# Accurate Machine Learning Predictions of Coercivity in High-Performance
Permanent Magnets

Churna Bhandari  The Ames National Laboratory, U.S. Department of Energy, Iowa
State University, Ames, Iowa 50011, USA  Gavin N. Nop  The Ames National
Laboratory, U.S. Department of Energy, Iowa State University, Ames, Iowa
50011, USA  Department of Mathematics, Iowa State University, Ames, Iowa
50011, USA  Jonathan D.H. Smith  The Ames National Laboratory, U.S. Department
of Energy, Iowa State University, Ames, Iowa 50011, USA  Department of
Mathematics, Iowa State University, Ames, Iowa 50011, USA  Durga Paudyal  The
Ames National Laboratory, U.S. Department of Energy, Iowa State University,
Ames, Iowa 50011, USA  Department of Electrical and Computer Engineering, Iowa
State University, Ames, Iowa 50011, USA

###### Abstract

Increased demand for high-performance permanent magnets in the electric
vehicle and wind turbine industries has prompted the search for cost-effective
alternatives. Nevertheless, the discovery of new magnetic materials with the
desired intrinsic and extrinsic permanent magnet properties presents a
significant challenge. Traditional Density Functional Theory (DFT) accurately
predicts intrinsic permanent magnet properties such as magnetic moments,
magneto-crystalline anisotropy constants, and exchange interactions. However,
it cannot compute extrinsic macroscopic properties, such as coercivity
($H_{c}$), which are influenced by factors like microscopic defects and
internal grain structures. Although micromagnetic simulation helps compute
$H_{c}$, it overestimates the values almost by an order of magnitude due to
Brown’s paradox. To circumvent these limitations, we employ Machine Learning
(ML) methods in an extensive database obtained from experiments, DFT
calculations, and micromagnetic modeling. Our novel ML approach is
computationally much faster than the micromagnetic simulation program, the
mumax3. We successfully utilize it to predict $H_{c}$ values for materials
like cerium-doped $\mathrm{Nd}_{2}\mathrm{Fe}_{14}\mathrm{B}$, and
subsequently compare the predicted values with experimental results.
Remarkably, our ML model accurately identifies uniaxial magnetic anisotropy as
the primary contributor to $H_{c}$. With DFT calculations, we predict the Nd-
site dependent magnetic anisotropy behavior in
$\mathrm{Nd}_{2}\mathrm{Fe}_{14}\mathrm{B}$, confirming $4f$-site planar and
$4g$-site uniaxial to crystalline $c$-direction in good agreement with
experiment. The Green’s function atomic sphere approximation calculated a
Curie temperature ($T_{\rm C}$) for
$\mathrm{Nd}_{2}\mathrm{Fe}_{14}\mathrm{B}$ that also agrees well with
experiment.

##  I Introduction

With the rapid advance of computational capabilities, there is considerable
research interest in machine learning (ML) methods for predicting material
properties using extensive databases1, 2. Of specific interest is the
remarkable speed of these techniques, which outperform traditional first-
principle methods like density functional theory (DFT) by an order of
magnitude. ML methods can be used effectively to investigate material
properties under extreme conditions such as complex structures, temperature,
pressure, and stress that are challenging for DFT. Recent applications of ML
have enabled accurate explorations of material properties3, 4, 5, 6, however
the determination of coercivity ($H_{c}$) has remained problematic for
micromagnetic simulation, since Brown’s paradox7, 8, 9 commonly leads to
overestimates of $H_{c}$ for permanent magnets. Modern efforts to apply ML to
micromagnetic models have yielded reasonable results 10, however they are
limited by applying exclusively to micromagnetic models.

ML requires datasets that include information about different materials and
their properties, such as crystal structure, micromagnetic grain boundaries,
saturation magnetization ($M_{s}$), uniaxial magnetocrystalline anisotropy
constant ($K_{u}$), exchange stiffness constant ($A_{ex}$), and Curie
temperature ($T_{\rm C}$). To build predictive models based on our dataset, we
exploit both classical ML and Artificial Neural Network (ANN) algorithms.
These models establish patterns and relationships between the independent and
target material properties. The models are trained on subsets of the known
data and tested on the complementary subset to test model accuracy and
reliability.

In this work, we study over $8770$ magnetic materials, including, to our
knowledge, the largest collected database of $300$ experimentally known
materials. Further, the experimental material database is augmented with
micromagnetic computations. First, we compute $H_{c}$ of known materials using
input parameters $M_{s}$, $K_{u}$, and $A_{ex}$ with $\mathrm{mumax}^{3}$ , a
micromagnetic numerical solver to augment ML results. Next, we use the
$\mathrm{mumax}^{3}$ generated datasets with sensible values of input
parameters to train supervised ML models 11, 12 to predict $H_{c}$ ML models
to predict $H_{c}$. In experimental materials, we find the non-linear models
such as Decision Tree (DT) and Random Forest (RF) 13 produce the best results
with $R^{2}\sim 0.87$, which is slightly improved with Extreme Gradient
Boosting (XGB) regressor, beating even the state-of-the-art Light Gradient
Boosted Machine (LGBM). Besides, we employ ab initio in cerium (Ce)-doped
Nd2Fe14B; $2:14:1$ materials to demonstrate the complete pipeline enabled by
the new ML toolkit. First, for pure no-magnet, the site contribution to
magnetic anisotropy is analyzed with the help of DFT calculations. Second, we
compute $T_{\rm C}$ for pure compound using Green’s function method. Third, we
employ DFT computed parameters to ML in predicting $H_{c}$ of Ce-doped
compositions.

Using state-of-the-art algorithms, not only for micromagnetic simulations, but
also for the most extensive database of experimentally measured micromagnetic
properties yet compiled, we find the following key results. i) From both real
and hypothetical data set ML trainings, we accurately confirm $K_{u}$ as a
leading contribution to the $H_{c}$ followed by $M_{s}$. ii) ML predicts that
$H_{c}$ is directly related to $K_{u}$ and $A_{ex}$ i.e., $H_{c}\propto K_{u}$
and $H_{c}\propto A_{ex}$ and inversely with $M_{s}$ i.e.
$H_{c}\propto\dfrac{1}{M_{s}}$. iii) From DFT calculations, the intrinsic
properties are shown to have site-dependent magnetic anisotropy, in
particular; the Nd $4f$-site is planar and the $4g$-site is uniaxially aligned
with the crystalline $c$-direction. iv) The predicted $T\rm{{}_{C}}$ with
Green’s function for Atomic Sphere Approximation (ASA) is in good agreement
with the experiment. v) The computed $H_{c}$ matches with experiment which
demonstrates the power of the ML methodology.

##  II Micromagnetism

Micromagnetics focuses on the study of magnetic behaviors at the sub-
millimeter scale, particularly for ferromagnetic materials. During its early
formalization, the field emphasized qualitative aspects of magnetism, such as
the role of domain structures, domain walls, and magnetic vortices. The
transition from hand calculations to computer simulations in micromagnetics
has been a significant advancement, augmenting the rough analytical approach
with detailed examinations of the forces at play inside practical materials
14.

Effects, such as grain pinning and grain walls, play profound effects in
determining the coercivity of a material. Naively, the estimate
$\dfrac{2K_{u}}{M_{s}}=H_{c}$ gives an upper bound on $H_{c}$. However, this
calculation disregards effects such as shape anisotropy and multi-grain
structures in materials, giving rise to a massive overestimate of coercivity
from theory alone, known as Brown’s paradox. Simulations have enabled the
study of the effects of these imperfections on the microscale magnetic
behavior, enhancing the understanding of fundamental magnetic phenomena.
Addtional complex effects as a result of the conditions under manufacture
hinder micromagnetic simulations from an accurate prediction of coercivity for
practical materials, though micomagnetic simulations is still crucual to
acheiving good understanding of micromagnetic materials.

###  II.1 Theory

Magnetodynamics is described by a nonlinear partial differential equation of
the spacio-temporal magnetization vector $M({\bf r},t)$. The time evolution of
$M({\bf r},t)$ is given by Landau–Lifshitz–Gilbert (LLG) equation

$\displaystyle\dfrac{\partial M({\bf r},t)}{\partial t}$ $\displaystyle=$
$\displaystyle\dfrac{\gamma}{1+\alpha^{2}}\big{(}M({\bf r},t)\times
H_{eff}({\bf r},t)$ (1) $\displaystyle-\alpha M({\bf r},t)\times[M({\bf
r},t)\times H_{eff}({\bf r},t)]\big{)},$

for $M_{s}$, i.e., the maximum magnetic moment per unit volume that a material
can achieve, $H_{eff}$ is the effective static magnetic field, $\gamma$ is the
gyromagnetic ratio, and $\alpha$ is the damping parameter, quantifying the
rate at which the magnetization relaxes back to equilibrium. In Eq. 1, the
first term is the precession of themagnetic moment around the external
magnetic field and the second term is damping, which relaxes the magnetic
moment to the equilibrium. For time-independent scenarios, such as the
computation of a hysteresis loop, the $\alpha$ term may be set to $0$.

The $H_{eff}$ consists of several field terms viz., the magnetostatic field
($H_{ms}$), the exchange field $(H_{ex})$, the applied field ($H$), the
demagnetization field ($H_{dem}$) due to dipolar field, the anisotropy field
($H_{ani}$), and the thermal field. The $H_{ms}$ is a long-range magnetic
field

$H_{ms}({\bf
r})=\dfrac{1}{4\pi}\int\nabla\nabla^{{}^{\prime}}\dfrac{1}{|r-r^{{}^{\prime}}|}.M({\bf
r^{{}^{\prime}}})dr^{{}^{\prime}}.$ (2)

The exchange field is the Laplacian of the magnetization ($m$), which is
obtained from the classical Heisenberg model

$H_{ex}=\dfrac{2A_{ex}}{\mu_{0}M_{s}}\nabla^{2}m.$ (3)

Although this expression is originally deduced for localized spins, it is
still valid for itinerant systems to the first order approximation. $A_{ex}$
is a measure of the strength of magnetic exchange interaction, which is
directly proportional to magnetic exchange coupling ($J$) and inversely
proportional to lattice constant ($a$) as given

$\displaystyle A_{ex}=\dfrac{JS^{2}}{a}n,$ (4)

where $S$ is the spin quantum number, and $n$ is the number of magnetic ions
per unit cell. Equivalently, $A_{ex}$ can be approximately expressed in
$T_{\rm C}$ as

$\displaystyle A_{ex}\sim\dfrac{3T_{\rm C}}{2za},$ (5)

where $z$ is the number of the nearest neighbors of magnetic ions

The third term is a uniaxial magnetic anisotropic energy
$\phi_{ani}=K_{u}\sin^{2}\theta,$ where $K_{u}$ is the uniaxial anisotropy
constant and $\theta$ is the angle between the local magnetization and
uniaxial anisotropy axis. The anisotropy field is

$H_{aniso}=\dfrac{2K_{u}}{\mu_{0}M_{s}}$ (6)

where $A_{ex}$ is exchange stiffness constant and $\mu_{0}$ permittivity of
medium.

Figure 1: (Left) a discretized magnetic structure into mesh points showing
different micromagnetic structures. (Right) the colors within each
microstructure refer to different spin orientations. The snapshot of spin
texture is captured during the relaxation of a magnet. The circular nature of
the coloring indicates a closed magnetic current induced at the center of the
two-dimensional 2D plane.

The micromagnetic equations are solved in the continuum approximation $M({\bf
r},t)=M_{s}(r)m({\bf r},t)$15, 16, 17. $H_{eff}$ is deduced from the magnetic
free energy functional $F[m]$ as $H_{eff}=\dfrac{1}{\mu_{0}M_{s}}\dfrac{\delta
F[m]}{\delta m}$ where

$\displaystyle F[m]$ $\displaystyle=$
$\displaystyle\int_{V}\Big{[}A_{ex}(\nabla m)^{2}-\mu_{0}M.H_{ext}$ (7)
$\displaystyle-\dfrac{\mu_{0}}{2}M.H_{dem}+f_{DMI}(m)+\cdots\Big{]}d^{3}r,$

where $f_{DMI}$ is Dzyaloshinskii–Moriya interaction (DMI). Additional terms
may be added as needed to count for additional interactions. The $F[m]$ is
minimized with respect to $m$ using the steepest descent algorithm as
implemented in the micomagnetic program 17. Then the hysteresis loop is
obtained by evaluating $m$ in each equilibrium magnetic state for different
values of external applied magnetic field from which $H_{c}$ is extracted at
$m=0$.

###  II.2 Experimental Materials

The experimental database of $300$ materials was collected from Refs. [18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38,
39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57,
58, 59, 60, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73]. Magnetic
materials were included if $K_{u}$, $M_{s}$, and $H_{c}$ were explicitly
included in experimental results. For $A_{ex}$, which is rarely directly
measured, three options were taken. The first was a direct reporting of the
stiffness exchange, which was rare. The second was to derive the stiffness
exchange from the Curie-stiffness relation in Eq. 5 (for R2Fe14B, R is rare-
earth element and Ce-doped Nd2Fe14B63, binary alloys61, SmCo5; $1:5$
compositions58, La/Pr/Co-doped hexaferrites53, and other element doped
hexferrites49, 74, 48, 52) from experimental $T_{\rm C}$ and lattice constant
$a$75. For R2Fe14B family, only experimental anisotropy field ($H_{a}$) is
available from which we estimated $H_{c}\sim H_{a}/4$ and similar
approximation is used for other materials76. Finally, some materials were
interpolated by referring to varieties of similar compounds, particularly for
alloy mixes.

Materials demonstrated significant variation in $H_{c}$ based on experimental
conditions, primarily dependent on the fabrication method. For instance,
oxygen content directly near iron (Fe) magnets affected $H_{c}$ as oxygen
modified the material composition during the hardening process. The
crystalline defects, such as nucleation and pining of the domain walls, play
critical roles in the $H_{c}$ mechanism. The nucleation field77 (magnetic
field at which the atomic spin ceases to align along the magnetic easy axis)
lowers the measured $H_{c}$, while the pinning does the opposite.
Additionally, cooling rates had significant effects on material grain size,
though grain size was not typically measured.

Several dozen materials that had grain size recorded were noted in the
database. Grain size is known to be heavily predictive of a material’s $H_{c}$
78. However, correlating the grain measurements from different sources made it
apparent that a single number cannot completely represent the structure of
grains and that additional inaccuracies may be possible depending on the
measurement techniques employed.

In order to better train ML networks, an additional supplementary database was
generated with a computational study of micromagnetism. In our study, we
include $H_{ms}$, $H_{ex}$, $H$, $H_{dem}$, and $H_{aniso}$ in the $H_{eff}$.
We consider both the cuboid and the uniaxial approximation for all materials
in the simulations and supply $A_{ex}$, $K_{u}$, and $M_{s}$ as inputs for
exchange, anisotropy fields, and saturation magnetization. The minimums and
maximums of the $A_{ex}$, $K_{u}$, and $M_{s}$ coordinates of the $300$
experimental materials were taken as the bounds of the cuboid from which
additional materials for micromagnetic prediction were uniformly sampled.
Following this, we delineate two measures of $H_{c}$. The first is the
experimentally measured coercivity, $H_{c}$(exp), which only the experimental
materials possess. The second is the computationally determined $H_{c}$,
calculated for hypothetical materials.

For the purpose of the micromagnetic numerical methodology, simple structures
were chosen for the magnetic samples, as inducing additional grain structure
imposes a scaling on the $H_{c}$, which can be replicated directly with ML.
The magnetic mesh was chosen to be $32\times 32\times 32$ at
$1~{}\,\mathrm{nm}$ scale. The volumetric quantities, such as magnetization
and magnetic field, are approximated to the center of the cell, while the
coupling parameters, like exchange stiffness constants, are located on the
faces of cells. Figure 1 shows a $\mathrm{mumax}^{3}$ -generated model
micromagnetic structure in which the colors within each microstructure
represent the different spin orientations. As experimental grains varied
significantly in scale, shape, and internal structure, this was a necessary
simplification in material treatment.

###  II.3 Computational Results

In the micromagnetic simulation, we assign the spins in randomly distributed
gains as shown in Fig. 1. The top figure (top panel) shows the randomized
sizes of grains projected in the 2D plane. In general, the size of grain
varies from 1 few nanometers to 100 nanometers, depending on the experimental
situation, e.g., temperature, defects, and chemical dopants. The spin
alignments and the inter-grain exchange-stiffness constant in each grain also
vary. During the relaxation, the spins reorient due to the applied magnetic
field, and Fig. 1 shows a snapshot (bottom panel) in the intermediate state.
The circular nature of the spin texture indicates that a closed magnetic
current has been established inside. At the critical field, a perfect spin
reversal occurs in magnetic materials.

First, we compute the $H_{c}$ for experimentally known systems, and we list
the comparison of the calculated and experimental values for selected
materials in 1. Theoretically, the upper limit for $H_{c}$ is anisotropy field
is $H_{a}=\dfrac{2K_{u}}{M_{s}}$; however, the true experimental value is an
order of magnitude smaller due to Brown’s paradox 7, largely due to the
uncertainty in the coercivity mechanism in permanent magnets. Experimentally
measured $H_{c}$ fits well with the expression $H_{c}=cH_{a}-N_{eff}4\pi
M_{s}$79, 80, where (i) the first term represents field required to nucleate a
reverse domain which reduces the $H_{a}$ and (ii) the second term is
demagnetization field and $N_{eff}$ is demagnetization factor. They reported
the renormalization factor $c\sim 0.25(0.37)$ and $N_{eff}\sim 0.26(1)$ for
Nd2Fe14B(sintered).

Table 1: $H_{c}$ in T obtained by solving the LLG equation with $\mathrm{mumax}^{3}$ with input parameters: $M_{s}$ in A/m, $A$ in pJ/m, and $K_{u}$ in MJ/m3 and its comparison with experiment in different rare-earth permanent magnets. For realistic comparisons, the $\mathrm{mumax}^{3}$ values are scaled by a factor of $c=0.25$ in column $cH_{c}$. $H_{c}$(diff) refers difference between $cH_{c}$ and $H_{c}$(exp). Material | $M_{s}$ | $A_{ex}$ | $K_{u}$ |  $H_{c}$(exp) |  $H_{c}$($\mathrm{mumax}^{3}$ ) | $cH_{c}$ |  $H_{c}$(diff)
---|---|---|---|---|---|---|---
La2Fe14B | 1098169.11 | 7.41 | 1.4 | 0.60 | 2.11 | 0.53 | 0.03
Ce2Fe14B | 931056.42 | 6.00 | 1.70 | 0.78 | 3.14 | 0.79 | 0.25
Pr2Fe14B | 1241408.56 | 7.94 | 4.66 | 2.25 | 6.46 | 1.62 | 0.08
Nd2Fe14B | 1273239.54 | 8.23 | 4.65 | 2.19 | 6.25 | 1.56 | 0.32
Gd2Fe14B | 708239.50 | 9.33 | 0.85 | 0.23 | 2.40 | 0.60 | 0.37
Tb2Fe14B | 557042.30 | 8.78 | 6.13 | 2.93 | 21.62 | 5.41 | 2.45
Dy2Fe14B | 565000.05 | 9.42 | 4.24 | 1.70 | 14.79 | 3.70 | 1.99
Ho2Fe14B | 644577.52 | 8.14 | 2.42 | 2.25 | 7.26 | 1.81 | 1.07
Lu2Fe14B | 931056.42 | 7.66 | 1.21 | 0.78 | 2.30 | 0.58 | 0.28
Y2Fe14B | 1122042.35 | 8.01 | 1.46 | 0.78 | 2.16 | 0.54 | 0.35
Th2Fe14B | 1122042.35 | 6.77 | 1.46 | 0.78 | 2.10 | 0.53 | 0.36
La2Co14B | 792712.73 | 1.50 | 1.19 | 0.34 | 3.06 | 0.77 | 0.42
Pr2Co14B | 1040205.38 | 1.51 | 5.20 | 2.50 | 9.43 | 2.35 | 0.85
Nd2Co14B | 1076367.74 | 1.51 | 2.42 | 3.69 | 4.22 | 1.05 | -2.64
Gd2Co14B | 229678.24 | 1.52 | 1.03 | 0.30 | 9.48 | 2.37 | 2.07
Y2Co14B | 847717.44 | 1.52 | 1.19 | 0.34 | 2.84 | 0.71 | 0.37
SmCo5 | 860000.00 | 1.20 | 1.72 | 7.50 | 38.80 | 9.70 | 2.20
YCo5 | 779400.38 | 7.11 | 5.50 | 3.90 | 13.33 | 3.33 | -0.57
LaCo5 | 712013.69 | 6.47 | 6.30 | 5.25 | 16.89 | 4.22 | -1.03
CeCo5 | 597033.47 | 5.13 | 6.40 | 5.70 | 20.66 | 5.17 | -0.53
PrCo5 | 939005.17 | 6.96 | 8.10 | 5.32 | 16.08 | 4.021 | -1.30
NdCo5 | 933607.03 | 7.09 | 0.24 | 0.15 | 0.47 | 0.12 | -0.03

The $\mathrm{mumax}^{3}$ calculated values differ from the experiment by
potentially larger factors of up to $\sim 5$. Interestingly,
$\mathrm{mumax}^{3}$ values are very similar to the experimental $H_{a}$.
Moreover, for some materials, we used the estimated values for $M_{s}$,
$A_{ex}$, and $K_{u}$ using the empirical relations as discussed in section
II.2, which may result in partial errors. $H_{c}$ depends non-linearly on
grain size and domain wall width or particle size in soft magnetic
materials81, 82, 83, 84, 85. In neo-magnet Ramesh et al. report that $H_{c}$
is negatively correlated with the grain size86 resulting from higher grain
surface area hosting more defects, which are not included in our
$\mathrm{mumax}^{3}$ calculations for defect-free materials. Besides, the
grain boundary is reported to affect the demagnetization factor87, which
further reduces the $H_{c}$.

The experimental features for 210 materials are known, while the remaining
have some uncertainties. The latter subset shows larger discrepancies in the
$\mathrm{mumax}^{3}$ $H_{c}$ results, which is worth investigating
experimentally and theoretically. We show a comparison (only for selected key
magnetic materials) between the $\mathrm{mumax}^{3}$ and experimental in Table
1. As all the $\mathrm{mumax}^{3}$ $H_{c}$ is overestimated, the only
meaningful comparison is obtained with the scaled coercivity $cH_{c}$, where
$c=0.25$ fits well with the experiment for 200 materials. The $cH_{c}$ is
generally good match for $2:14:1$ compositions, except for Co based Nd2Co14B
(-2.64) and Gd2Co14B (2.07). However, we note that the scaling factor may
differ for different compositions. Overall, the $H_{c}$ dependency on
independent features is similar in theory and experiment with some exceptions,
e.g., La2Fe14B. For $1:5$ compositions, we find a similar $H_{c}$ trend
between theory and experiment with independent features.

##  III Machine Learning

###  III.1 Classical machine learning algorithm

ML is a variety of advanced statistical techniques to create a correspondence
between a space of independent variables, $X$, to a space of dependent
variables, $Y$, by taking a function representing a sequence of observations
on a limited subset of the data, $X_{o}\subset X$ with $f_{o}:X_{o}\rightarrow
Y$ to construct a more general function $f:X\rightarrow Y$ where f represents
$f_{o}$. This ground truth list of observations may also be represented as a
list of tuples, $\\{x_{i}\rightarrow y_{i}\\}_{i}\subset X_{o}\times Y$. The
method by which $f$ is constructed from $f_{o}$ is referred to as ML, and we
can write it symbolically as $f=ML(f_{o})=ML(\\{x_{i}\rightarrow
y_{i}\\}_{i})$ where we potentially may subscript $ML$ to indicate the
algorithm or hyperparameters in use.

The goal of $ML$ is to represent the structure underlying $f_{o}$ as $f$. This
is usually quantified by splitting the observations into the training set
$f_{o}^{\prime}$ and testing set $f_{o}^{*}$ so that $f_{o}^{\prime}\cap
f_{o}^{*}=\emptyset$, then the function is constructed relative to a norm,
termed a loss function, so that $||f_{o}^{\prime}-ML(f_{o}^{\prime})||$ is
minimized in some appropriately defined subspace of $X\rightarrow Y$ to avoid
overfitting, and then the ability of $f=ML(f_{o}^{\prime})$ is measured by
evaluating $||f_{o}^{*}-f||$ using the same class of norms. In this paper, we
choose NORM HERE(loss function).

We used the scikit-learn library for the Python programming language for
predictive data analysis 88. The dataset (both experimental and mumax) is
split into the 70:30 train test data set for the model training. We note that
the performances do not change drastically by changing the split ratio to
80:20. The classical ML models used in this paper are the linear models:
1\. Linear regression
2\. Lasso regularization
3\. Ridge regularization
and the non-linear regression models:
1\. Decision tree (DT) regression
2\. Random forest(RF)regression
3\. Gradient boost (GB) regression
4\. XGBoost(XGB) regression

Typically, these regression ML algorithms use the mean squared loss (MSE)
internally to measure the quality of each model against the experimental
$H_{c}$. In ML training with cross-validation, the training dataset is further
divided into two parts - the train and the validation datasets. Most ML models
have arbitrary hyperparameters, which are not modified directly during
training. The validation dataset allows a meta-training of the ML models by
random or grid search on the training data and evaluation on the validation
set without contaminating the testing set by predicting ML hyperparameter
choice on test set evaluation. The most coming cross-validation method is
k-fold validation, in which the dataset is randomly split into $k-fold$ among
which one data set is chosen as validation dataset.

###  III.2 Deep neural network

We use ANN for $H_{c}$ prediction, which is suitable for handling the high
complexity in the problem.

Figure 2: Schematic diagram showing artificial deep neural network (only a
portion used in the calculations) with 7 neurons in input layers, 4 neurons in
each two deep (hidden) layers, and a neuron for regression in the output
layer.

Figure 2 shows a schematic diagram (a portion used in the calculations) for
ANN consisting of input layers (7 neurons), two hidden layers, each containing
4 neurons, and the output layer with one neuron. A non-linear activation
function ‘leaky-relu’ ($f(z)=max(0,z)+0.01*min(0,z)$) is used between the
input and hidden layers to counter typical convergence problems found in small
ANN’s. We tested two gradient descent algorithm-based benchmark optimizers (1)
RMSProp89 and (ii) Adam90, as implemented in tensorflow91 in the Keras
API92.The latter produces better results, and only the predictions made with
this approach are included in the paper.

####  III.2.1 Performance comparison

For the regression model, given an ML fit, we use two common statistical
measures of error applied to the $H_{c}$ predictions: (i) Mean Square Error
(MSE):

$MSE=\frac{1}{N}\sum_{i}^{N}(y_{i}-y_{{pred}_{i}})^{2}$ (8)

(ii) R-squared ($R^{2}$):

$R^{2}=1-\frac{MSE}{V(y)},$ (9)

where $V(y)$ is variance written as

$V(y)=\frac{1}{N}\sum_{i}^{N}(y_{i}-\langle y\rangle)^{2},$ (10)

where $\langle y\rangle$ is the mean of predicted values. Equivalently, it can
be written as $R^{2}=1-RSS/TSS$, where RSS is Residual Sum of Square
$RSS=\sum_{i}^{N}(y_{i}-y_{{pred}_{i}})^{2}$ and $TSS$ is Total Sum of Square
$TSS=\sum_{i}^{N}(y_{i}-\langle y\rangle)^{2}$.

The MSE measure of error is usually incorporated into the loss function - a
norm used to measure the performance of the ML algorithm, written as
$||\cdot||$. However, in addition to the MSE, which is defined on the data
points, typically, first or second-order norms of the target function
parameters are also considered to penalize overfitting for parametrized
machine learning.

###  III.3 Results and discusssion

####  III.3.1 Machine learning on experimental data

Figure 3: The distribution of $H_{c}$ as a function of $M_{s}$, $A_{ex}$, and $K_{u}$ in real magnetic materials. All the quantities are given in normalized scale with mim-max scalar. The color map shows the magnitude of $H_{c}$ within the triangle. $H_{c}$ has the largest value (yellow color projection) near the $K_{u}$ corner of the triangle, indicating that it is directly related to $K_{u}$. Similarly, it is related inversely with $M_{s}$, and very little with $A_{ex}$. Table 2: Comparison of $R^{2}$ and MSE metrics in different ML models in test data sets (experimental materials). The non-linear regression models show better values of $R^{2}$ and MSE over linear models. Model | $R^{2}$ | MSE
---|---|---
Linear regression | 0.62 | 0.64
Lasso | 0.38 | 1.07
Ridge | 0.63 | 0.64
Lasso-CV | 0.61 | 0.68
Ridge-CV | 0.63 | 0.66
Elasticnet | 0.62 | 0.66
DT regressor | 0.89 | 0.19
DT pruned | 0.84 | 0.27
RF regressor | 0.87 | 0.23
Gradient Boosting (GB) regressor | 0.80 | 0.34
Tuned GB regressor | 0.87 | 0.22
XGB regressor | 0.87 | 0.23
Tuned XGB regressor | 0.89 | 0.18
ANN (Adam) | 0.64 | 0.62
LGBM | 0.70 | 0.60
finetuned ANN | 0.85 | 0.25

In this section, we explore the experimental data and the relation between
target and independent variables. We visualize $H_{c}$ as a function of
$K_{u}$, $M_{s}$, and $A_{ex}$, which is shown in Fig. 3. As the actual values
of variables differ by several orders of magnitudes, for clarity, they are
scaled using the sci-kit minmaxscalar, in which the scaled feature variable
$x_{scaled}$ in the range (0,1) is obtained using the following relation

$x_{scaled}=\dfrac{(x-x_{min})}{(x_{max}-x_{min})},$ (11)

where $x_{min}$ and $x_{max}$ are minimum and maximum values of feature
variable $x$. Figure already shows how $H_{c}$ is dependent on the independent
features. In particular, $H_{c}$ varies directly with $K_{u}$, inversely with
$M_{s}$, and very little with $A_{ex}$.

Additionally, for the ML results, all independent and dependent variables were
scaled by the logarithm. The $H_{c}$ distribution is right skewed because
there are a few high-performing magnetic materials. This was to adjust for
gradient-based algorithms, such as ANN, where additive relationships are
easier to model than multiplicative ones. As a result, all measures of model
performance are given in terms of the $\ln$ of the $H_{c}$ rather than using
it directly. As a result of this change, performance increases of the order
$0.01$ were observed.

Table 2 shows the performances of various ML model training. The linear
models, including the regularization yield, show poor performance as
$R^{2}<<1$ and MSE is much larger. This is expected in a linear model, where
the relation between the dependent and independent variables is complex. The
non-linear model DT and RF performs better with $R^{2}$ values of 0.89 and
0.87. The more advanced X gradient boost regressor93 shows $R^{2}$ of 0.87
similar performance. Although we find slightly overfitting of the training
dataset, the $R^{2}$ score falls within the acceptable range.

Figure 4: ML predicted ($H_{c}$ predicted) and experimental coercivity
($H_{c}$ experiment) of magnetic materials obtained with tuned XGBtop and
ANNbottom. Only selected key materials are labeled in the figure. All these
non-linear regressors yield similar results, although ANN seems fair, which is
expected for smaller data sizes.

In ANN model, we find similar $R^{2}$ (see Table 2 is as in other classical
models. We tested both RMSProp and Adam optimizers, the latter shows better
prediction over former, and it has $R^{2}$ of $\sim 0.80$.  We used the ANN
architecture with (dense layers=64, activation function=leaky_relu, dense
layers=32, activation function=leaky_relu, dense layers=16, activation
function=leaky_relu, and output layer=1, and learning rate=0.007, epochs=1000,
batch size=24) showing less predictive than RF or RF regressors. But ANN with
Adam optimizer yields a similar performance to RF. Given the size of
experimental data, ANN underperforms, which is attributed to smaller data
size.
Feature importances
The model performance can be further validated from the $H_{c}$ dependency on
the independent features. We compute the importance of RF features by
utilizing the model-agnostic interpretive features of scikit-learn on the RF
and XGB regression models 94. The feature importance is computed in two ways:
(i) Gini importance or the mean decrease impurity95 and (ii) mean decrease
accuracy, while later is not implemented in scikit-learn. We used method (i)
Gini index, which is a measure of purity and impurity in a node and is defined
as

${\rm{\it Gini~{}index}}=1-\sum_{i}p_{i}^{2}$ (12)

where $p_{i}$ is the probability of class $i$ in the data. Alternatively, the
maximal information about the classification/regression can be quantified by
the information gain as

${\rm{\it Information~{}gain}}=H(Y)-H(Y|X),$ (13)

where $H(Y)=\sum_{i}p_{i}log(p_{i})$ is the Shannon entropy of Y, X, and Y are
the independent and dependent variables, and $H(Y|X)$ is the entropy of $Y$
for given $X$.

Figure 5: Histogram of feature importances contributing to $H_{c}$ in the XGB
model. The $K_{u}$ is a leading contributor, followed by $M_{s}$, and $A_{ex}$
to the $H_{c}$.

The nonlinear models, DT, RF, and the tuned XGB algorithms, yield the best
performance. Figure 5 depicts the relative importance of the independent
parameters to $H_{c}$ as computed by the RF model. ML accurately predicts
$K_{u}$ as a leading contributing feature to $H_{c}$. More surprisingly, the
effect of $M_{s}$ is a factor of $\sim 4$ larger than that of $K_{u}$. As the
logarithm was applied to all inputs and outputs, the upper bound on $H_{c}$
reads as $\ln{H_{c}}=\ln{2}+\ln{K_{u}}-\ln{M_{s}}$, giving equal weight to
$K_{u}$ and $M_{s}$. However, as shown in Fig. 3, the uneven distribution of
materials along the input parameter axes combined with a constant noise level
may explain the unequal weighting. Finally, note that $A_{ex}$ has a
comparatively minimal effect on $H_{c}$.

###  III.4 Machine learning prediction from micromagnetic data

We trained different sizes of data for convergence from 200 to 12000 by
computing the error metric Mean Absolute Error (MAE):
$\text{MAE}=\sum_{i}\dfrac{|y_{i}-y_{pred_{i}}|}{N}$.

Figure 6: Mean absolute error MAE convergence for $H_{c}$ with respect to
training data set size in XGB model. The $x$-axis is in the scale of $10^{4}$.

Figure 6 shows the convergence of MAE (obtained with XGB model) vs. train data
set size and indicates that the model performs well for data points larger
than 6000. The distribution for $8770$ data set is right skewed as shown in
Fig.7 with peak around 0.2-0.4 T. It is similar to the experimental materials
(not shown here), which is realistic distribution as most of the materials
have values around the peak and a few materials, such as $1:5$ compositions
have larger values $2-6$ T. For better ML training, logarithmic transformation
is appropriate for skewed data, which is employed in our model fits.

Figure 7: Distribution of coercivity $H_{c}$ (in T) in training 8770 data
points. The $y$-axis is in the scale of $10^{3}$. The data points are right
skewed.

In order to explore the relation between $H_{c}$ and independent variables, we
trained $8770$ $\mathrm{mumax}^{3}$ -generated data. We use the standard
Pearson’s correlation coefficient to find the correlation between the
variables defined as

$\displaystyle r=\dfrac{\sum(X_{1}-\langle X_{1}\rangle)(X_{2}-\langle
X_{2}\rangle)}{\sqrt{\sum(X_{1}-\langle X_{1}\rangle)^{2}\sum(X_{2}-\langle
X_{2}\rangle)^{2}}},$ (14)

where $X_{i}$ and $\langle X\rangle$ are variable $i$ and its average value.
The positive (negative) value of $r$ indicates a positive (negative)
correlation between the two variables. The larger the value of $r$ higher the
correlation is. We show the correlation among the variables using a heatmap in
Fig.8

Figure 8: Heat map showing the correlation between the $H_{c}$, $A_{ex}$, $K_{u}$, and $H_{c}$. The color palette shows the correlation coefficient between the variables. $H_{c}$ shows the highest correction with $K_{u}$ and lowest correlation with $A_{ex}$. The correlation of $H_{c}$ is negative with both features, while it is opposite with $M_{s}$. $M_{s}$ shows slightly negative correlation with $K_{u}$ and $A_{ex}$. $K_{u}$ shows slightly positive correlation with $A_{ex}$. Table 3: Mumax data on hypothetical materials: Comparison of metrics $R^{2}$ and MSE in the test data set $H_{c}$ predictions for different ML models. The DT produces $R^{2}$ of 0.98, and the tuned XGB regressor also yields a similar result. The fine-tuned model hyperparameters are given in Appendix A. Model | $R^{2}$ | $MSE$
---|---|---
Linear regression | 0.95 | 0.22
Lasso | 0.54 | 2.13
Ridge | 0.95 | 0.22
Lasso-cv | 0.95 | 0.25
Ridge-cv | 0.95 | 0.22
Elasticnet | 0.94 | 0.26
DT regressor | 0.97 | 0.13
DT pruned | 0.97 | 0.15
RF regressor | 0.98 | 0.08
GB regressor | 0.98 | 0.09
Tuned GB regressor | 0.98 | 0.08
XGB regressor | 0.98 | 0.98
Tuned XGB regressor | 0.98 | 0.08
ANN | 0.98 | 0.09

Quite interestingly, the $r$ between $H_{c}$ and $K_{u}$ is the largest
(0.66), followed by $M_{s}$ (-0.62), and $A_{ex}$ (0.2). This suggests that
$H_{c}$ is heavily dependent on $K_{u}$. Although we find the changes in the
magnitude of $r$ with the randomly chosen distribution of parameters, the sign
of $r$ remains the same. For instance, the negative correlation between
$H_{c}$ and $M_{s}$ is consistent with the theoretical
$H_{c}\propto\dfrac{1}{M_{s}}$. Likewise, the $r$ between $H_{c}$ and
$K_{u}/A_{ex}$ confirms that $H_{c}\propto K_{u}$ and $H_{c}\propto A_{ex}$.
These results are consistent with the trend shown by real materials as
demonstrated in Fig. 3.

Next, we discuss the model performances in $\mathrm{mumax}^{3}$ data as given
in Table 3. Overall, both linear and non-linear models exhibit good
performance as $R^{2}$ score is above $0.9$. This is expected in a linear
regression model where multivariate statistics analysis demonstrates very
little multicollinearity in randomly generated independent variables. The
multicollinearity can be measured with the Variance Inflation Factor (VIF).
The VIF is computed as $VIF_{i}=\dfrac{1}{1-R_{i}^{2}}$, where $i$ labels
$ith$ independent variable and $R_{i}^{2}$ is the $R^{2}$ of $ith$ variable on
remaining ones, measures the multicollinearity. The computed VIF factors are
$1.53$ for $M_{s}$, $4.36$ for $A_{ex}$, and $1.53$ for $K_{u}$ and are well
within the acceptable limit.

In particular, more advanced decision tree regressors improve the
performances. We picked tuned XGB as a benchmark model with hyperparameters
given in Appendix A for $H_{c}$ prediction in experimental data. The deviance
in MSE of test and train data sets is shown in Appendix A Fig. 11. The more
advanced ANN also exhibits a similar performance as other non-linear
regressors. We use the relu activation function ($f(z)=max(0,z)$) here, and
ANN is as follows: (dense layers=64, activation function=relu, dense
layers=32, activation function=relu, dense layers=16, activation
function=relu, and output layer=1, and learning rate=0.007, epochs=1000, batch
size=56). To get a deeper insight into ML predictability, we compare the
experimental, the scaled-$\mathrm{mumax}^{3}$ , and the ML-predicted $H_{c}$
for the real materials. The $H_{c}$ data are given in Table 4. In general, ML-
predicted values are smaller than the $\mathrm{mumax}^{3}$ -computed scaled
values. For $2:14:1$ rare-earth magnets, ML improves the results. For $1:5$
rare-earth magnets, it slightly underestimates the experiment. This is
reasonable since we use the same scaling factor for all magnetic materials,
which may depend on the materials.

We computed the Root Mean Square Error (RMSE) as

$\mathrm{RMSE}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(y_{i}-\hat{y}_{i})^{2}},$ (15)

where $y$ and $\hat{y}$ refers to experimental and ML/$\mathrm{mumax}^{3}$
predicted values. Remarkably, the RMSE between experimental and ML predicted
values of $\sim 1$ is smaller than the RMSE between experimental and the
scaled-$\mathrm{mumax}^{3}$ values of $\sim 1.13$. indicating that ML helps to
correct the $\mathrm{mumax}^{3}$ -predictions. It also shows how the $H_{c}$
is correlated with other features. Moreover, ML is an order of magnitude
computationally faster than $\mathrm{mumax}^{3}$ , and we can use the trained
model for predicting $H_{c}$ new materials without any $\mathrm{mumax}^{3}$
calculations.

Table 4: $H_{c}$ comparisons among experiment, ML prediction, and scaled $\mathrm{mumax}^{3}$ for different magnetic materials. ML (tuned XGB and ANN) prediction is obtained by training the hypothetical materials using the random independent variables $M_{s}$, $A_{ex}$, and $K_{u}$, and the $\mathrm{mumax}^{3}$ computed $H_{c}$. RMSE and MAE quantify the error between experiment ML and the $\mathrm{mumax}^{3}$ scaled $cH_{c}$. Material |  $H_{c}$ (exp) |  $H_{c}$ (XGB) |  $H_{c}$ (ANN) | $cH_{c}$
---|---|---|---|---
La2Fe14B | 0.50 | 0.42 | 0.45 | 0.53
Ce2Fe14B | 0.54 | 0.66 | 0.72 | 0.79
Pr2Fe14B | 1.54 | 1.29 | 1.30 | 1.62
Nd2Fe14B | 1.24 | 1.23 | 1.26 | 1.56
Gd2Fe14B | 0.23 | 0.43 | 0.51 | 0.60
Tb2Fe14B | 2.93 | 3.28 | 4.18 | 5.41
Dy2Fe14B | 1.70 | 2.22 | 3.10 | 3.70
Ho2Fe14B | 0.75 | 1.21 | 1.27 | 1.82
Lu2Fe14B | 0.29 | 0.46 | 0.49 | 0.58
Y2Fe14B | 0.19 | 0.47 | 0.46 | 0.54
Th2Fe14B | 0.17 | 0.46 | 0.46 | 0.53
La2Co14B | 0.34 | 0.55 | 0.63 | 0.77
Pr2Co14B | 2.5 | 1.82 | 1.87 | 2.36
Nd2Co14B | 3.69 | 0.81 | 0.83 | 1.06
Gd2Co14B | 0.30 | 0.54 | 1.24 | 2.37
Y2Co14B | 0.34 | 0.52 | 0.57 | 0.71
NdLaCeF14B | 0.65 | 1.02 | 0.91 | 1.39
LaCeYFe14B | 0.41 | 0.80 | 0.79 | 1.39
NdPr2Fe14B | 0.8 | 1.13 | 1.13 | 1.39
Sm2Co17 | 1.25 | 1.26 | 1.28 | 1.62
Sm2Fe17N3 | 2.3 | 2.46 | 2.47 | 3.19
SmCo5 | 7.50 | 5.58 | 5.86 | 9.70
YCo5 | 3.9 | 2.63 | 2.68 | 3.33
LaCo5 | 5.25 | 3.34 | 3.40 | 4.22
CeCo5 | 5.7 | 3.37 | 4.02 | 4.13
PrCo5 | 5.32 | 3.28 | 3.14 | 4.02
NdCo5 | 0.15 | 0.11 | 0.11 | 0.12
RMSE |  | 1.00 | 1.01 | 1.13
MAE |  | 0.64 | 0.70 | 0.84

##  IV Machine learning application: DFT + ML

The computational modeling consists of two-fold calculations. DFT calculations
for Ce-doped Nd2Fe14B and the ML $H_{c}$ prediction, which is made by using
the trained ML on micromagnetic generated databases in DFT computed input
parameters.

###  IV.1 Density functional methods and crystal structure

Figure 9: Crystal structure of Nd2Fe14B occupied by Nd at two non-equivalent
sites ($4f~{}{\rm and}~{}4g$), Fe at six inequivalent sites
($16k,~{}16k,~{}8j,~{}8j,~{}4c,~{}{\rm and}~{}4e$ (labeled by Wyckoff
positions), and one B at $4f$site.

We used the Vienna Simulation Package (VASP) 96, 97 with the Projector
Augmented Wave (PAW) formalism in the Generalized Gradient Approximation (GGA)
of Perdew-Burke-Ernzerhof (PBE) semi-local exchange-correlation functionals98,
99 including onsite electron-electron correlation Dubarev’s Hubbard100
$U_{eff}=U-J$ for the $4f$ states of Nd, and spin-orbit interaction. We used
the kinetic energy cut-off of 520 eV for the plane wave expansion and $4\times
4\times 2$ k-mesh for the Brillouin zone integration. We used experimental
structure in the calculations to avoid the overestimate of bond lengths and
lattice constants in the PBE.

The crystal structure of Nd2Fe14B is tetragonal with 68 atoms (4 formula
units) with space group $P42/mnm$ (136) and lattice constants $a=b=8.80$ and
$c=12.20$ Å, $\alpha=\beta=\gamma=90^{\circ}$ at
$\mathrm{T}=285\,~{}\mathrm{K}$20, 101. For full Ce-doped, we use the
experimental parameters from Ref. [102] The crystal structure of $2:14:1$ neo-
magnet is shown in Fig. 9, which consists of rare-earth Nd in the 2
inequivalent sites $4f$ and $4g$, transition element Fe in the 6 inequivalent
sites at $16k$, $16k$, $8j$, $8j$, $4e$, and $4c$, and finally B at the $4f$
sites. Fully self-consistent spin-orbit calculations, i.e., PBE \+ $U$ \+
Spin-Orbit Coupling (SOC), were performed to obtain magnetic anisotropy energy
and spin and orbital magnetic moments.

###  IV.2 Magnetic properties

Here we discuss the PBE +$U$ +SOC results for Nd2Fe14B and ML for $H_{c}$
prediction. Hubbard $U$ correction is necessary for highly correlated $3d$ and
$4f$ elements103, 104, 105, 106 to accurately predict $K_{u}$. We performed
calculations for various values of $U$ from 4-7 eV for Nd($4f$) orbitals.
Table 5 shows the spin and orbital magnetic moments of Nd, Fe, and B at
different crystallographic sites for different values of $U_{eff}$. Non-
magnetic B atom has negligible magnetic moments. Nd exhibits Nd3+ state with
spin magnetic moment ($\mu_{s}$) of $\sim-3.3$ and strongly quenched orbital
moment ($\mu_{l}$) of $\sim 1.5$ $\mu_{B}$ due to crystalline electric field.
The spin and orbital moments have the opposite sign, consistent with Hund’s
rule for less than half-filled $4f$-shell. The net magnetic moment is very
robust and does not vary significantly with $U$.

Table 5: Spin and orbital magnetic moments ($\mu_{s}$ and $\mu_{l}$) of individual atoms in different Wyckoff positions and total magnetic spin and orbital magnetic moment ($\mu_{st}$ and $\mu_{lt}$) per unit cell (in $\mu_{B}$) in ${\rm Nd_{2}Fe_{14}B}$ computed with PBE $+U+$ SOC method for various values of $U$ for Nd($4f$) states. The Fe moments at two $16k$ and $8j$ sites are given as a pair, which differ slightly. The $\mu_{l}$ of pair at $8j$-sites do not differ and only one value is given representing both atoms. atom$\rightarrow$ | Nd($4f$) | Nd($4g$) | Fe($16k$) | Fe($8j$) | Fe($4e$)) | Fe($4c$) | B($4g$) | total |
---|---|---|---|---|---|---|---|---|---
$U\downarrow$ | $\mu_{s}$ | $\mu_{l}$ | $\mu_{s}$ | $\mu_{l}$ | $\mu_{s}$ | $\mu_{l}$ | $\mu_{s}$ | $\mu_{l}$ | $\mu_{s}$ | $\mu_{l}$ | $\mu_{s}$ | $\mu_{l}$ | $\mu_{s}$ | $\mu_{l}$ | $\mu_{st}$ | $\mu_{lt}$
4 eV | -3.26 | 1.52 | -3.28 | 1.53 | 2.27,2.36 | 0.043,0.049 | 2.29,2.71 | 0.044 | 2.03 | 0.045 | 2.48 | 0.054 | -0.17 | 0.00 | 105.24 | 14.75
5 eV | -3.26 | 1.52 | -3.28 | 1.53 | 2.27,2.36 | 0.042,0.048 | 2.29,2.70 | 0.044 | 2.03 | 0.045 | 2.47 | 0.052 | -0.17 | 0.00 | 105.25 | 14.75
7 eV | -3.27 | 1.52 | -3.28 | 1.53 | 2.27,2.36 | 0.042,0.048 | 2.29,2.71 | 0.042 | 2.03 | 0.044 | 2.48 | 0.049 | -0.17 | 0.00 | 105.25 | 14.68

Table 6 lists the $K_{u}$, $M_{s}$, and $T_{\rm C}$ and their comparison with
experimental values. $K_{u}$ is positively correlated with $U$. The
experimental $K_{u}$ is of 4.5 MJ/m3107 at 300 K, which corresponds to the
computed value for $U\sim 4$ eV. $T_{\rm C}$ is computed with static Green’s
function (GF) as implemented in ASA in the Local Density Approximation (LDA)
108 for the exchange-correlation functional in Linearized Muffin-Tin Orbital
(LMTO) program109, 110. The pair exchange interaction $J_{RR^{{}^{\prime}}}$
between magnetic ions between $R$ and $R^{{}^{\prime}}$ site is computed from
Lichtenstein formula111 form which $T_{\rm C}$ is estimated. The calculated
Weiss mean-field theory112 value 718.1 K is larger than the experiment, which
is expected in the Mean-Field Approximation (MFA). According to a spin-waves
theory by Tyablikov113, the Random-Phase Approximation (RPA) corrects the
value, bringing it down to 539.1 K. The experimental value lies within the MFA
and RPA limits.

Table 6: $K_{u}$ in MJ/m3, $M_{s}$ in $\mu_{B}$/f.u. with PBE $+U$\+ SOC, and $T_{\rm C}$ in Kelvin (K) with Green’s function-ASA (LDA) calculations and comparison with experiment. $U$ | $K_{u}$ | $M_{s}$ |  $M_{s}$(Expt 20) | $T_{\rm C}$ |  $T_{\rm C}$(Expt20)
---|---|---|---|---|---
4 eV | 6.16 | 29.98 | 37.7111at low temperature, (4), 32.5222at high temperature (295 K) | 718.1333Mean Field Approximation (MFA), 539.1444Random Phase Approximation (RPA) | 585
5 eV | 8.56 | 30.00 | $\cdots$ |  | $\cdots$
7 eV | 11.58 | 30.01 | $\cdots$ |  | $\cdots$

Next, we discuss the site-resolved spin-orbit anisotropy energy which is
computed as $E_{aniso}=E_{100}-E_{001}$104, where
$E_{100}~{}\rm{and}~{}E_{001}$ are PBE$+U+$ SOC computed atomic site energies
for spin-quantization along [100] and [001] directions. Generally,
$4f$-elements contribute to $K_{u}$, and $3d$-elements contribute to the
magnetic moments in rare-earth-based magnets. The spin moments of rare-earth
ion is anti-parallel to the spin moment of $3d$ ion, which is also the case
for neo-magnet. The site contribution of Nd-element to MAE is given in Table
7. In our PBE \+ $U$ \+ SOC calculations, we did not impose symmetry, which
means all the atoms are inequivalent due to the $P_{1}$ crystal symmetry. All
eight Nd atoms split into eight different sites. We find that the individual
on-site energy differs significantly for the atoms belonging to the same
crystallographic site. Interestingly, for at least one Nd (Nd1) at $4f$ has a
negative contribution to MAE as given in Table 7, indicating a planar
anisotropy. Theoretically, we can infer that Nd has a tendency to be planar at
the $4f$ site, differing from the $4g$ site, which is strictly uniaxial along
the crystalline $c$ direction. These results are consistent with experimental
results reported in Ref. 114.

Table 7: Site resolved spin-orbit anisotropy energy ($E_{aniso}$) in meV of Nd contribution to MAE. In PBE $+U+$SOC calculations the crystalline symmetry lowered to $P_{1}$ point group which results in all the atoms split, including the Nd atoms into eight sites. The negative for Nd at $4f$ site indicates it produces a planar contribution to MAE consistent with the experiment114. Atom | Wyckoff-position | $E_{aniso}$
---|---|---
Nd1 | ${4f}$ | 1.3780
Nd2 | ${4f}$ | -1.1914
Nd3 | ${4f}$ | 0.5336
Nd4 | ${4f}$ | 6.2204
Nd5 | ${4g}$ | 0.4865
Nd6 | ${4g}$ | 3.8064
Nd7 | ${4g}$ | 6.2094
Nd8 | ${4g}$ | 3.0622
Table 8: Total spin, orbital, and spin + orbital magnetic moments $\mu_{s}$, $\mu_{l}$, and $\mu_{s}+\mu_{l}$ in $\mu_{B}$/cell, and magnetic anisotropy constant $K_{u}$ in MJ/m3 of Ce2Fe14B and Nd2Fe14B. Properties | NdFe14B | Ce2Fe14B
---|---|---
$\mu_{s}$ | 105.24 | 120.73
$\mu_{l}$ | 14.75 | 4.55
$\mu_{s}+\mu_{l}$ | 119.99 | 125.28
$K_{u}$ | 6.16 | 1.21

Table 8 shows the computed values of the magnetic moments and magnetic
anisotropy in Ce-substituted neo-magnet at Nd-$4f$ sites as Ce-atom carries
small spin magnetic moment $\sim 1\mu_{B}$ as compared to Nd $\sim 3\mu_{B}$
which is negative with respect to Fe-moments. Moreover, the orbital moment
does not fully cancel the spin moment in Nd, while in Ce, the net spin +
orbital moment vanishes. Therefore, there is an increase in the net magnetic
moment with Ce. On the other hand, magnetic anisotropy is significantly
reduced because the Ce-site contribution is much smaller than Nd. The
calculated value of $K_{u}$ is slightly underestimated with the experiment
(see Table I). This is reasonable given the choice of $U$ value used for Ce
and $Nd$ $4f$-states (we used $U_{eff}=2$ eV for Ce and $4$ eV for Nd) in the
calculations.

Figure 10: Hysteresis curve for Nd1-xCexFe12B ($x=0,~{}0.5,~{}1.0)$)
calculated with the $\mathrm{mumax}^{3}$ program. Ce leads to the reduction in
the coercivity.

###  IV.3 ML coercivity prediction

We discuss the $\mathrm{mumax}^{3}$ $H_{c}$ for Ce-doped ${\rm
Nd_{2}Fe_{14}B}$ utilizing DFT computed $M_{s}$, $K_{u}$, and $A_{ex}$ from
Table 1. For a half Ce-doped structure, we failed to converge the self-
consistent calculations, and we used the average value of full Ce-doped and
pure material to estimate the input parameters for $H_{c}$. We note that the
experimentally measured values of $H_{c}$ range from 1-3.5 T depending on the
sample preparations for neo-magnet. From ML, we find $H_{c}$ values of 1.92
for Nd2Fe14B, 0.99 for CeNd2Fe14B, and 0.33 T for Ce2Fe14B. The reduction
trend of $H_{c}$ with Ce is similar to that found experimentally in Ref. 115.
Figure 10 shows the hysteresis loop for Ce-doped Nd2Fe12B. As discussed
earlier, mumax3 overestimates the value of $H_{c}$; however, the $H_{c}$ trend
is similar to experiment115 and ML predictions.

##  V Conclusion

Using the micromagnetic simulation, we study magnetization as a function of
applied magnetic fields for real and hypothetical magnetic materials. We
primarily focus on the $1:5$ and $2:14:1$ rare-earth-based permanent magnets,
such as neo-magnet (Nd2Fe14B) with randomly oriented grains. From the magnetic
hysteresis loop, we estimate $H_{c}$, which is typically overestimated by a
factor $\sim 5$ for experimentally known materials due to Brown’s paradox.
With ML, we explore the correlation between $H_{c}$ and other properties and
find that $H_{c}$ primarily depends on $K_{u}$ and is directly proportional to
it. It is inversely related to $M_{s}$ and very little with $A_{ex}$.
Interestingly, the other properties of the material under consideration show a
non-linear relationship with $H_{c}$. In $\mathrm{mumax}^{3}$ generated
hypothetical materials, $H_{c}$ roughly depends in the same proportion to
$M_{s}$ and $K_{u}$ but oppositely. From ML training of $\mathrm{mumax}^{3}$
data, we predict the $H_{c}$, which is found to match better with the
experiment. This, indeed, is the strength of ML, which learns the pattern of
data and their relations between the independent and target features, enabling
a more accurate prediction of the target variable. The $\mathrm{mumax}^{3}$
computed $H_{c}$ also shows the non-linear relations with independent
variables based on the model performance. Besides, it is computationally much
faster than $\mathrm{mumax}^{3}$ . We compute the intrinsic properties and
$T_{\rm C}$ of Nd2Fe14B from the DFT calculations. We find Nd site depicts
site-dependent magnetic anisotropy behavior, e.g., the $4f$ site prefers
planar, while the $4g$ tends to be uniaxial, consistent with the experimental
observation. It suggests the possibility of tuning rare-earth magnetic
properties by substituting non-critical elements at specific sites. For
instance, Ce-doping at $4f$ sites shows a reduction in $H_{c}$ as confirmed by
ML prediction. Moreover, our ML study, incorporating a large number of
experimentally verified rare-earth-based magnetic materials and the
$\mathrm{mumax}^{3}$ -simulated materials, establishes a robust computational
foundation for predicting novel magnetic materials and optimizing their
properties.

###### Acknowledgements.

This work is supported by the Critical Materials Innovation Hub, an Energy
Innovation Hub funded by the US Department of Energy, Office of Energy
Efficiency and Renewable Energy, Advanced Materials and Manufacturing
Technologies Office. The Ames National Laboratory is operated for the U.S.
Department of Energy by Iowa State University of Science and Technology under
Contract No. DE-AC02-07CH11358. GNN acknowledges support from the U.S.
Department of Energy, Office of Science, Office of Workforce Development for
Teachers and Scientists, Office of Science Graduate Student Research (SCGSR)
program. The SCGSR program is administered by the Oak Ridge Institute for
Science and Education for the DOE under contract number DE‐SC0014664.

##  Appendix A ML model and hyperparameter tuning

In this appendix we provide the information about the hyperparameters used in
fine tuning of ML models. For experimental data, we used the hyper parameters
as given in Table 9 for tuned XGB model training:

Table 9: Hyperparamters used for XGB model training of experimental magnetic
materials. n_estimators=500
---
min_weight_fraction_leaf=0
max_depth=5
learning_rate=0.006
Table 10: Fined tuned hyperparameters for different ML models used in training of $\mathrm{mumax}^{3}$ simulated data. Model | Hyperparameters
---|---
Tuned RF | n_estimators=1200, max_depth=5,
| random_state=1,
| min_samples_leaf=3, bootstrap=True,
| max_features=’auto’
Tuned XGB | n_estimators=1200, max_depth= 5,
| min_weight_fraction_leaf=0,
| learning_rate=0.007
Tuned XGB | n_estimators= 1200, max_depth= 5,
| min_child_weight=0, learning_rate=0.007,
| random_state=42

In $\mathrm{mumax}^{3}$ data training we use the following hyperparameters
given in Table 10.

Figure 11: Deviance as a function of boosting iterations for
$\mathrm{mumax}^{3}$ dataset with XGB model. Deviance is MSE between the
actual and predicted data set computed in each iteration. Both the train and
test data set are well converged after 600 boosting iterations.

The hyperparameter used in the fine-tuned GBR model training is shown in
Fig.11. The Deviance (MSE) is very well converged in both test and train data
set beyond 600 boosting iterations, confirming a very good model performance.

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