# Analysis of a Skyrme energy density functional with deep learning

N. Hizawa  K. Hagino  Department of Physics, Kyoto University, Kyoto 606-8502,
Japan  K. Yoshida  Research Center for Nuclear Physics, Osaka University,
Ibaraki, Osaka 567-0047 Japan  RIKEN Nishina Center for Accelerator-Based
Science, Wako, Saitama 351-0198, Japan

###### Abstract

Over the past decade, machine learning has been successfully applied in
various fields of science. In this study, we employ a deep learning method to
analyze a Skyrme energy density functional (Skyrme-EDF), that is a Kohn-Sham
type functional commonly used in nuclear physics. Our goal is to construct an
orbital-free functional that reproduces the results of the Skyrme-EDF. To this
end, we first compute energies and densities of a nucleus with the Skyrme
Kohn-Sham + Bardeen-Cooper-Schrieffer method by introducing a set of external
fields. Those are then used as training data for deep learning to construct a
functional which depends only on the density distribution. Applying this
scheme to the 24Mg nucleus with two distinct random external fields, we
successfully obtain a new functional which reproduces the binding energy of
the original Skyrme-EDF with an accuracy of about 0.04 MeV. The rate at which
the neural network outputs the energy for a given density is about
$10^{5}$–$10^{6}$ times faster than the Kohn-Sham scheme, demonstrating a
promising potential for applications to heavy and superheavy nuclei, including
the dynamics of fission.

##  I Introduction

Recent progress of deep learning is quite remarkable. It has actually gained
popularity in various fields of science and technology, such as natural
language processing, computer vision, and speech recognition [1, 2, 3, 4, 5].
In several fields of physics, such as condensed matter physics, a multitude of
ideas to utilize machine learning are arising. For instance, in Ref. [6], an
energy density functional (EDF) for electron systems that depends solely on an
electron number density was constructed using the method developed in Ref.
[7], in which an attempt was made with a neural network to predict the
solution of a two-dimensional Schrödinger equation in a random potential.
Other applications have already existed also for a variety of problems,
including those in spin systems [8] and superconducting systems [9].

In contrast, an application of deep learning to nuclear physics has still been
in its early stage [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
24, 25]. We mention that nuclear physics continues to face numerous unresolved
challenges that call for innovative solutions, including a description of
large amplitude collective motions. Such problems may be solved efficiently by
applying the machine learning techniques, developed in other fields of
physics.

In particular, the recent application of deep learning to the Kohn-Sham type
DFT [6] mentioned above could be readlily applied also to nuclear physics. In
nuclear physics, phenomenological models for a functional have often been
employed [26]. The resultant Kohn-Sham type energy density functional (KS-EDF)
is not an explicit functional of the particle number density only, but is
parameterized together with other local densities, such as the kinetic energy
density, the spin-orbit density, and the pair density when considering
explicitly the nucleonic superfluidity. To calculate observables using the KS-
EDF, such as the binding energy of a nucleus, one needs to solve a self-
consistent differential equation of the same form as that in the mean-field
theory many times, which is computationally expensive especially for heavy
systems. Therefore, it is desirable to develop an orbital-free EDF (OF-EDF)
theory that does not depend on Kohn-Sham orbitals. Deep learning can be a
powerful tool for that purpose [6]. Such theory will be based on a functional
that depends sorely on the particle number density. Notice that this is
totally consistent with the original philosophy of the density functional
theory (DFT).

The aim of this paper is to apply the method developed in Ref. [6] to a
nuclear system and construct a deep-learning-based nuclear OF-EDF that
reproduces results of the Skyrme-EDF. In applying the method of Ref. [6] to a
nuclear system, one has to take into account several aspects that make nuclear
systems different from electron systems. One obvious difference is that a
nucleus is a self-bound attractive system. In a electron system without
phonons, the only interaction between electrons is the repulsive Coulomb
force, which causes two electrons to distribute as far as possible. In marked
contrast, nucleons tend to get closer to each other due to a short-ranged
attractive nuclear force, and thus the mechanism which determines the density
distribution is quite different between electron and nucleon systems [27]. In
addition, for electron systems, the KS-EDFs, which are inspired by the
Hartree-Fock method, works well in general. On the other hand, in nuclear
systems, superfluidity plays a crucial role in open-shell nuclei, and
observables are better explained using a KS-EDF that is inspired by the
Hartree-Fock-Bardeen-Cooper-Schrieffer (BCS) or Hartree-Fock-Bogoliubov method
rather than by the Hartree-Fock method. This leads to a technical difference
in that a nuclear KS-EDF depends also on the pair density. It will be
intriguing to investigate how well the deep learning method works in such
attraction-dominated nuclear systems.

The paper is organized as follows. In Sec. II, we introduce the KS-EDF which
we employ, and define a protocol for deep learning. We also discuss how to
generate data sets to train neural networks. In Sec. III, we carry out the
deep learning for the 24Mg nucleus and discuss how well the data sets can be
learned. We then summarize the paper in Sec. IV and discuss future
perspectives.

##  II Formulation

###  II.1 Skyrme EDF

We first introduce a Kohn-Sham type energy density functional (KS-EDF) for
training on a neural network. Throughout this study, we consistently employ
the following Skyrme-type EDF [28]:

$E_{\mathrm{tot}}=E_{\mathrm{kin}}+E_{\mathrm{int}}+E_{\mathrm{pair}}+E_{\mathrm{CoM}},$
(1)

with

$\displaystyle E_{\mathrm{kin}}[\tau]$
$\displaystyle=\frac{\hbar^{2}}{2m}\left(1-\frac{1}{A}\right)\sum_{q}\int
d^{3}r\,\tau_{q}(\bm{r}),$ (2) $\displaystyle
E_{\mathrm{int}}[\rho,\tau,\bm{J}]$ $\displaystyle=\int
d^{3}r\biggl{\\{}\frac{b_{0}}{2}\rho^{2}-\frac{b^{\prime}_{0}}{2}\sum_{q}\rho_{q}^{2}$
$\displaystyle+\frac{b_{3}}{3}\rho^{\alpha+2}-\frac{b^{\prime}_{3}}{3}\rho^{\alpha}\sum_{q}\rho_{q}^{2}+b_{1}\rho\tau-b^{\prime}_{1}\sum_{q}\rho_{q}\tau_{q}$
$\displaystyle\qquad\qquad-\frac{b_{2}}{2}\rho\nabla^{2}\rho+\frac{b^{\prime}_{2}}{2}\sum_{q}\rho_{q}\nabla^{2}\rho_{q}$
$\displaystyle-
b_{4}\rho\nabla\cdot\bm{J}-b^{\prime}_{4}\sum_{q}\rho_{q}\nabla\cdot\bm{J}_{q}\biggr{\\}},$
(3) $\displaystyle E_{\mathrm{pair}}[\rho,\tilde{\rho}]$
$\displaystyle=\sum_{q}\frac{V^{(q)}_{0}}{4}\int
d^{3}r\left\\{1-\left(\frac{\rho}{\rho_{0}}\right)^{\gamma}\right\\}\tilde{\rho}_{q}^{2},$
(4)

and

$\displaystyle E_{\mathrm{CoM}}[\rho]$ $\displaystyle=\frac{C}{2}\left(\int
d^{3}r\,z\rho(\bm{r})\right)^{2},$ (5)

where $m$ is the nucleon mass and $A$ is the mass number of a nucleus.
$E_{\mathrm{kin}},~{}E_{\mathrm{int}},~{}E_{\mathrm{pair}}$, and
$E_{\mathrm{CoM}}$ are the kinetic energy, the interaction energy, the pairing
energy, and a cost function for the center-of-mass, respectively.
$\rho,\tau,\bm{J}$, and $\tilde{\rho}$ are the particle number density, the
kinetic density, the spin density, and the pair density, respectively, in
which the subscript $q$ refers to neutron or proton. Those are defined as

$\displaystyle\rho(\mbox{\boldmath$r$})$
$\displaystyle=2\sum_{q}\sum_{k>0}v_{q,k}^{2}|\varphi_{q,k}(\bm{r})|^{2},$ (6)
$\displaystyle\tau(\mbox{\boldmath$r$})$
$\displaystyle=2\sum_{q}\sum_{k>0}v_{q,k}^{2}|\nabla\varphi_{q,k}(\bm{r})|^{2},$
(7) $J(\mbox{\boldmath$r$})$
$\displaystyle=2\sum_{q}\sum_{k>0}v_{q,k}^{2}\varphi^{*}_{q,k}(\bm{r})\left(-i\nabla\times\bm{\sigma}\right)\varphi_{q,k}(\bm{r}),$
(8) $\displaystyle\tilde{\rho}(\mbox{\boldmath$r$})$
$\displaystyle=-2\sum_{q}\sum_{k>0}u_{q,k}v_{q,k}|\varphi_{q,k}(\bm{r})|^{2},$
(9)

where $\varphi_{q,k}(\bm{r})$ is the $k$-th Kohn-Sham orbrbital in a spinor
form with isospin $q$, and $v_{q,k}^{2}=1-u_{q,k}^{2}$ is the occupation
probability for the $k$-th orbital. Notice that we take the BCS approximation
for the treatment of the pairing correlation.

In the interaction part of the functional, $b_{i}$ and
$b^{\prime}_{i}~{}(i=1\text{--}4)$ as well as $\alpha$ are the Skyrme
parameters. In this paper, we use the SLy4 parameter set [29] for these
parameters. For simplicity, we ignore the Coulomb interaction, even though the
entire Coulomb interaction term can be explicitly described as a functional of
the proton number density if the Slater approximation is introduced to the
exchange term.

For the pairing part, we employ a surface-type functinal of the Density-
Dependent Delta-Interaction (DDDI) [30], which contains the parameters
$V_{0}^{(q)},\rho_{0}$, and $\gamma$. In this study, we take $\gamma=1$ and
$\rho_{0}=0.16\,\mathrm{fm}^{-3}$, and determine $V_{0}^{(q)}$ so that the
average pairing gap coincides with the empirical pairing gap,
$\Delta_{q}=12/\sqrt{A}$ MeV [31, 28]. The zero-range pairing interaction has
to be supplemented with an energy cut-off. In this paper, the sharp cut-off
energy of 60 MeV is introduced to the single particle energy of the Kohn-Sham
orbitals. The resultant strengths for the pairing are
$V_{0}^{(n)}=V_{0}^{(p)}=-683.344$ MeV fm3.

In addition to the ordinal Skyrme EDF, we introduce a functional
$E_{\mathrm{CoM}}[\rho]$ to fix the center-of-mass position in the $z$
direction. This is necessary as we introduce external fields (see Sec. II C
below) to generate various density distributions. By fixing the center-of-mass
position, one can prevent a nucleus from localizing around the edges of the
box, which is useful to generate various deformed states in a small box. In
this study, we take $0.625\,\mathrm{MeV/fm^{2}}$ for the value of $C$.

In this paper, we consider only the 24Mg nucleus. This choice of a nucleus is
convenient, as this nucleus has equal numbers of protons and neutrons, and
thus the proton and the neutron densities coincide to each other when the
Coulomb interaction is ignored. Furthermore, we impose the axial symmetry and
the time-reversal symmetry on the system, enabling the local densities to be
expressed in the cylindrical coordinates $(r,z)$ [32]. Notice that Ref. [6]
also used a two-dimensional EDF for electron systems. With these
simplifications, in principle, the EDF of the system should be able to be
expressed solely with the nucleon number density $\rho(r,z)$, which can be
considered as a monochromatic image.

We solve the Kohn-Sham equations for this EDF by introducing various external
fields to obtain a set of ground state energies and nucleon number densities.
The explcit forms of the external fields are specified in Sec. II C. We solve
the Khon-Sham equations by discretizing the real space, with the mesh size of
0.8 fm in both the $r$ and $z$ directions. We take 10 grid points in the $r$
direction and $20$ points in the $z$-direction, with which the density
$\rho(r,z)$ can be considered as a $10\times 20$-dimensional vector in our
calculations. We choose the box boundary condition and include the
$z$-component of angular momentum up to $9/2$.

###  II.2 Neural network

In this paper, we carry out a regression analysis of $E=E[\rho]$ using a set
of the particle number density and the energy $D=\\{E^{(i)},\rho^{(i)}\\}_{i}$
generated by the KS-DFT. To this end, we utilize a neural network with fully-
connected layers for the fitting function. The fundamental structure of a
neural network involves a repetition of linear and nonlinear transformations
on the input vector; fully-connected layers signify that all the neurons in
the previous layer are connected to all the neutrons in the next layer.

We mention that neural networks composed solely of fully-connected layers may
encounter an issue of an excessive number of parameters when the dimension of
an input vector is large. To avoid this problem, a convolutional neural
network (CNN) is often employed, which has demonstrated a remarkable success
in the field of computer vision [33, 34]. In fact, in the previous application
of deep learning to KS-DFT [6], the input size of a vector has as large as
$256\times 256$ dimension, and thus the CNN was employed. However, the
dimension of our studies in this paper is much smaller, with $10\times 20$
dimension. Therefore, we do not need to introduce the CNN, and a simpler
neural network consisting of the fully-connected layers, as depicted in Fig.
1, is employed in this study (see the caption for the details).

We use the Adam optimizer [35], which has three tunable parameters. Among the
three parameters, we set a learning rate to be $10^{-4}$ and the others to be
default value of the Keras API [36]. The batch size is 128, namely we divide
training data into subsets, each of which contains 128 components. In each
update of the fitting parameters, we do it only within each subset to minimize
a loss function, for which we take a mean square loss function. To avoid the
problem of overfitting, we adopt the early stopping strategy and stop the
learning at the 500th epoch. We decrease the learning rate sequentially to
$10^{-5}$ (at epoch = 101), $10^{-6}$ (at epoch = 201), $5.0\times 10^{-7}$
(at epoch = 301), and $10^{-7}$ (at epoch = 401).

Figure 1:  A neural network employed in this work to learn the Skyrme-EDF,
$E[\rho]$. It consists of 10 hidden layers, all of which are fully connected.
Their activation functions are the ReLU, and the sigmoid activation function
is employed for the output layer. The number of neurons in each layer is
listed below the layers.

###  II.3 External fields

For a given EDF, one can make a correspondence between the particle number
density and the energy of the ground state for a specific external field. This
property will be used to construct a data set to be trained for an OF-EDF. For
this purpose, a diverse range of external fields is required. In this
subsection, we introduce two methods to generate the external potentials used
in this study. The basic idea of these methods is adapted from the previous
studies [7, 6] on two-dimensional systems, but we modify them for the axial-
symmetric systems.

####  II.3.1 Simple Harmonic Oscillators (SHO)

The first method is to use external fields based on a Simple Harmonic
Oscillator (SHO). As the name implies, this is a deformed harmonic oscillator
potential shifted in the $z$-direction:

$v^{(i)}_{\mathrm{SHO}}(r,z)=\frac{1}{2}k^{(i)}_{r}r^{2}+\frac{1}{2}k^{(i)}_{z}(z-z^{(i)}_{0})^{2}.$
(10)

The parameters in the range of $0\leq k_{r},k_{z}\leq
1.1\,\mathrm{MeV/fm^{2}}$, and $-1.6\,\mathrm{fm}\leq z_{0}\leq
1.6\,\mathrm{fm}$ in the potential are generated from uniform random
parameters 111Unless otherwise noticed, all the random numbers used in this
paper are uniform random numbers..

The SHO potentials would be able to encompass only a small portion of a domain
of the external fields to be used in the Skyrme-EDF. However, for practical
calculations, only a limited variety of external fields, such as a quadrupole
moment, has frequently been utilized, if a constrained field is regarded as an
external field in a broad sense. It is therefore still useful to examine the
effectiveness of the learning process with the SHO potentials.

####  II.3.2 Random Potentials (RND)

Figure 2:  Histograms for the results of the Skyrme EDF calculations with
250,000 different external fields based on the SHO (the top panels) and the
RND (the bottom panels) fields. The left and right panels show the binding
energy and the pairing energy in units MeV, respectively. It can be observed
that the structure in the shape of the histograms is washed out to a large
extent for the RND external fields, which are more random than the SHO cases.

As the second method, we introduce a Random Potential (RND). This is a highly
random potential with many random numbers:

$v^{(i)}_{\mathrm{SHO}}(r,z)=m(r,z)\times\mathrm{sr}^{(i)}(r,z),$ (11)

where $m(r,z)$ and $\mathrm{sr}^{(i)}(r,z)$ are defined as,

$m(r,z)=e^{-4.0\max\\{0,\,\sqrt{r^{2}+z^{2}}-r_{0}\\}^{2}/r_{0}^{2}},$ (12)

and

$\mathrm{sr}^{(i)}(r,z)=\sum_{r^{\prime},z^{\prime}}s^{(i)}(r,z;r^{\prime},z^{\prime})\,\mathrm{rnd}^{(i)}(r^{\prime},z^{\prime}),$
(13)

respectively, with

$s^{(i)}(r,z;r^{\prime},z^{\prime})=e^{-\\{(r-r^{\prime})^{2}+(z-z^{\prime})^{2}\\}/\mu_{2}^{(i)}(r^{\prime},z^{\prime})}.$
(14)

The meaning of $\mathrm{rnd}^{(i)}(r,z)$ in Eq. (13) and $\mu^{(i)}_{2}(r,z)$
in Eq. (14) is as follows. First, for each grid point $(r,z)$, a random number
within the range of $[v_{\rm min},v_{\rm max}]$ is generated and labeled as
$\mathrm{rnd}^{(i)}(r,z)$. Since the potential with those random numbers is
too irregular to be used as a potential, it is smoothed with a Gaussian
filter, denoted as $s^{(i)}$, as in Eq. (13). At this stage, the square of the
Gaussian width $\mu^{(i)}_{2}(r,z)$ in Eq. (14) is randomly generated within
the range of $[\mu_{2\rm min},\mu_{2\rm max}]$ to prevent the external field
from acquiring scale information due to the standard deviation of the
Gaussian. Finally, a mask defined by Eq.(12) is applied in Eq.(11) to
circumvent a numerical instability caused by a reduction of the external field
near the boundary. In this study, we take $r_{0}=1.4\times
1.2A^{1/3}\,\mathrm{fm}$, $v_{\rm min}=-1.1$ MeV, $v_{\rm max}=1.1$ MeV,
$\mu_{2\rm min}=$ 0.8 fm2, and $\mu_{2\rm max}=$ 1.2 fm2.

In Refs. [7, 6], random $\\{0,1\\}$ binary data were utilized for
$\mathrm{rnd}^{(i)}(r,z)$. For electronic systems, such a choice would be
plausible because the potential primarily arises from the Coulomb potential
due to a nucleus. On the other hand, in nuclear systems, it would be a highly
non-trivial question to ask which potential is useful to describe static and
dynamical properties of atomic nuclei. While many calculations employ a
phenomenological deformed mean-field potential with e.g., a qudrupole
deformation to study deformed nuclei, it is not obvious whether such choice is
optimal. Therefore, in this study, we use random real numbers for
$\mathrm{rnd}^{(i)}(r,z)$ to generate more diverse external fields than in the
previous studies. Additionally, since the constraint on the center-of-mass
position is included in the definition of the KS-EDF (1), a different mask
function $m$ from that in the previous studies is also introduced.

##  III Results

###  III.1 Generation of a dataset

Figure 3:  Comparisons of the Kohn-Sham method (the horizontal axes) and the
predicted results from the neural network (the vertical axis) for $E[\rho]$
and $E[v]$ for the RND (the top panels) and SHO (the bottom panels) external
fields, all given in units of MeV. From the left to the right panels, the
training results are shown for the binding energy, the pairing energy, and the
energy of the external fields. The results for 20,000 test data points are
plotted in each figure, in which densely populated (under populated) points
are displayed in red (blue).

Let us now apply the deep learning protocol discussed in the previous section
to the Skyrme EDF. We first prepare 250,000 data sets for each of the SHO and
the RND external fields. For each calculation, the outputs are i) the nucleon
number density $\rho$, ii) the kinetic energy $E_{\mathrm{kin}}$, iii) the
interaction energy $E_{\mathrm{int}}$, iv) the pairing energy
$E_{\mathrm{pair}}$, and v) the energy for the external field
$E_{\mathrm{ex}}$. The binding energy $E_{\mathrm{bin}}$ is also computed as a
sum of $E_{\mathrm{kin}},~{}E_{\mathrm{int}}$, and $E_{\mathrm{pair}}$, as
$E_{\mathrm{bin}}=E_{\mathrm{kin}}+E_{\mathrm{int}}+E_{\mathrm{pair}}$. Figure
2 displays the distribution of $E_{\mathrm{bin}}$ and $E_{\mathrm{pair}}$ for
each of the SHO and the RND external fields. The distributions of the other
components of the energy are summarized in Appendix A (see Fig. 8). In order
to use these data for deep learning, we reject those outside the regions given
in Tab. 1. From the remaining data, we select 200,000 data for training. Out
of those 200,000 data, we adopt 90 % of them for training data, while the rest
for test data, which are not used for training.

Table 1: The lower and the upper cut-off energies, in units MeV, for the two different types of the external fields, SHO and RND. For each leraning, only the data within the intervals are employed. The value of cutoffs are determined so that approximately all the data shown in Figs. 2 and 8 can be included.  | SHO | RND
---|---|---
type | lower | upper | lower | upper
$E_{\rm bin}$ | $-\infty$ |  $-$217.5 | $-\infty$ |  $-$217.5
$E_{\rm kin}$ | 395.0 | 450.0 | 360.0 | 420.0
$E_{\rm int}$ |  $-$650.0 |  $-$600.0 |  $-$630.0 |  $-$550.0
$E_{\rm pair}$ |  $-$22.0 | +$\infty$ |  $-$35.0 | $+\infty$
$E_{\rm ex}$ | $-\infty$ | 120.0 |  $-$70.0 | 50.0

###  III.2 $\rho\to E[\rho]$

We first discuss the results for each energy as an objective variable with the
nucleon number density as an explanatory variable. In other words, we
construct the OF-EDF, which yields the energies from a density distribution as
an input. In DFT, apart from an external field, there would be an ambiguity to
divide the functional into components: $E_{\mathrm{kin}},E_{\mathrm{int}}$,
and $E_{\mathrm{pair}}$ themselves may not have strict physical meanings.
Nevertheless, these components can be employed as indicators at least for
qualitative discussions, and we thus follow Ref. [6] to examine the subparts
of the EDF. In particular, it is interesting to investigate the pairing energy
$E_{\mathrm{pair}}$, as it qualitatively verifies whether we can learn the
effect of superfluidity, or the pair density, with deep learning. Notice that
this was not addressed in the previous study in Ref. [6].

The top panels in Fig. 3 compare the results of the Kohn-Sham method (the
horizontal axes) with the neural network predictions (the vertical axes) for
the test data with the RND external fields not used in learning. The results
with the SHO external fields (see the third top panels) are found to be more
accurate 222This is a natural outcome from the simplicity of the SHO external
fields.. Figure 3 shows only $E_{\mathrm{bin}}$, $E_{\mathrm{pair}}$, and
$E_{\mathrm{ex}}$, while the other components of the energy are displayed in
Appendix A (see Fig. 9). If the learning is perfect, the distribution should
be diagonal: actually this is almost the case for all the energies except for
the energy of the external fields plotted in the rightmost figure.

We have found that the large error in $E_{\rm ex}$ was not improved by
changing the learning method, such as a CNN model (see Appendix). This may be
due to the fact that the particle number densities with different external
fields tend to have a similar shape because of the saturation property, which
results in an information loss in the process of compressing information on
the external fields into the density distributions. Of course, according to
the principle of DFT, ideally there should be no loss of information because
there is a bijection between an appropriately defined density and an external
field. However, in actual calculations, information on the detailed structure
of external fields may be lost due to several numerical errors such as
rounding errors, finite difference errors, and errors associated with a
convergence criterion in self-consistent calculations. It is then natural that
the prediction error becomes large when one attempts to recover the external
field information from such a density distribution. The inaccuracy in
predicting the energy of external fields was reported also in the previous
study [6], but the inaccuracy seems more pronounced in atomic nuclei, which
are systems with an attractive interaction. As we will show in the next
subsection, this problem can be improved by using the external fields as
explanatory variables.

To quantitatively evaluate the errors, we calculate the mean absolute error
(MAE) for each learning, which are summarized in Tab. 2. It is remarkable that
the MAE for the binding energy is as small as 0.0051 MeV for the SHO external
fields and 0.0433 MeV for the RND external fields, which are much more
accurate than the accuracy required e.g., for a fission barrier of heavy
nuclei as well as for nuclear masses. For instance, for the latter, the
accuracy of 100 keV is required for the r-process studies [37]. The MAE for
the pairing energy is 0.0233 MeV for the SHO and 0.1567 MeV for the RND. These
values indicate that the particle number density predicts well the
contribution of the pairing correlation, even though the error is slightly
larger than that for the binding energy.

Finally, let us discuss a computational time. For the 24Mg nucleus, it
typically takes about a minute to solve the Skyrme-EDF with the Kohn-Sham
method and obtain a single training data point. In marked contrast, the time
to predict the energy with the neural network used in this paper from a given
density is much shorter, about 0.1 ms. The difference in the computational
speed will become larger for heavy nuclei. This makes a great advantage of
using the deep learning method e.g., in plotting a multi-dimensional potential
energy surface for nuclear fission studies of heavy nuclei.

Table 2: The mean absolute error (MAEs) for each learning with the SHO and the RND external fields. The units are MeV for $E[\rho]$ and $E[v]$, while the MAE for $\rho[v]$ is dimensionless (see Eq. (16)).  | SHO | RND
---|---|---
type | $E[\rho]$ | $E[v]$ | $E[\rho]$ | $E[v]$
$E_{\rm bin}$ | 0.0051 | 0.0054 | 0.0433 | 0.0237
$E_{\rm kin}$ | 0.0165 | 0.0071 | 0.1131 | 0.0900
$E_{\rm int}$ | 0.0105 | 0.0182 | 0.0431 | 0.1499
$E_{\rm pair}$ | 0.0233 | 0.0261 | 0.1567 | 0.1411
$E_{\rm ex}$ | 0.0318 | 0.0105 | 6.6973 | 0.1338
| $\rho[v]$ | $\rho[v]$
| 0.1107 | 0.4101

###  III.3 $v\to E[v]$

While it is somewhat tangential to the topic of DFT, there is a certain demand
in electronic systems for a functional that directly predicts the energy from
a given external field. Because of this, in the previous study [6], an energy
functional $E[v]$ was constructed following the same procedure as that to
construct a functional $E[\rho]$. Even though it is unclear whether such a
functional is useful in nuclear physics, it may be worth investigating whether
a functionl $E[v]$ can be constructed in connection to the discussion in Ref.
[6]. We therefore carry out similar calculations using the same neural network
and dataset as those in the previous subsection, but with the external fields
as the explanatory variables.

Figure 4:  A neural network with the encoder-decoder structure employed in
this work for a mapping from an external $v$ to a particle number density
$\rho$. It consists of 10 hidden layers, all of which are fully-connected.
Their activation functions are ReLU, and the softmax activation function is
employed for the output layer.

The MAEs for $E[v]$ are summarized in Tab. 2, which shows that the MAE for
$E[v]$ tends to be decreased compared to that for $E[\rho]$. This is because
the external field contains more information than the density distribution.
This is particularly true for learning the energy from the external fields. On
the other hand, the accuracy gets lowered for the binding energy with the SHO
external fields. To investigate the origin for this, the lower panels in Fig.
3 show comparisons between $E[v]$ from the Skyrme KS calcaulations and the
result of the deep learning. We find that the points with large errors are due
to external fields that have small amplitudes, that is, almost flat
potentials. Since many SHO potentials used in the dataset have a large
curvature, it is diffult to learn information about external fields with a
small curvature. Such a problem is less likely to occur in fermionic systems
when density distributions are used as the explanatory variables, leading to a
somewhat better accuracy.

Figure 5:  The absolute error of the density distribution directly generated
by a deep learning from a given external field $v$. It is plotted as a
function of the binding energy from the corresponding Kohn-Sham calculation.
The left and the right panels show the results with the SHO and the RND
external fields, respectively. The densely populated points are displayed in
red, while the underpopulated points are shown in blue.  Figure 6:  Examples
of the predicted densities (the bottom panels) generated directly from the RND
external potentials shown in the top panels. For a comparision, the
corresponding Khon-Sham densities are also plotted in the middle panels. The
units of the color coordinate are MeV for the external potentials and
$\mathrm{fm}^{-3}$ for the densities. In each panel, the horizontal axis
denotes the $r$ coordinate while the vertical axis denotes the $z$ coordinate,
whose scales are shown in the left bottom panel.

###  III.4 $v\to\rho[v]$

Observables are in general calculated in DFT with a particle number density,
which is obtained with a given functional. That is, a functional has to be
known in advance in obtaining a particle number density. As demonstrated in
Ref. [6], if a neural network can directly predict the density for a given
external field, the calculation speed will be significantly improved. We
therefore carry out deep learning for the nuclear system with the external
fields as the explanatory variables and density distributions as the objective
variables. To this end, we have to take into account the fact that the
densities are normalized to the particle number, that is, $\int
d^{3}r\,\rho=A$. The softmax function, which is commonly used in
classification problems, enables one to require the normalization condition.
We shall employ this approach in this study for the output layer. For the
axial symmetric system, the following relationship exists with a discretized
spatial mesh:

$\frac{2\pi}{A}\iint
rdrdz\,\rho(r,z)\simeq\sum_{i,j}\rho(r_{i},z_{j})\frac{2\pi r_{i}\Delta
r\Delta z}{A}=1,$ (15)

where $\Delta r=\Delta z=0.8\,\mathrm{fm}$ are the mesh width.

Figure 7:  A verification of generalization performance for the present deep
learning. The left and right panels show the results with the RND and the SHO
external fields, respectively. The horizontal axes denote the energies
obtained with the Kohn-Sham calculations. On the other hand, the vertical axes
denote $E_{\rm SHO}[\rho_{\rm RND}]$ (the left panel) and $E_{\rm
RND}[\rho_{\rm SHO}]$ (the right panel), that is, the predictions of deep
learning trained with the SHO (the left panel) and the RND (the right panel)
external fields. Both the taining and test data (200,000 data in total) are
plotted in each panels because the RND (SHO) dataset are not used in training
$E_{\rm SHO}[\rho]$ ($E_{\rm RND}[\rho])$.

Thus, by selecting $2\pi r_{i}\Delta r\Delta z\rho(r_{i},z_{j})/A$ as the
objective variable, the normalization is automatically imposed. In this study,
we use a neural network with an encoder-decoder structure for training, as is
shown in Fig. 4. The MAE for the learning of $\rho[v]$ is defined as

$\mathrm{MAE}=\overline{2\pi\iint
rdrdz\,|\rho_{\mathrm{pred}}(r,z)-\rho_{\mathrm{ans}}(r,z)|},$ (16)

where $\rho_{\mathrm{pred}}(r,z)$ and $\rho_{\mathrm{ans}}(r,z)$ denote a
predicted density and a Kohn-Sham result, respectively. Here, the bar symbol
represents the average over the test data. We apply the same cut-off energies
to the training data as those for the binding energy (see Tab. 1).

Figure 5 shows the error for each test data point plotted as a function of the
corresponding binding energy from the Kohn-Sham calculation. Their average
corresponds to the MAE (16), which is 0.1107 for the SHO external fields and
0.4101 for the RND external fields. Figure 6 presents the images of the
predicted densities for a few randomly selected data points for the RND
external fields, in comparison to the corresponding Kohn-Sham densities. These
examples clearly show that our neural networks successfully reproduce the
Konh-Sham densities.

###  III.5 Generalization performance

We have so far introduced the two types of external fields and constructed the
two independent datasets. For each dataset, we have successfully provided
predictions for the training data with sufficient accuracy; however, this does
not guarantee performance for unknown data. For instance, a neural network
trained with the RND data does not necessarily yield accurate predictions for
the SHO data. This is because the RND and the SHO external fields yield
density profiles in a different way to each other. In general, such
generalization performance is a critical concern in applying a trained neural
network to another dataset.

To investigate this issue in the context of nuclear physics, let us consider
$E_{\rm SHO}[\rho_{\rm RND}]$ and $E_{\rm RND}[\rho_{\rm SHO}]$, where
$\rho_{\rm SHO}$ and $\rho_{\rm RND}$ are the Kohn-Sham densities obtained
with the SHO and the RND external fields, respectively, and $E_{\rm SHO}$ and
$E_{\rm RND}$ are the functionals trained with $\rho_{\rm SHO}$ and $\rho_{\rm
RND}$, respectively. In Sec. III B, we have investigated $E_{\rm
SHO}[\rho_{\rm SHO}]$ and $E_{\rm RND}[\rho_{\rm RND}]$, but here we are
interested in the performance of the functionals when the densities obtained
with the other types of external fields are used as inputs. The left panel in
Fig. 7 compares the binding energies obtained with the Kohn-Sham calculations
with the RND external fields with $E_{\rm SHO}[\rho_{\rm RND}]$. The right
panel shows similar quantities, but by inverting RND and SHO, that is a
comparison between the Kohn-Sham calculations with the SHO external potentials
and $E_{\rm RND}[\rho_{\rm SHO}]$. One can see that the performance of the
neural network trained with the SHO external fields, $E_{\rm SHO}$, is quite
poor in reproducing the RND test data with large randomness. On the other
hand, the neural network trained with the RND external fields, $E_{\rm RND}$,
successfully predicts the SHO test data, although the errors are larger than
those for $E_{\rm RND}[\rho_{\rm RND}]$ shown in Fig. 3. The MAEs between
Kohn-Sham results and predictions are $1.1523$ MeV for $E_{\rm SHO}[\rho_{\rm
RND}]$ and $0.122$ MeV for $E_{\rm RND}[\rho_{\rm SHO}]$. A similar conclusion
has been obtained also in Ref. [6]. Therefore we can conclude that the RND
potentials which we adopted are random enough for deep learning.

##  IV Summary and future perspectives

Starting from a Skyrme functional, we have successfully constructed an energy
density functional (EDF) which depends only on a particle number density. This
functional does not require Kohn-Sham orbitals, and thus can be regarded as an
orbital-free EDF (OF-EDF). To this end, we have applied deep learning, in
which the density distributions obtained with two types of random external
fields (SHO and RND) were mapped on the energy with a neural network. The
resultant EDF was found to predict various energies for the original Skyrme
EDF with reasonable accuracy, except for the energy of the RND external
fields, whose accuracy could however be considerably improved when the
energies were predicted with deep learning in which the external fields
themselves were directly learned. The latter feature is more pronounced in
systems with an attractive interaction than in electron systems. We have also
found that deep learning with less random SHO external potentials has smaller
errors as compared to that with the RND external fields.

In this paper, we have employed simple supervised learning. However, there are
various methods of machine learning besides this. For example, generative
models such as a generative adversarial network (GAN) [38, 39] and a diffusion
model [40, 3] may provide efficient ways to generate the particle number
densities, that is the input for deep learning used in this work to construct
an OF-EDF. These methods maybe useful alternatives for future application of
the deep learning method discussed in this paper.

In nuclear physics, a triaxial deformation often plays an important role,
particularly in nuclear fission. In that occasion, one needs to deal with
3-dimensional densities, accounting also for spin and isospin indices. We
mention that traditional neural networks, comprising fully-connected layers,
tend not to perform efficiently with such 3-dimensional data, primarily
because the data size tends to become huge when the data are converted to
1-dimensional data. On the other hand, CNNs have shown adaptability to data of
general dimensions. With the Keras API [36], 3D CNNs can be conveniently
implemented, making a straightfoward extension of the present work to 3D
cases. Furthermore, the Vision Transformer (ViT) [4], which has recently
demonstrated success in image recognition tasks, can also be extended to
3-dimensional data. With those schemes, the dimensionality of the density
itself is not a crucial issue in learning EDFs, without incurring additional
costs for preparing training data.

One of the big advantages of using deep learning methods is that energies can
be rapidly computed once test data are prepared and trained. With such low-
cost calculations, numerical experiments will become much easier than before.
We mention that, as objectives of research become more and more sophisticated,
the number of DFT calculations required to publish a single research paper has
in general been increased in these days. A typical example is a calculation
for fission barriers in a multi-dimensional space. Even though computer
performance continues to be improved, a computational cost of research has in
general been increased, and it has been more complicated than before to test
an idea with a numerical experiment. Fast computational methods like the one
developed in this work, particularly when they are provided in a convenient
format such as a Python library, can significantly shorten the time required
to test and validate ideas. If a numerical accuracy is an issue, one may
revalidate ideas obtained with deep learning by using the traditional Kohn-
Sham scheme. This could be interpreted as an application of the idea of
materials informatics (MI) [41] to a theoretical research.

A potential problem in performing supervised learning is that one has to
collect a large set of training data. In this work, we have chosen a
relatively light nucleus, 24Mg, and imposed axial symmetry, and thus we have
treated a relatively low-cost system. However, heavy and superheavy nuclei,
such as uranium isotopes, will be very costly in terms of data collection,
especially when no symmetry is imposed, even though those nuclei have
attracted lots of attention in nuclear physics, as e.g. a finding the optimal
pathway in fission has still been a big theoretical challenge. In this regard,
we would like to point out that a data collection needs not be performed
individually; it could actually be done collaboratively by many researchers. A
lot of good quality data, which are ready to be used in deep learning, may
have already existed for some selected nuclei. Therefore, we believe that it
is desirable to establish a framework in the nuclear theory community to
collect numerical data and/or to carry out numerical calculations with unified
hyperparameters such as a mesh size. Such a collaborative approach will help
advance research more efficiently and effectively, benefiting the whole
nuclear physics community.

## Acknowledgments

We thank G. Colò for useful discussions. This work was supported by JSPS
KAKENHI (Grant Nos. 21J22348, JP19K03824, JP19K03861, JP19K03872, and
JP23K03414).

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##  Appendix A Figures for $E_{\rm kin}$ and $E_{\rm int}$

In this Appendix, we show figures for $E_{\rm kin}$ and $E_{\rm int}$ which
are not shown in Sec. III. We also show a figure for a histogram for $E_{\rm
ex}$. The conclusions remain the same as those for Figs. 2 and 3.

Figure 8:  Same as Fig. 2, but for the kinetic energy, the interaction energy,
and the energy for the external field.  Figure 9:  Same as Fig. 8, but for the
kinetic and the interaction energies.

##  Appendix B Convolutional Neural Network

In the previous study [6], the convolutional neural network (CNN) was used for
calculating an OF-EDF. An efficient structure in the CNN enables computers to
recognize images, which reduces the trainable parameters, and the CNN works
well in learning with large-size images. Since the inputted image size is
small enough in this study, we can use the neural network only consisting of
the fully-connected layer. We can in principle perform the training with CNNs
as well, but we find that the results are not significantly different (see
Table V). To this end, we use the CNN listed in Tabs. 3 and 4. The learning
methods are the same as for the fully-connected layer. Considering that the
CNN is computationally more expensive than the fully-connected layer, the
benefit of using the CNN does not seem to be substantial, at least for the
system studied in this paper.

Of course, there are many choices and hyperparameters in deep learning.
Therefore, we do not mean to claim that there is no sufficient benefit from
using CNNs. However, if one needs better and more accurate training results,
skill and experience are required. In that occasion, one way to proceed is to
ask professionals to submit their ideas in a competition on sites such as
Kaggle [42], for example. Hopefully, such approach could help identify optimal
machine learning methods and hyperparameters. For example, the IceCube held a
competition on Kaggle [43].

Table 3: A CNN employed in this work to learn $E[\rho]$ and $E[v]$. The type of each layer is expressed with the language of Keras API [36], with which one can reproduce this neural network easily. In each layer, all arguments not mentioned are default values.  layer | type
---|---
input | Input(shape=(10, 20, 1))
1 | Conv2D(filters=32, kernel_size=3, activation=’relu’)
2 | Conv2D(filters=64, kernel_size=3, activation=’relu’)
3–8 | Conv2D(filters=64, kernel_size=4, padding=’same’, activation=’relu’)
9 | Conv2D(filters=128, kernel_size=3, activation=’relu’)
10–13 | Conv2D(filters=128, kernel_size=3, padding=’same’, activation=’relu’)
14 | Flatten()
15 | Dense(units=128, activation=’relu’)
output | Dense(units=1, activation=’sigmoid’)
Table 4: A CNN employed in this work to learn $\rho[v]$. The neural network has encoder-decoder structure. Layer 1–4 are reducing convolutional layers, and Layer 5–7 are non-reducing convolutional layers, where the size of images are (2, 12). We use Layer 8–11 as deconvolution layers to enlarge them up to (10, 20). In each layer, all arguments not mentioned are default values.  layer | type
---|---
input | Input(shape=(10, 20, 1))
1 | Conv2D(filters=32, kernel_size=3, activation=’relu’)
2 | Conv2D(filters=64, kernel_size=3, activation=’relu’)
3 | Conv2D(filters=128, kernel_size=3, activation=’relu’)
4 | Conv2D(filters=256, kernel_size=3, activation=’relu’)
5–7 | Conv2D(filters=256, kernel_size=4, padding=’same’, activation=’relu’)
8 | Conv2DTranspose(filters=256, kernel_size=3, activation=’relu’)
9 | Conv2DTranspose(filters=128, kernel_size=3, activation=’relu’)
10 | Conv2DTranspose(filters=64, kernel_size=3, activation=’relu’)
11 | Conv2DTranspose(filters=32, kernel_size=3, activation=’relu’)
12 | Flatten()
13 | Dense(units=200, activation=’softmax’)
output | Reshape(target_shape=(10, 20, 1))
Table 5: The mean absolute errors (MAEs) for each learning with the SHO and the RND external fields using the CNNs. The units are MeV for $E[\rho]$ and $E[v]$, while the MAE for $\rho[v]$ is dimensionless.  | SHO | RND
---|---|---
type | $E[\rho]$ | $E[v]$ | $E[\rho]$ | $E[v]$
$E_{\rm bin}$ | 0.0049 | 0.0055 | 0.0336 | 0.0245
$E_{\rm kin}$ | 0.0151 | 0.0079 | 0.1010 | 0.1134
$E_{\rm int}$ | 0.0119 | 0.0191 | 0.0484 | 0.1619
$E_{\rm pair}$ | 0.0179 | 0.0250 | 0.1505 | 0.1852
$E_{\rm ex}$ | 0.0220 | 0.0108 | 4.7059 | 0.0889
| $\rho[v]$ | $\rho[v]$
| 0.1092 | 0.3484