Battery lifetime prediction across diverse ageing conditions with inter-cell deep learning

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## nature machine intelligence

Article [https://doi.org/10.1038/s42256-024-00972-x](https://doi.org/10.1038/s42256-024-00972-x)
# Battery lifetime prediction across diverse ageing conditions with inter-cell deep learning

Received: 6 December 2023

Accepted: 12 December 2024

Published online: 15 January 2025

Check for updates

Han Zhang 1,2,4 , Yuqi Li 1,3,4 , Shun Zheng 1 , Ziheng Lu 1 , Xiaofan Gui 1 ,
Wei Xu 2 & Jiang Bian 1

Accurately predicting battery lifetime in early cycles holds tremendous
value in real-world applications. However, this task poses significant
challenges due to diverse factors influencing complex battery capacity
degradation, such as cycling protocols, ambient temperatures and
electrode materials. Moreover, cycling under specific conditions is both
resource-intensive and time-consuming. Existing predictive models,
primarily developed and validated within a restricted set of ageing
conditions, thus raise doubts regarding their extensive applicability. Here
we introduce BatLiNet, a deep learning framework tailored to predict
battery lifetime reliably across a variety of ageing conditions. The distinctive
design is integrating an inter-cell learning mechanism to predict the
lifetime differences between two battery cells. This mechanism, when
combined with conventional single-cell learning, enhances the stability of
lifetime predictions for a target cell under varied ageing conditions. Our
experimental results, derived from a broad spectrum of ageing conditions,
demonstrate BatLiNet’s superior accuracy and robustness compared to
existing models. BatLiNet also exhibits transferring capabilities across
different battery chemistries, benefitting scenarios with limited resources.
We expect this study could promote exploration of cross-cell insights and
facilitate battery research across comprehensive ageing factors.

With high energy densities and low production costs, lithium-ion batteries have become widely adopted across modern industries, driving the growth of renewable energy solutions and electric vehicles 1–3 .
Nevertheless, the capacity of lithium-ion batteries inevitably fades
with cyclic operations due to their intrinsic electrochemical mechanisms 4 . Unexpected rapid degradation not only leads to poor user
experiences, such as range anxiety for electric vehicles, but can also
affect the operation of essential facilities, such as the stability of power
grids. To proactively mitigate these side effects, accurately predicting battery lifetime in early cycles has been identified as a critical

task 5–8, where the lifetime is typically measured in cycle life, which is
defined as the number of charge–discharge cycles until the capacity
of a battery cell drops to 80% of its nominal capacity 9,10 . Predicting
battery lifetime in early cycles is rather challenging because numerous factors, including but not limited to cycling protocols, ambient
temperatures and electrode materials, collectively influence the
complex battery ageing process. Additionally, achieving comprehensive coverage of various ageing conditions is prohibitively costly
and time-consuming, as testing a cell under a specific condition is a
lengthy process.

1Microsoft Research, Beijing, China. 2Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China. 3Present address:
Department of Materials Science and Engineering, Stanford University, Stanford, CA, USA. 4 These authors contributed equally: Han Zhang, Yuqi Li.

[e-mail: shun.zheng@microsoft.com; jiang.bian@microsoft.com](mailto:shun.zheng@microsoft.com)

a b
Package structures

1.05

1.00

0.95

0.90

0.85

0.80

1.05

1.00

0.95

0.90

4,049

3,267

2,486

1,704

923

Electrode

materials

5 3

401
240

This study
MATR
HUST

SNL
CALCE

Total
cells

0 250 500 750 1,000 1,250 1,500

Cycles

100

3
168

Cycling
protocols

Ambient
temperatures

0.85 142
0 20 40 60 80 100

Cycles

Fig. 1 | Diverse ageing conditions covered in this study and complex degradation behaviours associated. a, We compare the broad coverage of different ageing
factors in this study against typical datasets, including MATR 9, HUST 12, SNL 19 and CALCE 13–15 . b, The varying capacity degradation behaviours over long- and short-term
cycles.

Existing methods for battery lifetime prediction have been developed and validated under limited ageing conditions, such as testing only lithium-iron-phosphate (LFP) cathode materials and using
a certain group of cycling protocols 9–12 . Data characteristics under
these restricted conditions affect influence a thorough examination of
feature designs and modelling techniques, potentially limiting the success and generalization of their conclusions. It remains questionable
whether these methods perform well across broad ageing conditions.
Moreover, focusing on limited ageing conditions restricts the research
development of leveraging historical data collected under different
conditions. This limitation separates battery datasets emphasizing
different ageing factors as isolated islands, hindering the development
of general modelling approaches.

In this study, we introduce BatLiNet, a deep learning framework
designed for reliably predicting battery lifetime across diverse ageing conditions, such as variations in cycling protocols, ambient temperatures and even battery chemistries. At its core, the framework
introduces inter-cell learning that contrasts pairs of battery cells to
discern lifetime differences, a significant leap from traditional models that focus solely on individual cells. By combining this approach
with conventional single-cell learning, BatLiNet not only captures the
unique degradation patterns of individual batteries but also places
these patterns within a broader, comparative ageing landscape.

In our experiments covering various ageing complexities, BatLiNet
consistently outperforms traditional models, delivering precise and
robust lifetime predictions. A noteworthy observation is BatLiNet’s
ability to reduce the mean absolute percentage error (MAPE) of its
single-cell learning counterpart by more than 40% on average, underscoring the significance of inter-cell learning. Additionally, BatLiNet
exhibits intriguing capabilities to transfer learning across different
battery chemistries. These findings underscore the importance of considering a broad range of ageing conditions in battery data modelling
and also reveal the untapped potential of leveraging the connections
across different ageing conditions.

Results
To gain a comprehensive understanding of various ageing factors
and their influences on battery lifetime, we aggregate most publicly
available datasets, including MATR 9, HUST 12, CLO 11, CALCE 13–15, HNEI 16,
UL-PUR 17, RWTH 18 and SNL 19 . Figure 1 compares the ageing conditions
examined in this study with typical datasets, highlighting the associated complex degradation behaviours. Supplementary Note 1 provides
additional detailed comparisons across all datasets.

In this study, we derive five datasets reflecting different variations
in ageing conditions. First, we inherit two evaluation sets of the MATR
dataset, labelled MATR-1 and MATR-2, to maintain a consistent comparison with existing models. Besides, we consider the HUST dataset,

using similar LFP cells to MATR, to investigate the adaptation of current
models to different cycling protocols. Furthermore, we include other
datasets by merging all datasets into a MIX set because of limited cells
under other conditions, different ambient temperatures, packing
structures and cathode active materials, such as lithium nickel cobalt
manganese oxide (NMC), lithium cobalt oxide (LCO) and lithium nickel
cobalt aluminium oxide (NCA). We develop two prediction setups
based on the MIX dataset: MIX-100 and MIX-20. MIX-100 examines a
typical early prediction scenario where models must predict the 80%
end-of-life point using only the first 100 cycles of data. MIX-20 presents
a more challenging task, wherein models must forecast the number of
cycles before the capacity degrades to 90% of the initial value using
just the first 20 cycles. It is worth noting that batteries that reached
their end-of-life prematurely during the early cycles were excluded
from the study.

The BatLiNet framework
The inherent conflict between numerous ageing conditions and limited
cell cycling coverage poses a significant challenge to predictive modelling. Existing models primarily focus on intra-cell learning, capturing early variations of a single cell to predict its long-term lifetime 9,10 .
However, this approach struggles with data scarcity due to limited cell
samples available. In contrast, our BatLiNet framework, presented in
Fig. 2, integrates inter-cell learning, which is essentially an auxiliary task
of predicting lifetime differences given a pair of cells. We designate a
cell with an unknown lifetime as the ‘target cell’ and use a fully cycled
cell from existing databases as the ‘reference cell’.

Figure 2a depicts the process of constructing effective features for
intra- and inter-cell learning, where we use the voltage–capacity curve
during the discharge stage as a typical cycle-level feature. Traditional
approaches for intra-cell learning operate on a single cell and obtain
effective intra-cell features by calculating the cycle-level feature differences between a pair of early cycles 9,10 . We refer to these operations
as calculating intra-cell differences. In contrast, we calculate inter-cell
differences by contrasting the cycle-level features between a target
cell and a reference cell in the same cycle. The inter-cell differences
aim to represent the connections between cells cycled under different conditions.

Figure 2b illuminates the potential and challenges of both intraand inter-cell learning across various ageing conditions. The upper plot
illustrates a near-linear relationship between the standard deviation
of intra-cell difference curves and the corresponding lifetime for different LFP cell groups. While the linearity is consistent with previous
studies 9, this correlation highlights the complexities introduced by
diverse ageing conditions, as shown by the scattered points, outliers
and distinct clusters due to additional cells undergoing varied protocols and temperatures. Conversely, the lower plot delves into inter-cell

a b

Inter-cell difference

40.0
30.5
21.0

11.5

2.0

38.3
28.5
18.8

9.0

–0.8

×102

10 –5 10 –4 10 –3 10 –2

SD of inter-cell difference curves

×102

All points use the
same reference cell

×10–2

2.3 4.7 7.2 9.6 12.2

SD of inter-cell difference curves

MATR-2017-05

MATR-2017-06

MATR-2018-04

MATR-2019-01

HUST

SNL-Temp-18C

SNL-Temp-26C

SNL-Temp-36C

4,049

3,268

2,487

1,707

926

146

3,826

3,045

2,264

1,484

703

–77

100

Cycles

_V_

Intra-cell difference

Target

cell

_V_

∆

_V_

2 _Q_
0 0.5 1.0

–0.5

–0.4

–0.6

_Q_

0 0.5 1.0

∆

_V_

_V_ _V_

_Q_ 3 _Q_ 2 _Q_ –0.6 _Q_

0 0.5 1.0

2
0 0.5 1.0

0 0.5 1.0

Embedding

space

Difference curves

Reference

cell

10 100

Cycles

Final prediction of

target lifetime _y_

Final prediction of

Mean

_y_

Unknown target lifetime _y_

Known reference lifetimes

Intra-cell difference
with a specific cycle

_V_ ∆ _V_

∆ _V_ ∆ _V_

Reference

signals

2 _Q_
0 0.5 1.0

_V_ ∆ _V_

Intra-cell
difference

encoder _f_

_Q_

_Q_
_Q_

–0.5

0 0.5 1.0

_V_

_V_
_V_

Target cell cycle _t_, _xt_

Direct prediction of

target lifetime _y_

Median

Shared
architecture

Intra-cell difference

_x_     - _xt_

Unbiased

Aligned linear
layer _w_

2 _Q_ _Q_
0 0.5 1.0

Target cell _x_

_Q_

_VV_

_V_ _V_ _VV_ + _y’_

× _N_

Prediction of lifetime

difference _y_  - _y’_

_~~Q~~_

_Q_

Inter-cell difference
with a reference cell

_y’_

–0.4

–0.6

_Q_

Inter-cell
difference
encoder _g_

_Q_

_Q_

0 0.5 1.0 × _N_

Reference cell _x’_

0 0.5 1.0

Inter-cell difference

_x_    - _x’_

× _N_

Fig. 2 | An overview of the BatLiNet framework. a, The feature construction for
intra- and inter-cell learning given the curve illustrating the relationship between
_Q_ (current capacity normalized by nominal capacity) and _V_ (voltage in volts) as

learning by mapping the correlation between inter-cell differences and
associated lifetime differences. Here, the same LFP cells serve as target
cells, and a shared LCO cell acts as a reference. Despite the challenges
posed by diverse ageing factors, a clear connection is evident. This
suggests that a simple feature computed on inter-cell difference curves
can differentiate lifetime differences, even with a reference cell from a
different battery chemistry. Supplementary Note 2 further supports
the feasibility of inter-cell learning within or across different battery
chemistries with more data-driven evidence.

Moreover, these nonlinear patterns have indicated the necessity
of leveraging deep learning for effective representation learning.
Figure 2c concludes the data flow of BatLiNet. Given a target cell, we
feed its cycle-level features into two separate branches. The first branch
calculates intra-cell difference curves by comparing features in all early
cycles with that of a reference cycle and uses a specific neural network
_f_ for intra-cell learning. In addition, the second branch includes a reference cell to obtain inter-cell difference curves and further feeds them
into another neural network _g_ for inter-cell learning. After the encoding procedures of _f_ and _g_, the resulting representations are mapped
to an aligned linear space, allowing us to leverage a shared linear layer
to emit outputs for these two branches. The rationale for separated
encoding followed by a shared linear layer is detailed in the Methods.
During training, we can simply select any pair of cells from a training set
to stimulate robust inter-cell modelling across various combinations

a per-cycle feature map. b, The correlations between constructed features and
prediction targets for both intra-cell (the upper half) and inter-cell (the lower
half) learning. c, The data flow of BatLiNet.

of ageing conditions. To ensure robust prediction during inference,
we randomly select a batch of reference cells from the training set
and use the average of outputs from intra- and inter-cell branches as
the final prediction.

Figure 3 offers a comparison of BatLiNet with state-of-the-art baselines across different datasets, using two metrics: root mean squared
error (r.m.s.e.) and MAPE. We can observe that basic linear models relying on hand-crafted features, such as the ‘Var.’, ‘Dis.’ and ‘Full’ models 9,
demonstrated commendable performance on their initial datasets.
However, their performance degraded significantly when faced with
datasets encompassing complex ageing conditions. This trend is also
evident in some advanced statistical learning techniques applied
to raw capacity–voltage curves 10, such as ridge regression 20, partial
least-squares regression 21, principal component regression 22 and support vector machine 23 . An exception to this pattern is random forest 24,
which consistently delivered relatively robust predictions, even on the
MIX-100 and MIX-20 datasets. Despite this, random forest generally
yielded suboptimal performances across all datasets under evaluation.

Compared with statistical methods, deep learning models operating in the feature space developed by ref. 10 are susceptible to high
variability resulting from different random initializations, including
multi-layer perceptron 25, long-short-term memory network 26 and
convolutional neural network (CNN) 27 . This may be due to the limited
number of cells available for learning, combined with diverse ageing

11 13 11 10
8

MATR-1

MATR-2

HUST

MIX-100

MIX-20

400

200

400

200

600

400

200

600

400

200

1,000

500

399

162
138 86 100 125 100 97 140 140 118 125 63

511

15 15 12 12
10
6

12 11 12 11 11 11

300

202 207

15 16
11 11 11

233
237
162

196 173 214 188 176 193

1,047

~~398~~ 435 431 444 441 482 420
322 335 ~~344~~ ~~345~~

431
~~344~~ ~~345~~

444 441 482

~~2~~ 68

18

19 18 ~~16~~ ~~16~~ 18 20

100

~~17~~
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521

395 384 371
331

573

265 252
168

895

30 28 26
22
18 15

150

461

257
214

28

16 15
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102

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47
39

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600

437

837
707

594

207

75
60

~~519~~
376

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290

53
33

~~1~~ 9

Fig. 3 | Performance comparisons of BatLiNet against linear models along with
statistical models and deep learning models. Linear models (Var., Dis. and Full)
developed in ref. 9, and statistical models (ridge regression, partial least-squares
regression, principal component regression, support vector machine, random
forest) and deep learning models (multi-layer perceptron (MLP), long-shortterm memory (LSTM), convolutional neural network (CNN)) from ref. 10. These
comparisons span five datasets, MATR-1, MATR-2, HUST, MIX-100 and MIX-20,

patterns and significant noise. BatLiNet also belongs to the deep learning family and uses the same CNN architecture for feature encoding. We
observe that it has significantly mitigated the high variability associated with deep learning baselines, consistently delivering highly competitive results across diverse ageing conditions. Specifically, BatLiNet
reduces the r.m.s.e. of the best-performing baseline (which varies by
dataset) by 36.5, 6.8, 20.1, 27.4 and 40.1%, respectively. When compared
with its single-cell learning counterpart, CNN, BatLiNet can reduce the
average MAPE by up to 40%. It is worth noting that even for BatLiNet,
the MAPE score on the MIX-100 and MIX-20 datasets is distinctly higher
than that on previous datasets such as MATR-1 and MATR-2. We believe
this highlights the significance of considering comprehensive ageing
factors in battery data modelling.

Furthermore, we delve into investigating cross-chemistry transfer of BatLiNet. More specifically, we aim to confirm if BatLiNet can
enhance the transfer from a resource-rich chemistry to other chemistries that have limited cells. This arrangement can highlight its potential in expediting precise lifetime predictions for future chemistries.
In our experiments, we use 275 LFP cells as the rich-resource battery
chemistry and aim to investigate how well we can leverage these LFP
cells to develop models for the remaining 37 LCO cells, 22 NCA cells
and 69 NMC cells. For consistent evaluation, we randomly sampled
21 LCO cells, 14 NCA cells and 53 NMC cells to form respective test
sets. Then, from the remaining cells, we sampled 1, 2, 4, 8 and 16 cells
as the available data for model training, corresponding to different
low-resource conditions for each target battery chemistry. Figure 4
presents the overall comparisons of these models. We can observe that

and cover two evaluation metrics, r.m.s.e. and MAPE. For deep learning models,
we run various experiments with eight unique initializations ( _n_ = 8 trials). The bar
plot displays the mean and standard deviation values. The dashed line represents
a simple baseline, using the average lifetime of training cells as a fixed prediction
to indicate the difficulty of learning. PCR, principal component regression; PLSR,
partial least-squares regression; RF, random forest; Ridge, ridge regression; SVM,
support vector machine.

in most cases, BatLiNet has demonstrated distinct error reductions
over other learning models. For instance, direct learning cannot leverage information from LFP cells and thus struggle when the number of
target cells is extremely rare. Parameter-based transfer learning relies
on pretrained parameters to enable implicit transfer, which we also
find could encounter generalization challenges in certain cases, such
as transferring from LFP cells to only one NCA or LCO cell. In contrast,
BatLiNet enables an explicit transfer via inter-cell learning, which delivers more robust and accurate predictions in most cases.

Finally, we explain the reference selection strategy in BatLiNet.
Figure 5 presents extended results on the MIX-100 dataset. Figure 5a
shows that the choice of reference cells for a target cell can significantly
affect prediction errors, as seen from the gap between the best and
worst choices across different test set cells. While selecting the optimal
reference cell is challenging, we mitigate performance variations by
sampling a batch of reference cells and using their median predictions, as shown in Fig. 5b. Figure 5c reveals that this batch sampling
mechanism does result in a significant reduction in inference speed as
the batch size increases exponentially. However, given the extensive
battery cycling time, these differences in inference speed are negligible.
We find that using a batch of 64 reference cells is effective in maintaining a reasonable inference speed yet delivering robust predictions.

Discussion
We introduce BatLiNet, a deep learning framework that integrates
inter- and intra-cell learning to improve battery lifetime predictions
across varied ageing conditions. Inter-cell learning complements

Directly training on target cells
Pretraining on LFP cells and fine-tuning
BatLiNet

of different signals across different cycles and charge–discharge
stages. Besides, due to the redundant recording of electrical signals,
we can obtain these _Q_ -indexed series by performing interpolations on
raw time-indexed signals and easily control the feature dimensions by
adjusting interpolation granularity. After this step, we can obtain four
types of processed series indexed by _Q_, _V_ c( _Q_ ), _V_ d( _Q_ ), _I_ c( _Q_ ) and _I_ d( _Q_ ),
where _Q_ varies from 0 to 1 with a prespecified step, _V_ denotes voltage,
_I_ denotes current and d and c are superscripts to denote discharge
and charge stages, respectively. We further derive two additional
signals to characterize the connections between the charge and discharge stages: (1) Δ _V_ ( _Q_ ) = _V_ c( _Q_ ) − _V_ d( _Q_ ), meaning the gap between
charge voltages and discharge voltages, and (2) _R_ ( _Q_ ) = ( _V_ c( _Q_ ) − _V_ d( _Q_ ))/
( _I_ c( _Q_ ) − _I_ d( _Q_ )), corresponding to the status of internal resistance as
_Q_ varies. These cycle-level features serve as the basic elements for
calculating intra- and inter-cell differences.

There are two critical processing steps to ensure the effectiveness
of calculating these differences from cycle-level feature maps. This first
is to normalize the capacity of all batteries by their respective norminal
capacities, ensuring the range of _Q_ falls within [0, 1], to adapt to various battery types with very different nominal capacities. The second
is to eliminate the data noise introduced by abrupt changes in current
and voltage signals due to the variations in cycling protocols. We use
a rolling-median-based filter to alleviate this issue:

M = rolling_median ( r, _w_ ), (1)

Δ M = abs(rolling_median ( M, _w_ )), (2)

150

100

125

100

89
65

NCA–graphite cells

51
46 35 32 40 31

52 55

1 2 4 8

LCO–graphite cells

83
75 89

39 42 44 46 42 40 35
26 29 23 25 23 23

100

1 2 4 8 16

NMC–graphite cells

33 3 ~~5~~ 35 30 34 28 28 29 27 26 24 22

1 2 4 8 16

Number of target cells for training

_r_ new _t_ = {

_r_ new _t_

_Mt_ if Δ _Mt_ - 3 × median (Δ M ),

_rt_ otherwise.

(3)

Fig. 4 | Performance comparison of BatLiNet when transferring from 275
LFP cells to limited NCA, LCO and NMC target cells. We simulate various lowresource scenarios by varying the number of available target cells for training
(1, 2, 4, 8, 16). BatLiNet’s inter-cell learning is compared to two baselines: a CNN
model trained directly on target cells and a CNN model pretrained on LFP cells
then fine-tuned. Each case and model is tested over eight unique initializations
and data splits ( _n_ = 8 trials), with bar charts showing average error and error bars
indicating standard deviation across runs.

intra-cell learning by enabling knowledge transfer across different
ageing conditions.

This approach has potential applications in other critical prediction tasks, such as estimating battery state of charge and health 5,28,
adapting to diverse ageing conditions such as fast-charging protocols 29 and emerging chemistries such as solid-state and sodium-ion
batteries 30–32, predicting calendar life under real-world conditions for
more accurate lifetime estimates 33 .

While this study demonstrates the data-driven feasibility of
inter-cell learning across various conditions, a deeper physical understanding and modelling remain essential 34 . Future research should
prioritize a comprehensive approach to understanding and modelling
diverse ageing factors to advance battery research, development and
application.

Methods
Building per-cycle feature maps
We use the normalized capacity ( _Q_ ∈ [0, 1]) as a unified index to align
different electrical signals in different cycles and compute their differentiation within the same stage or between charge and discharge
stages. In this way, we do not need to worry about the variable lengths

Here r is the raw signal and _w_ represents the window size of the rolling operations. These multi-faceted feature maps will both intra- and
inter-cell learning as separate channels.

Unifying intra- and inter-cell learning
Given the intra-cell feature of a battery cell, denoted x, and its cycle
life _y_, we want to optimize the following objective to obtain a perfect
cycle life predictor.

where Δ x = x - x ′, Δ _y_ = _y_ - _y_ ′, ( x, _y_ ) and ( x ′, _y_ ′ ) are independently sampled from _D_ and _gϕ_ is a function parameterized by _ϕ_ that operates in
the space of Δ x .

min _θ_ ??( x, _y_ )∈ _D_ ‖ ‖

min

‖ ‖ _fθ_ ( x ) − _y_ ‖ ‖

‖ ‖

2 , (4)

where _D_ denotes the data distribution, _fθ_ is an encoding function parameterized by _θ_ and w _f_ is the weight of the last prediction layer. In practice,
we need to perform the empirical risk minimization over limited training instances, that is

min

_θ_

_N_
∑

_i_ =1

‖‖ _fθ_ ( x _i_ ) − _yi_ ‖ ‖

2 . (5)

‖ ‖

However, battery lifetime prediction is a special task that suffers from the data-scarce challenge due to the necessity of nonlinear
modelling and the huge cost of obtaining data labels. In this scenario,
instantiating _fθ_ as a neural network has a high risk of overfitting. To
alleviate the data-scarce issue, we consider modelling the differences
between two distinct cells:

min _θ_ ??(Δ x,Δ _y_ ) ‖ ‖

2 , (6)

min

‖ ‖ _gϕ_ (Δ x ) − Δ _y_ ‖ ‖ 22

‖ ‖ 2

2,000

1,500

1,000

500

8,000

6,000

4,000

2,000

‖ ‖

contrastive objective (6) because ??(Δ x,Δ _y_ ) ‖ ‖

contrastive objective (6) because ??(Δ x,Δ _y_ ) ‖ ‖ w ∗ _T_ Δ x − Δ _y_ ‖ ‖2 can be decom-

posed into

‖ ‖ w ∗ _T_ Δ x − Δ _y_ ‖ ‖

Median reference
Best reference
Worst reference

Cells in the MIX-100 test set

150 200 250 300

r.m.s.e.

3,457.1

1,925.5

1,015.5
521.9 265.4

1 2 4 8 16 32 64

No. of reference cells

2 . (7)

??( x, _y_ )∈ _D_ ‖ ‖

‖ ‖ w ∗ _T_ x − _y_ ‖ ‖

‖ ‖

22 + ??( x ′, _y_ ′)∈ _D_ ‖ ‖ w ∗ _T_ x ′ − _y_ ′‖ ‖22

‖ ‖2

8,736.3

5,794.2

Batch size

Inference speed

In the nonlinear setting, such as using neural networks as function approximators, it is intractable to establish the exact connection
between _fθ_ and _gϕ_, which leads to two separate optimization procedures. Inspired by the same optimality of the objectives (4) and (6)
under the linear setting, we propose to share the last linear layer of _fθ_
and _gθ_ when using neural networks, that is

_fθ_ ( x ) = w _T_ _hθ_ ( x ), _gϕ_ (Δ x ) = w _T_ _hϕ_ (Δ x ), (8)

where w is the shared parameter, _hθ_ ( ⋅ ) and _hϕ_ ( ⋅ ) are two separate
neural networks parameterized by _θ_ and _ϕ_, respectively. In this way,
we can connect the optimization of the objectives (4) and (6) via the
shared parameter _w_ and enjoy the complementarity between intra- and
inter-cell modelling.

Moreover, we also need to perform empirical risk minimization to
mimic the desired objective (6). Specifically, given that we only have _N_
training instances, we use _N_ ( _N_ - 1) pairs of instances to substitute the
expectation over independently sampled instance pairs. Together
with the original regression, we have the following joint empirical risk
minimization problem:

where Δ x _i_, _j_ = x _i_ - x _j_, Δ _yi_, _j_ = _yi_ - _yj_ and _λ_ is a hyper-parameter to balance
two regression objectives.

After the optimization stage, we can leverage the neural networks
for encoding intra- and inter-cell differences to make predictions. Given
an unseen instance x, we have

_c_ = w _Thϕ_ ( x - x ′) + _y_ ′, (10)

500

400

300

200

100

_y_

_o_ = w _Thθ_ ( x ),̂ _y_

̂ _o_ and ̂ _y_

where ̂ _yo_ and ̂ _yc_ are the predictions in the original and contrastive space,

respectively, and ( x ′, _y_ ′ ) can be sampled from the training instances.
Last, we predict the lifetime as ̂ _y_ = _α_ ̂ _yo_ + (1 − _α_ )̂ _yc_, where _α_ is a hyper

where ̂ _y_ ̂

̂ ̂ _o_ + (1 − _α_ )̂ _y_

Fig. 5 | Analysis of reference cell selection in BatLiNet. a, Best, worst and
median prediction errors for all test cells in the MIX-100 dataset, using a single
reference cell traversing all training cells. Shaded areas represent the standard
deviation of prediction errors from randomly selecting a reference cell. b, Effects
of increasing reference cell numbers on predictions in the MIX-100 dataset, with
eight random selections per case ( _n_ = 8 trials). Every box plot displays medians
as centre lines, the 25th and 75th percentiles as lower and upper quartiles, with
whiskers extending to 1.5 times the interquartile range, including outlier points.
c, Changes in inference throughput as the number of reference cells increases,
tested on an NVIDIA GTX 4090 graphical processing unit.

The inter-cell learning formulation (6) assumes that the differences in feature representations for any pair of battery cells hold a
unified relationship with the differences in their lifetimes. Specifically,
we can establish a clear connection between _fθ_ and _gϕ_ in the linear setting. For example, if _y_ is zero-centred (can be done via preprocessing),
and the optimal solution for the original objective (4) is _fθ_ ∗ ( _x_ ) = w ∗ _T_ x,
then it is easy to verify that _gϕ_ ∗ = _fθ_ ∗ is also the optimal solution for the

Last, we predict the lifetime as ̂ _y_ = _α_ ̂ _yo_ + (1 − _α_ )̂ _yc_, where _α_ is a hyper
parameter to balance the two types of prediction.

Data availability
All datasets used in this study are publicly accessible and include MATR
[and CLO (https://data.matr.io/1/), HUST (https://data.mendeley.com/](https://data.matr.io/1/)
[datasets/nsc7hnsg4s/2), CALCE (https://web.calce.umd.edu/batteries/](https://data.mendeley.com/datasets/nsc7hnsg4s/2)
[data) and RWTH (https://publications.rwth-aachen.de/record/818642/](https://web.calce.umd.edu/batteries/data)
[files). HNEI, SNL and UL-PUR datasets are hosted by BatteryArchive](https://publications.rwth-aachen.de/record/818642/files)
[(https://www.batteryarchive.org/study_summaries.html). Moreover,](https://www.batteryarchive.org/study_summaries.html)
we leverage an open-source tool, BatteryML 35, to preprocess these
datasets into a unified format to support experiments reported in this
article.

Code availability
Our code is available on CodeOcean 36, including instructions to train
all models from scratch and to perform direct inference with our provided checkpoints.

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Acknowledgements
The authors received no specific funding for this work.

Author contributions
H.Z. led the development of BatLiNet and conducted most
experiments. Y.L. provided valuable insights into cross-chemistry and
facilitated the experiment analysis. S.Z. led the overall research, built
the initial code base and introduced the concept of inter-cell learning.
S.Z., H.Z. and Y.L. wrote the paper. Z.L. contributed to refining the
manuscript and inspiring cross-chemistry explorations. X.G. ensured
the reproduction of all experiments and refined paper presentation.
Both W.X. and J.B. participated in the discussion of ideas and analysis
of results. J.B. provided overarching supervision for the research and
strong support for the entire team.

Competing interests
The authors declare no competing interests.

Additional information
Supplementary information The online version contains
supplementary material available at
[https://doi.org/10.1038/s42256-024-00972-x.](https://doi.org/10.1038/s42256-024-00972-x)

Correspondence and requests for materials should be addressed to
Shun Zheng or Jiang Bian.

Peer review information _Nature Machine Intelligence_ thanks Kandler
Smith, Qingyu Yan and the other, anonymous, reviewer(s) for their
contribution to the peer review of this work.

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