Optimizing Cycle Life Prediction of Lithium-ion Batteries via a Physics-Informed Model
Authors: Nathan Sun
Daniel Nicolae
Sara Sameer
Karena Yan
Venue: Accepted by TMLR ()

Abstract: Accurately measuring the cycle lifetime of commercial lithium-ion batteries is crucial for performance and technology development. We introduce a novel hybrid approach combining a physics-based equation with a self-attention model to predict the cycle lifetimes of commercial lithium iron phosphate graphite cells via early-cycle data. After fitting capacity loss curves to this physics-based equation, we then use a self-attention layer to reconstruct entire battery capacity loss curves. Our model exhibits comparable performances to existing models while predicting more information: the entire capacity loss curve instead of cycle life. This provides more robustness and interpretability: our model does not need to be retrained for a different notion of end-of-life and is backed by physical intuition.

Published in Transactions on Machine Learning Research (04/2025)
Optimizing Cycle Life Prediction of Lithium-ion Batteries via a Physics-Informed Model
Constantin-Danie! Nicolae"
cdn27@com.ac.
University of Cambridge
Sara Sameer" National University of Computer and Emerging Sciences
surusameer:99 &ggmail com
Nathan Sun*t Harvard University
naun@Bollege.harvordd..e
Karena Yan'
kyan@princeton..dd edu
Haroord University
Reviewed on OpenReview: https: // opt.r/veew.eee...m net/ forum? id- IweZ9Wsa
Abstract
Accurately measuring the cycle lifetime of commercial lithium-ion balteries crucial for performance and technology development. We introduce novel hybrid approach combin- ing a bhssics-based equation with self-attention model to predict the cycle lifetimes of commercial lithium iron phosphate graphite cells via early-cycle data. After fitting capacity loss curves to this physics-based equation, we then use self-attention layer to reconstruct entire battery capacity loss curves. Our model exhibits comparable performances to existing models while predicting more information: the entire capacily loss curve instead of cycle life. This provides more robustness and interpretability; our model does not need to be retrained for a different notion of end-of-life and backed by physical intuition.
1 Introduction
Predicting the cycle life of a lithium-ion battery remains challenging due to the complexity of the chemical
side effects responsible for degrading the performance of battery as is repeatedly cycled. Tn particular, it is well known that solid electrolyte interphase (SEI) formation crucially affects cycle life and occurs within
the first. few charging/ srigiinn/dsscagiinngg cycles (von Kolzenberg Broussely et al. 2001) Accurately predicting the cycle life of batteries while accounting for all these side chemical processes is important for maintaining battery performance.
al. 2020; Wang al. 2011 Spotnitz 2003;
Recently, Severson et al. (019) presented a state-of-the-arl dataset containing 124 lithium-ion batteries with 72 different fast-charging policies and showed that a regularized linear regression model predicting cycle lifetimes performs very well on batteries with different charge policies. They also successfully showed that this prediction can be obtained within the first hundred charging cycles. Their method of using early- cycle data for prediction has great practical implications, since one need not wait to charge battery for many cycles before knowing its lifetime.
Previous literature on predicting cycle lifetimes of batteries is rich. Data-driven models (Yao et al. 2022
Celik et al. 2022 Abu-Seif et al. 2022 Xing et al. 2023), which focus on using machine learning techniques to identify trends in how batteries degrade, have been thoroughly studied, with models ranging from linear
models (Severson el. al. 2019) to neural networks (Celik al.) 2022 Strange Dos Reis 2021 Su el al.
*All authors contributed equally.
1 "Corresponding author.

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2021 to support vector regression (Zhu et al. 2022]. These types of models are agnostic to the mecha- nisms of degradation, but require significant effort to fine-tune, as their performance depends heavily on hyperparamettr optimization, feature selection, and the quality of the training data. Slysics-based models, which rely upon knowledge in how a battery degrades over time, have also beon well-studied. These methods
usually rely upon cell chemistry (Wright et al.) 2003; Rahman & Lin 2022 Yang et al. 2017] or analyzing how the material in the electrodes changes over time (Christensen & Newman 20004 Pinson Bazant, 2012). However, since batteries can be ddincerddsse in a variety of environments, this hinders how descriptive physical models by themselves can be. Recently, hybrid models combining physics knowledge
with a data-driven approach have been proposed (Xu et al. 2022. Cordoba-Arenas et al. 2015 Pang et al.
2022 Saxena et al. 2022) to combine the advantages of both approaches.
We propose a hybrid model combining a physics- based equation and a self-attention mechanism for predic- tion. The latter is inspired by the recent rise of transformers to predict sequential data (Nguyen et al., 2022; Vaswani et al., 2017). and the former uses physics insights to capture more information on the behavior of the capacily loss curves of lithium-ion batteries. Although transformers have already found applications in predicting the cycle life of batteries (Xu et al. 2023), combining physical equation with self-attention to predict complete capacity loss curves has not been thoroughly studied to the best of our knowledge.
Using the hybrid model, we are able to predict battery capacity loss curves. Cycle life is defined as the point where the effective capacity of the battery has dropped to a percentage of its nominal capacity, In the literature, 80% is typically the threshold, but other studies have considered 85% (Schauser et al. 2022; Sours et al. 2024). With entire capacity loss curves, we are able predict cycle life for all other thresholds without retraining, while models that are specifically trained with the 80% threshold must be retrained. This accounts for conditions that may require different threshold for cycle life, such as the make of the battery and the presence of non-ideal operating conditions.
The paper is organized as follows. In Section 2 we introduce data processing techniques and extract relevant information from discharge curves. We then introduce a hybrid model utilizing an Arrhenius Law to model capacity loss and utilize self-attention to predict cycle life from early-cycle discharge data. Finally, we compare our errors with Severson al. (2019) and conclude with future directions.
2 Data Processing
We utilize the public dataset provided by Severson et al. (019)) This dataset comprises 124 lithium-in phosphate/graphite battery cells, each with a nominal capacity of 1.1 ampere-hours (Ah) (A12 2014). The cells are cycled (repeatedly
until end of life, which we take as the point where the effective capacity of the battery has dropped to 80% of its nominal value. The ratio of the effective and nominal capacity is commonly referred to as the state of health. All battery cells in this dataset are cycled under fast-charging conditions in a constant-tempent coustant-eemperature temperatre environment; however, the charging policy dictating the specific charge rate schedule differs from cell to cell.
In our work, we preserve the beinn/test/seconddee direct comparison between our result and theirs. The primary test set was obtained using the same batch of cells as the train set and similar charge policies; therefore we use it to evaluate the model's ability to interpolate in the input space. On the other hand, the secondary test set was obtained from a different batch of cells and using significantly different scheduling, and we use it to examine the model's ability to extrapolate. Ensuring our model can generalize to different batteries makes for an advantage over prior work [Saxena et al. 2022 Strange & Dos Reis 2021 that uses random split of cells into train/ /aaavvvvaaioo///e /test sets.
test data split in Severson et al. (2019), allowing for
For each cell, four forms of data are recorded:
1. Cycle life: The number of charge/dischane cycles until the state of health drops to 80%, ranging from 150 to 2,300.
2. Charge policy: The schedule of charge rates followed during cell cycling.
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Fully discharged
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Time
Voltage (V)
Figure (a) Discharged capacity, Qa((), and voltage, V((), measured for one battery the course of charge-discharg cycle. The discharge portion of the cycle is boxed in black. (b) The sercharge-voltagee curve, Qa(V), for the same battery. Note that the x-axis in (b) is flipped to reflect voltage decrcasing over the course of the discharging process.
3. Cycle summary features: Features calculated for each cycle, such as the state of health, internal resistance (IR), average cell temperature (Tavg), and maximum cell temperature (Tmax).
Full cycle data: Measurements taken over the course of each cycle, such as voltage (V), discharged capacity (Qa), and lemperature (T)-
Voltage (V) and discharged capacity (Qa) are particularly relevant for lifetime prediction and are depicted for example charge-discharge cycle in Figure aa.
The discharge-voltage curve for a cycle, denoted Qa(V), is constructed by plotting Qa against V for the discharge portion of the cycle, as boxed in Figure a (a) One such curve is shown in Figure b)). According to manufacturer specifications, a battery cell is considered fully charged when its voltage reaches 3.6V and fully discharged when reaches 2.0V (A12 2014). An analogous curve can be constructed for each cycle of battery's operation.
Choosing the voltage to be our coordinate for the discharge curves results in highly irregular set of evaluation points for Qa(V). As illustrated in Figure [b). the V values sampled at regular timesteps are sparser in the intervals [3.2V, 3.6V] and [2.0V, 3.0V] and denser in the interval [3.0V, 3.2V]. Moreover than being irregular, the points at which Q2(V) were to be evaluated are also different from one cycle to another. This makes direct comparisons between different cycles difficult. To standardize the discharge curves, we utilize radial basis function interpolation, a standard method used for unstructured inputs. In 1D, this consists of an interpolant given by Equation I
Qa(V)= (o(V))pm(V)
Σ > -VI)
(1)
The datapoints therefore interpolated by weighted sum of radial basis functions, with the origins at the interpolation nodes V,, and by Pm (V) = + V +... cm Vm a polynomial of degree m. The cocfficients of Pm and the weights A₃ are found by solving a set of linear equations. The form of & dictates the final form of the interpolant. The role of the polynomial term is to model any possible trend the data, which the RBFs are unable to do due to their symmetric, radial nature.
The evolution of Qv) over cycles can be exploited predict battery lifetime. Severson al. 2019 observe that more dramatic early-cycle curve sagging occurs for batteries with low lifetimes than for batteries with
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high lifetimes, visualized in Figure 2aa, b). To capture the phenomenon of curve sagging, Severson et al. (2019) propose taking the discharge-voltage curves for cycles 100 and 10 and computing the difference between the two. This new curve, calculated as QV) (V), is denoted AQ100-10(V), Figure cc,d) shows that AQ100-10(V) succinctly capture the difference in curve sagging behavior between two batteries of different cycle lives, and Figure e) shows a clear linkage between curve shape and cycle life across all batteries the dataset. In particular, batteries with shorter cycle lives exhibit more ample dips in the AQ 10(V) curve.
Now statistical quantities, such the variance, minimum, and mean, of AQ100-10 are calculated to further condense information of cycle life for each battery. A simple variance- based model would, for instance, use Var(AQQno) 1000000(oV)))) as an input to predict the cycle life for a single battery.
3 Model
3.1
Physics-Based Model
It well known that as a lithium-ion battery is cycled, other chemical processes occur in the battery that affects long term cyclability. Notably, the SET (solid electrode interphase) is formed on the surface of the anode within the first. five charging cycles, impeding electron movement in the battery. Although poorly understood, it is believed that SEI formation has an impact capacity loss (von Kolzenberg et al. 2020). Hence it is important to consider the impact of SEI formation when studying capacity loss. In response, we construct a model of the capacity curves in the Severson et al. (2019) dataset via an Arrhenius Law, which is commonly associated with chemical processes akin to SEI formation (Wang et al.
2011 Spotnitz 2003
Broussely et al., 2001).
We now introduce an equation describing the process of capacity loss. Wang et al. for different charge rates, the true capacity loss Qloes can be approximated as
2011 have shown that
Ep RT
@x0 = D exp
x
(2)
for constants and D, where Quess represents the percentage of capacity loss, Eg is the activation energy, R the gas constant, T' the absolute temperature, and æ the cycle number. Note that the temperature dependence in the exponential term resembles the Arrhenius Law, which is commonly associated with thermally activated chemical process, such as SEI layer formation during cycling ( (Wang et al. 2011 1 Spotnitz,
2003 Broussely et al. [001]. Equation (2) aims to account for SEI formation that produces undesired side
effects in batterics. In addition, there is also a power law with respect to the cycle number, consistent with
previous findings of the rate of lithium consumption at the negative electrode (Wang et al.) 2011 Spotnitz, 2003 Wright et al. 2002 Broussely et al.] 2001) In short, this equation soeks describe SEI formation and is consistent with the resulting mechanism of active lithium consumption in the presence of the SEI layer.
We adapt Equation 2 in three ways, while maintaining its physical interpretation. In our dataset, we observe that cells typically show an initial capacity loss even at cycle O, which can be attributed to various factors, including calendar aging during storage and formation cycles. To account for this, we introduce a constant term that represents this initial capacity loss. We use the nominal capacity as our reference than the measured capacity at cycle O, as this provides a consistent baseline across all cells and allows for more robust comparison between different colls, particularly when initial conditions may vary due to storage time or formation cycles.
Next, we observe that the average temperatures reached during testing did not vary greatly across cycles. Treating T as invariant allows us to incorporate the original exponential factor from Equation (2) into the constant D. This has the advantage that. E,, need not be known anymore.
Two adaptations have boen made the original Equation (2) from Wang al. (011). Firstly, replaced the dependency time with dependency on cycle number. This is acceptable since all cycles last equal amounts of time. Secondly, changed the notation of the parameters conflict with notations we introduce next.
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a.
b.
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Figure 2: (a, b) Evolution of the discharge-voltage curve over cycles for two batteries with different cycle lives. Curves from cycles evenly spaced between and 100 are plotted and distinguished by saturation. As cycle number increases, the curve progressively sags more for the battery with lower lifetime. (c, d) AQ100-10 for the same two batteries. (e) AQ100-10(V) plotted for all batteries the dataset, with shade corresponding to cycle life.

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Lastly, we note that typical values of the constant D have very large order of magnitude, which introduces numerical instability and overflow issues in further calculations. We mitigate this by predicting A = In D instead of D itself.
Thus, we may reduce Equation 2 to one of the form given by Equation By defining C' to be equal to the initial capacity loss (loss(0); we arrive a functional form with just two parameters, A and B.
Q (2) = A +C
We then fit the capacity loss curves in Severson et al. (2019) to Equation (3) via least squares regression. Figure 3 aa-c) illustrates three capacity loss curves with substantially different cycle lives and their best fit curves. Not only are the curves themselves a good fit, but the cycle lives predicted by our best fit curves are remarkably similar to the actual cycle lives. Cycle life é is the point where Quess = 0.2, calculated from the best fit. curve as
= 1= C
(4)
where 0.2 is used as the threshold capacity loss indicating end of life. Figure 3 d) plots true cycle life against predicted cycle life for all batteries in the dataset and demonstrates high goodness of fit, with R 0.994 and a RMSE of 28.6 cycles. Additional evidence of goodness of fit can be found in Appendix A
As seen in Figure d) and Appendix A our equation very accurately models the cycle lives of batteries, and henceforth we assume that the ground truth of capacity loss follows Equation 3 given values A and B. In other words, we assume that Qlaes Ri Qlaes-
Now the other half of our hybrid model involves training self-attention model to predict Å and B from cycle imput data of the first 100 cycles, which will be explained in detail in the section below. Call this model f. Then given vector of early-cycle statistical quantities, output variables are (Á, B) f(x) and our predicted cycle life i = (e-A((0 A(02 2 - C)JL/B This model is illustrated Figure
3.2 Self-Attention for Regression
We endeavor to predict the parameters of the capacity loss curve based on early-cycle data, when the full capacity loss curve not known. When limited to carly-cycle capacity loss data, we cannot fit and extrapolate the full curve with good fidelity using least squares. Instead, we employ a single self-attention block to learn complex, nonlinear relations between the capacity loss curve and other measurements available during battery's early operation. As presented in Nguyen et al. (2022), there exists equivalence between the self-attention mechanism and support vector regression (SVR) problem formulation. This implics that any problem where an SVR model performs well can potentially be improved by employing an atlention-based architecture.
Let the input sequence of N features derived from carly-cycle operation data for the i-th battery be denotod 2, (NIT E RNx1, where each =(j), represents a measurement feature such as voltage, current, 2; = |2 (1) ?
and temperature We compute the standard quory matrix Q, key matrix K, and value matrix from the sell-attention mechanism via the following transformations:
Q: 2, W
K, = 2, WT
V. z, WWY
where weight matrices We Wx DD1 and Wy E R.x1 are learnable layers, D is hyperparameter determining the embedding dimension, and D, is the output dimension. Note that for the multi-output regression task of predicting two variables, D. = 2. We define the self-attention output H; as:
Q.K VD
H = softmar
V A,V,
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where the softmax function above is applied to each row of the matrix Q.K.
vD to obtain the attention matrix
A,, To collapse the output H, E RNDD, to vector, we append an averaging layer to the self-attention mechanism that takes the mean along the columns of H,
A [B
Yr H m
where 1 11T ERI. The result y; D.x1 is the vector of predicted parameters for the capacity loss curve. These parameters are then used to predict cycle life:
1 = [e- A. (0.2 (02-CC))/B
(5)
3.3 Feature Selection
Finally, select. the best features for prediction from the available cycle summary features and discharge curve characteristies described in Section 2 From the cycle summary data, extract capacity fade char- acteristics by calculating slopes over different cycle windows. From the discharge-voltage curves Qa(V), we derive statistical features including the variance, minimum, and mean of the AQ curves, following the approach introduced by Severson et al. (2019). Additionally, we calculate logarithmic transformations of these quantities to better capture the nonlinear relationships in the data.
Our main task and B. To accomplish this, analyze the correlation between each feature and the target variables. We prioritize features with strong positive negative correlations, they are more likely to provide accurate predictions. We use Spearman's 's correlation coefficient to ascertain which features are most correlated with À and B. In our analysis, we select the top five features with the highest correlation coefficients. The results of our correlation analysis using this approach are visually represented in Figure 5 and the full correlation coefficients can be lound in Appendix B
carefully select the features that are most closely related to our target variables, Á
Note that these features are:
1. DeltaQ LootaaQ_oogVrrr Vars:
The log of the variance of the AQ100-10(V) curve.
DeltaQ logMin: The log of the min of the AQ100-10(V) curve.
3. logMeen: The log of the mean of the AQ100-10(V) curve.
Capacity fade rate from cycle 2 to 100. This is the rate that the capacity fade curve decreases from cycle 2 to cycle 100.
fade 2 100:
5. haape__appctt_____ 991 _1__10001 Capacity fade rate from cycles 91 to 100. This is the rate that the capacity [ade curve decreases from cycle 91 to cycle 100.
These live features were used train the sell-attention model as explained in Section 3.2
3.4 Model Training
Our interest lies in minimizing the root mean squared error (RMSE) of cycle life predictions, defined for a set of n batterics as:
RMSE; Σ 14 i,.
(6)
where 1, is the true cycle life of the i-th battery. However, given the complexity of the expression for cycle life, attaining optimal convergence is a nontrivial numerical task. To this end, we employ a two-stage procedure of coarse training on curve parameters followed by fine-tuning on cycle life.
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In the first stage, we train the RMSE of parameter loss, defined as
RMSE,
1 n
wa(A; A.)² wB(B; B.)2).
where WA and wg are tunable hyperparameters. The parameter loss function is smoother and leads to fewer numerical issues than Equation contrast, we observe that training only with the original cycle life loss function leads to exploding gradients and inconsistent behavior.
G producing stable results when training from a random initialization.
However, parameter loss is not always indicative of the accuracy of cycle life predictions. Due to the high nonlincarity of the capacity loss curve, it is possible for two sets of parameter estimates to incur cqual parameter loss but produce dramatically different cycle life predictions. Consequently, we follow up coarse training with a fine-tuning stage under low learning rate that trains on cycle life loss.
The two-stage training procodure can be summarized as follows:
Coarse training on parameter loss (Equation 7 This stage guides the model from a random initialization toward approximate parameters, eliminating numerical issues that would otherwise occur in Stage 2.
2. Fine-tuning on cycle life loss (Equation ) ). As many combinations of Å and ß can lead to similar paramoter losses, this stage tunes the approximate parameters from Stage to produce the most accurate cycle life predictions.
Training took place with 800 epochs and learning rate of 10-3 in step 1 and with 3000 epochs and a learning based upon the existing litcrature,
rate of 5 x 10-5 in step 2. The choice of optimal hyperparameters and such parameters can be consulted in the Github repository associated with our work, available upon request. A full comparison of the relevant training curves can be found in Appendix C and Appendix D
4 Results
We initiate the model training phase with bascline model using a regularized linear framework. For the baseline model, take sklearn's ElasticNet model that performed the best in Severson et al. (2019).
However, to ensure comparable results, we train the baseline model on the features chosen in Section 3.3 We further change the output of the baseline model to predict Á and B in order reconstruct the entire capacity curve. The results of the baseline model are shown in Figure
For hyperparameter tuning, we vary alpha and 11_ratio on a logscale over 10 and 10a and 10-5 and 10²; respectively. We find that the best model had primary test RMSE of 398.98 cycles and sccondary test RMSE of 455.4 cycles. We see that in this case, the linear assumptions of elastic net limit its ability to capture nonlinear degradation patterns and parameterize capacity loss curves.
Next, we trained the hybrid model (also referred to as the self-attention model) to predict the parameters À and B. The results of our model evaluation are presented in Figure 6 which compares the performance of the elastic net baseline with that of the self-attention model terms of the train RMSE, primary test RMSE, and secondary test RMSE. Results from the self-attention model are divided into two stages, Stage and Stage 2, corresponding to the two phases of training.
Note that the errors achieved by self-attention after two stages of training are significantly better than the elastic net baseline; further, they are comparable with the original RMSE errors in Severson et al. (2019 After two training stages, we were able to improve the primary test RMSE from 398.87 to 127.83 cycles and the secondary test RMSE from 455.4 to 179.92 cycles.
Figure 7 illustrates the true versus predicted cycle lives (a) as well as true and predicted capacity curves for sample batteries (b-d) using the fully trained self-artention model. We notice that the predicted capacity
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loss curves do indeed fit the behavior of the ground truth curves across train, primary test, and secondary test batteries. Further, recall that the full model in Severson et al. (2019) utilizing multiple features achieves a primary test RMSE of 118 and a secondary test RMSE of 214. Our primary test error is quite similar, but our secondary test error improves upon theirs by 30 cycles (over a 15% improvement). Given that our model captures more information on how a battery degrades over time, we conclude that our model serves as an appealing alternative to predict battery cycle life, espocially in cases where one wishes to have the flcxibility to consider a different definition of cycle life.
5 Conclusion and Future Directions
Our research presents a novel technique understanding and predicting battery capacity loss curves using physics-informed model. Our focus on reconstructing these curves rather than on just cycle life prediction offers distinct advantages by providing more robust and interpretable predictions without sacrificing the accuracy achieved in prior work (Severson et al., 2019). Parameterizing the ground truth capacity loss curves by fitting an equation inspired from an Arrhenius Law, we train a self-attention model that recovers these parameters and reconstructs the full capacity curves, achieving similar errors to Severson et al. (2019). This approach flexible to the definition of end of life, offering the advantage of predicting the cycle at which any percent of capacity is lost.
However, there are a few other directions that polentially further improve upon work. Notably, the battery cycle data used is time series, and hence utilizing entire time series prediction inputs is way to incorporate more information and features. This method could potentially improve results given the correct training policy, hyperparameters, and machine learning archilectures.
Our method not only provides precise capacity loss forecasts but also incorporates knowledge of the mech- anisms underlying battery degradation. The results of this study provide support for the effectiveness of hybrid models, since they combine the best aspects of data-driven and physics-basee baseed methodologies. This has the potential to ultimately improve battery life estimation for use in electric vehicles and other socially significant applications.
Data Availability
The datasets used this paper are available at httis::/data.matr..o//
Code Availability
The Github repository can be found at http:s//ggthho.ccmmaattta//////p
Disclosures
C.D Nicolae, S. Sameer, N. Sun, and K. Yan are listed as inventors on a patent application related to this work.
Acknowledgments
This research was conducted at UCLA as part the RIPS 2023 REU and supported by IPAM and the Toyota Research Institute. The authors are grateful to Tan Nguyen and Alex Pham for their mentorship and continued guidance. They also thank Lingyun Ding for fruitful discussions and his continued input. A special thanks to Susana Serna for her continued guidance and feodback navigating this project. Lastly, the authors are grateful to the anonymous referecs for their excellent feedback and comments.
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Ashish Vaswani, Noam Shazoer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaisor, and Illia Polosukhin. Attention is all you need. Advances in neural information processing systems, 30, 2017.
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Lars von Kolzenberg, Armulf Latz, and Birger Horstmann. Solid-clectrolyte interphase during battery cycling: Theory of growth regimes. ChemSusChem, 13((5):39013300) -3910,, 2020.
John Wang, Ping Liu, Jocelyn Hiiks-Garnee Elena Sherman, Souren Soukiazian, Mark Verbrugge, Harshad Tataria, James Musser, and Peler Finamore. Cycle-life model for graphite-L.FePO4 cells. Journal of Power Sources, (96($))3342-3944) 2011.
R.B. Wright, C.G. Motloch, J.R. Belt, J.P. Christophersen, C.D. Ho, R.A. Richardson, Bloom, S.A. Jones, V.S. Battaglia, G.T. Henriksen, et al. Calendar-and cycle-life studies of advanced technology development program generation lithium-ion batteries. Journal of Power Sources, 110(2):445 470, 2002.
R.B. Wright, J.P. Christophersen, C.G. Motloch, J.R. Belt, C.D. Ho, V.S. Battaglia, J.A. Barnes, T.Q. Duong, and R.A. Sutula. Power fade and capacity fade resulting from cyclo-life testing of advancod technology development program lithium-ion batteries. Journal of Power Sources, 119:865 869, 2003.
Wei W. Xing, Akeel A. Shah, Nadir Shah, Yinpeng Wu, Qian Xu, Aphichart Rodchanarowan, Puiki Leung, Xun Zhu, and Qiang Liao. Data-driven prediction of Li-ion battery degradation using predicted features. Processes, 1177)6678 2023.
Le Xu, Zhongwei Deng, Yi Xie, Xianke Lin, and Xiaosong Hu. novel hybrid physics- based and data-driven approach for degradation trajectory prediction in Li-ion batteries. IEEE Transactions on Tronsportation Electrification, 2022.
Ruilong Xu, Yujie Wang, and Zonghai Chen. A hybrid approach to predict battery health combined with attention-based transformer and online correction. Journal of Energy Storage, 65:107365, 2023.
Xiao-Guang Yang, Yongjun Leng, Guangsheng Zhang, Shanhai Ge, and Chao- Yang Wang. Modeling of lithium plating induced aging of lithium-ion batteries: Transition from linear to nonlinear aging. Journal of Power Sources, 360:28-40, 2017.
Jiwei Yao, Kody Powell, and Tao Gao. A two-stage deep learning framework for early-stage lifetime prediction for lithium-ion batteries with consideration of features from multiple cycles. Frontiers in Eneryy Research, 10:1059126, 2022.
Jiangong Zhu, Yixiu Wang, Yuan Huang, R. Bhushan Gopaluni, Yankai Cao, Michael Heere, Martin J. Mühlbauer, Liuda Mereacre, Haifeng Dai, Xinhua Liu, et al. Data-driven capacity estimation of commercial lithium-ion batteries from voltage relaxation. Nature communications, 13(1):2261, 2022.
A Fitted Capacity Loss Curves
This section presents the fitted capacity loss curves against the true capacity loss Qloes. One may be concerned that the fitted capacity loss curves only capture the cycle life instead of the full process of capacity loss. Our roconstruction of the capacity loss curves demonstrate goodness of fit to the entire capacity loss curve, with an average R² of 0.976 (standard deviation 0.0644) across all batteries in the dataset. Figure & plots the R² of all fitted capacity loss curves compared the true curves for all batteries, and Figure shows the fitted true capacity loss curves for all batteries.
B Feature correlation scores with parameters À and B
This section presents the correlation scores between the candidate features and equation parameters Á and B. The scores of the five features with the highest correlation are bolded in Table 1 detailed discussion of feature extraction methodology and their physical significance is provided in Section 2
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Feature Name
Feature Description
A
B
DeltaQ logVar
Variance of logarithmic changes in capacity difference
0.46
-0.23
DeltaQ logMin
Minimum logarithmic change in capacity difference
0.48
-0.25
DeltaQ logMean
Mean logarithmic change in capacily difference
0.48
-0.25
DeltaQ logSkew
Skewness of logarithmic changes in capacity difference
0.28
-0.18
DeltaQ logKurt
Kurtosis of logarithmic changes in capacity difference
0.34
-0.20
Slope capacity fade 2-100
Slope of capacity fade between cycles 2 and 100
-0.50
0.56
@lope_cppacit,, fade 91-100
Slope of capacity fade between cycles 91 and 100
-0.57
0.53
QD Max 2
Maximum discharged capacity relative to initial capacity
0.32
-0.15
QD_
2
Discharged capacity measured at the second cycle
0.22
-0.18
cactrcppttppppttttttpeee fade_ f-10
Intercept of the capacity fade between cycles 91 and 100
-0.30
0.26
Intercept capacity fade 91-100
Intercept of the capacity fade between cycles 91 and 100
0.15
-0.12
Min IR
Minimum internal resistance measured during cycling
-0.20
-0.21
IR 100-2
Difference in internal resistance between cycles 100 and 2
0.19
-0.17
Table 1: Feature correlation scores with Å and B
C
Training curves in two-stage training procedure
Figure 10 presents the training curves after each stage of the two-stage training procedure described in
Section 3.4 As expected, the curves after Stage 1 provide a rough estimate of the capacity loss curve, and
is refined after Stage 2 of training.
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b.
0.20
0.25
Raw data
Raw data
Fit curve
Fit curve
Capacity Loss
Capacity Loss
0.20
0.15
Cycle life: 703
Cycle life: 148
0.15
0.10
0.10
0.05
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0
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600
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150
Cycle Number
Cycle Number
C.
d.
0.20
2500
Raw data
R
0.994
Fit curve
2000
Capacity Loss
0.15
Fit cycle life
Cycle life: 1028
1500
0.10
1000
500
0.05
0
0
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500
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0
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Cycle Number
True cycle life
Figure 3: (a-c) Capacity loss model fit to three capacity loss curves using least-squares. The three curves reflect batteries with substantially different lifetimes, demonstrating the ability of the model to generalize. R² for each individual battery is displayed, and average R² across all 124 batteries in the dataset 0.976. (d) Cycle lives derived from the fitted capacity loss curves, plotted against true cycle lives. We observe R² 0.994, demonstrating high goodness of fit.
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Input data (from first 100 cycles; butteries
Var(AQ): Min(A),
Var(AQ)) Min(A)
features
Self-attenntin
Wy
Predicted
Ground truth
A,
parameters
yf
curve parameters
Bj
R
Stage Parameter loes
B
sevrieingg layer
Ground capacity
Predicted capacity loss curves
cure
Pred. lne
Pred lif 1949
8156
0.08
0.00
100
200
0.00
500
Humbor
(4
Slage Cycle life loss
I,
Predicted cycle lives
Figure 4: Schematic of the physics- based model. One half utilizes an Arrhenius Law-inspired model to capture capacity loss curves. The other half utilizes a self-attention layer to predict Arrhenius Law paramoters from early-cycle data.
0.4
0.46
0.48
0.48
-0.50
-0.57
0.2
0.0
-0.2
B
-0.23
-0.25
-0.25
0.56
0.53
-0.4
DeltaQ logVars
DeltaQ logMin
DeltaQ logMean
slope capacity fade 2_100
slope capacity fade 91 91_100 100
Figure 5: Feature correlation scores of the five features most correlated with Á and B.
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455
Elastic Net
Self Attn Stage
417
399
400
Self Attn Stage 2
347
326
RMSE (cycles)
300
281
200
180
128
113
100
0
Train
Test
Secondary Test
Figure 6: Comparison of RMSE for cycle life predictions made by the clastic net basclinc model, the self- allention model after the first stage of training, and the self-attention model after both stages of training.
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a.
Cycle Life Prediction
b.
Curve Prediction (Train)
2500
0.20
Train
Ground truth
Test
Predicted curve
2000
Secondary test
0.15
R2: 0.958
Capacity Loss
Predicted
True life: 444
1500
Predicted life: 494
0.10
1000
0.05
500
0.00
0
500
1000
1500
2000
2500
0
100
200
300
400
Truth
Cycle Number
Curve Prediction (Test)
d.
Curve Prediction (Secondary test)
0.20
Ground truth
0.20
Ground truth
Predicted curve
Predicted curve
0.15
R 0.812
0.15
R²: 0.863
Capacity Loss
Capacity Loss
True life: 880
True life: 1156
Predicted life: 916
Predicted life: 1240
0.10
0.10
0.05
0.05
0.00
0.00
0
200
400
600
800
0
250
500
750
1000
Cycle Number
Cycle Number
Figure 7: (a) Predicted versus true cycle lives using the self-attention model. (b-d) Predicted versus true capacity curves using the self-attention model for example batteries from the (b) train, (c) test, and (d) secondary test datascts.
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40
20
Figure 8: Distribution of R² for each fitted capacity loss curve, excluding one outlier at 0.295.
08
1000
1900
1000
730
110
Figure 9: Fitted vs true capacity loss curves via Equation
3
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Secondary
9000
1040
Pi00000d
Preticddd
500
1000
1520
2000
200
1030
1300
2003
Trahh
Troh
(a) Predicted and true cycle lives after Stage 1
(b) Predicted and true cycle lives after Stage 2
Figure 10: Predicted and true cycle lives after each stage of the two-stage training procedure.
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D Training curves in tto-stage training procedure
0.51
028
1800
1i0e
1200
120
(a) Predicted versus true capacity curves after stage 1 of training.
6004
1200
1000
1000
(b) Predicted versus true capacity curves after stage 2 of training.
Figure 11: Predieted versus true capacity curves after each stage of the two-stage training procedure.
19


Reviews:
[{"review_id": "TU9WztdD8x", "reviewer": "Reviewer_6Pu7", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n+ In general, a well written and easy to read paper.\n+ Good selection of evaluation metrics.\n+ In general, a good evaluation of the proposed model.\n+ The results show good prediction accuracy after the second-stage training.\n+ Good appendix with complimentary information.\n+ Good and up-to-date list of references and related work.\n\n**Weaknesses**\n- I'm surprised about the correlations between A_hat and B_hat, and features are relatively low, but the model still seems to work well. Maybe a deeper analysis and discussion of that?\n- The evaluation is done using only one dataset, with relatively few batteries, all of them low voltage and relatively low capacity. I know that the availability of datasets always is an issue, but it would have been nice to at least have a discussion about the applicability of the results for high-voltage and high-capacity batteries.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes and evaluates a hybrid approach to predict the battery cycle life time and the battery capacity loss curves for commercial lithium-ion batteries. The hybrid approach uses a combination of a physics-based equation and a self-attention model. The hybrid approach seems novel, although batteries as an application context is out of my competence domains. \n\nThe paper contains two main novelties, as far as I can see:\n* The combination of physics-based equation and an attention mechanism.\n* The prediction of not only the battery cycle lifetime, but also the capacity loss curve over time (charging cycles).", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I think the authors have done a good job in addressing the other reviewers' comments. Thus, I have no required changes.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "TgbTgBkGEC", "reviewer": "Reviewer_pbnc", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**strengths**\n\n- the paper is well written and easy to follow, the problem statement is well presented, and the description of the components of the hybrid approach (physics + self-attention) is also clear\n- the topic of using machine learning-augmented approaches towards prediction of physical properties (of batteries, in this case, but potentially of other physical systems as well) may be of interest to the audience of this journal\n\n**weaknesses**\n\nWhile the problem and approach are well introduced, the paper loses sharpness when presenting architectural details and results.\n\n- Section 3.2: the architecture is not entirely clear. In my understanding, the authors implement self-attention mechanism without several of the common building blocks found in current transformers (e.g., normalization, shortcuts). Does this model consists of a *single* self-attention block, that is, it consists of a total of 3 linear modules (Q, K, and V) plus an averaging operation (m)? Are other components in use? It would be useful to have a clear schematic of the architecture, maybe added to Figure 4.\n- Section 3.3: five features are selected to train the self-attention model, based on their correlation score with the two physics-based parameters. The authors should report which features were originally taken into considered and what was their correlation. Also, the select five features are not properly defined, only named, so their meaning is unclear.\n- Section 3.4: the training process is not well described. How many examples and training steps were taken? Can the authors provide the training / validation curves, to verify that the model is training correctly?\n- Section 4: the authors present values for a baseline \"similar to the model that performed the best in Severson et al.\" [1] which is based on ElasticNet. It is not clear how these new baseline values compare with [1], as the authors use RMSE but [1] reports test error in the main text. Similarly, it is later stated that a \"the full model in Severson et al. (2019) utilizing multiple features achieves a primary test RMSE of 118 and a secondary test RMSE of 214.\" Can the authors set up a direct comparison of the baselines and clarify what models taken from [1] they compare against?\n- Related to the previous comment, it is not clear why Figure 6 reports data for that particular baseline but not for the strongest one provided by [1]. Can the authors clarify this point?\n- Overall, my understanding is that the predictive capabilities on cycle lifetime of the proposed hybrid approach appear to be on par or, in some cases, moderately better than the best prediction obtained with the baseline approach. The authors stress that their hybrid approach provides not only a single estimate of lifetime, but the expected capacity loss curve as well. I notice from Figure 7 that there are definitely discrepancies between the estimated model and the ground truth. Can the authors outline some specific scenarios where knowledge of the capacity loss curve would be beneficial? In such scenarios, what discrepancy between prediction and actual behavior would be acceptable?\n\n** other comments **\n\n- typo in Section 2: \"For each cell, three forms of data are recorded\" is followed by four types of data\n- the authors mention that a Github repository is available upon request. I recommend adding the link to the paper, after the anonymous review phase has been completed.\n\n\n[1] Severson, et al. Data-driven prediction of battery cycle life before capacity degradation. Nature Energy, 4(5):383–391, 2019.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper sets up a framework to predict the cycle lifetime of lithium-ion batteries. First, a physics-based equation is used to fit the capacity loss as a function of number of life cycles. Then, a self-attention model is trained in a two-stage process to predict the parameters derived from the fit, taking as input features obtained from the early life of the battery (up to cycle 100). The authors show that on a given dataset of lithium-ion battery cells lifetime characteristics, this hybrid physics-attention approach can match or improve predictive performance of competing approaches not based on self-attention.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "see previous section", "broader_impact_concerns": "not addressed in the manuscript. No further concern on my end.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "uCoRm7sefr", "reviewer": "Reviewer_FBYN", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The combination of physics-informed and data-driven cycle life prediction feels well motivated.\n\nWeaknesses:\n- The exposition of the approach and subsequent evaluation feels incomplete. I provide specific comments in the request changes section.\n- I'm somewhat uncertain about TMLR being the correct audience for this work as it feels more relevant to the battery community but I will leave it to the editors to weigh in on this point.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "Optimizing cycle life prediction of lithium-ion batteries via a physics-informed model presents a modeling approach to predicting capacity fade as a function of cycle time. The authors first motivate the use of a structured (parameterized) model using all data showing that good fits can be obtained then transition to predict the parameters of the model using other derived features measured only during early cycling. They chose an approach based on self-attention for the later. The approaches are evaluated on a large LiFePO4 graphite dataset.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Introduction:\n- The phrase \"difficult to fine-tune\" should be clarified as I was not led to believe that the authors meant this in a numerical sense but I also didn't follow the point they are trying to make.\n\nPhysics-based Model:\n- The phrase \"Note that the temperature highly resembles...\" didn't make sense to me but I wonder if there is a typo? Please clarify.\n- I found \"accounts for SEI formation\" to be too strong of a statement and suggest rephrasing to something along the lines of \"aims to account for SEI formation\"\n- I think the authors mean to say that they predict e^B not lnB such that A is the contributions of the temperature term and e^B? It is confusing that the exponent in equation 2 for cycle life in eqn 2 is z and then B in eqn 3 as it appears that B != B.\n- \"Ground truth capacity loss\" is confusing. I think the authors are saying that \\hat{Q}(0)_loss != 0 because of calendar aging? This is not a \"reduction\" of equation 2 but instead of different model specification. It would be good for the authors to comment on the choice of always using the nominal capacity as opposed to the measured capacity at cycle 0 in defining capacity loss.\n- In Figure 3, I would argue that the model didn't \"generalize\" as a different model is fit for each curve using all of the data. However, the parameterization is able to generalize. I think it would be useful for the authors to present RMSE as a function of cycle life to help the reader understand possible biases in the parameterized model.\n\nSelf-attention for regression\n- I think the ‘x’ in the description of the self-attention section is not the same as ‘x’ in the physics-based model section and therefore should use another term. I also think the authors use a multidimensional sequence of inputs to predict A and B. Please clarify.\n- The equation for cycle life sometimes uses Q(0) and sometimes uses C\n\nFeature selection\n- The candidate features are not described nor is the procedure for proposing candidates. This discussion should be added.\n\nModel training\n- Did you try training using log l_i for improved numerics?\n\nResults\n- The results section is missing an analysis of using a linear model to predict A and B and comparing. As presented, I feel the contribution of the physics-informed model vs. the attention-based parameterization is unclear. I think understanding the relative contributions of the framework is critical to the manuscript.\n- I find the results in figure 6 confusing - the linear model is worse than Severson et al. because it only considers the five features selected in the Spearman’s analysis?\n\nSmall comments:\n- In the data processing section, there is a typo where \"three forms\" of data are mentioned before a list of four items", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: All reviewers voted for acceptance. The authors addressed the concerns raised by the reviewers, including clarifying the architecture, feature selection, training, etc. One reviewer (FBYN) asked for one additional figure showing the RMSE as function of the cycle number (as oppose to cycle life) as well as providing evidence that better supports the claim that the capacity fade curve is recovered and provide a comparison to the individual fits vs those recovered using the proposed features. I'd request the authors to provide this additional details in the final version of the paper.
Decision: Accept (poster)