# Toward High-Performance Energy and Power Battery Cells with Machine
Learning-based Optimization of Electrode Manufacturing

Marc Duquesnoy1,2,222text

& Chaoyue Liu1,222text

& Vishank Kumar3

& Elixabete Ayerbe2,4

& Alejandro A. Franco1,2,5,6,∗

###### Abstract

The optimization of the electrode manufacturing process is important for
upscaling the application of Lithium Ion Batteries (LIBs) to cater for growing
energy demand. In particular, LIB manufacturing is very important to be
optimized because it determines the practical performance of the cells when
the latter are being used in applications such as electric vehicles. In this
study, we tackled the issue of high-performance electrodes for desired battery
application conditions by proposing a powerful data-driven approach supported
by a deterministic machine learning (ML)-assisted pipeline for bi-objective
optimization of the electrochemical performance. This ML pipeline allows the
inverse design of the process parameters to adopt in order to manufacture
electrodes for energy or power applications. The latter work is an analogy to
our previous work that supported the optimization of the electrode
microstructures for kinetic, ionic, and electronic transport properties
improvement. An electrochemical pseudo-two-dimensional model is fed with the
electrode properties characterizing the electrode microstructures generated by
manufacturing simulations and used to simulate the electrochemical
performances. Secondly, the resulting dataset was used to train a
deterministic ML model to implement fast bi-objective optimizations to
identify optimal electrodes. Our results suggested a high amount of active
material, combined with intermediate values of solid content in the slurry and
calendering degree, to achieve the optimal electrodes.

1 Laboratoire de Reactivité et Chimie des Solides (LRCS), UMR CNRS 7314,
Université de Picardie Jules Verne, Hub de l´Energie, 15 rue Baudelocque,
Amiens Cedex, 80039, France

2 ALISTORE-European Research Institute, FR CNRS 3104, Hub de l’Energie, 15 rue
Baudelocque, Amiens Cedex, 80039, France

3 Umicore, Corporate Research & Development, Watertorenstraat 33, Olen, 2250,
Belgium

4 CIDETEC, Basque Research and Technology Alliance (BRTA), P∘ Miramón 196,
Donostia-San Sebastian, 20014, Spain

5 Institut Universitaire de France, 103 Boulevard Saint Michel, Paris, 75005,
France

6 Réseau sur le Stockage Electrochimique de l´Energie (RS2E), FR CNRS 3459,
Hub de l´Energie, 15 rue Baudelocque, Amiens Cedex, 80039, France

∗ Corresponding author<EMAIL_ADDRESS>
22footnotetext: These authors contributed equally to this work.

_K_ eywords Battery Cell Manufacturing $\cdot$ Bayesian Optimization $\cdot$
Machine Learning $\cdot$ Electrode $\cdot$ Numerical Simulation

##  1 Introduction

#####

In our modern society, the demand for batteries has surged due to the
widespread use of electric vehicles and portable electronic devices. Lithium-
ion batteries (LIBs) have emerged as the most powerful technology for a fast
energy transition [1, 2]. Driven by the increasing demand for high-performance
energy solutions with low-carbon emissions, the modern world is making efforts
to establish gigafactories and recycling solutions to significantly reduce the
production costs for LIBs and make it sustainable [3, 4]. The manufacturing
process is considered the most impactful part of battery design, and
optimizing this process is crucial for improving overall battery performance
[5]. This complex fabrication process involves numerous interlinked steps and
manufacturing parameters, resulting in various production approaches [6]. The
entire process includes electrode slurry preparation, coating and drying,
calendering, and the the cell assembly, electrolyte filling and formation [7].
Certain parameters, such as the type of material, the amount of material, and
the drying temperature applied to the slurry, have a significant impact on the
final battery performance and must be optimized throughout the entire
fabrication process. The electrode optimization in turn depends on the end
application that can be categorized as: 1) energy-oriented and 2) power-
oriented batteries [8, 9], which require different electrode design
strategies. In general, energy-oriented batteries favor higher material
loading, while power-oriented batteries favor lower material loading due to
the nonlinear relation between electrode energy density and applied current
[10, 11, 12, 13].

#####

The digitalization of the manufacturing process for LIBs has already
demonstrated progress in providing meaningful results to better understand the
impact of the manufacturing process chain on electrode properties, such as
conductivity, mass loading, and electrode porosity [14, 15, 16, 17, 18, 19].
This digitalization is supported by physics-based models and Machine Learning
(ML) models, which enable high throughput modeling and accelerate the
discovery of interlinked manufacturing steps [20, 21]. The ERC-funded ARTISTIC
project has become a pioneer in computational 3D-validated models and has
developed models for each step in the manufacturing process, with the models
sequentially linked with one another [22]. For example, the output from the
slurry model is used as the input for the drying model, which is followed by
the calendering process simulation. Coarse-Grained Molecular Dynamics (CGMD)
and the Discrete Element Method are used for the simulation of the slurry, its
drying and the calendering of the resulting electrode [23]. At the
electrochemical level, a 4D-resolved continuum model was used to perform
electrochemical and mechanical simulations of galvanostatic discharge-charge
and electrochemical impedance spectroscopy experiments [24], allowing us to
determine the relationship between performance and the corresponding
manufacturing parameters. Additionally, all of these models have been
experimentally validated using our battery prototyping platform, and their
practical application is enabled through an online calculator service
accessible to any user with an internet connection [25]. New applications of
ML techniques were also integrated in the ARTISTIC computational workflow,
supported on both classification and regression models [26, 21].

Figure 1: Figure 1: Schematic representation of the deterministic approach for
our electrochemical performance optimization based on the energy density and
power density for different battery applications. The workflow is similar to
the one used in Ref. [13] but here it takes into account the optimization of
cell performances instead of electrode textural properties.

#####

In general, CGMD simulations can generate an electrode slurry model in as
little as eight hours, but NEMD simulations can simulate the viscoelastic
behavior of the slurry structure in as little as five days. Due to this
restriction, physical modeling cannot be used for real-time optimization of
manufacturing processes [27, 28]. This limits the research on automatic
optimizations of other performance metrics in the interim [29]. Still,
physics-based simulations are very important, because they provide, in a high
fidelity fashion, an understanding based on physics of the reasons of why and
how manufacturing pararemeters impact electrode properties. Moreover, ML
algorithms have shown impressive potential in studying the effects of
manufacturing processes on electrode properties, such as predicting final
electrode properties using regression models [30], or estimating battery
state-of-health using common regression techniques [31]. These algorithms can
also support physics-based models in significantly reducing their overall
computational costs. For example, Functional Data Analysis has been applied by
us to viscosity simulations to reduce the time required for retrieving
simulation results [28]. Gao et al. developed a hybrid solution combining
modeling and data-driven approaches to accelerate electrode design [32], while
Thelen et al. integrated physical modeling and ML methods for primary
degradation models and battery cell capacity estimation [33]. Furthermore,
inverse design of battery manufacturing is a challenging yet promising
approach. Lv et al. focused on response surface methodology to optimize
titanium and vanadium that is co-doped in LiFePO4/C [34, 35]. Bayesian
Optimization (BO) has been used to solve complex optimization problems with a
probabilistic approach [36], and has been widely applied in domains such as
biology for sequence design [37], and cybersecurity for cyber-attack detection
[38]. However, R&D facilities are required to build even more appropriate
devices due to the rising demand for battery development. In that regard, BO
is in the spotlight for solving issues of this nature. It is worth noting that
maximizing cell energy is limited by high power usage, illustrating the need
for multi-objective optimization [39, 40]. Similarly, increasing electrode
thickness for energy cells can result in faster capacity fade and an expansion
of the tortuosity factor [41]. Consequently, the growing interest in inverse
design requires further exploration to provide numerical frameworks for
various optimization objectives.

#####

This study extends our previous work on inverse design of manufacturing
parameters to new energy- or power-like applications, and proposes a data-
driven approach to maximize electrochemical performance. To generate a
synthetic dataset of 3D calendered microstructures of LiNi1/3Mn1/3Co1/3O2
(NMC111) active material-based electrodes, we employed the chain of ARTISTIC
physics-based models that link input manufacturing conditions and output
electrode properties, as reported in our previous work [13]. These properties
were treated as input parameters of a Newman’s pseudo-two-dimensional (p2D)
model to produce electrochemical performances under galvanostatic discharge.
The obtained results offer an appropriate dataset to train and test
deterministic learning models, with primary objectives aimed at approximating
the complete computationally expensive electrode physics-based generation and
characterization and enabling interpolating the manufacturing inputs space in
order to analyze other conditions that were not included in the initial
synthetic experiments. By blending these functions into a BO framework for bi-
objective optimizations, we identified the best amount of active material (AM
%), slurry solid content (SC %), and calendering compression degree (CD %)
that can give an electrode with a relevant energy and power densities for
various cell applications as it can be illustrated in Figure 1. The bi-
objective optimizations were selected based on the specification of energy-
like or power-like applications at the electrode level that were
mathematically expressed for such NMC111 material, similar to concrete uses.
These potential applications are now being researched and developed, which
aims to find ways to optimize cell performance for a range of issues at the
lab and industrial levels. That is why our approach focuses on the comparison
of the bi-objectives to facilitate the design of battery problems and to
emphasize the importance of assessing an inverse design loop. This approach
opens up the possibility of developing autonomous battery manufacturing
processes.

##  2 Manufacturing physics-based modeling

### Data acquisition

#####

Due to the high computational cost required to simulate the electrochemical
performance for a continuum batch of various manufacturing conditions, we
utilized the synthetic dataset generated from our previous work [13] to obtain
a highly representative dataset of the manufacturing parameter space.
Specifically, we have generated quasi-random Sobol sequences with Saltelli
extension based on three parameters: the amount of active material (AM %), the
slurry solid content (SC %), and the electrode compression degree (CD %) [42].
These parameters are representative enough of the slurry preparation, drying,
and calendering processes, as important parameters to assess when
manufacturing electrodes [43, 44]. Our focus was to properly probe the input
manufacturing space and capture all of its sub-areas by varying these three
parameters. This design of experiments (DOE) was used as input values for
physics-based models to evaluate numerous electrode properties that
characterize the 3D microstructures [45]. More details on this can be found in
section 2 of our previous work [13]. It is worth mentioning that the DOE is
large enough to generate data for further machine learning (ML) regression
purposes while being efficient enough to avoid a huge computational cost
associated with generating all 3D microstructures for each manufacturing
condition. In fact, using formulation and drying as input parameters, the
production of a slurry and its matching dried electrode can take in computing
clusters roughly 150 hours. When using DOE approaches to map the behavior of
the electrodes under manufacturing circumstances as precisely as feasible, the
challenge of producing huge batches of synthetic data may be overcome. In the
end, the DOE ultimately includes 174 alternative situations that were deemed
relevant for the analysis of the manufacturing input space with the chosen
parameters in our earlier work, which was centered on enhancing electrode
textural properties.

Figure 2: Figure 2: Schematic representation of the link between the different
softwares for the handling of the 3D electrode microstructures for the data
acquisition generated by the physics-based modeling workflow. This starts with
the simulations from the slurry to the calendering, where the resulting
electrode is meshed in order to serve as input of the different codes for the
properties calculation. This ends by using the latter properties as inputs of
a p2D model to simulate the electrochemical performances to then extract the
gravimetric energy density and and the average gravimetric power density.

#####

In this work, we went beyond the study we previously reported by using the
electrode properties calculated from the 3D calendered microstructures as
input parameters of a Newman’s pseudo-two-dimensional (p2D) model in order to
simulate the electrochemical behavior of the electrode at 1C discharge like
Figure 2 illustrates. Six properties characterized from the synthetic
electrode microstructures were fed into the p2D model: electrode porosity,
tortuosity factor, mass loading, specific active surface area, electronic
conductivity, and thickness. In our electrochemistry simulation, the CBD phase
is considered to block Li+ transport. Therefore, the nanoporosity of CBD is
not taken into account for electrode porosity calculations and specific active
surface areas only consider the interface between AM and electrolyte. Both
values are calculated based on the voxelized microstructures (Figure 2).
Tortuosity factor and electronic conductivity are acquired by the open-source
code TauFactor [46] and the commercial code GeoDict respectively [47]. Since
the p2D model is a widely used model for LIBs study, we do not discuss much
more on details of the associated p2D modeling setup. The equations and
parameters used are listed in Table S1 and Table S2 in the Supporting
Information. Worth to be noted that the size distribution of AM particles is
taken into consideration in our CGMD model and p2D model, which is presented
as 6 groups of particles with different sizes. All cases have the same
particle size distribution profile. Volume fractions are reported in Table S2
in the Supporting Information individually. The specific active surface area
is also divided into six groups by the equation:

$\displaystyle
a_{v,n}=\frac{r_{n}^{2}n_{n}}{\sum_{m}r_{m}^{2}n_{m}}a_{v}\epsilon$ (1)

where $a_{v}$ is the total specific active area acquired from voxelized
structures, $\epsilon$ is the volume fraction, $r_{m}$ is the radius of the
particle and $n_{m}$ is the number of particles of each type.

Figure 3: Figure 3: 1C discharge curves resulting from the simulation of the
electrochemical performances using the Newman’s p2D model in COMSOL
Multiphysics 5.6 for the different electrode microstructures generated by the
ARTISTIC manufacturing simulator.

#####

Simulations of 1C discharge with a cut-off voltage of 3.2 V were performed for
all microstructures through parameter sweep. The discharge curves are
displayed in Figure 3. The difference in voltage plateaus and discharge
capacities shows the effect of microstructural characteristics on the
electrochemical performance. What leads to this difference is the
overpotential. The source of overpotential could be originated from four
aspects: mass transport in the solid phase, mass transport in the liquid
phase, ionic, electronic resistance, and Faradaic reaction overpotential [48].
Since we have the same particle size distribution for all cases, the solid
phase mass transport is under the same condition and therefore it is not the
source of differences. Mass transport in the liquid phase, however, is the
main source of capacity decrease. Figure S1 in the Supporting Information
shows the distributions of properties of all cases. The variety of porosities
and tortuosity factors shown in Figure S1 causes considerable differences in
Li+ concentration distribution in the electrode. Low concentration leads to
low exchange current density in the Butler-Volmer equation, which returns a
high Faradaic overpotential locally. Thus, the cases got lower discharge
capacities as the Li+ concentration gradient went higher.

#####

Electronic resistance was found to be not a determining factor in capacity
difference. As we have reported in our group’s previous work, an obvious
indicator can be used to identify if electronic conductivity is the limiting
factor [24]: if the particle SOC is found to be higher at the current
collector side, it means that the electrode has a low electronic conductivity
which causes capacity decrease. However, this phenomenon was not found in our
study, meaning that electronic resistance is not the main reason leading to
the capacity decrease. The specific reaction area is different due to the
different AM formulations in the synthetic electrodes. A higher AM ratio means
a more exposed AM surface to electrolyte due to less coverage of CBD. This
will lead to lower local surface Faradaic current, therefore lower reaction
overpotential.

#####

We recognize that the discharge profile of an electrode reflects a complex
combination of characteristics. Discharge capacity is insufficient to
comprehensively demonstrate the electrochemical performance, thus we use two
other descriptors, gravimetric energy density $E$ and average gravimetric
power density $P$, to measure and compare electrode performances.
Conventionally, $E$ and $P$ are referred in the community for the full battery
cell to describe the energy and power oriented cell design, but here we are
referring them in respect to a single electrode only as descriptors for
electrode performance assessment. Readers must avoid conflating the two
concepts.

$\displaystyle E=\frac{\int I_{app}Vdt}{M_{electrode}}$ (2) $\displaystyle
P=\frac{E}{t_{total}}$ (3)

where $I_{app}$ is the applied current, $V$ is the output voltage,
$M_{electrode}$ is the total mass of the electrode including mass of AM, CBD,
electrolyte and current collector. $E$ and $P$ will be the optimization
targets of our BO framework. The gravimetric energy density $E$ could be used
individually to determine the optimum case. By adding $P$ as another analysis
dimension, more information could be extracted enabling extending the
optimization to other C-rates. As we know, the gravimetric energy density $E$
can by itself determine the optimum electrode for a given current condition,
which in our case is 1C discharge. However, $E$ alone is not enough to capture
the characteristics of electrode under other conditions without doing
electrochemical simulations. Combining with $P$, the cases can be separated
into two groups that correspond to electrodes that are suitable for higher
C-rates and lower C-rates, respectively. In our work, we have an optimization
problem that find the optimal electrodes at 1C whereas two other scenarios
focus on an electrode that should be operated at low and ultra-low C-rates.
More detail discussion will be given in later sections.

### Deterministic learning

#####

The resulting synthetic dataset gathered enough manufacturing parameters and
associated electrode properties to train and validate deterministic learning,
serving as numerical functions to calculate the electrode performances at the
electrochemical level without the need to launch any new physics-based
modeling simulation. Indeed, such learning constitutes regression functions
trained on pre-defined data using the aforementioned DOE, improving the
predictive capabilities to forecast manufacturing conditions from the input
parameter space [49]. Therefore, the time needed to retrieve the electrode
performances is drastically reduced because it takes only a few seconds
compared to the usual case within the full physics-based model chain. This
possibility to inter/extrapolate for any relevant manufacturing condition
which is not represented within the DOE enables us to carefully initialize the
inverse design and save time when analyzing extra conditions. In that sense,
the combination of the DOE and predictive models capture the sensitivity of
the relationships between the selected properties and the manufacturing
parameters which is meaningful to optimize multiple properties simultaneously
[50]. We used a Gaussian process (GP) applied for non-linear regression
purposes to fit the latter two properties as a function of the manufacturing
parameters, due to its potential to infer a probability distribution over the
training dataset as a collection of random variables any finite number of
which have a joint Gaussian distribution [51, 52]. It is particularly
efficient for us since the training dataset is highly representative of the
entire manufacturing parameter space to estimate the prior distribution. The
associated algorithm assimilates any output property $y$ (i.e., the energy or
power density) written as $y=f(x)+\epsilon$ where $x$ is a set of input values
(i.e., a manufacturing condition) and $\epsilon$ a noise term with variance
$\sigma_{\epsilon}^{2}$.

#####

However, $f$ is an unknown function which will be here approximated, assuming
a Gaussian process

$\displaystyle f(x)\sim\mathcal{GP}(m(x),k(x,x^{{}^{\prime}}))$ (4)

where $\mathcal{GP}$ is defined by a mean $m(x)$ and a covariance function
$k(x,x^{{}^{\prime}})$ which depends on a kernel function written $k$ [53].
The choice of the kernel function depends on the relationships between input
variables and the smoothness that exists in the patterns of data. It is common
to use the radial basis function (RBF) [54], the constant kernel (CK), the
exponential sine squared kernel (ESSK) [55], or even a combination of any
existing kernels. For our case study, we choose the RBF whose free parameter
is equal to 1 because $k(x,x^{{}^{\prime}})$ has to decrease smoothly with
distance for sparse inputs, whereas having similar outputs for similar inputs.

#####

Under prior distribution from the training dataset $(x_{i},y_{i})_{\\{i\leq
n\\}}$ (where $n$ represents the number of training data points),
$y=[y_{1},y_{2},y_{3},...,y_{n}]_{\\{i\leq n\\}}$, and the kernel $k$, GP
assigns the posterior distribution by evaluating a joint Gaussian distribution
for any new testing data point $x^{*}$ as it follows matrix-wise spoken [56,
51]

$\displaystyle\left[{\begin{array}[]{c}y\\\ y^{*}\\\
\end{array}}\right]\sim\mathcal{N}(0,\left[{\begin{array}[]{cc}k(x,x)+\sigma_{\epsilon}^{2}I_{n}&k(x,x^{*})\\\
k(x,x^{*})&k(x^{*},x^{*})\\\ \end{array}}\right])$ (9)

where $k(x,x)$ is the $n$-order covariance based on the training set,
$k(x,x^{*})$ is the covariance based on the training set and the new testing
data point $x^{*}$, and $y$ is the set of output properties defined above.

#####

Therefore, it becomes feasible to predict the output corresponding to the new
testing data point $x^{*}$ according to the posterior probability distribution
which is also following a Gaussian distribution but with different mean and
covariance functions as written below:

$\displaystyle\begin{cases}y^{*}|x^{*}\sim\mathcal{N}(\mu,\sigma)\\\
\mu=k(x,x^{*})[k(x,x)+\sigma_{\epsilon}^{2}I_{n}]^{-1}y\\\
\sigma=k(x^{*},x^{*})-k(x^{*},x)[k(x,x)+\sigma_{\epsilon}^{2}I_{n}]^{-1}k(x,x^{*})\end{cases}$
(10)

#####

That is being said, the new output can be rewritten as a linear sum of
similarities between the new testing data point and each training data point
from the kernel application, weighted by coefficients
$w_{i}=[k(x_{i},x_{i})+\sigma_{\epsilon}^{2}I_{n}]^{-1}y_{i}$ that arise from
the numerical calculations of covariances matrix from equation (6) [56]. It
results a new equation for the mean of the posterior distribution
$\mu=\sum_{i=1}^{n}w_{i}\times k(x_{i},x^{*})$.

### Optimization process

#####

After training the deterministic learning for electrochemical performance prediction, we performed Bayesian bi-objective optimization loops (BO) by assessing a specific objective function (i.e., $C_{f}$) being dependent on the latter numerical functions [57, 58]. BO intended to optimize the energy and power density differently by formalizing the objective function into a single-weighted minimization problem according to the final battery application. BO is already well known for multi-objective optimizations, applying Gaussian process regressions as the model to numerically approximate $C_{f}$ [59]. BO represents an iterative loop over data that have been tested as potential candidates of the optimal conditions (i.e., $\mathcal{D}$), where the algorithm calculates a posterior distribution ($C_{f}$) | $\mathcal{D}$ to propose a new set of data (e.g., a set of manufacturing parameters) through an acquisition function which balances between the exploitation of prior parameter value combinations from $\mathcal{D}$ to establish nearby minima, and the exploration to identify minima far from prior parameter value combinations. In the end, the optimization loop stops when $C_{f}$ has reached values under a certain threshold, or after a particular number of steps that have been pre-defined when initializing the BO algorithm. Considering that optimizing battery manufacturing at the electrochemical level means maximizing the energy and power density for our study, we formalized a minimization problem with $C_{f}$ which is equivalent to find the best manufacturing parameter combination $\mathbf{x}^{*}$ that minimizes the objective function expressed like

$\displaystyle\mathbf{x}^{*}\hskip 2.84544pt=\hskip 2.84544ptargmin_{x}\hskip
2.56064pt(C_{f}(x))$ (11)

### Relative importance assessment

#####

For every bi-objective optimization performed in this study, both $E$ and $P$
are considered simultaneously. However, we granted different relative
importance (called weights in the following paragraphs) according to the
problem we aimed to solve. To obtain the best electrode, in other words, the
electrode with the highest energy density, we put full weight on $E$ and zero
on $P$. Then, more weight is assigned to $P$ to shift the optimum case to the
high $P$ direction. As a result, several sets of manufacturing parameters will
be acquired. The following equation (8) denotes how $C_{f}$ was built to
correspond to the different weighted power bi-objective optimizations [60].

$\displaystyle
C_{f}(x_{i})=w_{E}\times(1-y_{\\{x_{i},E\\}})^{2}+w_{P}\times(1-y_{\\{x_{i},P\\}})^{2}$
(12)

#####

Indeed in the BO loops, the deterministic learning predicts the $E$ and $P$
for a given set of manufacturing parameter values $x_{i}$ at each step, which
is quick and reduces the computational cost for any optimization. Then, we
applied a scalar fitness function to transpose the numerical predictions from
the deterministic learning (i.e., $y_{\\{x_{i},E\\}}$ and $y_{\\{x_{i},P\\}}$)
into the interval [0, 1] when calculating the two terms of the $C_{f}(x_{i})$
for a given set of manufacturing parameter values $x_{i}$ (the scalar fitness
function is reported in the Supporting Information). Overall, this gives
relative order of magnitude for standardized terms and then avoids a bias
induces by the values of $E$ and $P$. In the end, the maximization of the two
electrode performances is equivalent to minimize the terms
$1-y_{\\{x_{i},E\\}}$ and $1-y_{\\{x_{i},P\\}}$ to target the lowest $C_{f}$
as possible. The weight of each term (i.e., $w_{E}$ or $w_{P}$) will be given
by the type of battery-oriented problem (Table 1). Two extra cases are given
to show the trend of electrode performance change with weight assignment. All
these problems correspond to a tremendous practical value for labs and
industries. They have been chosen for a comparative purpose where other
possible battery-oriented problems can be formalized using different weights
with respect to real applications.

#####

Nevertheless, the assignment of weights must reflect accurate quantification
regarding the decision-making, which is difficult to evaluate in reality since
they may depend on the overall prediction error [61]. That is why it exists a
variety of proposals to determine the correct weights in a multi-criteria
approach [62]. One popular way is to give equal importance to all of the
properties to be optimized in the BO loop when the decision maker knows
nothing about the right weights, whereas unbalanced and ranked approaches seem
to be more convenient when the order of importance is at least prioritized
[63]. For instance, the rank sum weight method (RS), the rank exponent weight
method (RE), or the rank-order centroid weight method (ROC) are relevant
alternatives once the ranked attribute weights are pre-defined [63].

optimization problem |  Method |  Weights
---|---|---
optimal electrode at 1C |  RE |  {1, 0}
electrode should be operated at low C-rate |  equal |  {$\frac{1}{2}$, $\frac{1}{2}$}
electrode should be operated at ultra low C-rate |  RE |  {0, 1}
case ($*$) |  RE |  {$\frac{7}{8}$, $\frac{1}{8}$}
case ($**$) |  RE |  {$\frac{1}{8}$, $\frac{7}{8}$}
Table 1: Table 1 : Assessment of weights for the inverse design based on the
type of optimization problem. It is important to notice that the sum of
weights must be equal to 1. The RE weights method corresponds to the rank
exponent weight method to assess the weights for the bi-objective
optimizations. When the relative importance of both properties must be the
same, we said equal weights.

#####

Since we have two performance descriptors in our study, it is obvious to
define the order of the relative importance for the objective function
assessment. We selected the RE approach to assign the two weights (equation
(9)) as a generalization of the rank sum method since we decided to have a
steeper distribution of weights in order to correctly define the battery-
oriented optimizations with larger differences among weights [62]. Last but
not least, it is straightforward that extreme weight differences induce a bias
in the bi-objective (and in a multi-objective in general) optimization, being
similar to a single-objective optimization. Therefore, the BO can be
insensitive to the distribution of weights, predicting manufacturing parameter
conditions. As an example in our case study, the optimal electrode at 1C was
found with a single objective optimization similar to an extreme bi-objective
optimization on $E$.

$\displaystyle w_{i}=\frac{(2-i+1)^{p}}{\sum_{k=1}^{2}(2-k+1)^{p}}\quad
p\in\mathbb{N}$ (13)

### Validation of the deterministic learning

#####

The deterministic learning used the DOE which was split into a training and
testing dataset for regression training and validation respectively. 80 % of
the total raw data randomly selected were affected by the training dataset,
while the remaining 20 % went to the testing dataset. The ratio 80 %/20 % was
empirically chosen but is very common as a standard practice in ML [64]. The
R${}^{2}_{score}$ and the root mean square error in percentage (RMSE %) were
chosen as the validation metrics in order to analyze the goodness of fit for
both types of datasets. Table 2 reports the validation metrics for each
electrode performance included in the deterministic learning, where Figure 4
supports accurate results from the overall table. Indeed, an R${}^{2}_{score}$
close to 1 highlights good capabilities for the deterministic learning to
predict the energy and power density whereas the RMSE % explicit how far a
prediction is from its real synthetic value in average (expressed percentage).

Property |  RMSE % |  R${}^{2}_{score}$ |  CI95
---|---|---|---
Training process |  |  |
E |  4.986 |  0.980 |  [0.993; 0.995]
P |  0.968 |  0.986 |  [0.991; 0.993]
Testing process |  |  |
E |  3.490 |  0.975 |  [0.982; 0.985]
P |  1.002 |  0.984 |  [0.983; 0.986]
Table 2: Table 2: Validation metrics calculated over the training and testing
dataset, associated with the fitting of the energy performance $E$ and power
performance $P$. The 95 % confidence interval (CI95) were estimated with a
total of 75 random seeds of training/testing datasets for the uncertainty on
the R${}^{2}_{score}$. $E$ and $P$ are coming from equation 2 and equation 3
and correspond to the gravimetric energy density and average gravimetric power
density respectively.

#####

Even though the deterministic learning is obtained after training and testing
the GP regression models, we repeated such a process 75 times by shuffling
uniformly the training and testing dataset to calculate the distribution of
the R${}^{2}_{score}$ and determine the 95 % confidence interval (CI95) [65].
This random selection avoided data leakage and bias when splitting the
dataset. This interval contributes to the statistical analysis of how the
predictive capabilities of the deterministic learning behave concerning the
seed of the re-sampling procedure training/testing dataset. Therefore, CI95
estimates the variability of the metric and how precise it is likely to be
given a certain threshold. The larger the CI95 around the mean value, the
higher the variability of the deterministic learning to provide accurate
predictions regardless of the different training/testing datasets. Lastly, it
is usual to conclude that exists a bias in the dataset if the variability is
high (e.g., outliers in the training or testing dataset, too few data points,
wrong ML model assessment) [66]. Regarding our results, the CI95 includes
values close to 1 without a significant width and suggests that deterministic
learning can understand the different patterns of data from the training and
testing dataset in order to provide accurate predictions.

### Optimal manufacturing parameter values

#####

The BO framework was set with 300 iterations for each bi-objective
optimization as a cut-off to predict the most appropriate candidate for
manufacturing parameter combinations for each investigated optimization
problem. The function to minimize the posterior distribution within the BO
framework was a probabilistic choice based on the lower confidence bound,
negative expected improvement, and negative probability of improvement [67].
We initialized the manufacturing conditions generation by a Latin hypercube
sequence to start the optimization loop [68]. Table 3 provides the so-
predicted best manufacturing conditions associated with the different battery
cell applications. On one side, it can be seen that the AM % is always given
at a high value, which corresponds to the maximal value that the parameter can
take in the BO framework. On the other side, the SC %, and CD % are given at
intermediate values compared to the boundaries initialized by the BO
framework. These two parameters have been set in the interval [43; 72.8] and
[1.4; 38.8] respectively. Therefore, we were capable to obtain the optimal
values of the three manufacturing parameters that maximize the energy density
under applied C-rate and other C-rates by adjusting relative weights on $E$
and $P$ like Figure 5B compares. Figure S2 and S3 from the Supporting
Information support the comparison from Table 3 and Figure 5 to highlight how
each predicted manufacturing parameter, $P$, and $E$ are represented within
the overall distribution from the synthetic dataset. Even though extreme AM %
are always predicted based on the different optimization problems, the latter
have a significant impact on the differences in $E$ values rather than $P$
values according to the empirical skewed distribution. That means the bi-
objective optimization has to carefully assess the weights in order to avoid a
strong decrease of $E$ when optimizing at lower C-rates.

Figure 4: Figure 4: Illustration of the validation metrics for (A) the energy
performance and (B) the power performance, for the training and testing
datasets. For each case, the regression plot displays how values predicted by
the deterministic learning are close to real synthetic values. Associated
$R^{2}_{score}$ are reported in the third column of Table 2. The residuals as
a function of the real synthetic values show that it does not exist any bias
in the training/testing process from the deterministic learning. Lastly, the
histogram of absolute residuals displays the distribution differences between
predictions and real synthetic values, giving low error predictions on average
with some higher errors for a few amount of data points.

##  3 Results and discussions

### Interest of bi-objectives

#####

By introducing $P$ into the analysis, we are able to do multi-C-rate
optimization with simulation results with a single C-rate by assigning
different relative weight on $E$ and $P$ as in Table 1. As a reminder for the
readers, the shift of relative weight should not be connected with the concept
of energy-oriented or power-oriented cells. Our optimization work is centered
on energy density of electrode, with P as a helper descriptor for further
assessments. As we can see in Figure 5A, $E$ and $P$ shows a volcano-shaped
outline, where on the peak of the volcano lies the optimal case which has the
highest energy density. From the optimal point as the center, the energy
density of the electrode decreases monotonically on both sides. With the usage
of $P$, the decrease in energy density can be divided into two branches. On
the left branch (LB) of Figure 5A, $E$ and $P$ have a positive correlation,
while on the right branch (RB) they have a negative correlation. The different
$E$ vs. $P$ correlations observed in these two branches have different
origins. On LB, the decrease of $E$ comes from the low mass loading of the
electrode. Recalling equation (2), the calculation of $E$ takes into account
the weight of the current collector and electrolyte. When the electrode mass
loading is lower than a threshold, the energy density drops because of the
lower mass fraction of active material. Nevertheless, having a low mass
loading results in a low diffusion resistance of Li+. As a result, the
discharge times remain approximately at 3600 s. Therefore, according to
equations (2) and (3), $P$ and $E$ form a strong positive linear relation. RB
corresponds to high mass loading conditions. The high overpotential causes
capacity decrease of the electrode. According to equation (3), decreased
discharge time causes an increase of $P$. Therefore, we find a negative
relation between $E$ and $P$ on RB. From the analysis above, we can see that
mass loading is the key point in the $E$ and $P$ relationship. The conclusion
that can be drawn from here is that cases on LB are suitable for higher
C-rates, while cases on RB are suitable for lower C-rates. Therefore, in our
bi-objective optimization, we categorize the optimization problems as optimal
electrode, electrode that should be operated at low-C-rate and ultra low-C-
rate objectives, since we are exploring the RB where $P$ increases.

optimization problem |  AM % |  SC % |  CD %
---|---|---|---
optimal electrode at 1C |  RE |  {1, 0} |
optimal electrode at 1C |  96.8 |  50.8 |  12.8
electrode should be operated at low C-rate |  96.8 |  60.4 |  16.4
electrode should be operated at ultra low C-rate |  96.8 |  72.6 |  35.3
case ($*$) |  96.8 |  56.3 |  13.7
case ($**$) |  96.8 |  66.4 |  21.8
Table 3: Table 3: Validation metrics calculated over the training and testing
dataset, associated with the fitting of the energy performance $E$ and power
performance $P$. The 95 % confidence interval (CI95) were estimated with a
total of 75 random seeds of training/testing datasets for the uncertainty on
the R${}^{2}_{score}$. $E$ and $P$ are coming from equation 2 and equation 3
and correspond to the gravimetric energy density and average gravimetric power
density respectively.

### Optimized transport properties

#####

The optimization results in Table 3 show the best manufacturing parameters we
got for each scenario. Going in the direction of lower C-rates results in
higher SC % and CD %, while AM%̇ remains at the highest value. Figure S5 from
the Supporting Information shows the relation between manufacturing parameters
with electrode properties, from which we can see how manufacturing parameters
are linked with the electrochemical performances. The main effect of
increasing AM % is decreasing thickness and increasing specific surface area.
Decreased thickness shortens the diffusion distance of Li+ in the electrolyte,
and increased specific surface area reduces the local Faradaic current
density, which reduces the reaction overpotential [69]. Furthermore, increased
AM % reduces the mass fraction of CBD, which increases the calculated $E$
value directly. Thus, it shows a beneficial trend for both energy and power
density.

#####

SC % shows a clearer effect. Because we applied a certain range constraint on
coating comma gap, hence the solid content has a linear correlation resulted
in mass loading as can be seen in the same Figure S3. In the meantime,
electrode thickness also increases with higher solid content. This causes
discharge capacity decrease, which is directly linked with the decrease of
energy density shown in the plot. However, due to the fact that solid content
does not have an obvious effect on other properties, especially specific
surface area, the voltage plateau is not affected by solid content. Therefore,
based on equation (3), $P$ increases because of the decreased discharge time.
Naturally, we got the same trend in the optimized result.

#####

The compression degree shows a minor but similar trend as solid content.
Figure S3 shows that electrode compression has a complex impact on electrode
properties. On one hand, reduced porosity and increased tortuosity will cause
poor Li+ transport in the electrolyte phase, on the other hand, reduced
thickness is beneficial for the Li+ transport, and electronic conductivity is
also improved due to more compactness of the microstructures and more particle
cohesion. The final result depends on the sophisticated competition between
the factors. However, our optimization result shows that lower CD % is
preferred by the optimal scenario, and higher CD % by the lower C-rate cases.
This means that $E$ oriented optimization tends to avoid bad diffusivity of
Li+ brought by high SC % to make sure full utilization of the active material.
$P$ oriented optimization, on the other hand, tends to have a low discharge
time in order to promote average power output. Figure 5C exhibits the 3D
microstructures at the calendering level generated by the ARTISTIC physics-
based manufacturing simulator using the so-predicted best manufacturing
conditions from Table 3 as inputs of the physics-based models for each
optimization. These 3D volumes consider distinct bottom surface areas and
thicknesses due to the differences in terms of formulation and calendering
degree.

Figure 5: Figure 5: (A): Scatter plot comparing gravimetric energy density as
a function of average gravimetric power density calculated through the p2D
model at 1C discharge. The stars illustrate the different optimization
problems keeping the same colors as in (B) to simplify the legend. (B):
Barplot comparing the energy density (color-coded in darkcyan) and the power
density (color-coded in red) on the $y$ axis as a function of the type of
optimization problem. There are three bi-objective optimizations (i.e.,
optimal electrode, low-C-rate obj., and ultra low-C-rate obj. and two extra
optimizations to show the trend of $E$ and $P$ as a function of weights. (C):
Microstructures of the optimized electrodes. With a purpose of simplifying the
graphics, we narrow the two scenarios associated to an electrode that should
be manufactured at low and ultra-low C-rates with low C-rate obj. and ultra-
low C-rate obj. respectively.

## Conclusions

#####

In this study, we addressed smart deterministic-assisted bi-objective
optimizations of electrode performance in different energy-like or power-like
applications conditions. Our method predicts the ideal set of manufacturing
parameters to use to produce the electrode in light of the specified type of
battery applications. The entire method with support from physics-based and ML
models. We used previously created synthetic data that contained highly
representative process parameters for the slurry, drying, and calendering
steps as well as electrode properties that were extracted from the 3D
microstructures created by the ARTISTIC physics-based manufacturing process
simulator to perform the comparison of optimization problems.In order to
accurately predict the performance descriptors E and P as a function of the
amount of active material, the solid content in the slurry, and the
compression degree, the aforementioned data allowed for the development of a
deterministic learning algorithm that was trained and evaluated on the
synthetic dataset. This allowed for the analysis of additional manufacturing
conditions that were not present in the synthetic dataset. Regardless of the
convergence conditions, this deterministic learning allows the development of
many optimization loops with a generally short computing time. In the end, a
Bayesian optimization approach is used to quickly minimize the various
objective functions. Finally, we emphasized how $P$ separates the reduction in
energy density into two branches. Based on this, electrode optimization for
alternative C-rates, which corresponds to applications for energy- or power-
oriented battery cells, have been added to the bi-objective optimization. We
determined that the SC % and CD % should be lower (50.8 % and 12.8 %,
respectively) for the ideal electrode (i.e., optimization of $E$) than for
optimization instances of $P$ at low or very low C-rate.The overall workflow
follows the interest to study numerous battery optimizations conditions which
are practically important in industry, R&D, and labs without the need to
generate additional synthetic data from the physical modeling.Furthermore,
according to our potential scenario based on the data at hand, the definition
of a battery optimization problem only pertains to the distribution of weights
and how to balance them in order to achieve the specified electrochemical
performance.

#####

As a perspective, we aspire to shift our work to more manufacturing conditions
and parameters such as particle size distribution, drying conditions (slow and
fast drying rates). In addition to the research of NMC111, the general
methodology may be applied to different material chemistries while being
closely related to data analysis. The utilization of our complete
computational physics-based modeling workflow is possible as long as the
slurry and electrode can be handled as particle assemblies. Its ability to
direct experimental users in fabricating high-performance battery cells is
anticipated to be advantageous for battery manufacturing. The work presented
here, which was analyzed for the fabrication of LIBs, can be applied to
fabricate and optimize other electrochemical energy technologies (such as
solid-state batteries, zinc-air batteries, and fuel cells), in order to
improve their electrochemical performances. It is critical to specify
additional electrode qualities that will need to be improved before using
alternative technologies, such as lithium metal electrodes. Moreover, we are
convinced that our approach can be applied to study other synthetic datasets
using physics-based models focusing on aging mechanisms, such as lithium
plating or SEI formation [70]. Regarding the various batches of possible
applications, we believe that our approach paves the way towards the
development of software infrastructures enabling the gathering of significant
amount of data, leading to autonomous Machine Learning frameworks. Under this
scenario, this represents a major pillar to perform on-the-fly optimizations
of battery manufacturing processes, with the potential to merge synthetic and
experimental datasets to embed physical knowledge into new smart experimental
data acquisition techniques. We think that this ambitious but practical
strategy has the potential to be a turning point in battery research and
development and speed up the energy transition.

#### Materials availability

#####

This study did not generate new unique materials.

#### Data availability

#####

Codes and data are available upon reasonable request. The dataset will be made
also available through the computational portal in the ARTISTIC project
website.

## Acknowledgement

#####

A.A.F. and C.L. acknowledge the European Union’s Horizon 2020 research and
innovation program for the funding support through the European Research
Council (grant agreement 772873, ARTISTIC project). M.D., E.A. and A.A.F.
acknowledge the ALISTORE European Research Institute for funding support.
A.A.F. acknowledges the Institut Universitaire de France for the support. We
acknowledge Utkarsh Vijay at LRCS for the proofreading of the article and
useful comments.

## Supplementary materials

#####

The Supporting Information (SI) provides the empirical distributions of the
different electrode properties, the comparison between the different bi-
objective optimizations in terms of electrochemical performances and
manufacturing parameters as well as their representativity within the
synthetic dataset. SI includes also the scatter plots for pairwise
representations for the different electrode properties being inputs of the p2D
model from COMSOL Multiphysics and manufacturing parameters, as well as a
separate plot for the comparison of $E$ and $P$ as a function of manufacturing
parameters. SI includes two tables for technical details about the p2D, and a
table for the bounds used as prior information of the Bayesian optimizations.
Finally, SI includes the scalar fitness function.

## Author Contributions Statement

#####

Conceptualization, M.D., and A.A.F.; Methodology, M.D., and A.A.F.;
Investigation, M.D., and C.L.; Software, M.D., and C.L.; Formal Analysis,
M.D.; Writing - Original Draft, M.D., and C.L.; Writing - Review & Editing,
M.D., C.L., V.K., E.A., and A.A.F.; Funding Acquisition, E.A., and A.A.F.;
Resources, M.D., A.A.F.; Supervision, E.A., and A.A.F; Project Administration,
A.A.F.

## Competing Interests

#####

The authors declare that they have no known competing interest or personal
relationships that could have appeared to influence the work reported in this
article.

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