# Nuclear binding energy predictions using neural networks: Application of the
multilayer perceptron

Esra Yüksel<EMAIL_ADDRESS>Department of Physics, Faculty of Science
and Letters, Yildiz Technical University, Davutpasa Campus, 34220, Esenler,
Istanbul, Turkey  Derya Soydaner<EMAIL_ADDRESS>Department of
Statistics, Mimar Sinan Fine Arts University, Bomonti 34380, Istanbul, Turkey
Hüseyin Bahtiyar<EMAIL_ADDRESS>Department of Physics, Mimar
Sinan Fine Arts University, Bomonti 34380, Istanbul, Turkey

###### Abstract

In recent years, artificial neural networks and their applications for large
data sets have became a crucial part of scientific research. In this work, we
implement the Multilayer Perceptron (MLP), which is a class of feedforward
artificial neural network (ANN), to predict ground-state binding energies of
atomic nuclei. Two different MLP architectures with three and four hidden
layers are used to study their effects on the predictions. To train the MLP
architectures, two different inputs are used along with the latest atomic mass
table and changes in binding energy predictions are also analyzed in terms of
the changes in the input channel. It is seen that using appropriate MLP
architectures and putting more physical information in the input channels, MLP
can make fast and reliable predictions for binding energies of atomic nuclei,
which is also comparable to the microscopic energy density functionals.

##  I INTRODUCTION

One of the major research areas in nuclear physics is the nuclear mass
(binding energy) predictions, especially for nuclei far from the stability
line with extreme proton-neutron ratio. As the most fundamental property of
nuclei, accurate nuclear mass measurements and theoretical predictions have
vital importance not only for nuclear physics [1] but also for nuclear
astrophysics [2, 3, 4]. Alongside with other nuclear properties (i.e., charge
radii, separation energies, decay properties, etc.), the nuclear masses could
provide information about the nucleon-nucleon interaction, shell, and pairing
properties of nuclei, and its precise determination is also crucial for our
understanding of the formation of chemical elements heavier than iron in the
universe [5].

In the last decades, nuclear mass measurements have gained acceleration with
the developments in experimental facilities. According to the latest atomic
mass table AME2016 [6], the ground-state masses of 3435 nuclei have been
measured. However, measurement of the ground-state properties for nuclei close
to the drip lines still stands as a challenge. Also, there exist large
deviations in theoretical model calculations for nuclei close to the drip
lines. Therefore, further studies are needed to make reliable predictions in
these regions. Up to now, microscopic-macroscopic (mic-mac) [7, 8, 9] and
microscopic models [10, 11, 12, 13, 14, 15] have been employed for the mapping
of the nuclear landscape. Although the microscopic models are more complete in
terms of the physics behind them, much better results are obtained using the
mic-mac models in nuclear mass predictions since their constants are
determined using the experimental ground-state masses of nuclei. While the
root-mean-square (rms) deviation of nuclear masses is generally high (several
MeV) using the microscopic models, the FRDM2012 [9] predicts an rms deviation
0.57 MeV with respect to the AME2003 atomic mass table [16], and WS3 model
gives even lower rms deviation (0.336 MeV) for nuclear masses [8].

Considering the microscopic models, the first complete nuclear mass table was
based on the well-known Skyrme Hartree–Fock (HF) method in the non-
relativistic framework [17], and the root-mean-square error was obtained as
0.738 MeV using the 1995 Audi–Wapstra compilation [18]. With further
improvements, the HFB-31 mass model also gave a model error of 0.561 MeV for
the measured mass of 2353 nuclei [19]. Using the Gogny HFB method, the rms
deviation was obtained as 0.798 MeV with respect to the experimental
predictions of the 2149 nuclei [11]. A systematic study was also conducted on
6969 nuclei to predict nuclear properties using the relativistic mean-field
model, and rms deviation for nuclear masses was obtained as 2.1 MeV [20].
According to the latest microscopic mass model based on the relativistic
continuum Hartree-Bogoliubov (RCHB) theory [21], the root-mean-square
deviation of the binding energies with respect to the experimental data was
obtained at around several MeV. Although considerable progress has been
achieved with the mic-mac and microscopic models, the rms deviations are still
high and one needs more accurate results, especially for astrophysical
applications. Therefore, different approximations and models are required to
understand the discrepancies in the results as well as to make more precise
predictions.

In recent years, there has been an increasing amount of interest in artificial
neural networks [22, 23, 24, 25, 26, 27, 28], which is known as a
nonparametric estimator in Machine Learning (ML). Applications of neural
networks cover many areas in science as well as the different branches of
physics. Considering the variety and richness of the available experimental
data, nuclear physics is also a good candidate to study using neural networks.
Long ago, several studies were performed to predict nuclear properties using
various techniques [29, 30, 31]. Neural networks were used to predict nuclear
mass excess and neutron separation energies [30], and it is shown that neural
networks can be used as a new tool to predict the properties of atomic nuclei.
Later on, nuclear mass defect predictions were made using the neural networks,
showing that the neural networks can be considered as powerful tools to
explore nuclear properties alongside the theoretical models [31]. The ground-
state energies [23] and charge radii [22] of nuclei were also investigated
using artificial neural networks, and the usefulness of the method in the
predictions was shown. Recently, various machine learning algorithms were used
with the latest AME2016 data set to estimate the binding energies of atomic
nuclei [32]. Besides, it was shown that the deep neural networks can predict
the ground-state and excited energies as accurate as the nuclear energy
density functional with less computational cost [33]. In recent years, neural
network approaches were also used to train the mass residues of the
theoretical models to improve the predictive power of the models and achieved
considerable success [25, 26, 27, 28]. As far as we are concerned, there is no
work related to the application of the multilayer perceptron (MLP), which is a
class of feedforward artificial neural network (ANN), to nuclear physics data.
Therefore, it would be interesting to investigate the success of this model in
the predictions of nuclear properties.

In this work, we implement the multilayer perceptron to predict the total
binding energies (BE) of atomic nuclei. In our work, we first use the
experimental data [6] along with the proton (Z) and mass (A) numbers of the
selected nuclei as inputs. Then, we study the effects of the increasing number
of the hidden layers and inputs in the predictive power of the neural network.
Finally, we compare our results with the other microscopic and microscopic-
macroscopic models to evaluate the success of MLP in the binding energy
predictions compared to the other models.

##  II Multilayer Perceptron

In this study, we aim to create a model that makes ground-state binding energy
predictions for atomic nuclei by using input data. Inputs are the nuclear
properties that can affect the binding energies of nuclei. Our model takes
these properties as input and predicts the binding energies as the output.
Such problems, where the output is a numerical value, are known as the
regression problems. Regression is a supervised learning problem where there
is an input, X, an output, Y, and the task is to learn the mapping from the
input to the output [34]. To this end, in machine learning, we assume a model
as shown below:

$y=f(x|\theta)$ (1)

where $f(.)$ is the model and $\theta$ are its parameters. In our case, $y$
corresponds to the prediction for binding energy, and $f(.)$ is the regression
function. In the context of machine learning, the parameters, $\theta$, are
optimized by minimizing a loss function. Thus, the predictions are obtained as
close as possible to the correct values given in the input data.

We choose multilayer perceptron (MLP) as the model $f(.)$. MLP is a neural
network architecture that is mostly preferred to solve such regression
problems. In the training stage of an MLP, the backpropagation algorithm [35]
is used for computing the gradient. On the other side, another algorithm is
used to perform learning using this gradient [24]. The second algorithm is
used for optimization, which is usually called as the optimizer. In recent
years, a new type of algorithm, which is called the adaptive gradient method,
is preferred as the optimizer. In this study, we use Adam algorithm [36] to
train the MLP.

Basically, an MLP is a feedforward neural network with one or more than one
hidden layer between input and output layers. In the case of one hidden layer,
first, input $x$ is fed to the input layer. By using an activation function,
the activation propagates in the forward direction, and the values of the
hidden units z are computed. Each hidden unit usually applies a nonlinear
activation function to its weighted sum. After performing the forward pass, an
error is computed by using a loss function. By using this error, the weights
are updated in the backward pass [35].

However, an MLP with one hidden layer has limited capacity, and using an MLP
with multiple hidden layers can learn more complicated functions of the input.
That is the idea behind deep neural networks where each hidden layer combines
the values in its preceding layer and learns more complicated functions [34].
It is possible to have multiple hidden layers each with its own weights and
applying the activation function to its weighted sum. It should be noted that
different activation functions can be used in multilayer perceptrons, e.g.,
ReLU, tanh, sigmoid, etc. In this work, we implement both tanh and ReLU, which
are two commonly used activation functions in nuclear mass predictions [37,
32, 38, 26], and find that the ReLU function gives better predictions on the
test data. Therefore, we choose the ReLU function as the activation function
of the hidden layers in this work, which is also mostly preferred for the
hidden layers of deep neural networks [39]:

$\phi(x)=max(0,x)$ (2)

An MLP with three hidden layers is demonstrated in Fig. 1 where
$\textbf{w}_{1h}$, $\textbf{w}_{2l}$ and $\textbf{w}_{3k}$ are the weights
belonging to the first, second and third hidden layers, respectively. The
units on the first, second and third hidden layers are represented as
$z_{1h}$, $z_{2l}$ and $z_{3k}$, and v are the output layer weights. Such an
architecture is required four stages to compute the output. Firstly, input x
is fed to the input layer, the weighted sum is computed, and the activation
propagates in the forward direction. When the ReLU function is chosen as the
activation function, $z_{1h}$ is computed as shown below:

$\displaystyle z_{1h}$ $\displaystyle=$ $\displaystyle
ReLU(\textbf{w}_{1h}^{T}\textbf{x})$ (3) $\displaystyle=$ $\displaystyle
ReLU\Bigg{(}\sum_{j=1}^{d}w_{1hj}x_{j}+w_{1h0}\Bigg{)},h=1,...,H_{1}$

Figure 1: The structure of a multilayer perceptron with three hidden layers.

The computations for the second hidden layer are practiced similarly. At this
stage, the second hidden layer activations are computed by taking the first
hidden layer activations as their inputs. Then, the third hidden layer
activations are computed by taking the second hidden layer activations as
their inputs. In a regression problem, there is no nonlinearity in the output
layer. Therefore, the output $y$ is computed by taking the $z_{3}$ as input
[34]. Thus, the forward propagation is completed:

$\displaystyle z_{2l}$ $\displaystyle=$ $\displaystyle
ReLU(\textbf{w}_{2l}^{T}\textbf{z}_{1})$ (4) $\displaystyle=$ $\displaystyle
ReLU\Bigg{(}\sum_{h=0}^{H_{1}}w_{2lh}z_{1h}+w_{2l0}\Bigg{)},l=1,...,H_{2}$

$\displaystyle z_{3k}$ $\displaystyle=$ $\displaystyle
ReLU(\textbf{w}_{3k}^{T}\textbf{z}_{2})$ (5) $\displaystyle=$ $\displaystyle
ReLU\Bigg{(}\sum_{l=0}^{H_{2}}w_{3kl}z_{2l}+w_{3k0}\Bigg{)},k=1,...,H_{3}$

$y=\textbf{v}^{T}\textbf{z}_{3}=\sum_{k=1}^{H_{3}}v_{k}z_{3k}+v_{0}$ (6)

When the MLP goes deeper, one more step is added to these computations for
each one of additional hidden layer. In this study, we implement MLP
architectures of two different depths. Whereas the first one includes three
hidden layers, the other one includes four. Thus, we observe the effect of
depth on the binding energy predictions. As we predict the binding energy,
i.e. one single numerical value, only one unit exists in the output layer. In
order to determine the MLP architecture, we gradually increase the number of
hidden units according to the prediction performance on three data sets. Then,
we choose our final architectures. The MLP with three hidden layers includes
32,16 and 8 hidden units, respectively. It includes 769 (833) parameters for
the MLP model with two (four) inputs, which is much smaller than the number of
training data. On the other side, the MLP with four hidden layers includes
32,32,16 and 8 hidden units. It includes 1825 (1889) parameters for the MLP
model with two (four) inputs. In addition to the smaller number of parameters,
our architectures does not overfit because of two main reasons: Firstly, the
central challenge in machine learning is that we must perform well on new,
previously unseen inputs – not just those on which our model is trained [24].
As it is seen in our results below, our neural network can make good
predictions on test data. Secondly, overfitting occurs when the gap between
the training loss and test loss is too large [24]. However, in our
calculations, the gap between the training loss and test loss is too small.
For instance, it is found as 0.0023 (0.0029) for the training (test) data of
MLP architecture with four hidden layers using four inputs. Therefore, it is
seen that our neural network does not overfit, and it generalizes well on test
data of each dataset, and the losses of training and test data are so close to
each other.

Another important step of creating these architectures is to initialize the
layer weights. We initialize them with the Glorot normal initializer, also
known as Xavier normal initializer [40]. Besides, the input data is randomly
divided into two subsets as 70.0% for training and 30.0% for testing. We
prefer mean absolute error as the loss function on the training set X:

$E(\textbf{W},\textbf{v}|X)=\frac{\sum_{t=1}^{n}\left|r^{t}-f(x^{t})\right|}{n}$
(7)

where $r^{t}$ are the desired values and $f(x^{t})$ are predictions for the
binding energy.

We train our MLP architectures 800 epochs by using Adam optimization algorithm
to minimize mean absolute error. The name Adam is derived from adaptive moment
estimation. It is an adaptive gradient method that individually adapts the
learning rates of model parameters [36]. During training, this algorithm
computes the estimates of first and second moments of the gradients, and uses
decay constants to update them. Therefore, Adam algorithm requires
hyperparameters called decay constants in addition to the learning rate. In
this study, the initial learning rate is 0.001, and decay constants are 0.9
and 0.999, respectively.

##  III Results

Figure 2: Comparison of the experimental and predicted binding energy
differences for the testing set of the three layers MLP architecture using (Z,
A) (upper figure) and (Z, A, $\delta$, P) (lower figure) as inputs.

Figure 3: The same as in Fig. 2, but for the four layers MLP architecture.

In this work, the multilayer perceptron is used to predict ground-state
binding energies of atomic nuclei. Two different MLP architectures and inputs
are used to test the effect of the number of the hidden layers and inputs on
the results. In the input channels, we first use proton (Z) and mass (A)
numbers of nuclei as inputs along with the experimental data from the latest
AME2016 mass table [6] to predict binding energies. Therefore, this part of
the present work does not include any physical identity or theory, except the
proton and mass numbers. In the second part, we also include additional
physical inputs to improve the predictive power of our models. These inputs
carry information about the shell structure of nuclei, which in turn related
to nuclear binding energies [41, 42]. One of them is the pairing term
$\delta(Z,N)$, which is defined as

$\delta(Z,N)=\left[(-1)^{Z}+(-1)^{N}\right]/2.$ (8)

The pairing term becomes $+1$ ($-1$) for even-even (odd-odd) nuclei, and 0 for
other nuclei. A positive value for the pairing term indicates that the nucleus
is more bound while the opposite behavior is valid for a negative value [42].
Another input is the promiscuity factor (P) of nuclei [41] and it is given by

$P=\nu_{p}\nu_{n}/(\nu_{p}+\nu_{n}),$ (9)

where $\nu_{p(n)}$ is the difference between the actual proton (neutron)
number and the nearest magic number. The promiscuity factor is defined as a
measure of the valance proton-neutron ($p-n$) interactions [41]. In this work,
the proton and neutron magic numbers are taken as Z=8, 20, 28, 50, 82, 126 and
N=8, 20, 28, 50, 82, 126, 184, respectively. The latest AME2016 mass table
provides data for 3413 nuclei for A$\geq$8\. However, only 2479 of them are
obtained experimentally, and the others are calculated using the trend from
the mass surface (TMS) in the neighborhood. Although the properties of some
nuclei are not obtained directly from experiments, they are expected to
provide reasonable data and trends for unknown nuclei. As it is well-known,
having a large amount of data is crucial for training an artificial neural
network. Using a large collection of data, performance of an algorithm can be
improved significantly. In our model, we make predictions by using only proton
and mass numbers of nuclei alongside some shell effects in the input. Since we
do not provide much information to the model in the input channel, we need a
large amount of data to make reasonable predictions. Therefore, non-
experimental values from the AME2016 mass table are also used in the
calculations to increase the performance of the model. While training the
model, light nuclei with N$<$8 and A$<$10 is not taken into account and we
randomly divide our data set (3388 nuclei in total) into training (70.0%) and
testing sets (30.0%), as usual.

In Figs. 2 and 3, we display the binding energy (BE) differences between the
experimental data and the results from the MLP model using two different
architectures and inputs. The first architecture has three hidden layers
(32-16-8), and we only use the proton and mass numbers of nuclei (Z, A) in the
input channel (see Fig. 2(a)). Then, we also add pairing and promiscuity
factors (Z, A, $\delta$, P) to see their effects on the results (see Fig.
2(b)).

The predictions of the MLP with the three hidden layers are given in Fig. 2
for 1017 nuclei in the testing set. Although the root-mean-square deviation
($\sigma_{rms}$) with respect to the experimental data is high and obtained as
$\sigma_{rms}=3.98$ MeV, the MLP can make reasonable predictions for the
nuclear binding energies using only (Z, A) as inputs, and the results are
comparable with the predictions of the nuclear energy density functionals.
First thing to notice is the large deviation of the model results for light
and heavy nuclei, which can be related to the limited number of experimental
data in these regions. Adding more physical information to the input, we find
that the predictive power of the MLP increases considerably as can be seen
from Fig.2(b). By adding pairing and promiscuity factors to the input channel,
the root-mean-square deviation is improved by about 45.73% and obtained as
2.16 MeV for the testing set nuclei. It is also clear that the predictive
power of the MLP increases considerably for light and heavy nuclei. To see the
performance of the MLP in different regions of the nuclear landscape, we
divide the testing set into three parts as light nuclei (Z$<$20, 93 nuclei),
medium-heavy nuclei (20$\leq$Z$\leq$82, 696 nuclei), and super-heavy nuclei
(Z$\geq$82, 228 nuclei). Using (Z, A) as inputs, the $\sigma_{rms}$ values are
obtained as 8.60, 2.93, and 3.80 MeV for light, medium-heavy and super-heavy
nuclei, respectively. Increasing the number of the inputs in the MLP
architecture, namely using (Z, A, $\delta$, P) in the input channel, we obtain
important improvements in the $\sigma_{rms}$ values (see Table 1). For
instance, the $\sigma_{rms}$ value for light nuclei is decreased and found as
3.39 MeV, which corresponds to 60.58% improvement in the predictions. Besides,
the $\sigma_{rms}$ value for medium-heavy and super-heavy nuclei is decreased
and obtained as 2.10 and 1.61 MeV, respectively. Using MLP with four inputs,
the model predictions are improved by about 28.32% and 57.63% for medium-heavy
and super-heavy nuclei, respectively.

Table 1: The root-mean square deviations ($\sigma_{rms}$) in units of MeV for different MLP architectures and inputs. The results of the best MLP architectures are shown in bold. Since the number of the parameters are higher than the number of the training data set, the results of the two hidden layers (64-64) MLP architecture are not presented. | MLP | Input | Z$<$20 |  $20\leq$ Z$\leq 82$ | Z$>$82 | Testing set |
---|---|---|---|---|---|---|---
|  |  | 93 nuclei | 696 nuclei | 228 nuclei | 1017 nuclei |
| (32-32) | (A,Z) | 7.80 | 2.80 | 2.11 | 3.46 |
| (64-32) | (A,Z) | 2.93 | 2.35 | 4.50 | 3.01 |
| (32-32) | (A,Z,$\delta$,P) | 4.13 | 1.95 | 4.76 | 3.04 |
| (64-32) | (A,Z,$\delta$,P) | 5.00 | 1.88 | 3.07 | 2.61 |
| (32-16-8) | (A,Z) | 8.60 | 2.93 | 3.80 | 3.98 |
| (64-8-4) | (A,Z) | 6.42 | 4.07 | 4.83 | 4.51 |
| (64-16-8) | (A,Z) | 6.22 | 2.11 | 2.05 | 2.75 |
| (32-16-8) | (A,Z,$\delta$,P) | 3.39 | 2.10 | 1.61 | 2.16 |
| (64-8-4) | (A,Z,$\delta$,P) | 6.75 | 2.93 | 4.76 | 3.90 |
| (64-16-8) | (A,Z,$\delta$,P) | 4.60 | 1.89 | 2.47 | 2.40 |
| (32-16-8-4) | (A,Z) | 10.37 | 3.01 | 2.60 | 4.20 |
| (32-16-16-8) | (A,Z) | 1.98 | 2.73 | 2.37 | 2.60 |
| (32-32-16-8) | (A,Z) | 4.89 | 3.06 | 4.82 | 3.72 |
| (32-16-8-4) | (A,Z,$\delta$,P) | 5.03 | 3.22 | 3.24 | 3.43 |
| (32-16-16-8) | (A,Z,$\delta$,P) | 9.11 | 2.83 | 2.27 | 3.78 |
| (32-32-16-8) | (A,Z,$\delta$,P) | 3.03 | 1.58 | 1.94 | 1.84 |
| (32-16-8-8-8) | (A,Z) | 4.20 | 2.56 | 5.25 | 3.50 |
| (32-16-16-8-4) | (A,Z) | 4.21 | 2.83 | 4.88 | 3.52 |
| (64-32-16-16-8) | (A,Z) | 2.87 | 2.46 | 5.55 | 3.44 |
| (32-16-8-8-8) | (A,Z,$\delta$,P) | 5.56 | 2.17 | 1.35 | 2.54 |
| (32-16-16-8-4) | (A,Z,$\delta$,P) | 10.83 | 2.98 | 1.78 | 4.18 |
| (64-32-16-16-8) | (A,Z,$\delta$,P) | 3.83 | 7.47 | 2.47 | 4.92 |
Figure 4: Comparison of the experimental and predicted binding energy
differences for 956 nuclei between the MLP with three hidden layers and (a)
UNEDF1 [43, 44, 13], SKM* [45, 44, 13], (c) FRDM-2012 [9] and (d) WS3 [8]
models. The black dashed lines are given to guide the eye. Figure 5: The same
as in Fig. 4, but using the four layers MLP architecture.

It is known that increasing or decreasing the number of hidden layers can also
affect the predictive power of the neural networks. Therefore, we also
increase the number of the hidden layers to four in the MLP architecture and
the same calculations are repeated to see its effect on the results. In Fig.
3, the binding energy differences between the experimental data and MLP
predictions with three hidden layers are displayed for 1017 nuclei in the
testing set. By increasing the number of the hidden layer by one unit, the
predictive power of the MLP model increases, and the $\sigma_{rms}$ values are
obtained as 3.72 and 1.84 MeV with two (see Fig.3(a)) and four inputs (see
Fig.3(b)), respectively. Similar to the MLP with the three hidden layers, the
largest deviations in the binding energies are obtained for light and super-
heavy nuclei, and inclusion of the additional inputs increases the success of
the model in these regions. Using MLP with four hidden layers and two inputs
(Z, A), the $\sigma_{rms}$ deviations are obtained as 4.89, 3.06, and 4.82 MeV
for light, medium-heavy, and super-heavy nuclei, respectively. Adding the
pairing and promiscuity factors to the input channels, the results are
improved by about 38.03%, 48.37%, and 59.75% and the $\sigma_{rms}$ deviation
values are obtained as 3.03, 1.58, and 1.94 MeV for light, medium-heavy and
super-heavy nuclei (see Table 1), respectively.

We should mention that different MLP architectures with different number of
hidden layers or hidden units are also tested to make nuclear mass
predictions, and obtain the best MPL model. The results of these works are
also given in Table 1. We found that decreasing the number of the hidden
layers also decrease the predictive power of the results. On the other hand,
increasing the number of the hidden layers more than four also does not give
better results. In this work, the best results are obtained using the three
(32-16-8) and four (32-32-16-8) layers MLP architectures with four inputs. Our
results indicate that using the proper number of hidden layers and inputs in
the MLP architecture, we can make fast and reliable predictions for nuclear
properties. Since the predictions are better using the (Z, A, $\delta$, P) as
inputs, we always use the results with four inputs to compare with other mic-
mac and microscopic results in the rest of the paper.

In figures 4 and 5, we also display the binding energy differences between
experimental data and modeling results from the MLP architectures with three
and four hidden layers and using four (Z, A, $\delta$, P) inputs. Both mic-mac
(FRDM-2012 [9] and WS3 [8]) and microscopic (UNEDF1 [43, 44, 13] and SkM* [45,
44, 13]) results from theoretical database Explorer [44] are shown along with
the MLP predictions for comparison. It is known that the accuracy of the self-
consistent models is not high for the nuclear mass predictions and the root-
mean-square deviations are obtained at about several MeV. On the other side,
the mic-mac models (FRDM-2012 and WS3) are not self-consistent, and their
parameter constants are fitted using the available experimental data, which in
turn provide better estimations for the nuclear binding energies as can be
seen from Figs. 4 and 5. The first thing to notice is that the binding energy
differences are generally high for light and heavy nuclei in all model
predictions. Compared to the UNEDF1, the deviation of the SkM* results from
the experimental data is quite high, and increases with the increase in mass
number. For mic-mac models (FRDM-2012 and WS3), the highest deviations are
obtained for nuclei with A$\geq$250\. Comparing the results of MLP
architectures with three and four layers (see figs. 4 and 5), it is clear that
four layers MLP results are better than the three layers MLP. Besides, both
MLP architectures are as successful as the FRDM-2012 and WS3 models in the
description of nuclei with A$\geq$250\. Although we only use the proton-mass
numbers alongside pairing and promiscuity factors of nuclei as physical inputs
in the training of the MLP, the model gives promising results that they are
comparable to other microscopic and mic-mac models.

Table 2: The root-mean square deviations ($\sigma_{rms}$) in units of MeV for the common 956 nuclei using various models. In here, MLP1 and MLP2 represent three layers (32-16-8) and four layers (32-32-16-8) MLP architectures using (Z, A, $\delta$, P) in the input channel. The nuclear bindings energy results are taken from the nuclear energy density functionals UNEDF1 [43, 44, 13] and SkM* [45, 44, 13]. The results of the FRDM-2012 and WS3 models are taken from Refs. [9, 8]. | MLP1 | MLP2 | UNEDF1 | SKM* | FRDM-2012 | WS3
---|---|---|---|---|---|---
$\sigma_{rms}$ | 1.97 | 1.72 | 2.13 | 7.81 | 0.99 | 0.55

In Table 2, the root-mean-square deviations ($\sigma_{rms}$) are also given
for each model to get a better insight into the success of the models. The
best results are obtained for FRDM-2012 and WS3 models, and rms deviations are
obtained below 1.0 MeV. While the rms deviation is rather high using the SkM*
functional and obtained as 7.81 MeV, the UNEDF1 functional makes better
predictions and rms deviation is obtained as 2.13 MeV. The root-mean-square
deviations are still high using the self-consistent microscopic models in the
calculations. Besides, calculations using nuclear energy density functionals
are still computationally demanding. It is seen that the MLP model can make
reliable predictions that are comparable to the well-known microscopic models,
and the $\sigma_{rms}$ values are obtained as 1.97 and 1.72 MeV with three and
four layers MLP architectures.

##  IV Conclusions

We implement the multilayer perceptron (MLP) to make ground-state binding
energy predictions for atomic nuclei. Two different architectures and inputs
are used in the MLP model to study the performance of this neural network in
the binding energy predictions. In the first one, we only use the proton and
mass numbers of nuclei alongside with the latest experimental data, and no
physical input is included. Then, we also added two additional inputs: pairing
and promiscuity factors of nuclei to give more physical information to the
models. We find that using proper hidden layers and units with relevant
information in the input channels, the nuclear binding energy predictions
using the MLP improve considerably, especially for light and medium-heavy
nuclei. For 1017 nuclei in the testing set, the best root-mean-square
deviations are obtained as 2.16 MeV and 1.84 MeV for three and four layers MLP
architectures using (Z, A, $\delta$, P) inputs, respectively.

Our findings show that the MLP model can make reasonable predictions for
binding energies of atomic nuclei and the results are also comparable to other
models. Although the MLP does not include any physics theory behind it and
considered as a statistical model, it is seen that the model can make fast and
reliable predictions with a proper architecture and relevant inputs. In this
respect, the artificial neural networks can be seen as an alternative tool to
other mic-mac and microscopic models.

As future work, we plan to extend our calculations by including more physical
quantities in the input to better estimate the nuclear properties. Improving
the extrapolation abilities of the neural networks for very neutron-rich
nuclei is also another challenging task and remains as a future work. Besides,
the neural network approaches can be used to train the residues of nuclear
properties as it is done in Refs. [25, 26, 27, 28], which in turn can be
helpful to understand the missing physics behind the microscopic models.

## References

  * Lunney, Pearson, and Thibault [2003] D. Lunney, J. M. Pearson, and C. Thibault, “Recent trends in the determination of nuclear masses,” Rev. Mod. Phys. 75, 1021–1082 (2003).
  * Mumpower  _et al._ [2015a] M. R. Mumpower, R. Surman, D.-L. Fang, M. Beard, P. Möller, T. Kawano, and A. Aprahamian, “Impact of individual nuclear masses on $r$-process abundances,” Phys. Rev. C 92, 035807 (2015a).
  * Mumpower  _et al._ [2015b] M. Mumpower, R. Surman, D. L. Fang, M. Beard, and A. Aprahamian, “The impact of uncertain nuclear masses near closed shells on ther-process abundance pattern,” Journal of Physics G: Nuclear and Particle Physics 42, 034027 (2015b).
  * Schatz and Ong [2017] H. Schatz and W.-J. Ong, “Dependence of x-ray burst models on nuclear masses,” The Astrophysical Journal 844, 139 (2017).
  * Martin  _et al._ [2016] D. Martin, A. Arcones, W. Nazarewicz, and E. Olsen, “Impact of nuclear mass uncertainties on the $r$ process,” Phys. Rev. Lett. 116, 121101 (2016).
  * Wang  _et al._ [2017] M. Wang, G. Audi, F. G. Kondev, W. Huang, S. Naimi, and X. Xu, “The AME2016 atomic mass evaluation (II). tables, graphs and references,” Chinese Physics C 41, 030003 (2017).
  * Wang  _et al._ [2010] N. Wang, Z. Liang, M. Liu, and X. Wu, “Mirror nuclei constraint in nuclear mass formula,” Phys. Rev. C 82, 044304 (2010).
  * Liu  _et al._ [2011] M. Liu, N. Wang, Y. Deng, and X. Wu, “Further improvements on a global nuclear mass model,” Phys. Rev. C 84, 014333 (2011).
  * Möller  _et al._ [2012] P. Möller, W. D. Myers, H. Sagawa, and S. Yoshida, “New finite-range droplet mass model and equation-of-state parameters,” Phys. Rev. Lett. 108, 052501 (2012).
  * Goriely, Chamel, and Pearson [2009] S. Goriely, N. Chamel, and J. M. Pearson, “Skyrme-hartree-fock-bogoliubov nuclear mass formulas: Crossing the 0.6 mev accuracy threshold with microscopically deduced pairing,” Phys. Rev. Lett. 102, 152503 (2009).
  * Goriely  _et al._ [2009] S. Goriely, S. Hilaire, M. Girod, and S. Péru, “First gogny-hartree-fock-bogoliubov nuclear mass model,” Phys. Rev. Lett. 102, 242501 (2009).
  * Stoitsov  _et al._ [2006] M. Stoitsov, J. Dobaczewski, W. Nazarewicz, and P. Borycki, “Large-scale self-consistent nuclear mass calculations,” International Journal of Mass Spectrometry 251, 243 – 251 (2006).
  * Erler  _et al._ [2012] J. Erler, N. Birge, M. Kortelainen, W. Nazarewicz, E. Olsen, A. Perhac, and M. Stoitsov, “The limits of the nuclear landscape,” Nature 486, 509–512 (2012).
  * Afanasjev and Agbemava [2016] A. V. Afanasjev and S. E. Agbemava, “Covariant energy density functionals: Nuclear matter constraints and global ground state properties,” Phys. Rev. C 93, 054310 (2016).
  * Agbemava  _et al._ [2014] S. E. Agbemava, A. V. Afanasjev, D. Ray, and P. Ring, “Global performance of covariant energy density functionals: Ground state observables of even-even nuclei and the estimate of theoretical uncertainties,” Phys. Rev. C 89, 054320 (2014).
  * Audi, Wapstra, and Thibault [2003] G. Audi, A. Wapstra, and C. Thibault, “The ame2003 atomic mass evaluation: (ii). tables, graphs and references,” Nuclear Physics A 729, 337 – 676 (2003), the 2003 NUBASE and Atomic Mass Evaluations.
  * Goriely, Tondeur, and Pearson [2001] S. Goriely, F. Tondeur, and J. Pearson, “A hartree–fock nuclear mass table,” Atomic Data and Nuclear Data Tables 77, 311 – 381 (2001).
  * Audi and Wapstra [1995] G. Audi and A. Wapstra, “The 1995 update to the atomic mass evaluation,” Nuclear Physics A 595, 409 – 480 (1995).
  * Goriely, Chamel, and Pearson [2016] S. Goriely, N. Chamel, and J. M. Pearson, “Further explorations of skyrme-hartree-fock-bogoliubov mass formulas. xvi. inclusion of self-energy effects in pairing,” Phys. Rev. C 93, 034337 (2016).
  * Geng, Toki, and Meng [2005] L. Geng, H. Toki, and J. Meng, “Masses, Deformations and Charge Radii—Nuclear Ground-State Properties in the Relativistic Mean Field Model,” Progress of Theoretical Physics 113, 785–800 (2005).
  * Xia  _et al._ [2018] X. Xia, Y. Lim, P. Zhao, H. Liang, X. Qu, Y. Chen, H. Liu, L. Zhang, S. Zhang, Y. Kim, and J. Meng, “The limits of the nuclear landscape explored by the relativistic continuum hartree–bogoliubov theory,” Atomic Data and Nuclear Data Tables 121-122, 1 – 215 (2018).
  * Akkoyun  _et al._ [2013] S. Akkoyun, T. Bayram, S. O. Kara, and A. Sinan, “An artificial neural network application on nuclear charge radii,” Journal of Physics G: Nuclear and Particle Physics 40, 055106 (2013).
  * Bayram, Akkoyun, and Kara [2014] T. Bayram, S. Akkoyun, and S. O. Kara, “A study on ground-state energies of nuclei by using neural networks,” Annals of Nuclear Energy 63, 172 – 175 (2014).
  * Goodfellow, Bengio, and Courville [2016] I. Goodfellow, Y. Bengio, and A. Courville,  _Deep Learning_ (MIT Press, 2016).
  * Utama and Piekarewicz [2017] R. Utama and J. Piekarewicz, “Refining mass formulas for astrophysical applications: A bayesian neural network approach,” Phys. Rev. C 96, 044308 (2017).
  * Niu and Liang [2018] Z. Niu and H. Liang, “Nuclear mass predictions based on bayesian neural network approach with pairing and shell effects,” Physics Letters B 778, 48 – 53 (2018).
  * Neufcourt  _et al._ [2018] L. Neufcourt, Y. Cao, W. Nazarewicz, and F. Viens, “Bayesian approach to model-based extrapolation of nuclear observables,” Phys. Rev. C 98, 034318 (2018).
  * Neufcourt  _et al._ [2019] L. Neufcourt, Y. Cao, W. Nazarewicz, E. Olsen, and F. Viens, “Neutron drip line in the ca region from bayesian model averaging,” Phys. Rev. Lett. 122, 062502 (2019).
  * Gernoth  _et al._ [1993] K. Gernoth, J. Clark, J. Prater, and H. Bohr, “Neural network models of nuclear systematics,” Physics Letters B 300, 1 – 7 (1993).
  * Gazula, Clark, and Bohr [1992] S. Gazula, J. Clark, and H. Bohr, “Learning and prediction of nuclear stability by neural networks,” Nuclear Physics A 540, 1 – 26 (1992).
  * Athanassopoulos  _et al._ [2004] S. Athanassopoulos, E. Mavrommatis, K. Gernoth, and J. Clark, “Nuclear mass systematics using neural networks,” Nuclear Physics A 743, 222 – 235 (2004).
  * Anil, Malik, and Banerjee [2020] M. U. Anil, T. Malik, and K. Banerjee, “Nuclear binding energy predictions based on machine learning,” ArXiv:2004.14196 (2020).
  * Lasseri  _et al._ [2020] R.-D. Lasseri, D. Regnier, J.-P. Ebran, and A. Penon, “Taming nuclear complexity with a committee of multilayer neural networks,” Phys. Rev. Lett. 124, 162502 (2020).
  * Alpaydın [2014] E. Alpaydın,  _Introduction to Machine Learning_ (MIT Press, 2014).
  * Rumelhart, Hinton, and Williams [1986] D. E. Rumelhart, G. E. Hinton, and R. J. Williams, “Learning representations by back-propagating errors,” Nature 323, 533–536 (1986).
  * Kingma and Ba [2014] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” preprint ArXiv:1412.6980 (2014).
  * Idinil [2019] A. Idinil, “Statistical learnability of nuclear masses,” arXiv:1904.00057 (2019).
  * Zhang  _et al._ [2017] H. F. Zhang, L. H. Wang, J. P. Yin, P. H. Chen, and H. F. Zhang, “Performance of the levenberg–marquardt neural network approach in nuclear mass prediction,” Journal of Physics G: Nuclear and Particle Physics 44, 045110 (2017).
  * Glorot, Bordes, and Bengio [2011] X. Glorot, A. Bordes, and Y. Bengio, “Deep sparse rectifier neural networks,” 14th International Conference on Artificial Intelligence and Statistics , 315–323 (2011).
  * Glorot and Bengio [2010] X. Glorot and Y. Bengio, “Understanding the difficulty of training deep feedforward neural networks,” Proceedings of the thirteenth international conference on artificial intelligence and statistics , 249–256 (2010).
  * Casten and Zamfir [1996] R. F. Casten and N. V. Zamfir, “The evolution of nuclear structure: the scheme and related correlations,” Journal of Physics G: Nuclear and Particle Physics 22, 1521–1552 (1996).
  * Kirson [2008] M. W. Kirson, “Mutual influence of terms in a semi-empirical mass formula,” Nuclear Physics A 798, 29 – 60 (2008).
  * Kortelainen  _et al._ [2012] M. Kortelainen, J. McDonnell, W. Nazarewicz, P.-G. Reinhard, J. Sarich, N. Schunck, M. V. Stoitsov, and S. M. Wild, “Nuclear energy density optimization: Large deformations,” Phys. Rev. C 85, 024304 (2012).
  * mas [2020] “Mass explorer, http://massexplorer.frib.msu.edu/content/dftmasstables.html,”  (2020).
  * Bartel  _et al._ [1982] J. Bartel, P. Quentin, M. Brack, C. Guet, and H.-B. Håkansson, “Towards a better parametrisation of skyrme-like effective forces: A critical study of the skm force,” Nuclear Physics A 386, 79 – 100 (1982).