic states are also crucial to guiding safety management, which helps with early warning of battery safety conditions before thermal issues during operations.

The monitoring results for the cells, until SOH (current capacity divided by initial capacity) drops below 50%, are shown in Fig. 4c, d, where the total discharging curves are randomly divided into training and testing datasets with a 50% portion of each. The mechanistic leading correction model and compensation model show improved SOC and SOH monitoring for the full lifespan with MAE of less than 0.50% and 0.68%, respectively, as listed in Table S4. SHAP interpretation also indicates the most important impact of the prior estimation and reflects the importance of the mechanistic features. Due to the high linear relationships between the corrected and prior estimations, while the higher nonlinear relationships of the residuals between the prior and real estimations, the mechanistic features have higher impacts on the compensation model compared to the correction model. Both models show improved monitoring and reveal the performance of the proposed mechanistic leading residual learning models. More importantly, the potential of early thermal safety warning can be achieved during operation and aging.

Finally, cylindrical batteries from Ref. 41. are employed for the verification of the proposed models by further considering unseen working conditions. Three different validation cases are generated using cells with different synthetic profiles or real driving profiles-based aging. Specifically, as shown in Fig. 5a, cells with independent highway or urban loading profiles are divided into a training dataset and the validation 1 dataset. The loading profile of combined highway and urban loadings is included in the validation 2 dataset, and the cells with two city driving profiles are used in the validation 3 dataset. Therefore, we evaluated our mechanistic leading residual learning models with different real applications, considering unseen validation scenarios. Typical loadings and the related prior estimations are demonstrated in Fig. S21. The state monitoring and corresponding interpretation results are shown in Fig. 5b, c, respectively. More detailed results for the three validation cases are shown in Figs. S22–S24 and Tables S5–S6. MAE for the SOC estimation keeps below 1.5% and below 1.8% for SOH estimations for all three validation cases with both the correction model and the compensation model. All the cases indicate enhanced state monitoring performance compared to conventional model-based methods, even under unseen loadings for the whole lifespan. Similar interpretations can be summarized from the SHAP results, i.e., the prior estimations plays significant roles for both the correction model and compensation model, while the mechanistic features provide critical information and have higher impacts on the compensation model compared to the correction model. Though the prior estimations are the most significant information, mechanistic parameters are essential for the model, as shown in Fig. S25, where the errors may increase instead by only using prior estimation for the model input. Only one fresh OCV curve is used for the estimations instead of updating during aging, which would require more tests and need further calibrations in real-world applications, and the results indicate the residual models are effective to learn the change during aging for accuracy enhancement.

In real-world applications, the sampling frequency is usually reduced, e.g., 0.1 Hz, so that the estimation accuracy is further deteriorated. The prior estimations with enlarged errors can also be enhanced by our mechanistic leading residual learners. We down-sampled the original data with 0.1 Hz and re-tested the three validation cases to evaluate the robustness of our models in potential real application conditions. The down-sampled currents and original currents (1 Hz) for different loading profiles are shown in Fig. S26, where the currents with 0.1 Hz lost fidelity. Therefore, the prior estimations became worse compared to the results with 1 Hz sampled currents (Fig. 5b), as shown in Fig. S27. Based on the proposed correction model and compensation model, the accuracies are enhanced with MAE less than 1.92% and 3.58% for SOC and SOH monitoring (which are more than 3.4% and 5.5% of filtering based-prior estimations for all the three cases), respectively, demonstrating the effectiveness in low sampling frequency applications. Through model interpretation, we noticed that, though the prior estimations are still the most important information, the accuracy is worse in these cases, which makes the mechanistic features become more important for the model to enhance state monitoring.

We have verified the models using real-world driving cycles for potential real-world EV applications. To further evaluate the model extrapolation to other application scenarios, such as grid storage, we applied the same model trained based on the training cells in Fig. 5a for the verification of cells aged with three different kinds of periodic profiles. The results for the SOC and SOH monitoring are shown in Fig. S28, indicating the effectiveness of the proposed model for accuracy enhancement even under quite different unseen loading profiles and aging trajectories.

The mechanistic leading residual learners facilitate collaborative monitoring of different battery states throughout the entire lifespan, with capacity updating during each dynamic discharging cycle. To validate their effectiveness and the extrapolation capability of the proposed pipeline, we applied these learners for the monitoring of a cell from dataset 1 aged under hybrid discharging loads with varying temperature conditions to emulate the daily usage of EVs in seasonal scenarios, as shown in Fig. 6. The data from this cell is not included in the training dataset for unseen condition monitoring and the test is conducted every 10 cycles for demonstration consideration. Real-time SOC estimations at four distinct aging stages (Fig. 6a–d) across the lifespan demonstrate the model’s ability to track SOC variations during dynamic operations accurately. Note that without closed-loop state updating, the estimation errors are enlarged during operation, as compared in Table S7. The two residual learners’ effectiveness in monitoring SOC during dynamic operations over the full lifespan is further evidenced in Fig. 6e, confirming their capability to accurately track SOC curves in varying DODs, thereby ensuring safe operations. A notable advancement of our model is its ability to precisely track capacity degradation under random partial dynamic operation profiles, leveraging the mechanistic leading residual learners.

As illustrated in Fig. 6e, the residual learners successfully follow the degradation path during dynamic operations, enabling timely capacity updates for real-time SOC monitoring and guiding necessary maintenance. The effectiveness and the extrapolation capability are proven to be good for the proposed pipeline. Additional results and comparisons are presented in Fig. S29 and Tables S7–S8. Interestingly, the prior estimation of capacity may become more accurate sometimes. This is because we adopted the closed-loop update for SOC and capacity, so the prior estimation can be very accurate when the SOC estimations have high accuracy. Another interesting application verification is for the testing of the cell with a clearly higher discharge capacity than other cells. We need to normalize the prior capacity by dividing the fresh capacity instead of the nominal capacity here, and then effective monitoring can be achieved, as shown in Fig. S30. Finally, an application for highway loading, the cell with a short lifetime, is also achieved in Fig. S30. Therefore, our model is also easy to transfer under different application scenarios to help enhance the state monitoring.

Then, we show the application for dataset 3 with unseen testing conditions under real-world driving profiles. Note that we also use the dynamic cycle after the diagnostic test for verifications, so the degradation curve is composed of sparse points. By using the training cells in Fig. 5a to train the residual learners, the state monitoring under unseen working conditions (cell 92 aged using real-world driving protocols) entire the lifespan are enhanced, as shown in Fig. 6f. Unseen validation for exemplary cells working under another real city protocols (cell 95) and synthetic Urban + Highway (cell 88) protocols are also shown in Fig. 6f, demonstrating the stability and robustness of our model under varying unseen working conditions, which prove the good extrapolation capability of the proposed mechanistic leading residual learning pipelines. Numerical results listed in Table S9 indicate that the MAE for all the tests is below 1.8% for both SOC and SOH estimation. With enhanced SOC and SOH estimation accuracy through closed-loop operations possibly exceeding 50%, battery systems are poised for improved management and operational performance. This closed-loop structure not only supports effective state monitoring but also self-updates to maintain functionality and provide safety warnings, thereby optimizing the overall management of the battery system without the need for costly and time-consuming offline testing. Furthermore, we evaluate the computational cost for consideration of practical applications. The execution time for the full discharge at each SOH point in Fig. 6f is shown in Fig. S31. Due to the low dimension of features and with a simple machine learning model, only a slight additional time (less than 1 second for the full discharge) is required for the residual learning models for accuracy enhancement.

Overall, we employed our mechanistic leading residual learning with two pipelines for enhanced state monitoring during the whole lifespan with feature interpretations and validations using different battery types and verification cases, which indicates high generalizations of our models. Nevertheless, the verifications are still conducted separately for the three datasets, though varying conditions are considered. It is still hard for cross-battery material applications due to the large discrepancies in the mechanistic features of different batteries caused by the chemical properties. Therefore, future work needs to further investigate the improved approach enabling applications across different battery types, especially for lithium-ion phosphate batteries, which pose a significant challenge to address the impact of hysteresis and OCV flatness in state estimations. In addition, the proposed mechanistic leading hybrid modeling framework is worthy of investigation for improving the electrochemical model fidelity.

Methods
The experimental setup is aimed at emulating real dynamic aging. To this end, three datasets are included in this work. The first dataset includes three different dynamic loading profiles to age pouch batteries (MGLSPIM08HP 8Ah high power pouch cell) under both constant and varying temperature conditions. Specifically, the representative loading profiles for urban (UDDS), highway (HWFET), and hybrid (combination of UDDS and HWFET) working conditions are employed for discharging, while the multi-constant current charging is used for charging during the aging test34. The chemistry is nickel cobalt aluminum Oxide (NCA) and graphite, and the voltage limit is 2.8–4.2 V. The loading currents of each type of cycling test are shown in Fig. S2. Two different temperature conditions, including constant temperature (25°C) and variable temperature (15-25-35°C) are set for the testing environment. More details about the experiment and data processing are depicted in Note S5. In the second dataset, we use three large-format prismatic cells (L148N50B 50Ah), which are widely employed in EVs, for dynamic aging with WLTP profiles. The cathode chemistry is nickel-manganese-cobalt (NMC), the anode is graphite, and the voltage range is set as 2.75–4.3 V. One cell is charged with a constant current of 2C, and the other employs pulse currents (1 Hz, 50% duty cycle, 4C amplitude) for charging. The loading currents are shown in Fig. S5, and the aging test is conducted in a thermal chamber with a controlled 35°C. The sampling frequency is set as 10 Hz, which enables the tracking of variations during pulse loadings. The OCV test is conducted with a low constant current of C/20. The third dataset is from Stanford University, where the cylindrical cells are aged with both synthetic urban and highway dynamic loadings, and more practical real-world loading41. Periodical diagnostic tests are conducted where the real capacities can be obtained. The first dynamic discharge cycle after the diagnostic test is used for verification, where the real capacity can be obtained for verification from the diagnostic test. Furthermore, the sampling frequency is artificially reduced to 0.1 Hz to generate data for the robustness verification of our models in real-world applications.

Battery SOC and SOH, which represent the charge state in a short-time horizon and the health state in a long-time horizon, are monitored for the verification of the proposed mechanistic leading residual learning pipelines. The SOC definition is expressed as

SOC(t) = SOC(0) + 1/Cap * \int_{0}^{t} I dt (1)

where I is the loading current, t is the current time, and Cap depicts the battery capacity. With the equivalent circuit model, battery dynamic behavior is captured with the real-time identified model parameters. The parameter matrix identification can be described as

\theta = f(I, V) (2)

The relationship between battery SOC and OCV (identified from \theta) can be mapped through the OCV test, which is shown in Fig. S32, indicating that different battery types have different shapes of OCV curves. Then, the filtering approach is employed for the posterior real-time SOC estimation, which can be expressed as

SOC_{p,t} = g(SOC_{p,t}, \theta_t) (3)

By calculating the released capacity within one selected SOC window, the prior calculation of battery capacity and the corresponding SOH are obtained through

Cap_p = \int_{t1}^{t2} I dt / |SOC_{t2} - SOC_{t1}| (4)

SOH_p = Cap_p / Cap_n (5)

The features from the identified battery model parameters and the statistical features are extracted along with the prior estimations to construct the feature matrix \phi. Specifically, three types of features are extracted in this work. First, the prior estimations for the SOC and SOH from the filtering method serve as the main features for the two residual models. In addition, in order to inform the ML model of the mechanistic states, the features from the mechanistic model are derived. For SOC monitoring, the real-time identified resistance, polarization resistance and capacitance, and the OCV are used as the mechanistic features. For SOH monitoring, in consideration of the sequential information and the aging information derived from the dynamic discharging process, the features such as mean value, standard deviation, and maximum value of the mechanistic parameters and the estimated SOC are derived. The last category is the measured features, i.e., the measured current and voltage for the SOC model and the corresponding derived statistical features. The full description of the features is detailed in Note S2.

Then, the prior estimations and the extracted features are input for the ML model to get enhanced monitoring results. Two residual models are proposed in this work for state monitoring enhancement, and the structures are shown in Fig. S8. The first model is the correction model, and the function is to enhance the prior estimations directly with an improved estimation. The function can be expressed as

Z* = L_1(Z_p, \phi), z \in {SOC, SOH} (6)

where Z_p is the prior estimation and Z* is the enhanced estimation from the correction model. For the second residual model, the compensation model, the function of the ML is to estimate the residuals between the prior estimations and the real values to compensate for the prior estimations. The compensation model for enhanced state monitoring can be expressed as

Z** = Z_p + L_2(Z_p, \phi), z \in {SOC, SOH} (7)

where Z** is the enhanced state estimation through the compensation model. All the notations and abbreviations are summarized in Table S10.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This paper builds valuable battery data for SOC and SOH monitoring, covering diverse protocols and temperatures, and proposes a residual learning framework combining physics-based models with data-driven techniques. Both the dataset and the physics-informed approach could be valuable to the community and the related industrial applications. Below I present my remaining questions for the authors.\n1. I would like to learn more about data summary and statistics. For example, how many battery cells are tested in total? And how many in each charging and discharging condtitions? When you dischage a cell with different conditions (UDDS, HWFET, Hybrid), will one condition selected and continually used throughout all cycles or will these conditions be randomly selected in different cycles?\n2. In addition to random train-test splits, I wonder how will the method behave if only seeing certain discharge conditions during training. For example, if the model is only trained in data covering UDDS and HWFET conditions, how will it perform in the Hybrid condtion? This result can inform how this method could generalize in practice, where the discharge condition may be versatile.\n3. This work does not levearge time-series features. I wonder whether leverage some sequential information within a cycle (for SOC) or across cycles (for SOH) could be helpful.\n4. The overall learning framework is more like a stacking ensemble approach in machine learning, rather than the residual learning concept in the deep learning literature. Could you provide more thoughts or evidences on this? From my understanding, the data-driven learner is stacked on the outputs of the ECM model to produce final predictions.\n(Remarks on code availability)\nWould appreciate if a runnable version of the code could be provided through some online platforms, such as Code Ocean.", "reviewer_2": "This study proposes physics-inspired residual learners for battery state monitoring, By integrating prior knowledge from real-time filtering with physical and statistical features, the method significantly improves the accuracy and robustness of machine learning models. Several issues need to be addressed:\n1. The definition of Capn in the SOH calculation equation is unclear. Why does SOH exceed 100% in Figure 3e?\n2. Regarding the model comparison, what are the advantages of the proposed method compared to SOC and SOH joint estimation methods based on filtering algorithms? Has a comparison been conducted?\n3. When performing parameter identification based on the OCV curve, does the model update the OCV curve according to battery aging?\n4. The paper mentions that the sampling interval of the test data is 1s, but real-world applications may involve larger sampling intervals. Has the applicability of the model in such scenarios been considered?\n5. The study is applied only to NMC batteries, which have a relatively steep OCV curve. Can the proposed method be adapted to LFP batteries, which exhibit significantly different OCV characteristics?\nMinor comments:\n1. In Figure 3d, the denominator of Capp lacks an integral symbol.\n2. The paper includes numerous notations, but most are not explained. It is recommended to provide a notation table in the appendix.\n(Remarks on code availability)", "reviewer_3": "In this manuscript, the authors propose a framework for the joint estimation of SOC and SOH for battery cells undergoing continuous driving cycling under continuous and changing temperatures. The pipeline consists of ECM-based adaptive Kalman filtering for SOC estimation, which is enhanced by a first residual learner, giving a first SOH estimation that is improved upon by a second residual learner. While algorithm architecture, which seems to be the main focus of the paper, may be of interest, there are numerous concerns regarding formal aspects, experimental data, methodology, and result analysis that make me lean against publication.\nFormal aspects:\n1. Although the introduction provides some background on the application of ML for battery monitoring and feature selection, it is not specific enough in relation to the particular approach taken in this work. It is challenging to assess the novelty and contributions of this paper with respect to the vast wealth of literature on ML for battery monitoring and SOH prediction, and the motivation of this work is not fully justified as a result. Also, I assume this is not the first time residual learning has been applied to battery modeling and estimation, so authors should attempt to position their research more clearly in relation to gaps in the literature and describe how the presented approach constitutes a significant contribution to the state of the art.\n2. The manuscript relies too heavily on Supplementary Materials. While this is common practice for Nature journals, several details about the experimental procedures, modeling and feature selection are relevant enough to be included in the main document. For instance, it is not until the last page of the Supplementary Materials that the selected features are defined.\n3. The visualization of most figures should be improved. For example, in Fig. 2a-c, it is difficult to distinguish estimation from experimental results, and also, no grid is provided. This issue is also present in Fig. 5a-d, and Figs. S8 and S18. Fig. 5e is not informative. Some temperature plots in Figs. S4 and S5 are challenging to interpret. In Fig. S23, it is not possible to distinguish original data from fitted data.\nExperimental data:\n1. Important specifications of the battery cell are not provided, such as chemistry, voltage limits, current limits, temperature range and internal resistance.\n2. A detailed list of the test procedures should be provided indicating clearly which cells have been tested under which conditions.\n3. A major piece of information missing is the sample size, which is even more relevant if ML techniques are employed.\n4. While transparency regarding experimental difficulties is appreciated, some of the challenges experienced by the authors with voltage, current and temperature measurements (Note S4) are not such common occurrences in battery testing and may make the reader question the reliability of the presented experimental dataset.\nMethodology:\n1. The authors claim that the employed ML models are grounded on interpretable physical principles, but this is highly debatable. The only ‘physical’ element is the use of the 1-RC ECM as a battery model, which is not properly justified. Many authors have pointed out the shortcomings of RC models and their limited physical insight with respect to internal battery processes. Furthermore, if an ECM is employed in dynamic operation, a consideration of hysteresis voltage modeling should be made.\n2. It is unclear whether the full pseudo-OCV profile is determined, or only discrete points (GITT); Fig. S23 does not provide enough information. Also, I am skeptical that the full OCV profile can be accurately fitted with a low-order polynomial.\n3. What voltage is assigned SOC=0%, and according to what set of criteria? 3.2V is the lower cutoff according to Fig. S23, but 2.75V seems to be the limit judging from Figs. S3, S4 and S5. If the chemistry is NCA/Gr as it appears, usual voltage limits are 4.2V and 2.5V. This implies that the SOC validation range is narrower in reality.\n4. ECM parameters are determined online via a forgetting factor RLS algorithm, and it appears as though they are updated every timestep. This undermines any physical foundation that could be present in the first place.\n5. How is ‘real SOC’ determined? If it is through coulomb counting, then this defeats the purpose of the estimation algorithms.\n6. Similarly, if periodic RPTs are not conducted, how is ‘real SOH’ determined?\n7. There is a risk of circular dependency, as SOC estimation relies on battery capacity (Eq.1), which is updated as a function of the estimated SOC difference (Eq. 4). How is this prevented and how often is capacity updated?\n8. Remaining capacity is not a complete measure of battery health, and other indicators, such as degradation modes and impedance growth should be considered. Also, is the OCV profile assumed to remain unchanged during aging?\n9. It is stated that ‘the presented model can be extended to monitor other relevant states as required’, besides SOC and SOH (lines 115-117). What other states are implied here (temperature, lithium concentration, etc.)? The 1-RC ECM does not contain such information.\nResult analysis:\n1. A recurring observation is that Residual Learner 2 does not appear to improve estimation results with respect to Residual Learner 1, noted in Figs. 2g and 3g, as well as in Tables S1-S5. What is the explanation for this? Is RL2 needed?\n2. In Figs. 5f and S6, is the evolution of capacity with respect to cycle number realistic? The staggered profiles seem to match the fluctuations in ambient temperature, so these are not variations in actual capacity.\n3. The parallel resistance Rp is not included as a feature, only the time constant, but I would expect this parameter to carry more information about internal processes than the ohmic resistance R.\n4. How is this method an improvement over the standard methodology consisting of an adaptive EKF for SOC estimation and a RLS for SOH estimation? This has been proposed and validated extensively in the literature. The ML model seems like an overkill within this framework.\n5. Other papers that employ ML algorithms for SOH prediction (see Refs. [23] and [29]) make an effort to generalize their approaches to different chemistries, which remains the main challenge for ML-based methods.\n6. How is the risk of data leakage treated? Given that only drive cycles are tested, there is a risk that the ML model is learning features specific to those profiles that may not transfer well to real-world stochastic operation, which has recently been shown to decisively influence aging behavior (DOI: 10.1038/s41560-024-01675-8).\nIn summary, the contribution with respect to previous approaches is not clear, and the physical foundation is questionable. The groundtruth values of SOC and SOH are determined via coulomb counting, and the definition of SOH is incomplete. While combining physics-based models with ML is promising, the physical foundation must be more solid and the ML model should provide a qualitative advantage over established approaches in order for the results to be significant.\n(Remarks on code availability)\nThe code does not provide a README file with instructions for running, and does not contain sufficient comments. A sample of input data is not provided to verify performance."}, "author_response": "Dear Editor and Reviewers,\nThank you very much for the constructive feedback. We have revised the manuscript and its supplementary information to address all the issues raised. Please find below our detailed response and revisions. Reviewers’ comments are in blue font, author responses are in plain back font, and changes to the manuscript are in red.\nResponse to Reviewer #1 (Remarks to the Author):\nReviewer #1 Overall Comment: This paper builds valuable battery data for SOC and SOH monitoring, covering diverse protocols and temperatures, and proposes a residual learning framework combining physics-based models with data-driven techniques. Both the dataset and the physics-informed approach could be valuable to the community and the related industrial applications. Below I present my remaining questions for the authors.\nAuthor response: The authors appreciate the reviewer for the careful review and the positive recommendation. In the revised manuscript, we have added more descriptions regarding data generation, model development, and comparative verifications. The detailed responses are listed below, and the revised text is in red in the revised manuscript.\nReviewer #1 Comment 1: I would like to learn more about data summary and statistics. For example, how many battery cells are tested in total? And how many in each charging and discharging condtitions? When you dischage a cell with different conditions (UDDS, HWFET, Hybrid), will one condition selected and continually used throughout all cycles or will these conditions be randomly selected in different cycles?\nAuthor response: In the revised manuscript, we employed three datasets to verify and evaluate the proposed mechanistic leading residual learning pipelines. Specifically, 13 cells (pouch cells) are included in dataset 1 (the cells in the original manuscript), and 6 protocols with 2 different temperature conditions are used for the cycling test. Each cell is aged with one specific protocol throughout the whole test. In addition, we added another experimental dataset with 3 prismatic cells (dataset 2) to the revised manuscript for verification of the proposed models in state monitoring across the whole life span (SOH drops below 50%). Two different charging protocols (constant current and pulse current) are used for the charging, and WLTP dynamic loading is used for the discharging. Finally, 25 cells (cylindrical cells) from the recent public dataset are collected for several unseen verifications. These cells are tested using Highway, Urban, Highway + Urban, and two city profiles for dynamic discharging. In the revised manuscript, we have rewritten the data generation section for detailed descriptions of the three datasets used in this work. The revised text is in red below.\nData generation subsection: We employed three datasets to evaluate the proposed mechanistic leading residual learning models under dynamic loadings for the full life span. Three types of batteries, including pouch, prismatic, and cylindrical cells, which were cycled with urban, highway, and real-world dynamic profiles, are used for verifications under different scenarios.\nTo emulate the dynamic working and the varying environmental conditions encountered in real-world applications, the first dataset subjected 13 pouch batteries (nominal capacity of 8 Ah) to aging under dynamic discharge across the entire lifespan at different temperatures.30 Urban, highway, and hybrid profiles were employed and are illustrated in Fig. S2, while Fig. S3 presents the variations in voltage curves observed during the aging process. Capacity degradation curves relative to equivalent full cycles are shown in Fig. S4, demonstrating that urban dynamic loading profiles generally result in longer battery life compared to highway-based loading, with varying temperature conditions significantly affecting the degradation patterns. There are in total 27840 samples (represented as the dynamic discharge cycles) generated in dataset 1.\nBesides, 3 large-format prismatic cells with a nominal capacity of 50 Ah are used in the second aging test until they die at around 20% SOH with sudden death, i.e., rapid degradation in a few cycles. We employed WLTP (Worldwide Harmonized Light Vehicles Test Procedure) to discharge the cells and used both CC and pulse-constant current (PCC) for charging, as shown in Fig. S5. The pulse was set with an average rate the same as the CC (2 C), while with twice the amplitude. The frequency and duty cycle are 1 Hz and 50%, respectively. The capacity, current, voltage, and temperature curves during aging are shown in Fig. S6. The open circuit voltage (OCV) test was conducted using C/20 constant charging and discharging. The sampling frequency is 10 Hz, and the current and voltage variations for the pulses are tracked precisely. This dataset benefits wide applications, including state monitoring, life prediction, and thermal safety warning. The detailed information of the tested pouch and prismatic cell is listed in Table S1. The sample size, i.e., dynamic discharge curves until capacity drops below 50% of the fresh capacity, is 8590.\nFinally, datasets using 25 cylindrical cells from Ref. 37, which were aged with synthetic highway and urban, as well as real city driving profiles, are employed to verify our model for use under different unseen working conditions. Typical current and voltage curves during aging under different loading profiles are shown in Fig. S7, indicating varying discharge patterns observed among different cycling, which therefore cause diverse working scenarios and aging trajectories. We used the first dynamic test after the diagnostic test as the dynamic cycling data during aging, the C/2 discharge capacity before the dynamic cycle as the referenced capacity, and the C/40 discharge curve of a new cell as the OCV curve. There are 483 dynamic discharge curves generated for the model evaluation.\nReference: 37. Geslin, A. et al. Dynamic cycling enhances battery lifetime. Nat Energy, 10, 172–180 (2025).\nReviewer #1 Comment 2: In addition to random train-test splits, I wonder how will the method behave if only seeing certain discharge conditions during training. For example, if the model is only trained in data covering UDDS and HWFET conditions, how will it perform in the Hybrid condtion? This result can inform how this method could generalize in practice, where the discharge condition may be versatile.\nAuthor response: This work considers both random split and unseen tests for the evaluation of our method. Specifically, we first randomly split the discharge cycles with different data ratios to test the model’s performance. With a reduced ratio of training data such as10%, many unseen scenarios can also be tested, reflecting on the uncovered data distributions in Fig. S15. Then, we conducted several unseen tests using dataset 3, where the discharging protocols with Urban or Highway are used for training, and the protocols of Highway + Urban, and two real city protocols are used for testing for the extrapolation evaluations. Finally, in dataset 1, we also exclude the hybrid profile with varying temperature in the training scenarios and employ a continuous updating of SOC and SOH to evaluate the continuous monitoring of unseen conditions in the whole life span. This kind of application verification of unseen conditions has also been applied for the verification of the real-world driving (city 1 and city 2) protocols, and the results are shown in Fig. 6. We have added more descriptions of the unseen verifications in the revised manuscript, which are in red text below.\nVarying verifications subsection: Finally, cylindrical batteries from Ref. 37 are employed for the verification of the proposed models by considering unseen working conditions. Three different validation cases are generated using cells with different synthetic profiles or real driving profiles-based aging. Specifically, as shown in Fig. 5a, cells with independent highway or urban loading profiles are divided into a training dataset and the validation 1 dataset. The loading profile of combined highway and urban loadings is included in the validation 2 dataset, and the cells with two city driving profiles are used in the validation 3 dataset. Therefore, we evaluated our mechanistic leading residual learning models with different real applications, considering unseen validation scenarios. Typical loadings and the related prior estimations are demonstrated in Fig. S20. The state monitoring and corresponding interpretation results are shown in Figs. 5bc, respectively. More detailed results for the three validation cases are shown in Figs. S21–S23 and Tables S5–S6. MAE for the SOC estimation keeps below 1.5% and below 1.8% for SOH estimations for all three validation cases with both the correction model and the compensation model. All the cases indicate enhanced state monitoring performance compared to conventional model-based methods, even under unseen loadings for the whole life span. Similar interpretations can be summarized from the SHAP results, i.e., the prior estimations play significant roles for both the correction model and compensation model, while the mechanistic features provide critical information and have higher impacts on the compensation model compared to the correction model. Though the prior estimations are the most significant information, mechanistic parameters are essential for the model, as shown in Fig. S24, where the errors may increase instead by only using prior estimation for the model input.\nIn real-world applications, the sampling frequency is usually reduced, e.g., 0.1 Hz, so that the estimation accuracy is further deteriorated. The prior estimations with enlarged errors can also be enhanced by our mechanistic leading residual learners. We down-sampled the original data with 0.1 Hz and re-tested the three validation cases to evaluate the robustness of our models in potential real application conditions. The down-sampled currents and original currents (1 Hz) for different loading profiles are shown in Fig. S25, where the currents with 0.1 Hz lost fidelity. Therefore, the prior estimations became worse compared to the results with 1 Hz sampled currents (Fig. 5b), as shown in Fig. S26. Based on the proposed correction model and compensation model, the accuracies are enhanced with MAE less than 1.92% and 3.58% for SOC and SOH monitoring (which are more than 3.4% and 5.5% of filtering based-prior estimations for all the three cases), respectively, demonstrating the effectiveness in low sampling frequency applications. Through model interpretation, we noticed that, though the prior estimations are still the most important information, the accuracy is worse in these cases, which makes the mechanistic features become more important for the model to enhance state monitoring.\na c\nData Test Cells\nHighway 60, 62, 72\nTraining\nUrban 66, 68, 74, 76, 82, 84\nHighway 59, 64, 71\nValidation 1\nUrban 65, 67, 73, 75, 81\nHighway\nValidation 2 79, 80, 87, 88\n+ Urban\nCity 1 91, 92\nValidation 3\nCity 2 95, 96\nCorrection Compensation\nb\nFig. 5. Performance evaluation for cylindrical cells with unseen working conditions. a. Training and testing cells split strategies. b. SOC and SOH monitoring results for the three validation cases. c. Feature interpretation of the mechanistic leading models for the SOC and SOH monitoring.\nClosed-loop application entire life span subsection: The mechanistic leading residual learners facilitate collaborative monitoring of different battery states throughout the entire lifespan, with capacity updating during each dynamic discharging cycle. To validate their effectiveness and the extrapolation capability of the proposed pipeline, we applied these learners for the monitoring of a cell from dataset 1 aged under hybrid discharging loads with varying temperature conditions to emulate the daily usage of EVs in seasonal scenarios, as shown in Fig. 6. The data from this cell is not included in the training dataset for unseen condition monitoring.\na b c d\ne\nCorrection Compensation Real values\nCapacity updating during operations and aging\nf\nCity 1 City 2 Urban + Highway\nFig. 6. Application demonstrations for the continuous monitoring of battery charge and health status of the entire life span. a-d. SOC estimation at four different aging stages and the demonstration of random working SOC ranges used for capacity estimation. e. SOC and SOH curves tracking for the whole lifespan. f. Unseen applications for the closed-loop state monitoring under both real-world and synthetic hybrid loadings.\nThen, we show the application for dataset 3 with unseen testing conditions under real-world driving profiles. Note that we also use the dynamic cycle after the diagnostic test for verifications, so the degradation curve is composed of sparse points. By using the training cells in Fig. 5a to train the residual learners, the state monitoring under unseen working conditions (cell 92 aged using real-world driving protocols) entire the life span are enhanced, as shown in Fig. 6f. Unseen validation for exemplary cells working under another real city protocols (cell 95) and synthetic Urban + Highway (cell 80) protocols are also shown in Fig. 6f, demonstrating the stability and robustness of our model under varying unseen working conditions, which prove the good extrapolation capability of the proposed mechanistic leading residual learning pipelines.\nNumerical results listed in Table S9 indicate the MAE for all the tests is below 1.8% for both SOC and SOH. With enhanced SOC and SOH estimations through closed-loop operations, battery systems are poised for improved management and operational performance. This closed-loop structure not only supports effective state monitoring but also self-updates to maintain functionality and provide safety warnings, thereby optimizing the overall management of the battery system without the need for costly and time-consuming offline testing.\nReviewer #1 Comment 3: This work does not levearge time-series features. I wonder whether leverage some sequential information within a cycle (for SOC) or across cycles (for SOH) could be helpful.\nAuthor response: In this work, we calculated features for SOH estimation from sequential information, i.e., the statistical values from the dynamic discharge curves with random voltage ranges. Although the direct sequential inputs are not used, the calculated features, like the standard deviation, contain sequential information. In addition, the machine learning models used in this paper are random forests, so we use the features instead of the whole sequential input. Based on the SHAP analysis results, these statistical features derived from the sequential dynamic discharging information are effective for the performance improvement, as demonstrated in Figs. 3–5. In the revised manuscript, we added more descriptions for the feature extraction from the sequential dynamic discharges, which are highlighted in red.\nMechanistic leading residual learning pipelines for enhanced state monitoring subsection: To promote the real-time application of the pipelines, the mechanistic model adopts an equivalent circuit model in this work. For real-time battery SOC estimation, online parameter identification and adaptive filtering methods are employed (see Note S1 for a detailed procedure), where the SOC prior estimations and mechanistic model parameters (e.g., resistance and polarization features) are obtained. Coupled with SOC estimation and coulomb counting for released capacity calculation, the battery capacity can be derived to calculate the prior estimation of battery SOH. Meanwhile, measured parameters (e.g., voltage, current, and released capacity) and identified mechanistic parameters (e.g., identified resistance, OCV, and polarization resistance/capacitance) during dynamic discharging are used to calculate health features, which are composed of several statistical values including mean value, standard deviation, and min and max values of these sequential parameters from a random discharge phase. Detailed feature calculations are described in Methods and Notes S2–S3.\nMethods Section: The features from the identified battery model parameters and the statistical features are extracted along with the prior estimations to construct the feature matrix φ . Specifically, three types of features are extracted in this work. First, the prior estimations for the SOC and SOH from the filtering method serve as the main features for the two residual models. In addition, in order to inform the machine learning model of the mechanistic states, the features from the mechanistic model are derived. For SOC monitoring, the real-time identified resistance, polarization resistance and capacitance, and the OCV are used as the mechanistic features. For SOH monitoring, in consideration of the sequential information and the aging information derived from the dynamic discharging process, the features such as mean value, standard deviation, and maximum value of the mechanistic parameters and the estimated SOC are derived. The last category is the measured features, i.e., the measured current and voltage for the SOC model and the corresponding derived statistical features. The full description of the features is detailed in Note S2.\nReviewer #1 Comment 4: The overall learning framework is more like a stacking ensemble approach in machine learning, rather than the residual learning concept in the deep learning literature. Could you provide more thoughts or evidences on this? From my understanding, the data-driven learner is stacked on the outputs of the ECM model to produce final predictions.\nAuthor response: The mechanistic leading residual learning is proposed in this work for a hybrid modeling in battery state monitoring, and a closed-loop state updating framework is designed to ensure the effectiveness in the applications under full life. The machine learning part performs as the residual compensation for the prior estimations from the model-based filtering method. The prior estimations from the model-based method, together with the extracted mechanistic features, serve as the inputs for the following machine learning models. In order to make the functions of the two pipelines clearer, we defined the models as a correction model and another compensation model. The function of the correction model is to provide the final estimations directly, while the compensation model estimates the residuals between the prior estimations and the real values and compensates for the prior estimations. This is because the mechanistic model-based method has some limitations, such as the influence of inaccurate parameters and battery aging. So, the following machine learning part is expected to enhance the prior estimations while being inspired by the mechanistic features and the model-based estimations. It can be recognized as a sequential hybrid model consisting of the model-based part for prior estimation and the machine learning part for residual compensation. In the revised manuscript, we have added more descriptions of the two residual learning models, which are in red text below.\nMechanistic leading residual learning pipelines for enhanced state monitoring subsection:\nHere, we introduce mechanistic leading residual learners, incorporating a correction model and a compensation model to enhance the state monitoring results from prior estimations. As shown in Figs. 1b and S8, the prior estimations and extracted features serve as input for the machine learning to output an enhanced estimation and a residual for the prior estimation compensation. Therefore, the function of the correction model is to enhance the prior estimation by direct state predictions, while the compensation model enhances the estimations by adding the predicted residuals to the prior estimations. The prior estimation from the mechanistic model provides stable guidance, and those feature matrices help capture the expected battery states, condition indicators from the real battery responses, as well as the mechanistic features describing the internal conditions. Machine learning is built to compensate for the model uncertainties and other influences like changed OCV under varying conditions; therefore, the OCV and other model parameters do not require careful calibration under temperature and aging. The effectiveness of this method is demonstrated through two key states: real-time SOC and the more slowly varying SOH. Note that the compared prior estimations represent the conventional model-based filtering methods. The coulomb counting-based calculation is employed to obtain the referenced real SOC and SOH for the evaluations of the conventional model-based filtering method and the proposed residual learning models, while it is not used in any part of the modeling pipelines. Additionally, our model can be easily extended to monitor other relevant states as required, for example, the sensor-less temperature estimations based on the mechanistic features and the lamped-mass thermal model-based prior estimations.21\nMethods Section: Then, the prior estimations and the extracted features are input for the ML model to get enhanced monitoring results. Two residual models are proposed in this work for state monitoring enhancement, and the structures are shown in Fig. S8. The first model is the correction model, and the function is to enhance the prior estimations directly with an improved estimation. The function can be expressed as\nZ* = L (Z ,φ), z {SOC,SOH}, (6)\n1 p\nwhere Z is the prior estimation and Z* is the enhanced estimation from the correction model. For\np\n∈\nthe second residual model, the compensation model, the function of the machine learning is to estimate the residuals between the prior estimations and the real values to be compensated for the prior estimations. The compensation model for enhanced state monitoring can be expressed as\nZ** = Z +L (Z ,φ), z {SOC,SOH}, (7)\np p\n2\nwhere Z** is the enhanced state estimation through t∈he compensation model. All the notations and abbreviations are summarized in Table S10.\nReviewer #1 Comment 5: Would appreciate if a runnable version of the code could be provided through some online platforms, such as Code Ocean.\nAuthor response: We will provide the whole code package for the proposed model for open access after publication. In the revised manuscript, we also attached the runnable code for review.\nData Availability Section: Data from the pouch and prismatic cells and the main codes will be available at 10.5281/zenodo.15606247 after publication. The cylindrical cell dataset is available from Stanford University at https://purl.stanford.edu/td676xr4322.\nResponse to Reviewer #2 (Remarks to the Author):\nReviewer #2 Overall Comment: This study proposes physics-inspired residual learners for battery state monitoring, By integrating prior knowledge from real-time filtering with physical and statistical features, the method significantly improves the accuracy and robustness of machine learning models. Several issues need to be addressed:\nAuthor response: We thank the reviewer for the careful review and valuable comments. In the revised manuscript, we have added two more datasets with different battery types for the verifications, more detailed descriptions for the methods and comparison results, and a summary table for the abbreviations and notations. Moreover, we reduced the sampling frequency to 0.1 Hz to demonstrate the effectiveness of the proposed residual learning models in potential real-world applications, with notable accuracy improvement compared to the conventional model-based filtering methods. The detailed responses are listed below, and the revised texts are highlighted in red in the revised manuscript.\nReviewer #2 Comment 1: The definition of Capn in the SOH calculation equation is unclear. Why does SOH exceed 100% in Figure 3e?\nAuthor response: Since the SOH is calculated by dividing the capacity by the nominal capacity in Figure 3e (Figure 2d in the revised manuscript), and the inconsistency in battery manufacturing (reflected by Fig. S4 and Fig. S12), the calculated SOH could exceed 100%. As requested, we have added more descriptions of the SOH definition in the revised manuscript, which is in red text below.\nMechanistic leading residual learning pipelines for enhanced state monitoring subsection:\nThe predictions of the two residual learners, i.e., corrected SOH and compensations (Δ SOH) for the prior SOH respectively, shown in Figs. 2d e demonstrate the coefficient of determination (R2) larger than 0.98 and 0.84 respectively, here, the SOH is calculated by dividing the present capacity\n–\n(Cap ) by the nominal capacity (Cap ).\np n\nReviewer #2 Comment 2: Regarding the model comparison, what are the advantages of the proposed method compared to SOC and SOH joint estimation methods based on filtering algorithms? Has a comparison been conducted?\nAuthor response: The conventional filtering-based estimations are recognized as the prior estimations in our paper, which are compared with our models. The proposed mechanistic leading residual learning is designed to enhance the conventional model-based filtering methods. Two different types of residual learning pipelines, including a correction model and a compensation model, are designed to provide improved estimations. As demonstrated in Figs. 2a and f, Figs. 3d\nf, Figs. 4c d, and Fig. 5b in the revised manuscript, the comparisons between the conventional\n–\nfiltering methods and the enhanced estimations based on the proposed residual learning models\n–\nare given, which demonstrate the improved accuracy and reliability of the proposed method in varying verification scenarios across three different battery datasets. In the revised manuscript, we added an additional sentence for the clarification of the model-based filtering methods, which is in red text.\nMechanistic leading residual learning pipelines for enhanced state monitoring subsection:\nNote that the compared prior estimations represent the conventional model-based filtering methods.\nReviewer #2 Comment 3: When performing parameter identification based on the OCV curve, does the model update the OCV curve according to battery aging?\nAuthor response: We proposed the mechanistic leading residual learning to enhance the accuracy and reliability of the conventional model-based filtering methods. OCV curves serve as a key reference for SOC estimation in conventional model-based methods, which largely determines the accuracy. However, OCV curves vary with aging and temperature, thus influencing the estimation accuracy. With the proposed residual learning model, machine learning is used to compensate for the deficiencies of the estimations from model-based methods, such as inaccurate model parameters and the variation of OCV curves with aging and temperatures, which largely improves the model's robustness under different conditions. Therefore, we don’t need the OCV curves to be carefully measured regarding different aging and temperature conditions by using the proposed residual learning models. In the revised manuscript, we have added more descriptions of the proposed two pipelines and the clarifications of the OCV references, which are in red text below.\nMechanistic leading residual learning pipelines for enhanced state monitoring subsection:\nHere, we introduce mechanistic leading residual learners, incorporating a correction model and a compensation model to enhance the state monitoring results from prior estimations. As shown in Figs. 1b and S8, the prior estimations and extracted features serve as input for the machine learning to output an enhanced estimation and a residual for the prior estimation compensation. Therefore, the function of the correction model is to enhance the prior estimation by direct state predictions, while the compensation model enhances the estimations by adding the predicted residuals to the prior estimations. The prior estimation from the mechanistic model provides stable guidance, and those feature matrices help capture the expected battery states, condition indicators from the real battery responses, as well as the mechanistic features describing the internal conditions. Machine learning is built to compensate for the model uncertainties and other influences like changed OCV under varying conditions; therefore, the OCV and other model parameters do not require careful calibration under temperature and aging. The effectiveness of this method is demonstrated through two key states: real-time SOC and the more slowly varying SOH. Note that the compared prior estimations represent the conventional model-based filtering methods. The coulomb counting-based calculation is employed to obtain the referenced real SOC and SOH for the evaluations of the conventional model-based filtering method and the proposed residual learning models, while it is not used in any part of the modeling pipelines. Additionally, our model can be easily extended to monitor other relevant states as required, for example, the sensor-less temperature estimations based on the mechanistic features and the lamped-mass thermal model-based prior estimations.21.\nReviewer #2 Comment 4: The paper mentions that the sampling interval of the test data is 1s, but real-world applications may involve larger sampling intervals. Has the applicability of the model in such scenarios been considered?\nAuthor response: As requested, and considering the practical EV applications, we evaluated the proposed mechanistic-learning residual learning models by artificially reducing the sampling frequency to 0.1 Hz. To evaluate the model in the most relevant real-world applications, we provide verifications using the real-world driving protocols, i.e., the third dataset from Stanford University in this work. As shown in Fig. S25, the fidelity of the measured current was reduced a lot, which consequently reduced the accuracy of the model-based methods. Our residual learning models still perform effectively to improve the monitoring accuracy for both SOC and SOH, as shown in Fig. S26. We also added more descriptions of the evaluation results in the revised manuscript, which are highlighted in red text below.\nVarying verifications subsection: In real-world applications, the sampling frequency is usually reduced, e.g., 0.1 Hz, so that the estimation accuracy is further deteriorated. The prior estimations with enlarged errors can also be enhanced by our mechanistic leading residual learners. We down-sampled the original data with 0.1 Hz and re-tested the three validation cases to evaluate the robustness of our models in potential real application conditions. The down-sampled currents and original currents (1 Hz) for different loading profiles are shown in Fig. S25, where the currents with 0.1 Hz lost fidelity. Therefore, the prior estimations became worse compared to the results with 1 Hz sampled currents (Fig. 5b), as shown in Fig. S26. Based on the proposed correction model and compensation model, the accuracies are enhanced with MAE less than 1.92% and 3.58% for SOC and SOH monitoring (which are more than 3.4% and 5.5% of filtering based-prior estimations for all the three cases), respectively, demonstrating the effectiveness in low sampling frequency applications. Through model interpretation, we noticed that, though the prior estimations are still the most important information, the accuracy is worse in these cases, which makes the mechanistic features become more important for the model to enhance state monitoring.\nSupplementary Fig. S25. Original currents and down-sampled to 0.1 Hz currents for different loading profiles.\na c\nCorrection Compensation\nb\nSupplementary Fig. S26. Validation results with reduced current sampling frequency (0.1 Hz). a. MAE for the SOC and SOH monitoring results. b. RMSE for the SOC and SOH monitoring results. c. SHAP interpretation results for the two mechanistic leading residual learners.\nReviewer #2 Comment 5: The study is applied only to NMC batteries, which have a relatively steep OCV curve. Can the proposed method be adapted to LFP batteries, which exhibit significantly different OCV characteristics?\nAuthor response: While we don’t use the LFP batteries for the verifications, we employed three different battery types, including the pouch, prismatic, and cylindrical cells, for the verifications in the revised manuscript. The OCV curves of these three different battery types also show differences, as shown in Fig. S23. The verifications added in the revised manuscript indicate the effectiveness of the proposed models in state monitoring for different battery types. These batteries also have different anode and cathode materials. The added verification results are given below, and the discussion of future work on LFP battery verification is also added in the revised manuscript. The revised text is in red below.\nVarying verifications subsection: We further employed our model for the state monitoring of different battery types with different aging patterns caused by different aging conditions. The dynamic aging of large-format prismatic cells is verified. As shown in Fig. 4a, three large-format prismatic cells are aged with CC/PCC charging and WLTP dynamic discharging, which experience accelerated aging after SOH drops below the knee points. The average current of the pulses remains the same as the constant current, as shown in Fig. 4b. While different battery impedances cause the voltage to vary differently, and according to the heat generation, the cells have higher temperatures using pulse current compared to the constant current. Around 10℃ difference for the maximum temperature and 3.5℃ difference for the average temperature during cycling are seen for the two different charging protocols. These large thermal condition differences during cycling become the main cause of the different lifetimes for the cells under constant or pulse currents. Previous related works have revealed that pulse currents have an influence on battery lifetime. The battery aging is coupled and influenced by current profiles, temperature, current rates, etc.; thus, we advocate coupling mechanisms and battery types during evaluations for the lifetime extension strategy designs.\nWe continuously age the batteries until they die, where accelerated aging is seen. As shown in Fig. S19, the battery is heavily swelled after aging, and the thermal safety issue (largely and rapidly increased temperature) occurred as demonstrated in Fig. 5a. Interestingly, the battery capacity degradation accelerated earlier (around 100 cycles) than the thermal safety issue happened, and the identified mechanistic resistances also show earlier detection. Therefore, accurate estimations of the battery capacity and mechanistic states are also crucial to guide safety management, which helps with early warning of battery safety conditions before thermal issues during operations.\nThe monitoring results for the cells, until SOH (current capacity divided by initial capacity) drops below 50%, are shown in Figs. 4c d, where the total discharging curves are randomly divided into training and testing datasets with a 50% portion of each. The mechanistic leading correction model\n–\nand compensation model show improved SOC and SOH monitoring for the full life span with MAE of less than 0.50% and 0.68%, respectively, as listed in Table S4. SHAP interpretation also indicates the most important impact of the prior estimation and reflects the importance of the mechanistic features. Due to the high linear relationships between the corrected and prior estimations, while the higher nonlinear relationships of the residuals between the prior and real estimations, the mechanistic features have higher impacts on the compensation model compared to the correction model. Both models show improved monitoring and reveal the performance of the proposed mechanistic leading residual learning models. More importantly, the potential of early thermal safety warning can be achieved during operation and aging.\na b\nPulse Rest Pulse Pulse Rest Pulse\nDelay c\nΔT ≈ 10 oC\nThermal safety\nΔT ≈ 3.5 oC\nd\nFig. 4. Performance evaluation for prismatic cells. a. Full life span degradation curves, temperature curves, and resistance curves of the three cells during aging. b. Illustration of the constant and pulse loading currents and the corresponding voltage curves. c. SOC monitoring results and interpretations for the mechanistic leading correction and compensation models. d. SOH monitoring results and interpretations for the mechanistic leading correction and compensation models.\nFinally, cylindrical batteries from Ref. 37 are employed for the verification of the proposed models by considering unseen working conditions. Three different validation cases are generated using cells with different synthetic profiles or real driving profiles-based aging. Specifically, as shown in Fig. 5a, cells with independent highway or urban loading profiles are divided into a training dataset and the validation 1 dataset. The loading profile of combined highway and urban loadings is included in the validation 2 dataset, and the cells with two city driving profiles are used in the validation 3 dataset. Therefore, we evaluated our mechanistic leading residual learning models with different real applications, considering unseen validation scenarios. Typical loadings and the related prior estimations are demonstrated in Fig. S20. The state monitoring and corresponding interpretation results are shown in Figs. 5bc, respectively. More detailed results for the three validation cases are shown in Figs. S21–S23 and Tables S5–S6. MAE for the SOC estimation keeps below 1.5% and below 1.8% for SOH estimations for all three validation cases with both the correction model and the compensation model. All the cases indicate enhanced state monitoring performance compared to conventional model-based methods, even under unseen loadings for the whole life span. Similar interpretations can be summarized from the SHAP results, i.e., the prior estimations play significant roles for both the correction model and compensation model, while the mechanistic features provide critical information and have higher impacts on the compensation model compared to the correction model. Though the prior estimations are the most significant information, mechanistic parameters are essential for the model, as shown in Fig. S24, where the errors may increase instead by only using prior estimation for the model input.\na c\nData Test Cells\nHighway 60, 62, 72\nTraining\nUrban 66, 68, 74, 76, 82, 84\nHighway 59, 64, 71\nValidation 1\nUrban 65, 67, 73, 75, 81\nHighway\nValidation 2 79, 80, 87, 88\n+ Urban\nCity 1 91, 92\nValidation 3\nCity 2 95, 96\nCorrection Compensation\nb\nFig. 5. Performance evaluation for cylindrical cells with unseen working conditions. a. Training and testing cells split strategies. b. SOC and SOH monitoring results for the three validation cases. c. Feature interpretation of the mechanistic leading models for the SOC and SOH monitoring.\nIn real-world applications, the sampling frequency is usually reduced, e.g., 0.1 Hz, so that the estimation accuracy is further deteriorated. The prior estimations with enlarged errors can also be enhanced by our mechanistic leading residual learners. We down-sampled the original data with 0.1 Hz and re-tested the three validation cases to evaluate the robustness of our models in potential real application conditions. The down-sampled currents and original currents (1 Hz) for different loading profiles are shown in Fig. S25, where the currents with 0.1 Hz lost fidelity. Therefore, the prior estimations became worse compared to the results with 1 Hz sampled currents (Fig. 5b), as shown in Fig. S26. Based on the proposed correction model and compensation model, the accuracies are enhanced with MAE less than 1.92% and 3.58% for SOC and SOH monitoring (which are more than 3.4% and 5.5% of filtering based-prior estimations for all the three cases), respectively, demonstrating the effectiveness in low sampling frequency applications. Through model interpretation, we noticed that, though the prior estimations are still the most important information, the accuracy is worse in these cases, which makes the mechanistic features become more important for the model to enhance state monitoring.\nOverall, we employed our mechanistic leading residual learning with two pipelines for enhanced state monitoring during the whole life span with feature interpretations and validations using different battery types and verification cases, which indicates high generalizations of our models. Nevertheless, the verifications are still conducted separately for the three datasets, though varying conditions are considered. It is still hard for cross-battery material applications since the large discrepancies in the mechanistic features of different batteries. Therefore, the future work needs to further investigate the improved approach enabling applications across different battery types, especially for lithium-ion phosphate batteries.\nMethods Section: The relationship between battery SOC and OCV (identified from θ) can be mapped through the OCV test, which is shown in Fig. S23, indicating that different battery types have different shapes of OCV curves.\nReviewer #2 Minor Comment 1: In Figure 3d, the denominator of Capp lacks an integral symbol.\nAuthor response: The definition of Cap is based on the ampere-hour counting, where the\np\nnumerator is the accumulated charge, and the denominator is the corresponding SOC change.\nTherefore, there is an integral symbol in the numerator while only the absolute value of the SOC\nchange is in the denominator.\nReviewer #2 Minor Comment 2: The paper includes numerous notations, but most are not explained. It is recommended to provide a notation table in the appendix.\nAuthor response: As requested, we added an additional supplementary table to summarize all the notations and abbreviations used in this paper, which are also given below.\nMethods Section: All the notations and abbreviations are summarized in Table S10.\nSupplementary information:\nTable S10. Summary of notations and abbreviations.\nNotation/Abbreviation Meaning Notation/Abbreviation Meaning\nSOC State of charge CC Constant current\nSOH State of health PCC Pulse constant current\nML Machine learning OCV Open circuit voltage\nWorldwide Harmonized Light\nF / φ Feature matrix WLTP\nVehicles Test Procedure\nResiduals between the prior\nSOC Prior estimation of SOC ΔZ\np estimations and the real values\nSOC or SOH estimation\nSOC or SOH estimation based\nZ* based on the correction Z**\non the compensation model\nmodel\nRMSE Root mean square error MAE Mean absolute error\nNominal capacity / Fresh\nDOD Depth of discharge Cap\nn capacity\nSHapley Additive\nCap Present capacity SHAP\np exPlanations\nUrban Dynamometer Driving Highway Fuel Economy Test\nUDDS HWFET\nSchedule Cycle\nEVs Electric vehicles SOC Initial SOC\nRF Random forest DT Decision tree regressor\nGBDT Gradient boosting regressor ET Extra trees regressor\nECM Equivalent circuit model R Resistance\nR Polarization resistance C Polarization capacitor\np p\nResponse to Reviewer #3 (Remarks to the Author):\nReviewer #3 Overall Comment: In this manuscript, the authors propose a framework for the joint estimation of SOC and SOH for battery cells undergoing continuous driving cycling under constant and changing temperatures. The pipeline consists of ECM-based adaptive Kalman filtering for SOC estimation, which is enhanced by a first residual learner, giving a first SOH estimation that is improved upon by a second residual learner. While algorithm architecture, which seems to be the main focus of the paper, may be of interest, there are numerous concerns regarding formal aspects, experimental data, methodology, and result analysis that make me lean against publication.\nAuthor response: The authors appreciate the reviewer for the careful review and the detailed comments. The comments on the experiment data, methodology, and verifications help us improve the revised manuscript. We have revised the expressions of the proposed two residual learning pipelines as one correction model and one compensation model to interpret the function of each clearly. The two residual learning pipelines work in parallel instead of in series, while the functions are different. The closed-loop monitoring enables continuous monitoring of battery states throughout the entire life span, which is the most important demand of battery management. We have added two more datasets for the evaluation of the proposed models, especially for the unseen real-world loading profiles suggested by the reviewer. We have provided detailed responses to the comments and concerns from the reviewer below, and the revised text is in red in the revised manuscript and supplementary information.\nReviewer #3 Formal Comment 1: Although the introduction provides some background on the application of ML for battery monitoring and feature selection, it is not specific enough in relation to the particular approach taken in this work. It is challenging to assess the novelty and contributions of this paper with respect to the vast wealth of literature on ML for battery monitoring and SOH prediction, and the motivation of this work is not fully justified as a result. Also, I assume this is not the first time residual learning has been applied to battery modeling and estimation, so authors should attempt to position their research more clearly in relation to gaps in the literature and describe how the presented approach constitutes a significant contribution to the state of the art.\nAuthor response: Thank you for the great suggestion on the background and motivation analysis. As requested, we have added more background descriptions on machine learning based battery monitoring, especially for the most relevant works on residual learning. Although the residual learning has shown great improvement for the battery modeling (Ref. 19) and temperature estimation (Ref. 21), the verifications are conducted on several specific tests instead of for the applications in the whole life monitoring, which is the actual requirement of battery management systems. In addition, the previous works only focus on one kind of modeling pipeline for one specific task, while we generalize the residual learning pipelines with one correction model and one compensation model for clearer model function interpretations (as discussed in the responses to comment 14 below). The incorporation of feature importance analysis helps to understand the mechanistic features that enhance state monitoring, which has not been studied in previous hybrid modeling. In addition, the closed-loop design of the SOC and SOH monitoring enables continuous state monitoring for the whole life span without strict test and complex model calibrations. Finally, we demonstrate the importance of accurate state monitoring and the mechanistic feature extraction to help figure out the potential of thermal issues and battery swelling. In the revised manuscript, we have made the following revisions regarding the residual learning, feature extraction, and model verifications to highlight the gaps and motivations of our models. We also revised the main contribution summary correspondingly. The revised text is in red below.\nIntroduction Section: Widely adopted methods for battery state monitoring include model-based and data-driven approaches, each offering distinct advantages but also having certain limitations. Model-based methods are known for their robustness, particularly when incorporating high-fidelity models, though they tend to be more complex. In contrast, data-driven methods, which map the relationships between the features and battery states via machine learning (ML) models, offer greater flexibility but often suffer from limited generalization capabilities.11 Consequently, hybrid monitoring structures that combine model-based approaches with machine learning (ML) have gained popularity, aiming to enhance both accuracy and robustness.12–15 One straightforward method involves fusing the outputs of different sub-models to mitigate uncertainties associated with any single model.11,16 Another approach employs filtering techniques following initial estimates from data-driven methods to refine posterior estimates and improve accuracy.17 However, the implementation in SOC estimation adopted the real reference, i.e., coulomb counting, as the observation, which makes it hard to obtain accurate calculations in real applications. On the other hand, ML excels at modeling uncertain relationships, and the estimated errors from model-based methods vary and are hard to describe.18,19 Therefore, the sequential framework of the model-based method followed by ML was proposed.20,21 The main idea is that ML models the uncertainties and the errors of the mechanistic model-based methods, which have been employed for both hybrid battery modeling and temperature estimation. However, the specific model pipeline is designed for a specific purpose and only verified under several specific tests instead of continuous monitoring of battery states for the whole life span. Additionally, physics-informed neural network models have shown the potential to enhance physical interpretability and reduce data requirements in battery modeling.22,23 However, the sensitivity of these models to the tolerance of physical laws can greatly impact their predictive accuracy. Conversely, when reliable prior knowledge is incorporated, data-driven models demonstrate more stable improvements in performance.\nFeature extraction is one key factor that hugely influences the accuracy of the ML models.24–28 Methods for statistical calculation can extract highly effective features.24,25 However, these features are often sensitive to varying operating conditions, such as different current rates and temperatures, which can reduce model performance across diverse scenarios. While domain adaptation techniques can reduce the discrepancies between the feature distributions, thus enhancing the generalization of monitoring models under different conditions, they also increase model complexity and computational demands, hindering the application for real-time state estimations.29,30 Therefore, improving the suitability and coherence of features across different conditions is crucial for enhancing model accuracy and generalizability. Additionally, statistical features typically lack physical interpretability, complicating the explanation of ML models. Thus, incorporating features with physical meaning can significantly contribute to more interpretable monitoring systems.31,32 Recent research has increasingly focused on the interpretability of ML models, which provides deeper insights into their internal workings and facilitates better optimization.33–36\nMost of the existing work on SOC estimation has been limited to specific operational scenarios, often neglecting performance under real-world conditions where the model must function continuously across a battery's entire lifespan. Health predictions are similarly conducted under idealized conditions, assuming consistent availability of data from uniform charging and discharging cycles—an assumption that is difficult to meet in practical applications. Studies on health monitoring frequently consider ideal scenarios with stable working conditions, while in practice, the varying conditions pose significant challenges to maintaining the effectiveness of monitoring models. The continuous verifications of the SOC and SOH joint estimation model are expected to be conducted entire the whole life span, which is the practical demand of battery management system, instead of under several specific tests. Moreover, battery safety conditions are crucial for monitoring, which is key to avoiding hazards like thermal runaway. The potential of continuous state monitoring during dynamic aging to guide safety warnings worthy deeper studies.\nWe propose mechanistic leading residual learnings for continuous monitoring of battery states, specifically SOC and SOH, under varying dynamic working conditions throughout the entire lifespan. The general pipeline is adaptable for battery modeling and monitoring for different practical requirements. Using three datasets incorporating different battery types aging under different loading and temperature conditions for verifications, we demonstrate the effectiveness of our model working under different degradation patterns to emulate real-world usage scenarios. The proposed mechanistic leading residual models integrate a model-based approach with ML, where prior estimations and mechanistic features derived from the battery model are used as inputs for residual learners, including a correction model and a compensation model, for enhanced state monitoring. This general pipeline works effectively for both SOC and SOH and constructs a closed-loop framework to support the monitoring of the battery states for the entire life span, regardless of the varying loading profiles, temperature conditions, and working ranges (SOC windows) for representations of uncontrollable, randomly varying, and unseen working scenarios. The model is also beneficial for early thermal safety warnings during aging. Through interpretable ML and practical demonstrations, the proposed model enhances accuracy, robustness, and trustworthiness, thereby increasing its potential for real-world applications. The overall framework is illustrated in Fig. 1, where the pathway of the application and the flowchart of the pipeline for residual learning are demonstrated. Additional model details are provided in Fig. S1.\nReviewer #3 Formal Comment 2: The manuscript relies too heavily on Supplementary Materials. While this is common practice for Nature journals, several details about the experimental procedures, modeling and feature selection are relevant enough to be included in the main document. For instance, it is not until the last page of the Supplementary Materials that the selected features are defined.\nAuthor response: We thank the reviewer for the suggestion on the more detailed model descriptions, which helps the methods to be clearer and helps readers better understand the pipelines. Therefore, we have added more descriptions of the experiment, and the proposed method before the presentation of the verification results. The revised text is in red below.\nData generation subsection: We employed three datasets to evaluate the proposed mechanistic leading residual learning models under dynamic loadings for the full life span. Three types of batteries, including pouch, prismatic, and cylindrical cells, which were cycled with urban, highway, and real-world dynamic profiles, are used for verifications under different scenarios.\nTo emulate the dynamic working and the varying environmental conditions encountered in real-world applications, the first dataset subjected 13 pouch batteries (nominal capacity of 8 Ah) to aging under dynamic discharge across the entire lifespan at different temperatures.30 Urban, highway, and hybrid profiles were employed and are illustrated in Fig. S2, while Fig. S3 presents the variations in voltage curves observed during the aging process. Capacity degradation curves relative to equivalent full cycles are shown in Fig. S4, demonstrating that urban dynamic loading profiles generally result in longer battery life compared to highway-based loading, with varying temperature conditions significantly affecting the degradation patterns. There are in total 27840 samples (represented as the dynamic discharge cycles) generated in dataset 1.\nBesides, 3 large-format prismatic cells with a nominal capacity of 50 Ah are used in the second aging test until they die at around 20% SOH with sudden death, i.e., rapid degradation in a few cycles. We employed WLTP (Worldwide Harmonized Light Vehicles Test Procedure) to discharge the cells and used both CC and pulse-constant current (PCC) for charging, as shown in Fig. S5. The pulse was set with an average rate the same as the CC (2 C), while with twice the amplitude. The frequency and duty cycle are 1 Hz and 50%, respectively. The capacity, current, voltage, and temperature curves during aging are shown in Fig. S6. The open circuit voltage (OCV) test was conducted using C/20 constant charging and discharging. The sampling frequency is 10 Hz, and the current and voltage variations for the pulses are tracked precisely. This dataset benefits wide applications, including state monitoring, life prediction, and thermal safety warning. The detailed information of the tested pouch and prismatic cell is listed in Table S1. The sample size, i.e., dynamic discharge curves until capacity drops below 50% of the fresh capacity, is 8590.\nFinally, datasets using 25 cylindrical cells from Ref. 37, which were aged with synthetic highway and urban, as well as real city driving profiles, are employed to verify our model for use under different unseen working conditions. Typical current and voltage curves during aging under different loading profiles are shown in Fig. S7, indicating varying discharge patterns observed among different cycling, which therefore cause diverse working scenarios and aging trajectories. We used the first dynamic test after the diagnostic test as the dynamic cycling data during aging, the C/2 discharge capacity before the dynamic cycle as the referenced capacity, and the C/40 discharge curve of a new cell as the OCV curve. There are 483 dynamic discharge curves generated for the model evaluation.\nMechanistic leading residual learning pipelines for enhanced state monitoring subsection: To promote the real-time application of the pipelines, the mechanistic model adopts an equivalent circuit model in this work. For real-time battery SOC estimation, online parameter identification and adaptive filtering methods are employed (see Note S1 for a detailed procedure), where the SOC prior estimations and mechanistic model parameters (e.g., resistance and polarization features) are obtained. Coupled with SOC estimation and coulomb counting for released capacity calculation, the battery capacity can be derived to calculate the prior estimation of battery SOH. Meanwhile, measured parameters (e.g., voltage, current, and released capacity) and identified mechanistic parameters (e.g., identified resistance, OCV, and polarization resistance/capacitance) during dynamic discharging are used to calculate health features, which are composed of several statistical values including mean value, standard deviation, and min and max values of these sequential parameters from a random discharge phase. Detailed feature calculations are described in Methods and Notes S2–S3.\nHere, we introduce mechanistic leading residual learners, incorporating a correction model and a compensation model to enhance the state monitoring results from prior estimations. As shown in Figs. 1b and S8, the prior estimations and extracted features serve as input for the machine learning to output an enhanced estimation and a residual for the prior estimation compensation. Therefore, the function of the correction model is to enhance the prior estimation by direct state predictions, while the compensation model enhances the estimations by adding the predicted residuals to the prior estimations. The prior estimation from the mechanistic model provides stable guidance, and those feature matrices help capture the expected battery states, condition indicators from the real battery responses, as well as the mechanistic features describing the internal conditions. Machine learning is built to compensate for the model uncertainties and other influences like changed OCV under varying conditions; therefore, the OCV and other model parameters do not require careful calibration under temperature and aging. The effectiveness of this method is demonstrated through two key states: real-time SOC and the more slowly varying SOH. Note that the compared prior estimations represent the conventional model-based filtering methods. The coulomb counting-based calculation is employed to obtain the referenced real SOC and SOH for the evaluations of the conventional model-based filtering method and the proposed residual learning models, while it is not used in any part of the modeling pipelines. Additionally, our model can be easily extended to monitor other relevant states as required, for example, the sensor-less temperature estimations based on the mechanistic features and the lamped-mass thermal model-based prior estimations.21\nNote S2: In the proposed mechanistic-learning residual learning, we derived three types of features for the machine learning models, namely the prior estimations, the mechanistic features, and the measured features, respectively. The detailed descriptions are below.\nPrior estimation features: The prior estimations from the mechanistic model-based filtering are effective in tracking the real values in different scenarios for different battery types. Therefore, the prior estimations from the filtering method serve as the first type of features for the machine learning model.\nMechanistic features: The mechanistic states derived from the battery model reflect the electric and aging status. For example, the resistance primarily represents the intrinsic impedance of the battery, including contributions from electrode materials, ionic conduction resistance in the electrolyte, and contact resistance between current collectors and active materials. The polarization resistance and polarization capacitance describe dynamic polarization processes, where the polarization resistance ( ) accounts for charge transfer impedance and concentration polarization, while the polarization capacitance ( ) reflects ion adsorption/desorption at the electrode-electrolyte interface. The time constant ( ) governs the relaxation dynamics of the battery.\n𝑅𝑅𝑝𝑝\n𝐶𝐶𝑝𝑝\nIn addition, the battery OCV represents the equilibrium potential of the battery when no current is\n𝜏𝜏 = 𝑅𝑅𝑝𝑝 𝐶𝐶𝑝𝑝\napplied, serving as a fundamental indicator of the battery's SOC. More importantly, the variation of these parameters is highly correlated with battery aging. Therefore, these mechanistic features are used for battery state monitoring. Specifically, the real-time identified OCV, R, , and are\n𝑅𝑅𝑝𝑝 𝐶𝐶𝑝𝑝\nused for the SOC monitoring model. For SOH monitoring, to consider the information from the dynamic discharging process, the statistical features are derived from these mechanistic parameters. The mean, maximum value, and standard deviation of the R, , and are calculated. In addition, the SOC and OCV from the random discharge process also reflect the degradations. Therefore, the\n𝑅𝑅𝑝𝑝 𝐶𝐶𝑝𝑝\nmean, maximum and minimum values, and standard deviation of these two states are calculated as features.\nMeasured features: The real-time measured current and voltage are used as measured features for SOC monitoring models. The statistical features, including mean, maximum value, and standard deviation of the current, voltage, and released capacity from the partially dynamic discharge, are calculated for the SOH monitoring models.\nReviewer #3 Formal Comment 3: The visualization of most figures should be improved. For example, in Fig. 2a-c, it is difficult to distinguish estimation from experimental results, and also, no grid is provided. This issue is also present in Fig. 5a-d, and Figs. S8 and S18. Fig. 5e is not informative. Some temperature plots in Figs. S4 and S5 are challenging to interpret. In Fig. S23, it is not possible to distinguish original data from fitted data.\nAuthor response: In the revised manuscript, we have added more datasets for the verifications and therefore reorganized the figures. We revised the comparison figures to statistical figures to make them clearer. We did not draw grid lines to make the picture clearer, and this is also the method widely adopted by other related papers. As discussed, some temperature measurements have misalignment with the electric measurement, which makes the temperature figures staggered. We didn’t use the temperature as the input for the model, since on the one hand, to keep the model the same for all three datasets, and on the other hand, not all the cells have a temperature sensor in real-world applications. For the fitted OCV curve, we aim to illustrate that the fitted one overlapped well with the measurement, thus the fitted function can be used for the SOC estimations. In the revised figure, we also added the OCV fitting results for the other two added batteries. Please refer to the revised manuscript and the supplementary information for the revised figures.\nReviewer #3 Comment 1: Important specifications of the battery cell are not provided, such as chemistry, voltage limits, current limits, temperature range and internal resistance.\nAuthor response: As requested, more detailed information for the experiment cells is provided in the revised manuscript, and an additional supplementary table for the tested batteries is added in the supplementary information. The revised texts are in red below.\nData generation subsection: We employed three datasets to evaluate the proposed mechanistic leading residual learning models under dynamic loadings for the full life span. Three types of batteries, including pouch, prismatic, and cylindrical cells, which were cycled with urban, highway, and real-world dynamic profiles, are used for verifications under different scenarios.\nTo emulate the dynamic working and the varying environmental conditions encountered in real-world applications, the first dataset subjected 13 pouch batteries (nominal capacity of 8 Ah) to aging under dynamic discharge across the entire lifespan at different temperatures.30 Urban, highway, and hybrid profiles were employed and are illustrated in Fig. S2, while Fig. S3 presents the variations in voltage curves observed during the aging process. Capacity degradation curves relative to equivalent full cycles are shown in Fig. S4, demonstrating that urban dynamic loading profiles generally result in longer battery life compared to highway-based loading, with varying temperature conditions significantly affecting the degradation patterns. There are in total 27840 samples (represented as the dynamic discharge cycles) generated in dataset 1.\nBesides, 3 large-format prismatic cells with a nominal capacity of 50 Ah are used in the second aging test until they die at around 20% SOH with sudden death, i.e., rapid degradation in a few cycles. We employed WLTP (Worldwide Harmonized Light Vehicles Test Procedure) to discharge the cells and used both CC and pulse-constant current (PCC) for charging, as shown in Fig. S5. The pulse was set with an average rate the same as the CC (2 C), while with twice the amplitude. The frequency and duty cycle are 1 Hz and 50%, respectively. The capacity, current, voltage, and temperature curves during aging are shown in Fig. S6. The open circuit voltage (OCV) test was conducted using C/20 constant charging and discharging. The sampling frequency is 10 Hz, and the current and voltage variations for the pulses are tracked precisely. This dataset benefits wide applications, including state monitoring, life prediction, and thermal safety warning. The detailed information of the tested pouch and prismatic cell is listed in Table S1. The sample size, i.e., dynamic discharge curves until capacity drops below 50% of the fresh capacity, is 8590.\nFinally, datasets using 25 cylindrical cells from Ref. 37, which were aged with synthetic highway and urban, as well as real city driving profiles, are employed to verify our model for use under different unseen working conditions. Typical current and voltage curves during aging under different loading profiles are shown in Fig. S7, indicating varying discharge patterns observed among different cycling, which therefore cause diverse working scenarios and aging trajectories. We used the first dynamic test after the diagnostic test as the dynamic cycling data during aging, the C/2 discharge capacity before the dynamic cycle as the referenced capacity, and the C/40 discharge curve of a new cell as the OCV curve. There are 483 dynamic discharge curves generated for the model evaluation.\nMethods Section: The experimental setup is aimed at emulating real dynamic aging. To this end, three datasets are included in this work. The first dataset includes three different dynamic loading profiles to age pouch batteries (MGL SPIM08HP 8Ah high power pouch cell) under both constant and varying temperature conditions. Specifically, the representative loading profiles for urban (UDDS), highway (HWFET), and hybrid working conditions are employed for discharging, while the multi-constant current charging is used for charging during the aging test.30 The chemistry is nickel cobalt aluminum Oxide (NCA) and graphite, and the voltage limit is 2.8-4.2 V. The loading currents are shown in Fig. S2. Two different temperature conditions including constant temperature (25℃) and variable temperature (15-25-35℃) are set for the testing environment. More details about the experiment and data processing are depicted in Note S5. In the second dataset, we use three large-format prismatic cells (L148N50B 50 Ah), which are widely employed in EVs, for dynamic aging with WLTP profiles. The chemistry is nickel-manganese-cobalt (NMC), and the voltage range is set as 2.75-4.3 V. One cell is charged with constant current, and the other employs pulse currents (1 Hz, 50% duty cycle) for charging. The loading currents are shown in Fig. S5, and the aging test is conducted in a thermal chamber with a controlled 35℃. The sampling frequency is set as 10 Hz, which enables the tracking of variations during pulse loadings. The OCV test is conducted with a low current of C/20. The third dataset is from Stanford University, where the cylindrical cells are aged with both synthetic urban and highway dynamic loadings and more practical real-world loading.37 Periodical diagnostic tests are conducted where the real capacities can be obtained. The first dynamic discharge cycle after the diagnostic test is used for the verification where the real capacity can be obtained for verification from the diagnostic test. Furthermore, the sampling frequency is artificially reduced to 0.1 Hz to generate data for the robustness verification of our models in real-world applications.\nSupplementary Information:\nTable S1. Specifications of the test batteries.\nSpecification Pouch cell (dataset 1) Prismatic cell (dataset 2)\nNominal Capacity 8 Ah 50 Ah\nNominal Voltage 3.65 V 3.7 V\nUpper cut voltage 4.2 V 4.3 V\nLower cut voltage 2.75 V 2.75 V\nChemistry NCA - Graphite NCM - Graphite\nReviewer #3 Comment 2: A detailed list of the test procedures should be provided indicating clearly which cells have been tested under which conditions.\nAuthor response: As requested, we described the detailed experiment procedures for the cells in dataset 1 and dataset 2. In addition, the loading profiles and the voltage profiles during the aging tests for each cell are demonstrated in Figs. S2 S6. The third dataset, which used the open-access dataset from Stanford University (DOI: 10.1038/s41560-024-01675-8), is also described, and the\n–\nrepresentative loading and voltage profiles during aging are given in Fig. S7. Please refer to the response to the above comment for the detailed revised texts and Figs. S2 S6 for the visualization of each cell under its test procedure.\n–\nFig. S5. Loading profiles for aging the prismatic cells in dataset 2. a. The cycling protocol with CC charging and WLTP discharging. b. The cycling protocol with PCC charging and WLTP discharging.\na\nb\nFig. S6. Degradation curves of the prismatic cells in dataset 2. a. Charge and discharge capacity degradation curves. b. Current, voltage, and temperature curves of cell 1 during aging. c. Current, voltage, and temperature curves of cell 1 during aging. d. Current, voltage, and temperature curves of cell 1 during aging.\na\nb\nAged\nc\nd\nFresh\nFig. S7. Degradation curves of the cylindrical cells in dataset 3. a. Charge and discharge capacity degradation curves. b\nc\n. Current and voltage curves for cell 72 during degradation, which was aged with a long trip highway loading proile. . Current and voltage curves for cell 66 during degradation, which was aged with a synthetic urban (city 1) loading proile. . Current and voltage curves for cell 74 during degradation, which was aged with a synthetic urban (city 2) loading proile. . Current and voltage curves for cell 82 during degradation, which was aged with a synthetic urban (city 1+ city 2) loading proile. . Current and voltage curves for cell 79 during degradation, which was aged with a synthetic highway + urban loading proile. .\nh\nCurrent and voltage curves for cell 91 during degradation, which was aged with a real-world city 1 driving proile. . Current and voltage curves for cell 95 during degradation, which was aaged with a real-world city 2 driving proilbe.\nAged\nc d\ne f\ng h\nFresh\nReviewer #3 Comment 3: A major piece of information missing is the sample size, which is even more relevant if ML techniques are employed.\nAuthor response: As requested, the sample frequency and sample size for the tested datasets are added. Since for SOC estimation, each discharge curve has thousands of points, which means the sample size for the SOC estimations is very large. Therefore, the sample size is detailed as the samples of discharge curves in this paper, which is also equal to the SOH sample size. The revised text is in red below.\nData generation subsection: There are in total 27840 samples (represented as the discharge cycles) generated in dataset 1.\nThe sample size, i.e., dynamic discharge curves until capacity drops below 50% of the fresh capacity, is 8590.\nThere are 483 dynamic discharge curves generated for the model evaluation.\nReviewer #3 Comment 4: While transparency regarding experimental difficulties is appreciated, some of the challenges experienced by the authors with voltage, current and temperature measurements (Note S4) are not such common occurrences in battery testing and may make the reader question the reliability of the presented experimental dataset.\nAuthor response: It is commonly seen in real-world applications that the data quality is lower than the laboratory measurements, as discussed in many related papers using field datasets. Therefore, in real-world applications, people always need to consider the data preprocessing, where the abnormal data, such as outliers and missing data, are processed. In this work, there are some deficiencies (mainly the missing data) measurements of current, which are processed manually. The processing is conducted at the constant current stages while the data from dynamic loadings is kept. The constant current data filling is easy and reasonable. The proposed models are used in the random dynamic discharging with different loading profiles. Therefore, we believe this data processing is reasonable and necessary regarding real applications and does not influence the evaluations in this paper. There are also some misalignments with the temperature measurements, but we do not use temperature data for the model. This is because not all the cells have a temperature sensor in real-world applications, and we keep the same input for the model in different datasets to evaluate the generalization capability of our models. In addition, we also added the verifications of the proposed method in another two datasets (one experiment dataset and another public dataset from Stanford University (DOI: 10.1038/s41560-024-01675-8)), where the measurements are of good quality. The results proved the effectiveness of our models. In conclusion, we believe that data preprocessing is necessary and common in real-world applications and does not influence the effectiveness evaluation of the proposed models.\nReviewer #3 Comment 5: The authors claim that the employed ML models are grounded on interpretable physical principles, but this is highly debatable. The only ‘physical’ element is the use of the 1-RC ECM as a battery model, which is not properly justified. Many authors have pointed out the shortcomings of RC models and their limited physical insight with respect to internal battery processes. Furthermore, if an ECM is employed in dynamic operation, a consideration of hysteresis voltage modeling should be made.\nAuthor response: The authors thank the reviewer for the question regarding the mechanistic model used in this work. The main function of the mechanistic model in this work is to provide prior estimations of battery SOC and SOH based on conventional filtering methods and extract some mechanistic features, like resistance, polarization resistance, and polarization capacitance. Therefore, we are not aiming at improving the model accuracy while improving the model-based estimations based on the proposed residual learning models and considering the real-time solving in the battery management system. To this end, we adopted the ECM, which can be rapidly calculated during real applications. As the results verified through three different datasets consisting of different battery types, the ECM-based models provide stable estimations, though the accuracy is not high. These estimation errors are compensated by the proposed residual learning models, which largely improve the accuracy for both SOC and SOH under varying verification scenarios. Although the electrochemical meaning of ECM is lower than physical models like P2D and single particle models, it still reflects the ohmic resistance and polarization behaviors of the batteries. Moreover, the real-time identification of the mechanistic features is significant for machine learning, as reflected by the SHAP analysis (Figs. 3 5 and Figs. S21 23), to enable the enhancement of the state estimations. By also comparing with the results that exclude\n–\nthe mechanistic features, as shown in Figs. S11, S18, and S24, the results reveal that mechanistic features are necessary for the machine learning models for the accuracy enhancement. In the revised manuscript, we have revised our title and related content to reflect the main reasons for using the proposed pipeline. The revised text is in red below.\nTitle: Mechanistic Leading Residual Learning for Battery State Monitoring Entire Full Life\nMechanistic leading residual learning pipelines for enhanced state monitoring subsection: To promote the real-time application of the pipelines, the mechanistic model adopts an equivalent circuit model in this work. For real-time battery SOC estimation, online parameter identification and adaptive filtering methods are employed (see Note S1 for a detailed procedure), where the SOC prior estimations and mechanistic model parameters (e.g., resistance and polarization features) are obtained. Coupled with SOC estimation and coulomb counting for released capacity calculation, the battery capacity can be derived to calculate the prior estimation of battery SOH. Meanwhile, measured parameters (e.g., voltage, current, and released capacity) and identified mechanistic parameters (e.g., identified resistance, OCV, and polarization resistance/capacitance) during dynamic discharging are used to calculate health features, which are composed of several statistical values including mean value, standard deviation, and min and max values of these sequential parameters from a random discharge phase. Detailed feature calculations are described in Methods and Notes S2–S3.\nReviewer #3 Comment 6: It is unclear whether the full pseudo-OCV profile is determined, or only discrete points (GITT); Fig. S23 does not provide enough information. Also, I am skeptical that the full OCV profile can be accurately fitted with a low-order polynomial.\nAuthor response: In the work, both discrete points-based OCV and full pseudo-OCV (low C-rate discharge) are used. Specifically, the discrete points-based OCV is used in dataset 1, C/20 based full pseudo OCV is employed in dataset 2, and C/40 based full pseudo OCV is used in dataset 3. For real-time calculations and the derivatives in EKF-based SOC estimations, we adopted the polynomial fitting function for the OCV curve fitting, which is widely used in the battery state estimation based on filtering methods. As shown in Fig. S23, the fitting coefficients are all larger than 0.99 for the fitting results of the three battery types. Again, the machine learning based residual compensations are proposed for the enhancement of the filtering-based method, therefore, the main purpose of the model-based method is to ensure stable estimations and provide prior information for the machine learning models. The added fitting results of OCV curves in dataset 2 and dataset 3 are also given below.\nFig. S23. OCV curve measurements and fitting with polynomial function for model-based estimations. a. Measured and fitted OCV curves under different SOCs and temperatures. The fitting coefficient (R2) is higher than 0.998. b. Measured and fitted OCV curves under different SOCs with an average on the temperature. The fitting coefficient is also higher than 0.999. c. The measured and fitted OCV curves under C/20 for the prismatic cells, and the fitting coefficient is 0.998. d. The measured and fitted OCV curves under C/40 for the cylindrical cells, and the fitting coefficient is larger than 0.999.\na b\nataD\nlanigirO\nVCO\nataD\ndettiF\nc d\nReviewer #3 Comment 7: What voltage is assigned SOC=0%, and according to what set of criteria? 3.2V is the lower cutoff according to Fig. S23, but 2.75V seems to be the limit judging from Figs. S3, S4 and S5. If the chemistry is NCA/Gr as it appears, usual voltage limits are 4.2V and 2.5V. This implies that the SOC validation range is narrower in reality.\nAuthor response: As the discrete points-based OCV test is conducted for the pouch cell in dataset 1, the voltage after rest (following discharge) is used as the reference voltage, therefore, the voltage after relaxation is higher than 2.75 V while the cell is discharged until 2.75 V. For the full pseudo OCV test based on low C-rates (C/20 and C/40), the OCV drops below the voltage limit, as shown in the revised Fig. S23 in response to the above comment. In the cycling test, all the cells are discharged until the voltage drops below the lower voltage limit.\nReviewer #3 Comment 8: ECM parameters are determined online via a forgetting factor RLS algorithm, and it appears as though they are updated every timestep. This undermines any physical foundation that could be present in the first place.\nAuthor response: The mechanistic features, i.e., model parameters from ECM, are extracted every timestep since we employed these features for the SOC estimation, which is also performed every time step. Then, the statistical features of the sequential information of these features are calculated for use in SOH estimation. As demonstrated in the model interpretation results via SHAP analysis (Fig. 3 5) and the comparative results with the estimation outcomes by excluding these\n–\nmechanistic features, the importance and the necessity of these mechanistic features are proven. Indeed, there may be some outliers identified in the RLS algorithm, and we employed some regularizations that ensure the reasonable ranges of the parameter variations and adopted adaptive EKF for the model updating.\nReviewer #3 Comment 9: How is ‘real SOC’ determined? If it is through coulomb counting, then this defeats the purpose of the estimation algorithms.\nAuthor response: The ‘real SOC’ is determined by Coulomb counting, which is a widely adopted way for battery SOC estimation algorithm verifications. The ‘real SOC’ is only used for the comparisons between the estimations from the proposed models and the ‘real’ values to evaluate the estimation accuracy, while it is not used in any part of the modeling or the estimation pipelines. Although coulomb counting can be inaccurate in real-world applications, it can hold high accuracy through the experimental data, and meet the purpose for the model verifications and evaluations. Therefore, the Coulomb counting-based SOC calculation is reasonable for the verification purpose. In the revised manuscript, we added clarifications for the purpose of using Coulomb counting in this work, which is in red text below.\nMechanistic leading residual learning pipelines for enhanced state monitoring subsection:\nThe coulomb counting-based calculation is employed to obtain the referenced real SOC and SOH for the evaluations of the conventional model-based filtering method and the proposed residual learning models, while it is not used in any part of the modeling pipelines.\nReviewer #3 Comment 10: Similarly, if periodic RPTs are not conducted, how is ‘real SOH’ determined?\nAuthor response: The ‘real SOH’ is determined by the released capacity during the full dynamic discharging process, which determines the available capacity at the current cycle. Since we employed the same discharging protocols during the whole aging test, the SOH calculated by the discharge capacity is reasonable, and is also widely used in published papers, such as ‘Data-driven prediction of battery cycle life before capacity degradation’, ‘Closed-loop optimization of fast-charging protocols for batteries with machine learning’, and ‘Data-driven capacity estimation of commercial lithium-ion batteries from voltage relaxation’. In addition, we also adopted the public dataset from ‘DOI: 10.1038/s41560-024-01675-8’ for the evaluation of the proposed model, where the RTP capacity is used for the SOH calculation, and the results are demonstrated in the revised manuscript, and also given below.\nData generation subsection: Finally, datasets using 25 cylindrical cells from Ref. 37, which were aged with synthetic highway and urban, as well as real city driving profiles, are employed to verify our model for use under different unseen working conditions. Typical current and voltage curves during aging under different loading profiles are shown in Fig. S7, indicating varying discharge patterns observed among different cycling, which therefore cause diverse working scenarios and aging trajectories. We used the first dynamic test after the diagnostic test as the dynamic cycling data during aging, the C/2 discharge capacity before the dynamic cycle as the referenced capacity, and the C/40 discharge curve of a new cell as the OCV curve. There are 483 dynamic discharge curves generated for the model evaluation.\nVarying verifications subsection: Finally, cylindrical batteries from Ref. 37 are employed for the verification of the proposed models by considering unseen working conditions. Three different validation cases are generated using cells with different synthetic profiles or real driving profiles aging. Specifically, as shown in Fig. 5a, cells with independent highway or urban loading profiles are divided into a training dataset and the validation 1 dataset. The loading profile of combining highway and urban loadings is included in the validation 2 dataset, and the cells with two city driving profiles are used in the validation 3 dataset. Therefore, we evaluated our mechanistic leading residual learning models with different real applications, considering unseen validation scenarios. Typical loadings and the related prior estimations are demonstrated in Fig. S20. The state monitoring and corresponding interpretation results are shown in Figs. 5bc, respectively. More detailed results for the three validation cases are shown in Figs. S21–S23 and Tables S5–S6. MAE for the SOC estimation keeps below 1.5% for all three validation cases and keeps below 1.8% for SOH estimations for both the correction model and the compensation model. All the cases indicate enhanced state monitoring performance compared to conventional model-based methods, even under unseen loadings for the whole life span. Similar interpretations can be summarized from the SHAP results, i.e., the prior estimations play significant roles for both the correction model and compensation model, while the mechanistic features provide critical information and have higher impacts on the compensation model compared to the correction model. Though the prior estimations are the most significant information, mechanistic parameters are essential for the model, as shown in Fig. S24, where the errors may increase instead by only using prior estimation for the model input.\na c\nData Test Cells\nHighway 60, 62, 72\nTraining\nUrban 66, 68, 74, 76, 82, 84\nHighway 59, 64, 71\nValidation 1\nUrban 65, 67, 73, 75, 81\nHighway\nValidation 2 79, 80, 87, 88\n+ Urban\nCity 1 91, 92\nValidation 3\nCity 2 95, 96\nCorrection Compensation\nb\nFig. 5. Performance evaluation for cylindrical cells with unseen working conditions. a. Training and testing cells split strategies. b. SOC and SOH monitoring results for the three validation cases. c. Feature interpretation of the mechanistic leading models for the SOC and SOH monitoring.\nReviewer #3 Comment 11: There is a risk of circular dependency, as SOC estimation relies on battery capacity (Eq.1), which is updated as a function of the estimated SOC difference (Eq. 4). How is this prevented and how often is capacity updated?\nAuthor response: The updating of battery capacity is vitally important to enable accurate SOC estimation, especially for continuous monitoring of the entire life span. This is also considered in our work, where a closed-loop state updating framework is constructed. The real-time SOC estimation is used for capacity estimation, and the estimated capacity is updated for the SOC estimation in the following discharge cycle, as demonstrated in Fig. 1. Therefore, the continuous monitoring of SOC and SOH during the whole life span under dynamic operations is achieved in this paper, which is not verified in previously related papers. This is proved in the Closed-loop application entire life span section in the revised manuscript, with several test scenarios from the two datasets. The capacity is updated every cycle for dataset 1 and is periodically updated during each RPT test for dataset 3, since we verified the first dynamic discharge after each RPT in dataset 3. We added more descriptions in the revised manuscript and the results for dataset 3, which are also given below in red text.\nIntroduction Section: We propose mechanistic leading residual learnings for continuous monitoring of battery states, specifically SOC and SOH, under varying dynamic working conditions throughout the entire lifespan. The general pipeline is adaptable for battery modeling and monitoring for different practical requirements. Using three datasets incorporating different battery types aging under different loading and temperature conditions for verifications, we demonstrate the effectiveness of our model working under different degradation patterns to emulate real-world usage scenarios. The proposed mechanistic leading residual models integrate a model-based approach with ML, where prior estimations and mechanistic features derived from the battery model are used as inputs for residual learners, including a correction model and a compensation model, for enhanced state monitoring. This general pipeline works effectively for both SOC and SOH and constructs a closed-loop framework to support the monitoring of the battery states for the entire life span, regardless of the varying loading profiles, temperature conditions, and working ranges (SOC windows) for representations of uncontrollable, randomly varying, and unseen working scenarios. The model is also beneficial for early thermal safety warnings during aging. Through interpretable ML and practical demonstrations, the proposed model enhances accuracy, robustness, and trustworthiness, thereby increasing its potential for real-world applications. The overall framework is illustrated in Fig. 1, where the pathway of the application and the flowchart of the pipeline for residual learning are demonstrated. Additional model details are provided in Fig. S1.\nClosed-loop application entire life span subsection: The mechanistic leading residual learners facilitate collaborative monitoring of different battery states throughout the entire lifespan, with capacity updating during each dynamic discharging cycle.\nThen, we show the application for dataset 3 with unseen testing conditions under real-world driving profiles. Note that we also use the dynamic cycle after the diagnostic test for verifications, so the degradation curve is composed of sparse points. By using the training cells in Fig. 5a to train the residual learners, the state monitoring under unseen working conditions (cell 92 aged using real-world driving protocols) entire the life span are enhanced, as shown in Fig. 6f. Unseen validation for exemplary cells working under another real city protocols (cell 95) and synthetic Urban + Highway (cell 80) protocols are also shown in Fig. 6f, demonstrating the stability and robustness of our model under varying unseen working conditions, which prove the good extrapolation capability of the proposed mechanistic leading residual learning pipelines.\nNumerical results listed in Table S9 indicate the MAE for all the tests is below 1.8% for both SOC and SOH. With enhanced SOC and SOH estimations through closed-loop operations, battery systems are poised for improved management and operational performance. This closed-loop structure not only supports effective state monitoring but also self-updates to maintain functionality and provide safety warnings, thereby optimizing the overall management of the battery system without the need for costly and time-consuming offline testing.\na b c d\ne\nCorrection Compensation Real values\nCapacity updating during operations and aging\nf\nCity 1 City 2 Urban + Highway\nFig. 6. Application demonstrations for the continuous monitoring of battery charge and health status of the entire life span. a d. SOC estimation at four different aging stages and the demonstration of random working SOC ranges used for capacity estimation. e. SOC and SOH curves tracking for the whole lifesp–an. f. Unseen applications for the closed-loop state monitoring under both real-world and synthetic hybrid loadings.\nReviewer #3 Comment 12: Remaining capacity is not a complete measure of battery health, and other indicators, such as degradation modes and impedance growth should be considered. Also, is the OCV profile assumed to remain unchanged during aging?\nAuthor response: Battery aging mode is significant for comprehensive health diagnosis. Our recent paper proposed an interpretable and onboard battery health diagnosis model enabling aging model diagnosis during dynamic operations (arXiv:2503.07383). While the main focus of this work is on the macroscopic charge and health states monitoring, which are required in onboard battery management systems for control purposes. Therefore, we mainly consider capacity degradation monitoring in this work. Though the capacity-based SOH is monitored, we also find the extracted mechanistic features, like resistance, are also significant for describing battery health status, and for guiding warning of thermal issues and battery swelling. We demonstrated this in the revised manuscript by the prismatic cells, which have suffered rapid capacity degradation with increased temperatures and heavy swelling. We admit that degradation mode is significant to help understand the aging mechanisms, while it is not the main focus of this paper. In the revised manuscript, we added some outlooks for future work that incorporate aging mode estimations. One of the main advantages of the proposed machine learning based residual compensations is that the deficiencies from both the model and inaccurate measurements can be compensated. Though OCV at different aging stages helps improve the model accuracy, it is very hard to determine the real SOH value for OCV determination in real applications, i.e., there is still measurement inaccuracy existing. In addition, the OCV curves are also varied with different temperature conditions. Therefore, it will be challenging to consider the coupling influences of aging and temperature in real-world applications, and huge experiment costs will be required. Therefore, only one fresh OCV is required in this paper, while machine learning is expected for the deficiency compensations. The revised text is in red below.\nMechanistic leading residual learning pipelines for enhanced state monitoring subsection:\nHere, we introduce mechanistic leading residual learners, incorporating a correction model and a compensation model to enhance the state monitoring results from prior estimations. As shown in Figs. 1b and S8, the prior estimations and extracted features serve as input for the machine learning to output an enhanced estimation and a residual for the prior estimation compensation. Therefore, the function of the correction model is to enhance the prior estimation by direct state predictions, while the compensation model enhances the estimations by adding the predicted residuals to the prior estimations. The prior estimation from the mechanistic model provides stable guidance, and those feature matrices help capture the expected battery states, condition indicators from the real battery responses, as well as the mechanistic features describing the internal conditions. Machine learning is built to compensate for the model uncertainties and other influences like changed OCV under varying conditions; therefore, the OCV and other model parameters do not require careful calibration under temperature and aging. The effectiveness of this method is demonstrated through two key states: real-time SOC and the more slowly varying SOH. Note that the compared prior estimations represent the conventional model-based filtering methods. The coulomb counting-based calculation is employed to obtain the referenced real SOC and SOH for the evaluations of the conventional model-based filtering method and the proposed residual learning models, while it is not used in any part of the modeling pipelines. Additionally, our model can be easily extended to monitor other relevant states as required, for example, the sensor-less temperature estimations based on the mechanistic features and the lamped-mass thermal model-based prior estimations.21\nVarying verifications subsection: We further employed our model for the state monitoring of different battery types with different aging patterns caused by different aging conditions. The dynamic aging of large-format prismatic cells is verified. As shown in Fig. 4a, three large-format prismatic cells are aged with CC/PCC charging and WLTP dynamic discharging, which experience accelerated aging after SOH drops below the knee points. The average current of the pulses remains the same as the constant current, as shown in Fig. 4b. While different battery impedances cause the voltage to vary differently, and according to the heat generation, the cells have higher temperatures using pulse current compared to the constant current. Around 10℃ difference for the maximum temperature and 3.5℃ difference for the average temperature during cycling are seen for the two different charging protocols. These large thermal condition differences during cycling become the main cause of the different lifetimes for the cells under constant or pulse currents. Previous related works have revealed that pulse currents have an influence on battery lifetime. The battery aging is coupled and influenced by current profiles, temperature, current rates, etc., thus, we advocate coupling mechanisms and battery types during evaluations for the lifetime extension strategy designs.\nWe continuously age the batteries until they die, where accelerated aging is seen. As shown in Fig. S19, the battery is heavily swelled after aging, and the thermal safety issue (largely and rapidly increased temperature) occurred as demonstrated in Fig. 5a. Interestingly, the battery capacity degradation accelerated earlier (around 100 cycles) than the thermal safety issue happened, and the identified mechanistic resistances also show earlier detection. Therefore, accurate estimations of the battery capacity and mechanistic states are also crucial to guide safety management, which helps with early warning of battery safety conditions before thermal issues during operations.\nThe monitoring results for the cells, until SOH (current capacity divided by initial capacity) drops below 50%, are shown in Figs. 4c d, where the total discharging curves are randomly divided into training and testing datasets with a 50% portion of each. The mechanistic leading correction model\n–\nand compensation model show improved SOC and SOH monitoring for the full life span with MAE of less than 0.50% and 0.68%, respectively, as listed in Table S4. SHAP interpretation also indicates the most important impact of the prior estimation and reflects the importance of the mechanistic features. Due to the high linear relationships between the corrected and prior estimations, while the higher nonlinear relationships of the residuals between the prior and real estimations, the mechanistic features have higher impacts on the compensation model compared to the correction model. Both models show improved monitoring and reveal the performance of the proposed mechanistic leading residual learning models. More importantly, the potential of early thermal safety warning can be achieved during operation and aging.\na b\nPulse Rest Pulse Pulse Rest Pulse\nDelay c\nΔT ≈ 10 oC\nThermal safety\nΔT ≈ 3.5 oC\nd\nFig. 4. Performance evaluation for prismatic cells. a. Full life span degradation curves, temperature curves, and resistance curves of the three cells during aging. b. Illustration of the constant and pulse loading currents and the corresponding voltage curves. c. SOC monitoring results and interpretations for the mechanistic leading correction and compensation models. d. SOH monitoring results and interpretations for the mechanistic leading correction and compensation models.\nReviewer #3 Comment 13: It is stated that ‘the presented model can be extended to monitor other relevant states as required’, besides SOC and SOH (lines 115-117). What other states are implied here (temperature, lithium concentration, etc.)? The 1-RC ECM does not contain such information.\nAuthor response: In this work, we have demonstrated the mechanistic leading residual learnings for SOC and SOH estimation enhancements. This pipeline incorporates a mechanistic-model-based filtering estimation with a following residual compensation based on machine learning. The proposed pipelines are also suitable for other states, for example, SOE (state of energy), which incorporates SOC and voltage, can be easily adopted. In addition, battery temperature estimation can also be enhanced by the proposed method. In the authors’ previous paper (ref. 21), the mechanistic features, like resistance, are also used to achieve sensorless temperature estimations, since the internal resistance is highly related to battery thermal conditions. Therefore, we believe the proposed pipelines will be flexible for extending to other states' estimations, in case that conventional model-based filtering estimation can be obtained. In the revised manuscript, we have added descriptions for this statement, and the revised text is in red below.\nMechanistic leading residual learning pipelines for enhanced state monitoring subsection:\nAdditionally, our model can be easily extended to monitor other relevant states as required, for example, the sensor-less temperature estimations based on the resistance and the lamped-mass thermal model prior estimations. 21\nReviewer #3 Comment 14: A recurring observation is that Residual Learner 2 does not appear to improve estimation results with respect to Residual Learner 1, noted in Figs. 2g and 3g, as well as in Tables S1-S5. What is the explanation for this? Is RL2 needed?\nAuthor response: The mechanistic learning residual learnings in this work incorporate two pipelines (residual learner 1 and 2 in the original manuscript). The relationships between these two models are parallel instead of incremental, which means the residual learner 2 is not designed to improve the estimation results from residual learner 1. The functions of the two residual learners are different, while both of them achieve enhancements of state monitoring. In the revised manuscript, we redefine the two residual learners to be the correction model and compensation model, respectively, according to the functions achieved. The revised text is in red below.\nIntroduction Section: The proposed mechanistic leading residual models integrate a model-based approach with ML, where prior estimations and mechanistic features derived from the battery model are used as inputs for residual learners, including a correction model and a compensation model, for enhanced state monitoring.\nMechanistic leading residual learning pipelines for enhanced state monitoring subsection:\nHere, we introduce mechanistic leading residual learners, incorporating a correction model and a compensation model to enhance the state monitoring results from prior estimations. As shown in Figs. 1b and S8, the prior estimations and extracted features serve as input for the machine learning to output an enhanced estimation and a residual for the prior estimation compensation. Therefore, the function of the correction model is to enhance the prior estimation by direct state predictions, while the compensation model enhances the estimations by adding the predicted residuals to the prior estimations. The prior estimation from the mechanistic model provides stable guidance, and those feature matrices help capture the expected battery states, condition indicators from the real battery responses, as well as the mechanistic features describing the internal conditions.\nMethods Section: Then, the prior estimations and the extracted features are input for the ML model to get enhanced monitoring results. Two residual models are proposed in this work for state monitoring enhancement, and the structures are shown in Fig. S8. The first model is the correction model, and the function is to enhance the prior estimations directly with an improved estimation. The function can be expressed as\nZ* = L (Z ,φ), z {SOC,SOH}, (6)\n1 p\nwhere Z is the prior estimation and Z* is the enhanced estimation from the correction model. For\np\n∈\nthe second residual model, the compensation model, the function of the machine learning is to estimate the residuals between the prior estimations and the real values to be compensated for the prior estimations. The compensation model for enhanced state monitoring can be expressed as\nZ** = Z +L (Z ,φ), z {SOC,SOH}, (7)\np p\n2\nwhere Z** is the enhanced state estimation through t∈he compensation model. All the notations and abbreviations are summarized in Table S10.\nReviewer #3 Comment 15: In Figs. 5f and S6, is the evolution of capacity with respect to cycle number realistic? The staggered profiles seem to match the fluctuations in ambient temperature, so these are not variations in actual capacity.\nAuthor response: The available capacity under different loading, temperature, and aging conditions varies. The capacity with respect to the cycle number shown in this paper represents the available discharge capacity under the current conditions. For example, the discharge capacities at different ambient temperatures and aging conditions are different. Therefore, the accurate estimation of available capacity under specific conditions is important for real-world applications, and the battery management system is expected to estimate that to ensure safe operations. As we aim at the monitoring for onboard battery management system, we think the real capacity with respect to specific conditions is reasonable, which is also adopted by other related works (i.e., using real discharge capacity to represent the current conditions).\nReviewer #3 Comment 16: The parallel resistance Rp is not included as a feature, only the time constant, but I would expect this parameter to carry more information about internal processes than the ohmic resistance R.\nAuthor response: Thanks for the suggestions, and we agree that R and C have more direct\np p\nmeanings on the polarization than the time constant (R *C ). Therefore, in the revised manuscript,\np p\nwe replaced the feature of time constant to be R and C . As demonstrated in Figs 3 5, where the\np p\nfeature importance analysis reflects the importance of both R and C for the two residual learners\np p\n–\nfor both SOC and SOH estimation enhancements, as reflected in the revised text presented in the response to Comment 10 and 12.\nReviewer #3 Comment 17: How is this method an improvement over the standard methodology consisting of an adaptive EKF for SOC estimation and a RLS for SOH estimation? This has been proposed and validated extensively in the literature. The ML model seems like an overkill within this framework.\nAuthor response: The standard methodology, i.e., mechanistic model-based filtering methods, actually performs as the prior estimations and also the comparative methods throughout the manuscript. As demonstrated in Figs. 2 5, where the verifications of the proposed mechanistic-leading residual learnings enhanced both SOC and SOH monitoring compared to the standard\n–\nmethods. This is the main motivation of our method, i.e., to enhance the standard estimations via the sequential hybrid modeling. In the revised manuscript, we added the descriptions of the conventional methods to be compared in the results. The detailed comparative discussions are also presented in the responses to the above comments. The revised text is in red below.\nMechanistic leading residual learning pipelines for enhanced state monitoring subsection:\nNote that the compared prior estimations represent the conventional model-based filtering methods.\nThe coulomb counting-based calculation is employed to obtain the referenced real SOC and SOH for the evaluations of the conventional model-based filtering method and the proposed residual learning models, while it is not used in any part of the modeling pipelines.\nReviewer #3 Comment 18: Other papers that employ ML algorithms for SOH prediction (see Refs. [23] and [29]) make an effort to generalize their approaches to different chemistries, which remains the main challenge for ML-based methods.\nAuthor response: The generalization of ML-based methods for different chemistries is one open challenge in this field. In the revised manuscript, we added more discussions on the listed references and we added two different battery types (prismatic and cylindrical) with different chemistries for the verifications. The results show the effectiveness of the proposed pipelines. However, the cross-chemistry application is still an unsolved problem. Therefore, we outlined this challenge in the future perspectives, which is in red text below.\nIntroduction Section: Feature extraction is one key factor that hugely influences the accuracy of the ML models.24–28 Methods for statistical calculation can extract highly effective features.24,25 However, these features are often sensitive to varying operating conditions, such as different current rates and temperatures, which can reduce model performance across diverse scenarios. While domain adaptation techniques can reduce the discrepancies between the feature distributions, thus enhancing the generalization of monitoring models under different conditions, they also increase model complexity and computational demands, hindering the application for real-time state estimations.29,30 Therefore, improving the suitability and coherence of features across different conditions is crucial for enhancing model accuracy and generalizability. Additionally, statistical features typically lack physical interpretability, complicating the explanation of ML models. Thus, incorporating features with physical meaning can significantly contribute to more interpretable monitoring systems.31,32 Recent research has increasingly focused on the interpretability of ML models, which provides deeper insights into their internal workings and facilitates better optimization.33–36\nVarying verifications subsection: Overall, we employed our mechanistic leading residual learning with two pipelines for enhanced state monitoring during the whole life span with feature interpretations and validations using different battery types and verification cases, which indicates high generalizations of our models. Nevertheless, the verifications are still conducted separately for the three datasets, though varying conditions are considered. It is still hard for cross-battery material applications since the large discrepancies in the mechanistic features of different batteries. Therefore, the future work needs to further investigate the improved approach enabling applications across different battery types.\nReviewer #3 Comment 19: How is the risk of data leakage treated? Given that only drive cycles are tested, there is a risk that the ML model is learning features specific to those profiles that may not transfer well to real-world stochastic operation, which has recently been shown to decisively influence aging behavior (DOI: 10.1038/s41560-024-01675-8).\nAuthor response: Thank you for the suggestion of a good reference for the discussion of verification in real-world operations. In the revised manuscript, we have incorporated the dataset from the suggested reference for further verification of the model performance under unseen real-world loadings. We also adopted the capacity from the RPT test as the referenced ‘real capacity’ (as asked by the reviewer in comment 10). And the application for the full life monitoring of real-world city loadings is also demonstrated. Specifically, the data from the urban/highway is used for the model training, and the different cells in the same test conditions are employed as validation 1. In addition, the dataset from the cycling with urban + highway is used as validation 2. Furthermore, the cells with real-world city loadings are also verified as validation 3. We even verified the model's performance with reduced sampling frequency (0.1 Hz) to emulate the real-world EV applications. Therefore, we validated several unseen scenarios under real-world applications. The revised text is in red below.\nData generation subsection: Finally, datasets using 25 cylindrical cells from Ref. 37, which were aged with synthetic highway and urban, as well as real city driving profiles, are employed to verify our model for use under different unseen working conditions. Typical current and voltage curves during aging under different loading profiles are shown in Fig. S7, indicating varying discharge patterns observed among different cycling, which therefore cause diverse working scenarios and aging trajectories. We used the first dynamic test after the diagnostic test as the dynamic cycling data during aging, the C/2 discharge capacity before the dynamic cycle as the referenced capacity, and the C/40 discharge curve of a new cell as the OCV curve. There are 483 dynamic discharge curves generated for the model evaluation.\nVarying verifications subsection: Finally, cylindrical batteries from Ref. 37 are employed for the verification of the proposed models by considering unseen working conditions. Three different validation cases are generated using cells with different synthetic profiles or real driving profiles aging. Specifically, as shown in Fig. 5a, cells with independent highway or urban loading profiles are divided into a training dataset and the validation 1 dataset. The loading profile of combining highway and urban loadings is included in the validation 2 dataset, and the cells with two city driving profiles are used in the validation 3 dataset. Therefore, we evaluated our mechanistic leading residual learning models with different real applications, considering unseen validation scenarios. Typical loadings and the related prior estimations are demonstrated in Fig. S20. The state monitoring and corresponding interpretation results are shown in Figs. 5bc, respectively. More detailed results for the three validation cases are shown in Figs. S21–S23 and Tables S5–S6. MAE for the SOC estimation keeps below 1.5% for all three validation cases and keeps below 1.8% for SOH estimations for both the correction model and the compensation model. All the cases indicate enhanced state monitoring performance compared to conventional model-based methods, even under unseen loadings for the whole life span. Similar interpretations can be summarized from the SHAP results, i.e., the prior estimations play significant roles for both the correction model and compensation model, while the mechanistic features provide critical information and have higher impacts on the compensation model compared to the correction model. Though the prior estimations are the most significant information, mechanistic parameters are essential for the model, as shown in Fig. S24, where the errors may increase instead by only using prior estimation for the model input.\nIn real-world applications, the sampling frequency is usually reduced, e.g., 0.1 Hz, so that the estimation accuracy is further deteriorated. The prior estimations with enlarged errors can also be enhanced by our mechanistic leading residual learners. We down-sampled the original data with 0.1 Hz and re-tested the three validation cases to evaluate the robustness of our models in potential real application conditions. The down-sampled currents and original currents (1 Hz) for different loading profiles are shown in Fig. S25, where the currents with 0.1 Hz lost fidelity. Therefore, the prior estimations became worse compared to the results with 1 Hz sampled currents (Fig. 5b), as shown in Fig. S26. Based on the proposed correction model and compensation model, the accuracies are enhanced with MAE less than 1.92% and 3.58% for SOC and SOH monitoring (which are more than 3.4% and 5.5% of filtering based-prior estimations for all the three cases), respectively, demonstrating the effectiveness in low sampling frequency applications. Through model interpretation, we noticed that, though the prior estimations are still the most important information, the accuracy is worse in these cases, which makes the mechanistic features become more important for the model to enhance state monitoring.\na c\nData Test Cells\nHighway 60, 62, 72\nTraining\nUrban 66, 68, 74, 76, 82, 84\nHighway 59, 64, 71\nValidation 1\nUrban 65, 67, 73, 75, 81\nHighway\nValidation 2 79, 80, 87, 88\n+ Urban\nCity 1 91, 92\nValidation 3\nCity 2 95, 96\nCorrection Compensation\nb\nFig. 5. Performance evaluation for cylindrical cells with unseen working conditions. a. Training and testing cells split strategies. b. SOC and SOH monitoring results for the three validation cases. c. Feature interpretation of the mechanistic leading models for the SOC and SOH monitoring.\nClosed-loop application entire life span subsection: Then, we show the application for dataset 3 with unseen testing conditions under real-world driving profiles. Note that we also use the dynamic cycle after the diagnostic test for verifications, so the degradation curve is composed of sparse points. By using the training cells in Fig. 5a to train the residual learners, the state monitoring under unseen working conditions (cell 92 aged using real-world driving protocols) entire the life span are enhanced, as shown in Fig. 6f. Unseen validation for exemplary cells working under another real city protocols (cell 95) and synthetic Urban + Highway (cell 80) protocols are also shown in Fig. 6f, demonstrating the stability and robustness of our model under varying unseen working conditions, which prove the good extrapolation capability of the proposed mechanistic leading residual learning pipelines. Numerical results listed in Table S9 indicate the MAE for all the tests is below 1.8% for both SOC and SOH. With enhanced SOC and SOH estimations through closed-loop operations, battery systems are poised for improved management and operational performance. This closed-loop structure not only supports effective state monitoring but also self-updates to maintain functionality and provide safety warnings, thereby optimizing the overall management of the battery system without the need for costly and time-consuming offline testing.\na b c d\ne\nCorrection Compensation Real values\nCapacity updating during operations and aging\nf\nCity 1 City 2 Urban + Highway\nFig. 6. Application demonstrations for the continuous monitoring of battery charge and health status of the entire life span. a-d. SOC estimation at four different aging stages and the demonstration of random working SOC ranges used for capacity estimation. e. SOC and SOH curves tracking for the whole lifespan. f. Unseen applications for the closed-loop state monitoring under both real-world and synthetic hybrid loadings.\nReferences: 37. Geslin, A. et al. Dynamic cycling enhances battery lifetime. Nat Energy, 10, 172–180 (2025).\nReviewer #3 Comment 20: In summary, the contribution with respect to previous approaches is not clear, and the physical foundation is questionable. The groundtruth values of SOC and SOH are determined via coulomb counting, and the definition of SOH is incomplete. While combining physics-based models with ML is promising, the physical foundation must be more solid and the ML model should provide a qualitative advantage over established approaches in order for the results to be significant.\nAuthor response: We thank the reviewer for the careful review and the detailed comments regarding the experiment, methodology, and verifications. We have improved the manuscript according to the respective comments above. The experiments have been detailed, the descriptions of the methodology have been improved, and more verifications have been conducted with comparative evaluations. Please refer to the responses to the specific comments above.\nReviewer #3 Comment 21: The code does not provide a README file with instructions for running, and does not contain sufficient comments. A sample of input data is not provided to verify performance.\nAuthor response: As requested, we have provided an improved version of the code with a README file. In addition, the runnable code is demonstrated based on the standard dataset suggested by the reviewer (DOI: 10.1038/s41560-024-01675-8).\nMechanistic Leading Residual Learning for Battery State Monitoring Entire Full Life\nNCOMMS-24-82679A-Z\nDear Editor and Reviewers,\nThank you very much for the constructive feedback. We have revised the manuscript and its supplementary information to address all the issues raised. Please find below our detailed response and revisions. Reviewers’ comments are in blue font, author responses are in plain back font, and changes to the manuscript are in red.\nResponse to Reviewer #1 (Remarks to the Author):\nReviewer #1 Overall Comment: The provided responses and the revised manuscript look good to me. I recommend 'accept' for this work.\nAuthor response: The authors appreciate the reviewer's careful review and recommendation for acceptance of our manuscript.\nResponse to Reviewer #3 (Remarks to the Author):\nReviewer #3 Overall Comment: The efforts from the authors are acknowledged in relation to the use of additional datasets for validation and the improved description of methodology and experimental aspects. Nevertheless, there are some further aspects that authors are expected to address for the manuscript to meet the standards of Nature Communications.\nAuthor response: The authors appreciate the reviewer for the careful review, the acknowledgement of our revision, and the further comments to improve the manuscript. We have provided detailed responses to the comments from the reviewer below, and the revised text is in red in the revised manuscript and supplementary information.\nReviewer #3 Major Comment 1: Although authors make an effort to validate the methodology on the Stanford dataset (DOI: 10.1038/s41560-024-01675-8), it is noted that all the current profiles employed for validation are still drive cycles. A subsequent validation on unseen constant current and periodic profiles from the same dataset with different pulse frequencies and charge/discharge ratios would substantially enhance the credibility of the proposed methodology, due to their observed impact on aging trajectories.\nAuthor response: Thank you for the suggestion to enhance the verification with more cycling profiles. We mainly focus on enhanced state monitoring under dynamic cycling, which is a significant concern in real-world electric vehicle applications. The added verifications using the Stanford dataset also contain unseen conditions, e.g., the training using Highway and Urban synthetic profiles for the state estimation of cells aged with real-world city profiles. Constant current and periodic profiles, though they have a vital impact on aging trajectories, are not widely seen in real-world electric vehicle applications. Since periodic profiles are also seen in other applications such as grid energy storage, we added the applications to the verifications on the cells with periodic profiles to further evaluate the extrapolation capability of the proposed model for unseen conditions. Specifically, the model trained from the same training cells in Fig. 5a is applied to verify cells aged with “Charge-Discharge Ratio”, “High Current Amplitude”, and “Frequency” periodic profiles under both C/10 and C/5 from the Stanford dataset. We added a supplementary figure and the description text in red below.\nVarying verifications subsection: We have verified the models using real-world driving cycles for potential real-world EV applications. To further evaluate the model extrapolation to other application scenarios such as grid storage, we applied the same model trained based on the training cells in Fig. 5a for the verification of cells aged with three different kinds of periodic profiles. The results for the SOC and SOH monitoring are shown in Fig. S28, indicating the effectiveness of the proposed model for accuracy enhancement even under quite different unseen loading profiles and aging trajectories.\nFig. S28. Extrapolation verification to the periodic profiles with “Charge-Discharge Ratio”, “High Current Amplitude”, and “Frequency” under C/5 and C/10. a. SOC error distributions of the filtering-based method and the two residual learning models. b. RMSE and MAE of the SOC estimation results. c. SOH error distributions of the filtering-based method and the two residual learning models. d. RMSE and MAE of the SOH estimation results.\na b\nc d\nReviewer #3 Major Comment 2: The authors argue that 'only one fresh OCV is required in this paper, while machine learning is expected for the deficiency compensations'. This is a strong claim, especially when more advanced degradation stages are considered, which can substantially alter the shape of the OCV. The implications of this assumption on estimation accuracy should be fully discussed in order to properly justify this claim.\nAuthor response: We thank the reviewer for the careful review and questions. In this work, we only used the fresh OCV in estimations of both SOC and SOH over the whole lifespan, i.e., the whole cycling data from the datasets, even with SOH drops below 50% for dataset 2. We agree that the OCV curve shapes will change during aging, especially for the silicon-graphite anode cells in dataset 3. However, even for dataset 3, we also only used one OCV curve from the first RPT test for the estimations until the end of cycling, and the results indicate the good performance of the proposed models. In addition, the OCV also changed under different temperatures and other conditions in real applications, and we also proved our model under variable temperature conditions in dataset 1. Therefore, the proposed residual learning models are expected to learn the residuals caused by model-based methods, which include the one caused by the OCV variations during aging. This has been verified with the three datasets of different batteries aged under different conditions. It is also more promising for real-world applications with fewer degrees of freedom in variables and lower data availability. In addition, the authors also noticed the deficiencies of the OCV models during aging and are currently investigating similar residual compensations for improved OCV fitting during aging, which will be published in our upcoming paper. In the revised manuscript, we revised the following red text for clarification.\nMechanistic leading residual learning pipelines for enhanced state monitoring subsection:\nThe ML model is built to compensate for the error caused by model uncertainties and other influences such as OCV variations under varying aging and temperature conditions.\nVarying verifications subsection: Only one fresh OCV curve is used for the estimations instead of updating during aging, which would require more tests and need further calibrations in real-world applications, and the results indicate the residual models are effective to learn the change during aging for accuracy enhancement.\nReviewer #3 Major Comment 3: Authors compare the estimation accuracy of the proposed methodology to that of the conventional model-based filtering approach. However, a comparison in computational efficiency is lacking, despite it being the other main factor for real-time monitoring in a practical BMS application. The computational overhead due to the addition of the ML models should be quantified and the tradeoff between accuracy and computational complexity should be discussed.\nAuthor response: Thanks for the suggestion on computation cost evaluation regarding potential real applications. In our work, we avoid complex deep learning methods, instead employing a simpler machine learning method (Random Forest regression). This kind of machine learning method was expected to have fast execution after training with a low-dimensional feature vector for each estimation step. To evaluate this exectation, we include the computational costs in an additional Supplementary figure (S31) for the three application studies in Fig. 6F. The time for each point is the total estimation time of the full discharge curve running on a laptop (13th Gen Intel(R) CoreTM i9-13950HX, 2.20 GHz, 128 GB). As Figure S31 shows, the additional computation cost for machine learning is acceptable, with a relatively constant time because, when the model is trained, the input feature matrix keeps the same dimension for usage. The time difference between the three figures is mainly caused by different C-rates and battery aging. It shows that the additional computation time for more than 20,000 data points (loading profiles shown in Fig. S7) is less than 1 second, indicating the potential for real applications. We have inserted the below red text into the revised manuscript and the supplementary figure.\nClosed-loop application entire life span subsection: Furthermore, we evaluate the computational cost for consideration of practical application. The execution time for the full discharge at each SOH point in Fig. 6f is shown in Fig. S31. Due to the low dimension of features and with a simple machine learning model, only a slight additional time (less than 1 second for the full discharge) is required for the residual learning models for accuracy enhancement.\nFig. S31. Execution time for each discharge curve during the aging for the three cells aged under City 1, City 2, and Highway+Urban profiles is in a-c, respectively.\na b c\nReviewer #3 Minor Comment 1: Although the authors provide more context in the revised introduction, some background and shortcomings of the combination of Kalman Filters and Machine Learning would greatly aid in better positioning this work with respect to state of the art.\nAuthor response: As requested, more background and shortcomings of the combination of Kalman filters and machine learning have been inserted into the revised manuscript. The combinations of filtering and machine learning are discussed while citing more related publications, and a more specific motivation is added for better positioning of our work. The revised text is in red below.\nIntroduction Section: Model-based methods are usually constructed with filtering methods such as the Kalman filter for internal state estimation. In contrast, data-driven methods, which map the relationships between the features and battery states via machine learning (ML) models, offer greater flexibility but often suffer from limited generalization capabilities.11 Consequently, hybrid monitoring structures that combine model-based filtering approaches with ML have gained popularity, aiming to enhance both accuracy and robustness.12–15 One straightforward method involves fusing the outputs of different sub-models, e.g., filtering-based estimation and machine learning estimation, to mitigate uncertainties associated with any single model.11,16,17 Due to the large variances of the estimation from data-driven approaches, especially for SOC estimation, one popular approach employs filtering techniques afterwards to enhance the estimation from ML models.18–21 However, the implementation in SOC estimation usually adopts the real reference, i.e., coulomb counting, as the observation, which makes it hard to obtain accurate calculations in real applications. On the other hand, ML excels at modeling uncertain relationships, and the estimated errors from model-based methods vary and are hard to describe.22,23 Therefore, the sequential framework of the model-based method followed by ML was proposed, where ML is employed after the filtering method to improve the initial estimation from the model-based methods.24–26 The main idea is that ML models the uncertainties and the errors of the mechanistic model-based methods, which have been employed for both hybrid battery modeling and temperature estimation. Nevertheless, the specific pipeline is designed for a specific purpose and has only been verified under several specific tests, limiting its practical large-scale application. Systematic design for general hybrid pipelines integrating model-based filtering and ML methods with applications using dynamic cycling for the entire life span is a practical yet unsolved challenge.\nReviewer #3 Minor Comment 2: In Fig. 6f (urban+highway), the SOH estimation results are less accurate from 0 to 1000 EFCs. Could this be due to the fact that RPTs were conducted every 100 cycles? If so, authors should discuss the applicability of the methodology in real-world operation during which periodic calibration cycles or full discharges are not available or more sparsely distributed in time.\nAuthor response: Thanks for the careful review and questions. The proposed method adopted random and partial discharge curves for the SOH estimation, and the model is applicable to dynamic profiles. All these three aspects are the main concerns regarding real-world EV operations. In addition, as discussed above, only fresh OCV is required for modeling. This means the proposed model does not need the periodic calibration cycles or the full discharges for the SOH estimations, which aligns with the requirement for real-world applications. As for the errors for urban+highway profile in Fig. 6f, since we evaluate our models under unseen conditions with machine learning-based approaches, it is common that estimation errors vary. Instead of selecting estimations for specific cells with high accuracy, we give the estimations for randomly selected cells to demonstrate the generalization of the proposed model. We also give the estimations for the other three cells under the same urban+highway loading profiles below, and some cells show better accuracy, and some may have larger errors. Therefore, the errors are mainly caused by machine learning models in some scenarios, which are common for all kinds of machine learning-based approaches. The more important thing is the effectiveness of the proposed models under unseen loading profiles for their whole life with partial and random dynamic discharges.\nCell 80 Cell 79 Cell 87\nReviewer #3 Minor Comment 3: ECM parameters are updated via a regularized RLS algorithm; it would be helpful to show (maybe in a Supplementary Figure) how these evolve over a full discharge cycle at several aging stages in order to ensure that they do not take unreasonable values or exhibit unphysical fluctuations.\nAuthor response: In our manuscript, we have shown the identified resistance variations during aging to show the earlier thermal warning capabilities in Fig. 4a. In the revised manuscript, we also insert a supplementary figure (S10) to demonstrate the variations of the ECM parameters over the dynamic discharge cycle during the degradations under different loading profiles. The results have been inserted into the place where we first introduce the online identification of mechanistic parameters. The added text and figure are in red below.\nMechanistic leading residual learning pipelines for enhanced state monitoring subsection:\nExamples of the variations of the estimated model parameters during aging under three different dynamic profiles are shown in Fig. S10, indicating stable mechanistic feature extractions under different working and aging conditions.\nFig. S10. Variations of the estimated parameters over dynamic discharging of the mechanistic models during aging for a. UDDS discharge, b. HWFET discharge, and c. Hybrid discharge.\nFresh Aged\na\nb\nc\nReviewer #3 Minor Comment 4: Authors should clarify that the available capacity under specific conditions is what is meant by 'capacity', as opposed to the theoretical capacity that one would measure during an RPT under thermodynamic conditions.\nAuthor response: Thanks for the suggestion. We have inserted a statement on the estimated capacity in this work, which is the available capacity under specific working conditions, to avoid confusion with the standard capacity that is generally defined with the RPT test. The revised text is in red below.\nMechanistic leading residual learning pipelines for enhanced state monitoring subsection:\nThe capacity here represents the releasable discharge capacity under the corresponding discharging condition with specific loading profile, temperature, and aging conditions.\nReviewer #3 Minor Comment 5: As suggested by Reviewer 2, SOC estimation on NMC or NCA chemistries poses fewer challenges due to a greater slope of the OCV and smaller hysteresis voltage. The applicability of this methodology and potential challenges should be briefly discussed specifically for LFP, taking into account its particular features such as a flat OCV and considerable voltage hysteresis.\nAuthor response: The authors thank the reviewer for the question. As we responded to Reviewer 2’s comment in the first-round revision, we admit the challenge of state estimation of LFP batteries, especially due to their hysteresis and OCV flatness. In this work, we aimed at the systematic design of mechanistic-leading residual learning models with applications to state monitoring the entire life span. We verified the effectiveness of the proposed models using different batteries with varying aging conditions that emulate real-world applications. The extrapolation has also been verified for several unseen conditions. Though the verification of LFP batteries is missing, we have listed the challenge of the state monitoring of LFP batteries in the discussion and added more specifications regarding hysteresis and flatness. In addition, our group is currently studying a similar residual learning pipeline in battery modeling for LFP batteries to handle the challenges of hysteresis and flatness, similar to the model uncertainties discussed in this manuscript, and we have also inserted this statement as future work in the revised manuscript. The revised text is in red below.\nResults section: It is still hard for cross-battery material applications due to the large discrepancies in the mechanistic features of different batteries caused by the chemical properties. Therefore, future work needs to further investigate the improved approach enabling applications across different battery types, especially for lithium-ion phosphate batteries, which pose a significant challenge to address the impact of hysteresis and OCV flatness in state estimations. In addition, the proposed mechanistic leading hybrid modeling framework is worthy of investigation for improving the electrochemical model fidelity."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The provided responses and the revised manuscript look good to me. I recommend 'accept' for this work.", "reviewer_3": "The efforts from the authors are acknowledged in relation to the use of additional datasets for validation and the improved description of methodology and experimental aspects. Nevertheless, there are some further aspects that authors are expected to address for the manuscript to meet the standards of Nature Communications:\nMajor Comments:\n1- Although authors make an effort to validate the methodology on the Stanford dataset (DOI: 10.1038/s41560-024-01675-8), it is noted that all the current profiles employed for validation are still drive cycles. A subsequent validation on unseen constant current and periodic profiles from the same dataset with different pulse frequencies and charge/discharge ratios would substantially enhance the credibility of the proposed methodology, due to their observed impact on aging trajectories.\n2- The authors argue that 'only one fresh OCV is required in this paper, while machine learning is expected for the deficiency compensations'. This is a strong claim, especially when more advanced degradation stages are considered, which can substantially alter the shape of the OCV. The implications of this assumption on estimation accuracy should be fully discussed in order to properly justify this claim.\n3- Authors compare the estimation accuracy of the proposed methodology to that of the conventional model-based filtering approach. However, a comparison in computational efficiency is lacking, despite it being the other main factor for real-time monitoring in a practical BMS application. The computational overhead due to the addition of the ML models should be quantified and the tradeoff between accuracy and computational complexity should be discussed.\nMinor Comments:\n1- Although the authors provide more context in the revised introduction, some background and shortcomings of the combination of Kalman Filters and Machine Learning would greatly aid in better positioning this work with respect to the state of the art.\n2- In Fig. 6f (urban+highway), the SOH estimation results are less accurate from 0 to 1000 EFCs. Could this be due to the fact that RPTs were conducted every 100 cycles? If so, authors should discuss the applicability of the methodology in real-world operation during which periodic calibration cycles or full discharges are not available or more sparsely distributed in time.\n3- ECM parameters are updated via a regularized RLS algorithm; it would be helpful to show (maybe in a Supplementary Figure) how these evolve over a full discharge cycle at several aging stages in order to ensure that they do not take unreasonable values or exhibit unphysical fluctuations.\n4- Authors should clarify that the available capacity under specific conditions is what is meant by 'capacity', as opposed to the theoretical capacity that one would measure during an RPT under thermodynamic conditions.\n5- As suggested by Reviewer 2, SOC estimation on NMC or NCA chemistries poses fewer challenges due to a greater slope of the OCV and smaller hysteresis voltage. The applicability of this methodology and potential challenges should be briefly discussed specifically for LFP, taking into account its particular features such as a flat OCV and considerable voltage hysteresis."}, "author_response": "Mechanistic Leading Residual Learning for Battery State Monitoring Entire Full Life\nNCOMMS-24-82679A-Z\nDear Editor and Reviewers,\nThank you very much for the constructive feedback. We have revised the manuscript and its supplementary information to address all the issues raised. Please find below our detailed response and revisions. Reviewers’ comments are in blue font, author responses are in plain back font, and changes to the manuscript are in red.\nResponse to Reviewer #1 (Remarks to the Author):\nReviewer #1 Overall Comment: The provided responses and the revised manuscript look good to me. I recommend 'accept' for this work.\nAuthor response: The authors appreciate the reviewer's careful review and recommendation for acceptance of our manuscript.\nResponse to Reviewer #3 (Remarks to the Author):\nReviewer #3 Overall Comment: The efforts from the authors are acknowledged in relation to the use of additional datasets for validation and the improved description of methodology and experimental aspects. Nevertheless, there are some further aspects that authors are expected to address for the manuscript to meet the standards of Nature Communications.\nAuthor response: The authors appreciate the reviewer for the careful review, the acknowledgement of our revision, and the further comments to improve the manuscript. We have provided detailed responses to the comments from the reviewer below, and the revised text is in red in the revised manuscript and supplementary information.\nReviewer #3 Major Comment 1: Although authors make an effort to validate the methodology on the Stanford dataset (DOI: 10.1038/s41560-024-01675-8), it is noted that all the current profiles employed for validation are still drive cycles. A subsequent validation on unseen constant current and periodic profiles from the same dataset with different pulse frequencies and charge/discharge ratios would substantially enhance the credibility of the proposed methodology, due to their observed impact on aging trajectories.\nAuthor response: Thank you for the suggestion to enhance the verification with more cycling profiles. We mainly focus on enhanced state monitoring under dynamic cycling, which is a significant concern in real-world electric vehicle applications. The added verifications using the Stanford dataset also contain unseen conditions, e.g., the training using Highway and Urban synthetic profiles for the state estimation of cells aged with real-world city profiles. Constant current and periodic profiles, though they have a vital impact on aging trajectories, are not widely seen in real-world electric vehicle applications. Since periodic profiles are also seen in other applications such as grid energy storage, we added the applications to the verifications on the cells with periodic profiles to further evaluate the extrapolation capability of the proposed model for unseen conditions. Specifically, the model trained from the same training cells in Fig. 5a is applied to verify cells aged with “Charge-Discharge Ratio”, “High Current Amplitude”, and “Frequency” periodic profiles under both C/10 and C/5 from the Stanford dataset. We added a supplementary figure and the description text in red below.\nVarying verifications subsection: We have verified the models using real-world driving cycles for potential real-world EV applications. To further evaluate the model extrapolation to other application scenarios such as grid storage, we applied the same model trained based on the training cells in Fig. 5a for the verification of cells aged with three different kinds of periodic profiles. The results for the SOC and SOH monitoring are shown in Fig. S28, indicating the effectiveness of the proposed model for accuracy enhancement even under quite different unseen loading profiles and aging trajectories.\nFig. S28. Extrapolation verification to the periodic profiles with “Charge-Discharge Ratio”, “High Current Amplitude”, and “Frequency” under C/5 and C/10. a. SOC error distributions of the filtering-based method and the two residual learning models. b. RMSE and MAE of the SOC estimation results. c. SOH error distributions of the filtering-based method and the two residual learning models. d. RMSE and MAE of the SOH estimation results.\na b\nc d\nReviewer #3 Major Comment 2: The authors argue that 'only one fresh OCV is required in this paper, while machine learning is expected for the deficiency compensations'. This is a strong claim, especially when more advanced degradation stages are considered, which can substantially alter the shape of the OCV. The implications of this assumption on estimation accuracy should be fully discussed in order to properly justify this claim.\nAuthor response: We thank the reviewer for the careful review and questions. In this work, we only used the fresh OCV in estimations of both SOC and SOH over the whole lifespan, i.e., the whole cycling data from the datasets, even with SOH drops below 50% for dataset 2. We agree that the OCV curve shapes will change during aging, especially for the silicon-graphite anode cells in dataset 3. However, even for dataset 3, we also only used one OCV curve from the first RPT test for the estimations until the end of cycling, and the results indicate the good performance of the proposed models. In addition, the OCV also changed under different temperatures and other conditions in real applications, and we also proved our model under variable temperature conditions in dataset 1. Therefore, the proposed residual learning models are expected to learn the residuals caused by model-based methods, which include the one caused by the OCV variations during aging. This has been verified with the three datasets of different batteries aged under different conditions. It is also more promising for real-world applications with fewer degrees of freedom in variables and lower data availability. In addition, the authors also noticed the deficiencies of the OCV models during aging and are currently investigating similar residual compensations for improved OCV fitting during aging, which will be published in our upcoming paper. In the revised manuscript, we revised the following red text for clarification.\nMechanistic leading residual learning pipelines for enhanced state monitoring subsection:\nThe ML model is built to compensate for the error caused by model uncertainties and other influences such as OCV variations under varying aging and temperature conditions.\nVarying verifications subsection: Only one fresh OCV curve is used for the estimations instead of updating during aging, which would require more tests and need further calibrations in real-world applications, and the results indicate the residual models are effective to learn the change during aging for accuracy enhancement.\nReviewer #3 Major Comment 3: Authors compare the estimation accuracy of the proposed methodology to that of the conventional model-based filtering approach. However, a comparison in computational efficiency is lacking, despite it being the other main factor for real-time monitoring in a practical BMS application. The computational overhead due to the addition of the ML models should be quantified and the tradeoff between accuracy and computational complexity should be discussed.\nAuthor response: Thanks for the suggestion on computation cost evaluation regarding potential real applications. In our work, we avoid complex deep learning methods, instead employing a simpler machine learning method (Random Forest regression). This kind of machine learning method was expected to have fast execution after training with a low-dimensional feature vector for each estimation step. To evaluate this exectation, we include the computational costs in an additional Supplementary figure (S31) for the three application studies in Fig. 6F. The time for each point is the total estimation time of the full discharge curve running on a laptop (13th Gen Intel(R) CoreTM i9-13950HX, 2.20 GHz, 128 GB). As Figure S31 shows, the additional computation cost for machine learning is acceptable, with a relatively constant time because, when the model is trained, the input feature matrix keeps the same dimension for usage. The time difference between the three figures is mainly caused by different C-rates and battery aging. It shows that the additional computation time for more than 20,000 data points (loading profiles shown in Fig. S7) is less than 1 second, indicating the potential for real applications. We have inserted the below red text into the revised manuscript and the supplementary figure.\nClosed-loop application entire life span subsection: Furthermore, we evaluate the computational cost for consideration of practical application. The execution time for the full discharge at each SOH point in Fig. 6f is shown in Fig. S31. Due to the low dimension of features and with a simple machine learning model, only a slight additional time (less than 1 second for the full discharge) is required for the residual learning models for accuracy enhancement.\nFig. S31. Execution time for each discharge curve during the aging for the three cells aged under City 1, City 2, and Highway+Urban profiles is in a-c, respectively.\na b c\nReviewer #3 Minor Comment 1: Although the authors provide more context in the revised introduction, some background and shortcomings of the combination of Kalman Filters and Machine Learning would greatly aid in better positioning this work with respect to state of the art.\nAuthor response: As requested, more background and shortcomings of the combination of Kalman filters and machine learning have been inserted into the revised manuscript. The combinations of filtering and machine learning are discussed while citing more related publications, and a more specific motivation is added for better positioning of our work. The revised text is in red below.\nIntroduction Section: Model-based methods are usually constructed with filtering methods such as the Kalman filter for internal state estimation. In contrast, data-driven methods, which map the relationships between the features and battery states via machine learning (ML) models, offer greater flexibility but often suffer from limited generalization capabilities.11 Consequently, hybrid monitoring structures that combine model-based filtering approaches with ML have gained popularity, aiming to enhance both accuracy and robustness.12–15 One straightforward method involves fusing the outputs of different sub-models, e.g., filtering-based estimation and machine learning estimation, to mitigate uncertainties associated with any single model.11,16,17 Due to the large variances of the estimation from data-driven approaches, especially for SOC estimation, one popular approach employs filtering techniques afterwards to enhance the estimation from ML models.18–21 However, the implementation in SOC estimation usually adopts the real reference, i.e., coulomb counting, as the observation, which makes it hard to obtain accurate calculations in real applications. On the other hand, ML excels at modeling uncertain relationships, and the estimated errors from model-based methods vary and are hard to describe.22,23 Therefore, the sequential framework of the model-based method followed by ML was proposed, where ML is employed after the filtering method to improve the initial estimation from the model-based methods.24–26 The main idea is that ML models the uncertainties and the errors of the mechanistic model-based methods, which have been employed for both hybrid battery modeling and temperature estimation. Nevertheless, the specific pipeline is designed for a specific purpose and has only been verified under several specific tests, limiting its practical large-scale application. Systematic design for general hybrid pipelines integrating model-based filtering and ML methods with applications using dynamic cycling for the entire life span is a practical yet unsolved challenge.\nReviewer #3 Minor Comment 2: In Fig. 6f (urban+highway), the SOH estimation results are less accurate from 0 to 1000 EFCs. Could this be due to the fact that RPTs were conducted every 100 cycles? If so, authors should discuss the applicability of the methodology in real-world operation during which periodic calibration cycles or full discharges are not available or more sparsely distributed in time.\nAuthor response: Thanks for the careful review and questions. The proposed method adopted random and partial discharge curves for the SOH estimation, and the model is applicable to dynamic profiles. All these three aspects are the main concerns regarding real-world EV operations. In addition, as discussed above, only fresh OCV is required for modeling. This means the proposed model does not need the periodic calibration cycles or the full discharges for the SOH estimations, which aligns with the requirement for real-world applications. As for the errors for urban+highway profile in Fig. 6f, since we evaluate our models under unseen conditions with machine learning-based approaches, it is common that estimation errors vary. Instead of selecting estimations for specific cells with high accuracy, we give the estimations for randomly selected cells to demonstrate the generalization of the proposed model. We also give the estimations for the other three cells under the same urban+highway loading profiles below, and some cells show better accuracy, and some may have larger errors. Therefore, the errors are mainly caused by machine learning models in some scenarios, which are common for all kinds of machine learning-based approaches. The more important thing is the effectiveness of the proposed models under unseen loading profiles for their whole life with partial and random dynamic discharges.\nCell 80 Cell 79 Cell 87\nReviewer #3 Minor Comment 3: ECM parameters are updated via a regularized RLS algorithm; it would be helpful to show (maybe in a Supplementary Figure) how these evolve over a full discharge cycle at several aging stages in order to ensure that they do not take unreasonable values or exhibit unphysical fluctuations.\nAuthor response: In our manuscript, we have shown the identified resistance variations during aging to show the earlier thermal warning capabilities in Fig. 4a. In the revised manuscript, we also insert a supplementary figure (S10) to demonstrate the variations of the ECM parameters over the dynamic discharge cycle during the degradations under different loading profiles. The results have been inserted into the place where we first introduce the online identification of mechanistic parameters. The added text and figure are in red below.\nMechanistic leading residual learning pipelines for enhanced state monitoring subsection:\nExamples of the variations of the estimated model parameters during aging under three different dynamic profiles are shown in Fig. S10, indicating stable mechanistic feature extractions under different working and aging conditions.\nFig. S10. Variations of the estimated parameters over dynamic discharging of the mechanistic models during aging for a. UDDS discharge, b. HWFET discharge, and c. Hybrid discharge.\nFresh Aged\na\nb\nc\nReviewer #3 Minor Comment 4: Authors should clarify that the available capacity under specific conditions is what is meant by 'capacity', as opposed to the theoretical capacity that one would measure during an RPT under thermodynamic conditions.\nAuthor response: Thanks for the suggestion. We have inserted a statement on the estimated capacity in this work, which is the available capacity under specific working conditions, to avoid confusion with the standard capacity that is generally defined with the RPT test. The revised text is in red below.\nMechanistic leading residual learning pipelines for enhanced state monitoring subsection:\nThe capacity here represents the releasable discharge capacity under the corresponding discharging condition with specific loading profile, temperature, and aging conditions.\nReviewer #3 Minor Comment 5: As suggested by Reviewer 2, SOC estimation on NMC or NCA chemistries poses fewer challenges due to a greater slope of the OCV and smaller hysteresis voltage. The applicability of this methodology and potential challenges should be briefly discussed specifically for LFP, taking into account its particular features such as a flat OCV and considerable voltage hysteresis.\nAuthor response: The authors thank the reviewer for the question. As we responded to Reviewer 2’s comment in the first-round revision, we admit the challenge of state estimation of LFP batteries, especially due to their hysteresis and OCV flatness. In this work, we aimed at the systematic design of mechanistic-leading residual learning models with applications to state monitoring the entire life span. We verified the effectiveness of the proposed models using different batteries with varying aging conditions that emulate real-world applications. The extrapolation has also been verified for several unseen conditions. Though the verification of LFP batteries is missing, we have listed the challenge of the state monitoring of LFP batteries in the discussion and added more specifications regarding hysteresis and flatness. In addition, our group is currently studying a similar residual learning pipeline in battery modeling for LFP batteries to handle the challenges of hysteresis and flatness, similar to the model uncertainties discussed in this manuscript, and we have also inserted this statement as future work in the revised manuscript. The revised text is in red below.\nResults section: It is still hard for cross-battery material applications due to the large discrepancies in the mechanistic features of different batteries caused by the chemical properties. Therefore, future work needs to further investigate the improved approach enabling applications across different battery types, especially for lithium-ion phosphate batteries, which pose a significant challenge to address the impact of hysteresis and OCV flatness in state estimations. In addition, the proposed mechanistic leading hybrid modeling framework is worthy of investigation for improving the electrochemical model fidelity."}, {"round": 3, "reviewer_comments": {"reviewer_3": "The authors' responses to my comments are appropriate and clearly explained, and the additions with respect to validation on pulsed profiles and computational cost are satisfactory. Although I still have some concerns regarding the consideration of constant OCVs through aging and the fluctuations in parameter Cp within one cycle, I understand that the proposed methodology in its current form can be applied in a practical scenario with acceptable results and an affordable computational cost, and may therefore be of interest to potential readers of Nature Communications.\n(Remarks on code availability)"}, "author_response": ""}]

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