Nuclear binding energies in artiﬁcial neural networks Lin-Xing Zeng1Yu-Ying Yin1Xiao-Xu Dong1and Li-Sheng Geng2 1 3 4 1School of Physics Beihang University Beijing 102206 China

2025-05-06 0 0 567.19KB 18 页 10玖币

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Nuclear binding energies in artiﬁcial neural networks

Lin-Xing Zeng,1Yu-Ying Yin,1Xiao-Xu Dong,1and Li-Sheng Geng2, 1, 3, 4, ∗

1School of Physics, Beihang University, Beijing 102206, China

2Peng Huanwu Collaborative Center for Research and Education,

Beihang University, Beijing 100191, China

3Beijing Key Laboratory of Advanced Nuclear Materials and Physics,

Beihang University, Beijing 102206, China

4School of Physics and Microelectronics,

Zhengzhou University, Zhengzhou, Henan 450001, China

Abstract

The binding energy (BE) or mass is one of the most fundamental properties of an atomic nucleus. Precise

binding energies are vital inputs for many nuclear physics and nuclear astrophysics studies. However, due to

the complexity of atomic nuclei and of the non-perturbative strong interaction, up to now, no conventional

physical model can describe nuclear binding energies with a precision below 0.1 MeV, the accuracy needed

by nuclear astrophysical studies. In this work, artiﬁcial neural networks (ANNs), the so called “universal

approximators”, are used to calculate nuclear binding energies. We show that the ANN can describe all the

nuclei in AME2020 with a root-mean-square deviation (RMSD) around 0.2 MeV, which is better than the

best macroscopic-microscopic models, such as FRDM and WS4. The success of the ANN is mainly due

to the proper and essential input features we identify, which contain the most relevant physical informa-

tion, i.e., shell, paring, and isospin-asymmetry effects. We show that the well-trained ANN has excellent

extrapolation ability and can predict binding energies for those nuclei so far inaccessible experimentally. In

particular, we highlight the important role played by “feature engineering” for physical systems where data

are relatively scarce, such as nuclear binding energies.

∗E-mail: lisheng.geng@buaa.edu.cn

arXiv:2210.02906v1 [nucl-th] 6 Oct 2022

I. INTRODUCTION

The atomic nucleus is a quantum many-body system with an extremely complex structure [1].

As one of the most fundamental properties of atomic nuclei, binding energies (BE) can provide

crucial information on nuclear shapes [2], shell effects [3, 4] , pairing effects [5], and the dis-

appearance as well as emergence of magic numbers [4, 6]. In addition, binding energies are

essential inputs for superheavy nuclei syntheses [7] and nuclear astrophysical studies [8], e.g., the

r-process ([9, 10]), X-ray bursts [11], and etc. Therefore, reliable theoretical predictions and ex-

perimental measurements of nuclear binding energies have always been at the frontier of nuclear

physics [12–14].

In the latest atomic mass evaluation (AME 2020) [15], the masses of 3556 nuclei (including

measured and extrapolated) are compiled. However, various theoretical models predict that about

8000 to 10000 nuclei may exist [12, 13, 16, 17], including most of those relevant in nuclear ele-

ments syntheses. Therefore, reliable and accurate theoretical predictions are in urgent need. Some

of the most widely used theoretical models include the Weizsacker-Skyrme model (WS) [18–

22], the Relativistic Mean Field model (RMF) [17, 23], the Duﬂo-Zuker model (DZ) [24], the

Hartree-Fock-Bogoliubov model [25–28], the Finite-Range Droplet model (FRDM) [12, 29, 30],

and the RCHB [13] and DRHBc [31] models. Most of these models can describe the experimental

data with a root-mean-square deviation (RMSD) ranging from about 0.3 MeV to several MeV.

Among them, FRDM2012 [30] achieved an RMSD of 0.570 MeV while the Weizsacker-Skyrme

(WS4) [22] model gives the best description with an RMSD of 0.298 MeV. In general. the macro-

micro models, rather than the more “physical” microscopic models, perform better in describing

nuclear masses because their parameters are determined by ﬁtting to all the (then available) exper-

imental data.

In recent years, artiﬁcial neural networks (ANNs), as one of the most powerful machine learn-

ing methods, have been successfully applied in nuclear physics studies [9, 32], e.g., binding ener-

gies [33–37] , charge radii [38–41], α-decay half-lives [42] , β-decay half-lives [43], and ﬁssion

fragment yields [44–46].

The studies of nuclear binding energies (masses) can be divided into two categories, i.e., either

ﬁtting to the experimental data directly or to the residuals between experimental data and model

predictions. In Refs. [33, 37, 47–50], mass residuals are utilized to reﬁne the theoretical models.

In Refs. [33–35, 37], Bayesian neural networks are found to be able to describe nuclear binding

energies with an RMSD ranging from 0.266 to 0.850 MeV. The RMSD obtained in the WS4

supplemented with Light Gradient Boosting Machine (LightGBM) is 0.170 ±0.011 MeV [48].

The Bayesian machine learning (BML) method proposed in Ref. [51] achieves an RMSD of 84

keV, the ﬁrst crossing the 100 keV threshold.

However, there are fewer works that study the experimental data directly. In Refs. [36, 52],

feed-forward neural networks with different structures are explored. Ref. [36] yielded an RMSD

of 1.84 MeV for 1071 nuclei contained in AME2016 [53] as the test set. 1Ref. [52] applied the

data augmentation technique to expand the data set, and the RMSD decreased to 1.322 MeV for

the test set within the training data region and 1.495 MeV for the new nuclei beyond the training

data region. In Refs. [54, 55], mixed density networks with 12 physically motivated features [55]

or eight features constrained by the GK relation [54] are devised to describe nuclear mass excesses.

In the latter work [54], an RMSD of 0.316 MeV for the test set and 0.186 MeV for the training set

for nuclei with Z≥20 was achieved, whose performance is comparable to that of WS4 [22].

In this work, we develop an ANN with seven input features of most relevance. We ﬁnd that

among the 12 features studied in Ref. [55], only six of them are effective in our network. Mean-

while, we ﬁnd that taking GeLU [56] as the activation function enhances the predictive power of

the ANN. Our ANN provides a better description of nuclear binding energies than all the conven-

tional models and in addition shows good extrapolation ability.

This article is organized as follows. In section II, we explain how to construct the ANN and

determine the physically motivated input features. Results and discussions are presented in Section

III. A short summary and outlook is provided in Section IV.

II. THEORETICAL FORMALISM

In this section, we introduce the ANN and mass data we used in detail.

A. Artiﬁcial neural network

Generally speaking, an ANN is a supervised machine learning method which is also regarded as

a “universal approximator”. The ANN used in this work is a fully connected feed-forward neural

network, consisting of one input layer with seven features, two hidden layers, and one output layer

1The rather poor performance may be attributed to the fact that the MLP model has only been trained 800 epochs.

as shown in Fig. 1. The inputs Ijand outputs Ojof layer jare connected as follows

Oj=f(Wj·Ij+bj),(1)

where jruns over the input layer and the hidden layers, Wjare the weights, bjare the bias, and f

is the activation function to be speciﬁed. For the output layer, no activation function is needed.

Although in principle one could improve the description of BEs with either more hidden layers

or more nodes in each hidden layer, one often ends up with the over-ﬁtting problem. By trial and

error, we ﬁnd that with two hidden layers and about 800 parameters, our ANNs can well describe

the binding energies. For a ANN with Iinputs, two hidden layers, and one output, denoted as

[I, H1, H2, O], the number of parameters is (I+1)×H1+(H1+ 1)×H2+(H2 +1) ×O. Table

I lists the nodes and number of parameters of the different ANNs investigated in the present work.

Note that to better understand how the different input features affect the performance of ANNs, in

addition to the default ANN with seven features, we also study three other ANNs, where two, four,

and six features are used. For the activation function, we choose GeLU [56], which is found to

perform better than Tanh. For the loss function, we use the standard mean absolute error (MAE):

LOSS =PN

i=1 |BEth

i−BEexp

N.(2)

For numerical implementation, we use the optimized tensor library PYTORCH [57] and em-

ploy the Adam algorithm [58] with a learning rate 0.0001 and the decay constants 0.9and 0.999.

The weight matrices of our ANNs are initialized in PYTORCH with the same random seed.

TABLE I. Structure, number of parameters, and input features of the different ANNs studied in the present

work.

Model Structure Number of parameters Input features

ANN2 [2, 35, 19, 1] 809 Z,N

ANN4 [4, 35, 17, 1] 805 Z,N,ZEO,NEO

ANN6 [6, 32, 17, 1] 803 Z,N,ZEO,NEO,∆Z,∆N

ANN7 [7, 32, 16, 1] 801 Z,N,ZEO,NEO,∆Z,∆N, ASY

A supervised ANN maps inputs to the desired outputs. In the present case, the output is the

binding energy of a nucleus. As an atomic nucleus is completely determined by its proton and

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NuclearbindingenergiesinarticialneuralnetworksLin-XingZeng,1Yu-YingYin,1Xiao-XuDong,1andLi-ShengGeng2,1,3,4,1SchoolofPhysics,BeihangUniversity,Beijing102206,China2PengHuanwuCollaborativeCenterforResearchandEducation,BeihangUniversity,Beijing100191,China3BeijingKeyLaboratoryofAdvancedNuclearMateria...

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Nuclear binding energies in artiﬁcial neural networks Lin-Xing Zeng1Yu-Ying Yin1Xiao-Xu Dong1and Li-Sheng Geng2 1 3 4 1School of Physics Beihang University Beijing 102206 China

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