# Importance of physical information on the prediction of heavy-ion fusion
cross section with machine learning

Zhilong Li  College of Physics Science and Technology, Shenyang Normal
University, Shenyang 110034, Liaoning, China  School of Science, Huzhou
University, Huzhou 313000, China  Zepeng Gao  Sino-French Institute of Nuclear
Engineering and Technology, Sun Yat-sen University, Zhuhai 519082, China  Ling
Liu<EMAIL_ADDRESS>College of Physics Science and Technology, Shenyang
Normal University, Shenyang 110034, Liaoning, China  Yongjia Wang
<EMAIL_ADDRESS>School of Science, Huzhou University, Huzhou 313000,
China  Long Zhu  Sino-French Institute of Nuclear Engineering and Technology,
Sun Yat-sen University, Zhuhai 519082, China  Qingfeng Li  School of Science,
Huzhou University, Huzhou 313000, China  Institute of Modern Physics, Chinese
Academy of Science, Lanzhou 730000, China

(August 28, 2024)

###### Abstract

In this work, the Light Gradient Boosting Machine (LightGBM), which is a
modern decision tree based machine-learning algorithm, is used to study the
fusion cross section (CS) of heavy-ion reaction. Several basic quantities
(e.g., mass number and proton number of projectile and target) and the CS
obtained from phenomenological formula are fed into the LightGBM algorithm to
predict the CS. It is found that, on the validation set, the mean absolute
error (MAE) which measures the average magnitude of the absolute difference
between $log_{10}$ of the predicted CS and experimental CS is 0.129 by only
using the basic quantities as the input, this value is smaller than 0.154
obtained from the empirical coupled channel model. MAE can be further reduced
to 0.08 by including an physical-informed input feature. The MAE on the test
set (it consists of 280 data points from 18 reaction systems that not included
in the training set) is about 0.19 and 0.53 by including and excluding the
physical-informed feature, respectively. We further verify the LightGBM
predictions by comparing the CS of ${}^{40,48}{\rm Ca}$+${}^{78}{\rm Ni}$
obtained from the density-constrained time-dependent Hartree-Fock approach.
Our study demonstrates the importance of physical information in predicting
fusion cross section of heavy-ion reaction with machine learning.

##  I Introduction

The heavy ion fusion reaction is a process in which two colliding atomic
nuclei overcome the fusion barrier and then forms a excited compound nucleus.
It has important scientific and applied implications, including deeper
understanding of the properties and reactions of atomic nuclei, study of the
synthesis and properties of superheavy elements, exploring the origin of
elements, reference for the development and application of fusion energy. In
addition, it is one of the most important ways for us to explore the boundary
of nuclear landscape to get insights into the nuclear interactions. Therefore,
it has been a hot topic of research in the field of nuclear physics since 60
years ago back2014recent ; jiang2021heavy ; montagnoli2017recent ;
montagnoli2023recent ; sun2022effects . To perform heavy-ion reactions, a few
facilities have been established and some are under construction all over the
world, for example, the Cooler-Storage-Ring (CSR) xia2002heavy  and High
Intensity heavy-ion Accelerator Facility (HIAF) in China yang2013high ;
zhou2022status , rare-isotope beam accelerator complex (RAON) in Korea
hong2023status , the Facility for Rare Isotope Beams (FRIB) in the United
States, the Facility for An-tiproton and Ion Research (FAIR) in Gremany, the
Système de Production d’Ions Radioactifs (SPIRAL2) in France, the Radioactive
Isotope Beam Factory (RIBF) in Japan. By using various facilities, more than
1000 excitation functions for different projectile-target combinations have
been measured karpov2017nrv , but there are still many that have not been
measured, or have large errors.

The heavy-ion fusion reaction is a complex quantum many-body process, and it
involves the mutual coupling of nuclear structure and reaction dynamics
frobrich1984fusion ; balantekin1998quantum ; dasgupta1998measuring ;
canto2006fusion ; diaz2007effects ; zhang2011heavy ; canto2015recent ;
cheng2022non . Thus it is very difficult to study strictly with first
principles. Several theoretical models or empirical formulas have been
proposed to study the fusion cross section (CS) which is one of the most
important observables for studying heavy-ion reactions. Such as the coupled
channel calculations hagino1999program ; wen2020near , the time-dependent
Hartree-Fock (TDHF) theory plus solving Schrodinger equation simenel2018heavy
; simenel2012nuclear , and empirical coupled channel model wang2017systematics
; gautam2022comprehensive ; wong1973interaction ; lwin2017applicability ;
PhysRevC.103.024607 ; hill1953nuclear ; zhu2014orientation . But the cross
sections calculated by these methods are not completely compatible with the
experimental data.

In recent years, Machine learning (ML) methods have been widely and
successfully applied for analyzing data in many branches of science, such as
physics (see, e.g., Refs. carleo2019machine ; he2023machine ; he2023high ). In
the field of nuclear physics, ML has shown a strong ability in the study of
heavy ion collisions wang2023machine ; li2022determination ; pang2018equation
; wei2023event ; wang2021finding , properties of strongly interacting QCD
matter zhou2023exploring ; kuttan2021equation ; li2023deep ; du2022jet ,
nuclear spallation and projectile fragmentation reactions peng2022bayesian ;
ma2022precise ; song2023image , nuclear fission PhysRevLett.123.122501 ;
wang2021optimizing ; wang2022bayesian ; song2023image , nuclear masses
niu2018nuclear ; gao2021machine ; ming2022nuclear ; wu2021nuclear ;
zhao2022new ; le2023nuclear , $\beta$-decay half-lives and energy peng2022beta
; gao2023investigation ; munoz2023predicting , $\alpha$-decayli2022deep , the
charge radius of atomic nuclei dong2023nuclear ; su2023progress ;
wu2020calculation ; dong2022novel , nuclear density distribution
yang2021taming ; shang2022prediction , and the evaporation residual cross
sections for superheavy nuclei zhao2022improvement . Recently, a novel
artificial intelligence approach has been applied to model cross section data
dell2022modeling , in which phenomenological formulas for the calculation of
CS are derived based on a hybridization of genetic programming and artificial
neural networks. The derived phenomenological formulas can qualitatively
reproduce the trend but not the absolute value of the CS. ML is a rapidly
growing and flourishing field. Nowadays, a diverse array of ML algorithm has
been developed and continue to be refined to cover a wide variety of data
types and tasks. It is interesting to study if other ML algorithm can also
refine models that used in the calculations of CS, and more importantly,
whether physical insights into the heavy-ion fusion reaction can be derived.

The rest of this paper is organized as follows. In Section. II, we introduce
the methodology that we use in the present work, including the machine
learning algorithm, the dataset and the input features. The CS obtained with
machine learning algorithm are discussed in detail in Sect. III. The
conclusions are given in Sect. IV.

##  II Methodology

In the present work, the prediction of heavy-ion fusion cross section is a
supervised task and requires a machine learning algorithm, and a set of
labelled data with input and output variables. In this section, we introduce
briefly these items.

The machine learning algorithm we used in the present work is the Light
Gradient Boosting Machine (LightGBM) which was developed by Microsoft in 2016,
it is a gradient boosting framework that uses tree based learning algorithms
NIPS2017_6449f44a . LightGBM is becoming increasingly popular due to its
advantages including (1) faster training speed and higher efficiency, (2)
lower memory usage, (3) better accuracy, (4) support of parallel and graphics
processing unit learning, and (5) capability of handling large-scale data.
Moreover, LightGBM is a white-box model and has an excellent degree of
explainability because of its decision-tree-based nature. This is important
for studying a real physical problem, as explainability may improve our
knowledge about the relationship between the input features and the output. In
our previous works, the strong ability of LightGBM to refine nuclear mass
models gao2021machine  and mine physical information has been demonstrated
WANG2022137508 ; li2021application ; li2020application . Thus it is also
employed in the present work, and parameters in LightGBM are set to their
default values; we have checked that the results are insensitive to parameters
in LightGBM.

Table 1: Training set.  System | Data | Energy range | System | Data | Energy range | System | Data | Energy range
---|---|---|---|---|---|---|---|---
| points | (MeV) |  | pointts | (MeV) |  | pointts | (MeV)
${}^{12}{\rm C}$+${}^{89}{\rm Y}$ | 9 | 26-41 |  ${}^{19}{\rm F}$+${}^{93}{\rm Nb}$ | 12 | 43-60 |  ${}^{32}{\rm S}$+${}^{184}{\rm W}$ | 7 | 118-144
${}^{12}{\rm C}$+${}^{92}{\rm Zr}$ | 16 | 27-45 |  ${}^{19}{\rm F}$+${}^{139}{\rm La}$ | 13 | 61-115 |  ${}^{32}{\rm S}$+${}^{208}{\rm Pb}$ | 32 | 139-184
${}^{12}{\rm C}$+${}^{144}{\rm Sm}$ | 15 | 41-70 |  ${}^{19}{\rm F}$+${}^{208}{\rm Pb}$ | 33 | 73-145 |  ${}^{33}{\rm S}$+${}^{90}{\rm Zr}$ | 13 | 74-97
${}^{12}{\rm C}$+${}^{152}{\rm Sm}$ | 6 | 42-58 |  ${}^{27}{\rm Al}$+${}^{45}{\rm Sc}$ | 16 | 31-51 |  ${}^{33}{\rm S}$+${}^{91}{\rm Zr}$ | 16 | 72-97
${}^{12}{\rm C}$+${}^{154}{\rm Sm}$ | 12 | 43-58 |  ${}^{19}{\rm F}$+${}^{209}{\rm Bi}$ | 10 | 80-116 |  ${}^{33}{\rm S}$+${}^{92}{\rm Zr}$ | 16 | 72-97
${}^{12}{\rm C}$+${}^{181}{\rm Ta}$ | 12 | 50-92 |  ${}^{23}{\rm Na}$+${}^{48}{\rm Ti}$ | 11 | 32-46 |  ${}^{34}{\rm S}$+${}^{24}{\rm Mg}$ | 11 | 24-32
${}^{12}{\rm C}$+${}^{194}{\rm Pt}$ | 10 | 50-69 |  ${}^{27}{\rm Al}$+${}^{70}{\rm Ge}$ | 21 | 50-60 |  ${}^{34}{\rm S}$+${}^{25}{\rm Mg}$ | 19 | 24-33
${}^{12}{\rm C}$+${}^{198}{\rm Pt}$ | 10 | 50-69 |  ${}^{27}{\rm Al}$+${}^{72}{\rm Ge}$ | 17 | 49-61 |  ${}^{34}{\rm S}$+${}^{26}{\rm Mg}$ | 21 | 24-35
${}^{40}{\rm Ca}$+${}^{96}{\rm Zr}$ | 56 | 87-113 |  ${}^{27}{\rm Al}$+${}^{73}{\rm Ge}$ | 18 | 50-62 |  ${}^{34}{\rm S}$+${}^{89}{\rm Y}$ | 32 | 72-92
${}^{46}{\rm Ti}$+${}^{90}{\rm Zr}$ | 10 | 99-120 |  ${}^{27}{\rm Al}$+${}^{74}{\rm Ge}$ | 18 | 49-62 |  ${}^{34}{\rm S}$+${}^{168}{\rm Er}$ | 41 | 111-164
${}^{12}{\rm C}$+${}^{204}{\rm Pb}$ | 12 | 50-85 |  ${}^{27}{\rm Al}$+${}^{76}{\rm Ge}$ | 19 | 48-62 |  ${}^{35}{\rm Cl}$+${}^{24}{\rm Mg}$ | 12 | 26-36
${}^{12}{\rm C}$+${}^{206}{\rm Pb}$ | 8 | 54-81 |  ${}^{27}{\rm Al}$+${}^{197}{\rm Au}$ | 29 | 107-151 |  ${}^{35}{\rm Cl}$+${}^{25}{\rm Mg}$ | 12 | 27-38
${}^{12}{\rm C}$+${}^{208}{\rm Pb}$ | 12 | 54-89 |  ${}^{28}{\rm Si}$+${}^{28}{\rm Si}$ | 38 | 22-68 |  ${}^{35}{\rm Cl}$+${}^{26}{\rm Mg}$ | 12 | 27-38
${}^{12}{\rm C}$+${}^{237}{\rm Np}$ | 11 | 56-77 |  ${}^{28}{\rm Si}$+${}^{30}{\rm Si}$ | 22 | 22-49 |  ${}^{35}{\rm Cl}$+${}^{27}{\rm Al}$ | 11 | 30-75
${}^{12}{\rm C}$+${}^{238}{\rm U}$ | 21 | 60-119 |  ${}^{28}{\rm Si}$+${}^{68}{\rm Zn}$ | 12 | 50-71 |  ${}^{35}{\rm Cl}$+${}^{54}{\rm Cr}$ | 17 | 52-79
${}^{14}{\rm N}$+${}^{59}{\rm Co}$ | 8 | 25-45 |  ${}^{28}{\rm Si}$+${}^{90}{\rm Zr}$ | 13 | 65-93 |  ${}^{35}{\rm Cl}$+${}^{52}{\rm Cr}$ | 16 | 53-78
${}^{14}{\rm N}$+${}^{232}{\rm Th}$ | 9 | 67-87 |  ${}^{28}{\rm Si}$+${}^{92}{\rm Zr}$ | 22 | 65-89 |  ${}^{35}{\rm Cl}$+${}^{51}{\rm V}$ | 18 | 49-81
${}^{14}{\rm N}$+${}^{238}{\rm U}$ | 17 | 72-138 |  ${}^{28}{\rm Si}$+${}^{94}{\rm Zr}$ | 15 | 63-95 |  ${}^{35}{\rm Cl}$+${}^{50}{\rm Ti}$ | 22 | 47-75
${}^{15}{\rm N}$+${}^{56}{\rm Fe}$ | 10 | 25-37 |  ${}^{28}{\rm Si}$+${}^{93}{\rm Nb}$ | 13 | 68-92 |  ${}^{35}{\rm Cl}$+${}^{54}{\rm Fe}$ | 18 | 55-82
${}^{15}{\rm N}$+${}^{209}{\rm Bi}$ | 14 | 63-83 |  ${}^{28}{\rm Si}$+${}^{142}{\rm Ce}$ | 20 | 87-121 |  ${}^{35}{\rm Cl}$+${}^{58}{\rm Ni}$ | 9 | 60-88
${}^{16}{\rm O}$+${}^{58}{\rm Ni}$ | 6 | 31-48 |  ${}^{28}{\rm Si}$+${}^{154}{\rm Sm}$ | 21 | 90-123 |  ${}^{35}{\rm Cl}$+${}^{60}{\rm Ni}$ | 8 | 59-82
${}^{16}{\rm O}$+${}^{62}{\rm Ni}$ | 8 | 31-48 |  ${}^{28}{\rm Si}$+${}^{164}{\rm Er}$ | 12 | 108-151 |  ${}^{35}{\rm Cl}$+${}^{62}{\rm Ni}$ | 15 | 59-109
${}^{16}{\rm O}$+${}^{59}{\rm Co}$ | 4 | 26-40 |  ${}^{28}{\rm Si}$+${}^{198}{\rm Pt}$ | 11 | 114-167 |  ${}^{35}{\rm Cl}$+${}^{64}{\rm Ni}$ | 9 | 58-84
${}^{16}{\rm O}$+${}^{70}{\rm Ge}$ | 26 | 31-49 |  ${}^{28}{\rm Si}$+${}^{208}{\rm Pb}$ | 35 | 120-160 |  ${}^{35}{\rm Cl}$+${}^{90}{\rm Zr}$ | 7 | 86-119
${}^{16}{\rm O}$+${}^{72}{\rm Ge}$ | 29 | 31-52 |  ${}^{29}{\rm Al}$+${}^{197}{\rm Au}$ | 19 | 109-165 |  ${}^{35}{\rm Cl}$+${}^{92}{\rm Zr}$ | 20 | 77-98
${}^{16}{\rm O}$+${}^{73}{\rm Ge}$ | 27 | 32-53 |  ${}^{31}{\rm Al}$+${}^{197}{\rm Au}$ | 20 | 106-165 |  ${}^{36}{\rm S}$+${}^{48}{\rm Ca}$ | 25 | 36-61
${}^{16}{\rm O}$+${}^{74}{\rm Ge}$ | 29 | 31-56 |  ${}^{30}{\rm Si}$+${}^{30}{\rm Si}$ | 26 | 26-50 |  ${}^{36}{\rm S}$+${}^{64}{\rm Ni}$ | 24 | 51-83
${}^{16}{\rm O}$+${}^{76}{\rm Ge}$ | 32 | 31-56 |  ${}^{30}{\rm Si}$+${}^{238}{\rm U}$ | 9 | 124-170 |  ${}^{36}{\rm S}$+${}^{90}{\rm Zr}$ | 28 | 72-90
${}^{16}{\rm O}$+${}^{92}{\rm Zr}$ | 36 | 37-70 |  ${}^{32}{\rm S}$+${}^{27}{\rm Al}$ | 30 | 27-61 |  ${}^{36}{\rm S}$+${}^{96}{\rm Zr}$ | 33 | 71-88
${}^{16}{\rm O}$+${}^{112}{\rm Cd}$ | 14 | 45-60 |  ${}^{32}{\rm S}$+${}^{24}{\rm Mg}$ | 15 | 26-33 |  ${}^{36}{\rm S}$+${}^{92}{\rm Mo}$ | 5 | 76-86
${}^{16}{\rm O}$+${}^{144}{\rm Nd}$ | 13 | 54-81 |  ${}^{32}{\rm S}$+${}^{25}{\rm Mg}$ | 19 | 24-33 |  ${}^{36}{\rm S}$+${}^{94}{\rm Mo}$ | 8 | 75-87
${}^{16}{\rm O}$+${}^{148}{\rm Nd}$ | 6 | 58-72 |  ${}^{32}{\rm S}$+${}^{26}{\rm Mg}$ | 20 | 24-34 |  ${}^{36}{\rm S}$+${}^{96}{\rm Mo}$ | 7 | 71-88
${}^{16}{\rm O}$+${}^{150}{\rm Nd}$ | 7 | 55-73 |  ${}^{32}{\rm S}$+${}^{40}{\rm Ca}$ | 7 | 44-66 |  ${}^{36}{\rm S}$+${}^{98}{\rm Mo}$ | 7 | 75-88
${}^{16}{\rm O}$+${}^{144}{\rm Sm}$ | 28 | 55-91 |  ${}^{32}{\rm S}$+${}^{48}{\rm Ca}$ | 21 | 35-55 |  ${}^{36}{\rm S}$+${}^{100}{\rm Mo}$ | 9 | 74-90
${}^{16}{\rm O}$+${}^{147}{\rm Sm}$ | 8 | 55-68 |  ${}^{32}{\rm S}$+${}^{64}{\rm Ni}$ | 13 | 52-76 |  ${}^{36}{\rm S}$+${}^{100}{\rm Ru}$ | 8 | 77-90
${}^{16}{\rm O}$+${}^{148}{\rm Sm}$ | 25 | 53-75 |  ${}^{32}{\rm S}$+${}^{89}{\rm Y}$ | 23 | 72-96 |  ${}^{36}{\rm S}$+${}^{101}{\rm Ru}$ | 8 | 77-90
${}^{16}{\rm O}$+${}^{149}{\rm Sm}$ | 10 | 55-68 |  ${}^{32}{\rm S}$+${}^{90}{\rm Zr}$ | 41 | 72-96 |  ${}^{36}{\rm S}$+${}^{102}{\rm Ru}$ | 7 | 78-89
${}^{16}{\rm O}$+${}^{150}{\rm Sm}$ | 8 | 54-68 |  ${}^{32}{\rm S}$+${}^{94}{\rm Zr}$ | 50 | 70-95 |  ${}^{36}{\rm S}$+${}^{103}{\rm Rh}$ | 4 | 79-89
${}^{16}{\rm O}$+${}^{152}{\rm Sm}$ | 7 | 54-70 |  ${}^{32}{\rm S}$+${}^{96}{\rm Zr}$ | 50 | 69-98 |  ${}^{36}{\rm S}$+${}^{104}{\rm Pd}$ | 4 | 81-90
${}^{16}{\rm O}$+${}^{154}{\rm Sm}$ | 39 | 52-100 |  ${}^{32}{\rm S}$+${}^{94}{\rm Mo}$ | 9 | 76-90 |  ${}^{36}{\rm S}$+${}^{104}{\rm Ru}$ | 10 | 77-89
${}^{16}{\rm O}$+${}^{166}{\rm Er}$ | 17 | 58-94 |  ${}^{32}{\rm S}$+${}^{96}{\rm Mo}$ | 9 | 74-90 |  ${}^{36}{\rm S}$+${}^{105}{\rm Pd}$ | 4 | 81-90
${}^{16}{\rm O}$+${}^{176}{\rm Yb}$ | 15 | 59-89 |  ${}^{32}{\rm S}$+${}^{98}{\rm Mo}$ | 10 | 73-91 |  ${}^{36}{\rm S}$+${}^{106}{\rm Pd}$ | 5 | 80-90
${}^{16}{\rm O}$+${}^{186}{\rm W}$ | 35 | 62-83 |  ${}^{32}{\rm S}$+${}^{100}{\rm Mo}$ | 11 | 72-91 |  ${}^{36}{\rm S}$+${}^{108}{\rm Pd}$ | 6 | 79-95
${}^{16}{\rm O}$+${}^{204}{\rm Pb}$ | 12 | 66-98 |  ${}^{32}{\rm S}$+${}^{100}{\rm Ru}$ | 8 | 77-91 |  ${}^{36}{\rm S}$+${}^{110}{\rm Pd}$ | 6 | 79-95
${}^{16}{\rm O}$+${}^{208}{\rm Pb}$ | 38 | 65-110 |  ${}^{32}{\rm S}$+${}^{101}{\rm Ru}$ | 6 | 80-91 |  ${}^{36}{\rm S}$+${}^{204}{\rm Pb}$ | 26 | 138-179
${}^{16}{\rm O}$+${}^{209}{\rm Bi}$ | 14 | 71-95 |  ${}^{32}{\rm S}$+${}^{102}{\rm Ru}$ | 9 | 76-92 |  ${}^{36}{\rm S}$+${}^{238}{\rm U}$ | 5 | 147-173
${}^{16}{\rm O}$+${}^{232}{\rm Th}$ | 12 | 70-150 |  ${}^{32}{\rm S}$+${}^{103}{\rm Rh}$ | 8 | 78-92 |  ${}^{37}{\rm Cl}$+${}^{24}{\rm Mg}$ | 12 | 25-35
${}^{16}{\rm O}$+${}^{238}{\rm U}$ | 14 | 80-157 |  ${}^{32}{\rm S}$+${}^{104}{\rm Pd}$ | 6 | 81-92 |  ${}^{37}{\rm Cl}$+${}^{25}{\rm Mg}$ | 12 | 26-36
${}^{17}{\rm O}$+${}^{144}{\rm Sm}$ | 25 | 54-90 |  ${}^{32}{\rm S}$+${}^{104}{\rm Ru}$ | 9 | 77-92 |  ${}^{37}{\rm Cl}$+${}^{26}{\rm Mg}$ | 12 | 26-38
${}^{18}{\rm O}$+${}^{44}{\rm Ca}$ | 17 | 19-43 |  ${}^{32}{\rm S}$+${}^{105}{\rm Pd}$ | 5 | 81-92 |  ${}^{37}{\rm Cl}$+${}^{59}{\rm Co}$ | 13 | 55-71
${}^{18}{\rm O}$+${}^{74}{\rm Ge}$ | 26 | 30-50 |  ${}^{32}{\rm S}$+${}^{106}{\rm Pd}$ | 7 | 80-92 |  ${}^{37}{\rm Cl}$+${}^{70}{\rm Ge}$ | 12 | 62-76
${}^{18}{\rm O}$+${}^{112}{\rm Sn}$ | 17 | 42-78 |  ${}^{32}{\rm S}$+${}^{108}{\rm Pd}$ | 8 | 79-92 |  ${}^{37}{\rm Cl}$+${}^{72}{\rm Ge}$ | 14 | 63-77
${}^{18}{\rm O}$+${}^{118}{\rm Sn}$ | 17 | 44-79 |  ${}^{32}{\rm S}$+${}^{110}{\rm Pd}$ | 9 | 78-93 |  ${}^{37}{\rm Cl}$+${}^{73}{\rm Ge}$ | 12 | 63-77
${}^{18}{\rm O}$+${}^{124}{\rm Sn}$ | 17 | 44-79 |  ${}^{32}{\rm S}$+${}^{138}{\rm Ba}$ | 13 | 99-135 |  ${}^{37}{\rm Cl}$+${}^{74}{\rm Ge}$ | 9 | 63-75
${}^{18}{\rm O}$+${}^{192}{\rm Os}$ | 7 | 73-113 |  ${}^{32}{\rm S}$+${}^{154}{\rm Sm}$ | 9 | 101-128 |  ${}^{37}{\rm Cl}$+${}^{76}{\rm Ge}$ | 14 | 62-76
${}^{18}{\rm O}$+${}^{208}{\rm Pb}$ | 18 | 68-95 |  ${}^{32}{\rm S}$+${}^{182}{\rm W}$ | 15 | 122-170 |  ${}^{37}{\rm Cl}$+${}^{93}{\rm Nb}$ | 7 | 81-93
${}^{37}{\rm Cl}$+${}^{98}{\rm Mo}$ | 7 | 82-95 |  ${}^{40}{\rm Ca}$+${}^{194}{\rm Pt}$ | 31 | 159-200 |  ${}^{58}{\rm Ni}$+${}^{54}{\rm Fe}$ | 25 | 85-110
${}^{37}{\rm Cl}$+${}^{100}{\rm Mo}$ | 8 | 83-95 |  ${}^{40}{\rm Ca}$+${}^{197}{\rm Au}$ | 10 | 162-223 |  ${}^{58}{\rm Ni}$+${}^{58}{\rm Ni}$ | 17 | 95-109
${}^{40}{\rm Ar}$+${}^{112}{\rm Sn}$ | 15 | 96-126 |  ${}^{40}{\rm Ca}$+${}^{208}{\rm Pb}$ | 5 | 169-208 |  ${}^{58}{\rm Ni}$+${}^{64}{\rm Ni}$ | 27 | 89-110
${}^{40}{\rm Ar}$+${}^{116}{\rm Sn}$ | 14 | 95-128 |  ${}^{48}{\rm Ca}$+${}^{48}{\rm Ca}$ | 29 | 46-63 |  ${}^{58}{\rm Ni}$+${}^{124}{\rm Sn}$ | 6 | 143-162
${}^{40}{\rm Ar}$+${}^{122}{\rm Sn}$ | 17 | 94-130 |  ${}^{48}{\rm Ca}$+${}^{90}{\rm Zr}$ | 33 | 90-115 |  ${}^{64}{\rm Ni}$+${}^{64}{\rm Ni}$ | 13 | 89-107
${}^{40}{\rm Ar}$+${}^{144}{\rm Sm}$ | 11 | 116-140 |  ${}^{48}{\rm Ca}$+${}^{96}{\rm Zr}$ | 38 | 88-113 |  ${}^{64}{\rm Ni}$+${}^{124}{\rm Sn}$ | 10 | 144-170
${}^{40}{\rm Ar}$+${}^{148}{\rm Sm}$ | 12 | 112-150 |  ${}^{48}{\rm Ca}$+${}^{154}{\rm Sm}$ | 51 | 125-192 |  ${}^{124}{\rm Sn}$+${}^{40}{\rm Ca}$ | 22 | 107-150
${}^{40}{\rm Ar}$+${}^{154}{\rm Sm}$ | 12 | 108-145 |  ${}^{48}{\rm Ca}$+${}^{197}{\rm Au}$ | 8 | 164-245 |  ${}^{132}{\rm Sn}$+${}^{40}{\rm Ca}$ | 15 | 109-136
${}^{40}{\rm Ca}$+${}^{40}{\rm Ca}$ | 15 | 49-70 |  ${}^{48}{\rm Ca}$+${}^{208}{\rm Pb}$ | 5 | 171-206 |  ${}^{134}{\rm Te}$+${}^{40}{\rm Ca}$ | 10 | 115-140
${}^{40}{\rm Ca}$+${}^{44}{\rm Ca}$ | 15 | 47-75 |  ${}^{48}{\rm Ti}$+${}^{58}{\rm Fe}$ | 27 | 64-85 |  ${}^{124}{\rm Sn}$+${}^{48}{\rm Ca}$ | 13 | 109-170
${}^{40}{\rm Ca}$+${}^{46}{\rm Ti}$ | 25 | 54-80 |  ${}^{48}{\rm Ti}$+${}^{58}{\rm Ni}$ | 14 | 72-91 |  ${}^{132}{\rm Sn}$+${}^{48}{\rm Ca}$ | 12 | 109-170
${}^{40}{\rm Ca}$+${}^{48}{\rm Ca}$ | 14 | 48-67 |  ${}^{48}{\rm Ti}$+${}^{60}{\rm Ni}$ | 15 | 72-93 |  ${}^{132}{\rm Sn}$+${}^{58}{\rm Ni}$ | 10 | 152-200
${}^{40}{\rm Ca}$+${}^{48}{\rm Ti}$ | 28 | 53-82 |  ${}^{46}{\rm Ti}$+${}^{64}{\rm Ni}$ | 15 | 71-96 |  ${}^{132}{\rm Sn}$+${}^{64}{\rm Ni}$ | 9 | 147-200
${}^{40}{\rm Ca}$+${}^{50}{\rm Ti}$ | 30 | 53-83 |  ${}^{48}{\rm Ti}$+${}^{64}{\rm Ni}$ | 18 | 71-96 |  ${}^{208}{\rm Pb}$+${}^{26}{\rm Mg}$ | 7 | 109-190
${}^{40}{\rm Ca}$+${}^{58}{\rm Ni}$ | 21 | 67-95 |  ${}^{48}{\rm Ti}$+${}^{122}{\rm Sn}$ | 19 | 121-160 |  ${}^{50}{\rm Ti}$+${}^{90}{\rm Zr}$ | 8 | 100-118
${}^{40}{\rm Ca}$+${}^{64}{\rm Ni}$ | 26 | 63-94 |  ${}^{50}{\rm Ti}$+${}^{60}{\rm Ni}$ | 14 | 72-91 |  ${}^{40}{\rm Ca}$+${}^{192}{\rm Os}$ | 35 | 151-200
${}^{40}{\rm Ca}$+${}^{90}{\rm Zr}$ | 40 | 90-120 |  ${}^{46}{\rm Ti}$+${}^{93}{\rm Nb}$ | 10 | 100-120 |  |  |
${}^{40}{\rm Ca}$+${}^{94}{\rm Zr}$ | 59 | 88-115 |  ${}^{50}{\rm Ti}$+${}^{93}{\rm Nb}$ | 9 | 100-120 | Total systems | 220 |
|  |  |  |  |  | Total points | 3635 |
Table 2: Test set. System | Data | energy range | Ref. |
---|---|---|---|---
| points | (MeV) |  |
35Cl + 130Te | 16 | 90-125 | sahoo2020role  |
37Cl + 130Te | 17 | 90-125 | sahoo2019sub  |
37Cl + 68Zn | 11 | 60-90 | chauhan2020evaporation  |
16O + 61Ni | 8 | 25-41 | deb2022investigation  |
18O + 61Ni | 16 | 25-41 | deb2022investigation  |
18O + 62Ni | 15 | 25-42 | deb2022investigation  |
18O + 116Sn | 21 | 40-75 | kalita2021role  |
30Si + 156Gd | 14 | 90-116 | prajapat2022fusion  |
28Si + 100Mo | 13 | 65-98 | stefanini2021new  |
36S + 50Ti | 23 | 40-60 | colucci2019sub  |
36S + 51V | 22 | 40-60 | colucci2019sub  |
12C + 182W | 14 | 40-80 | sanila2022fusion  |
12C + 184W | 14 | 40-80 | sanila2022fusion  |
12C + 186W | 14 | 40-80 | sanila2022fusion  |
40Ca + 92Zr | 16 | 89-108 | stefanini2017new  |
48Ca + 116Cd | 20 | 104-130 | del2023influence  |
48Ca + 118Sn | 16 | 104-130 | del2023influence  |
48Ca + 120Te | 10 | 104-130 | del2023influence  |
Total systems | 18 |  |  |
Total points | 280 |  |  |

Data used in this work consists of three parts, the training set, the
validation set, and test set. Training set is used to adjust the parameters in
LightGBM, usually validation set is a part of training set which is used to
monitor and avoid overfitting. Test set is used to evaluate the actual
predictive power of LightGBM on unseen data. The training set is made of 3635
experimental data points from 220 reaction systems collected in
Ref.wang2017systematics , it will be randomly split into the training set and
validation set with a certain ratio. These experimental data were measured
before 2016. We note that the training set is built considering systems with
$12\leq Z_{1}\leq 48$ and $24\leq Z_{2}\leq 208$. In this way, we can neglect
too heavy systems, for which fusion-fission and quasi-fission can be the
dominant reaction modes, and also too light systems, where the presence of
break-up and transfer reactions complicate the analysis. The test set consists
of 280 data points from 18 reaction systems measured in recent experiments
sahoo2020role ; sahoo2019sub ; chauhan2020evaporation ; deb2022investigation ;
kalita2021role ; prajapat2022fusion ; stefanini2021new ; colucci2019sub ;
sanila2022fusion ; stefanini2017new ; del2023influence . We note here that
these 18 reaction systems in the test set have not appeared in the training
set. The reaction system, number of data points for each system, and the
energy range of the training and test sets are listed in Tab.LABEL:table1 and
2, respectively.

Usually, fusion cross sections at different energies varying over several
orders of magnitude, thus the logarithm of fusion cross section is set as the
output of LightGBM. The input quantities, including center-of-mass energy, the
charge, neutron, mass numbers, and binding energy of projectile, target, and
the compound nuclei, the fusion Q-value, as well as the one (two)-proton
(neutron) separation energies of the compound nuclei, are listed in Tab.3.
These quantities are chosen because they are basic features (BF) of a nucleus
and expected to relate to the fusion process. Usually, adding more related
features may benefit the performance of the trained ML model. We have tried to
add more features and different combinations of these BF quantities, but the
performance of the trained ML model is only slightly improved. To introduce
physical information related to heavy-ion fusion process, the fusion cross
sections calculated with the empirical coupled channel (ECC) model and Wong
formula are also used. Details of ECC model and the Wong formula can be found
in Ref. wang2017systematics . In addition, we introduce a simplified empirical
quantity Z1Z2/Ec.m. This quantity includes some important factors of heavy-ion
fusion reaction, e.g., Z1Z2 relates to the Coulomb barrier. These quantities
are physical-informed or physical-guided features, which are listed in Tab.4.

Table 3: Selection of basic features (BF). features |  Description
---|---
Ec.m |  collision center-of-mass energy (MeV)
Z1 |  charge of the first reaction partner
N1 |  number of neutrons of the first reaction
|  partner
A1 |  mass number of the first reaction partner
Z2 |  charge of the second reaction partner
N2 |  number of neutrons of the second reaction
|  partner
A2 |  mass number of the second reaction partner
Z3 |  charge of the compound nucleus
N3 |  number of neutrons of the compound
|  nucleus
A3 |  mass number of the compound nucleus
B1 |  binding energy of the first reaction partner
B2 |  binding energy of the second reaction
|  partner
B3 |  binding energy of the compound nucleus
Q |  fusion Q-value (MeV)
Sp |  one-proton separation energy of the
|  compound nucleus (MeV)
S2p |  two-proton separation energy of the
|  compound nucleus (MeV)
Sn |  one-neutron separation energy of the
|  compound nucleus (MeV)
S2n |  two-neutron separation energy of the
|  compound nucleus (MeV)
Table 4:  Physical-informed quantities features |  Description
---|---
sig_ECC |  $\log_{10}$ of CS obtained by ECC model
sig_W |  $\log_{10}$ of CS obtained by Wong formula
sig_E |  Z1Z2/Ec.m
Table 5:  Different modes with different input features. Mode name |  Input features
---|---
Mode_BF |  BF
Mode_ECC |  BF+sig_ECC
Mode_W |  BF+sig_W
Mode_E |  BF+sig_E

The main aim of this work is to establish the relationship between the
characteristic quantities of fusion reaction and the fusion cross section by
learning the training set with LightGBM. In the process of training, four
different input feature combinations are used and given in Tab.5. Mode_BF
represents the input feature is comprised of the 18 basic quantities as listed
in Tab.3. Mode_ECC represents the input feature is comprised of the 18 basic
quantities and sig_ECC. Mode_W and Mode_E represent the input features are
comprised of the 18 basic quantities as well as, sig_W and sig_E,
respectively. By comparing the performances of these different modes, one can
infer the importance of physical-informed features. The performance of ML
algorithm can be quantitatively evaluated via the mean absolute error (MAE),

$\text{ MAE
}=\frac{1}{N}\sum_{i=1}^{N}\mid\log_{10}\left(\sigma_{pred}\right)-\log_{10}\left(\sigma_{exp}\right)\mid$
(1)

Here N is the number of tested data points, $\sigma_{pred}$ and $\sigma_{exp}$
are the predicted and experimental cross section, respectively.

##  III Results

###  III.1 Performance on the training set

####  III.1.1 Effect of the size of training set

Figure 1:  (Color online) MAE on the validation data obtained with Mode_E from
100 runs. In each run, the 3635 data points were randomly split into training
and validation data sets at a ratio of 4:1 (blue), 1:1 (green), and 1:4
(orange) Figure 2:  (Color online) Density distribution of MAE on the
validation set. The results from 500 runs for each set are displayed. Dashed
lines denote a Gaussian fit to the distribution. The mean values and the
standard deviation of MAE values are 0.167, 0.103 and 0.081, and 0.008, 0.004
and 0.005 for the three sets with different ratios of training to test,
respectively. Figure 3:  (Color online) MAE on the validation data obtained
with Mode_E is plotted as a function of the percentage of the training set.

In this section, the effect of the ratio between the training set and the
validation set on the performance of LightGMB in the Mode_E is studied. For
this purpose, the 3635 data points listed in Table LABEL:table1 are randomly
split into training and validation sets 500 times with each ratio (i.e., 1:4,
1:1, and 4:1). The MAE and its density distribution are plotted in Figs. 1 and
2. As observed in Fig. 1, the MAE is the largest of all when the ratio of
training size to validation size is 1:4. The MAE for 2908 (the validation set,
i.e., 80% of the whole training set) data points predicted by LightGBM with
727 data points was 0.167$\pm$0.041, which means LightGBM can reproduce
experimental data within a factor of 100.167=1.47. This is comparable to some
physical models. With the training data set built from 2908 data points and
the remaining 727 data points constitute the validation data set, the MAE
reduces to 0.081$\pm$0.005, which is better than that of many physical models.
The value of MAE fluctuates in different runs, this is because when the
training and validation data sets are randomly selected, it may have a
probability that all data points from one reaction system are not included in
the training set, the prediction of CS for this system is a challenging task,
and resulting in a larger MAE. If a few data points of a reaction system are
selected in the training set, the prediction of remaining data on the
excitation function of this reaction system is much easier than that on a
excitation function without any data point.

By increasing the capacity of the training set, the model can learn more
information and reduce the MAE. However, the uncertainty of MAE on the
validation set increases with an increase in the percentage of the training
set, because the less tested data points, the larger the fluctuations, as
shown in Fig. 3. In the present work, to avoid either a large MAE value or
large uncertainty of MAE, the ratio between the training size and validation
size is chosen as 4:1 in the following discussion.

####  III.1.2 Comparison of different modes

Figure 4: (Color online) Density distribution of MAE for different modes. Results from 500 runs for each mode (Mode_ECC, Mode_E, and Mode_W) and from ECC model are displayed. Dashed lines denote a Gaussian fit to the distribution. In each run, the 3635 fusion reactions were randomly split into training and test sets at a ratio of 4:1  Table 6: The average MAE on the validation set obtained from different modes and from Wong formula and ECC model. The ratio of training set to test set is 4:1. Model |  MAE
---|---
Wong Formula |  2.38 $\pm$ 0.09
ECC model |  0.154 $\pm$ 0.008
Mode_BF |  0.129 $\pm$ 0.007
Mode_ECC |  0.068 $\pm$ 0.004
Mode_W |  0.081 $\pm$ 0.005
Mode_E |  0.081 $\pm$ 0.005

This section aims at demonstrating the impact of input features on the
performance of LightGBM. To do so, the density distribution of MAE for Mode_E,
Mode_BF, Mode_ECC, and Mode_W, together with the MAE from ECC model are
displayed Fig.4. The corresponding mean values and standard deviations are
listed in Table 6.

First, the MAE for Wong formula is about 2.381, which means the average
difference between the predicted CS and experimental CS are as large as two
orders of magnitude. This is understandable, as the training set consists of
many data points at deep sub-barrier energies, where Wong formula is
unreliable.

Second, MAE for Mode_BF is 0.129 which is smaller than that obtained with ECC
model. ECC model includes the effects of neutron transfer channels, the
couplings between the relative motion and intrinsic degrees of freedom, as
well as the nuclear deformation, it can give a reasonable fit to the fusion
excitation function in the vicinity of the Coulomb barrier. By feeding the
basic features of a reaction, LightGBM is able to achieve a better performance
on the prediction of CS than ECC model.

Third, the value of MAE can be significantly reduced by including the
physical-informed features in the input, this manifests the importance of
physical information on the prediction of CS with ML algorithm. MAE for
Mode_ECC is 0.068$\pm$0.004, which is the smallest of all. The values of MAE
for Mode_W and Mode_E are slightly larger than that of Mode_ECC. Considering
the fact that the calculations of CS with Wong formula or ECC model are much
complicated than Z1Z2/Ec.m, Mode_E is much favored over other modes.

###  III.2 Performance on the test set

Figure 5: (Color online) The nuclear fusion cross sections predicted by
different modes and by Wong formula. Black dots represent experimental data
taken from Refs. sahoo2020role ; sahoo2019sub ; chauhan2020evaporation ;
deb2022investigation ; kalita2021role ; prajapat2022fusion ; stefanini2021new
; colucci2019sub ; sanila2022fusion ; stefanini2017new ; del2023influence .

The performance of LightGBM is further validated using the test set which
consists of 280 data points from 18 reaction systems. Fig.5 displays the
comparison of the CS predicted by Mode_BF, Mode_E, and Mode_W with the recent
experimental data. It can be seen that the CS obtained with Wong formula is
close to experimental data only at high energies, and is too small at low
energies. This phenomenon is caused by the invalidation of the parabolic
approximation at energies well below the Coulomb barrier, has been widely
found and discussed in literature, see e.g., Refs. hagino2012subbarrier ;
jiang2021heavy . Both the CS and its energy-dependent behaviour predicted by
Mode_E and Mode_W are close to the experimental data, while in some reaction
systems, the energy-dependent behaviour obtained by Mode_BF is very different
compared to the experimental data. The values of MAE on the test set obtained
with Mode_E, Mode_W, and Mode_BF are 0.197$\pm$0.006, 0.187$\pm$0.005, and
0.526$\pm$0.013, respectively. This indicates that the physical-informed
features can guide machine learning algorithms for successfully capturing the
energy-dependent behaviour, then improve the performance. The MAE values on
the test set are larger than that on the validation set, this is
understandable because reaction systems in the test set are not included in
the training set.

###  III.3 Comparison with the DC-TDHF approach

The density-constrained (DC) time-dependent Hartree-Fock (TDHF) is a fully
microscopic approach, it provides a good description of the fusion excitation
function for many reaction systems umar2009density ; simenel2018heavy . To
further verify the performance of LightGBM, the nuclear fusion cross sections
for 40,48Ca + 78Ni obtained from DC-TDHF approach are compared with the
prediction of Mode_E, as shown in Fig.6. The uncertainties of DC-TDHF result
from different potentials PhysRevC.105.034601 . It can be seen that the
results predicted with Mode_E are in line with the DC-TDHF calculations.
However, in contrary to the observed enhancement of fusion cross sections of
40Ca + 78Ni at subbarrier energies in the DC-TDHF calculations, the CS of 40Ca
+ 78Ni predicted with Mode_E is smaller than that of 48Ca + 78Ni. As discussed
in Ref.PhysRevC.105.034601 , this enhancement for 40Ca + 78Ni is due to its
narrower width of the ion-ion potential. We note in
Refs.bourgin2016microscopic ; bourgin2014barrier  that Bourgin et al. reported
the fusion cross section of 40Ca + 64Ni system is higher than that of 40Ca +
58Ni, because the large neutron transfer probabilities in 40Ca + 64Ni result
in a lowering of the fusion threshold. The experimental data of 40Ca + 58,64Ni
are contained in the training set, thus LightGBM should learned from these
data and predicted a higher CS for 48Ca + 78Ni. Further studies regarding the
isospin-dependent fusion cross section are needed in order to classify the
roles of isospin in fusion dynamics.

Figure 6: (Color online) The nuclear fusion cross sections for 40Ca + 78Ni
(upper panel) and 48Ca + 78Ni (lower panel). Red points denote the predictions
with Mode_E. The shaded bands denote the calculated results from the density-
constrained time-dependent Hartree-Fock (TDHF) approach, taken from Ref.
PhysRevC.105.034601 .

###  III.4 Interpretability of the model

Figure 7: (Color online) Importance ranking for the input features obtained
with the SHAP package. Each row represents a feature, and the x-axis is the
SHAP value, which shows the importance of a feature for a particular
prediction. Each point represents a reaction, and the color represents the
feature value (with red being high and blue being low).

As a decision tree based algorithm, LightGBM has excellent interpretability.
This is important because one expects the ML algorithm to not only perform
well in refining the theoretical fusion cross section model, but also to
provide some fundamental physics that the theoretical fusion cross section
model does not have. Understanding what happens when the ML algorithm makes
predictions can help us further improve our knowledge of the relationship
between the input feature quantities and the predicted values. One possible
way to understand how LightGBM provides specific prediction is to find the
most important features that drive the model. To do this, one of the most
popular feature attribution methods, SHapley Additive Prediction
(SHAP)lundberg2017unified , is applied to obtain the importance ranking of
input features, as displayed in Fig 7. The top is the most important feature,
while the bottom is the least relevant feature for the prediction of the
fusion cross section in each mode. It is seen that physical-informed features
(sig_ECC, sig_W, and sig_E) in Mode_W, Mode_ECC, and Mode_E are ranked in the
top, and their SHAP values are significantly larger than others. Besides these
physical-informed features, the collision center-of-mass energy $E_{\rm c.m}$
and fusion Q-value are also ranked in the top-five. It is well known that
these two quantities are essential to the process of heavy-ion fusion
reaction. In addition, it can be seen that the neutron number (N3) and mass
number (A3) of the compound nucleus also exhibit high importance, this
indicates these two quantities are strongly related to the fusion cross
section, which can be further considered in modeling the fusion cross section.

##  IV Summary

To summarize, the underlying basic quantities and the physical-informed
quantities are fed to LightGBM to predict the cross section for heavy-ion
fusion reaction. The physical-informed quantities used in this work include
the fusion cross sections calculated with the empirical coupled channel (ECC)
model and Wong formula, as well as a simplified quantity Z1Z2/Ec.m. It is
found that, by using only basic quantities, LightGBM can reproduce
experimental data of cross section within a factor of 100.129=1.35, which is
better than 100.154=1.43 obtained from the coupled channel model. When the
physical-informed quantities are included in the input feature, the
performance of LightGBM can be significantly improved. The MAE on the test set
which consists of 118 data points from 8 reaction systems is about 0.66 by
only using basic quantities as the input, whereas it is reduced to about 0.2
if the physical-informed quantities are included. In addition, the trend of
the excitation function can be reproduced by LightGBM when the input feature
includes the physical-informed quantities. All together, our study
demonstrates the importance of physical information in predicting fusion cross
section of heavy-ion reaction with machine learning algorithm.

##  V Acknowledgement

The authors are grateful to the C3S2 computing center in Huzhou University for
calculation support. The work is supported in part by the National Natural
Science Foundation of China (Nos. U2032145, 12075327, 12335008), Fundamental
Research Funds for the Central Universities, Sun Yat-sen University under
Grant No. 23lgbj003, and Guangdong Major Project of Basic and Applied Basic
Research under Grant No. 2021B0301030006.

## References

  * (1) B. Back, H. Esbensen, C. Jiang, K. Rehm, Recent developments in heavy-ion fusion reactions, Reviews of Modern Physics 86 (1) (2014) 317.
  * (2) C. Jiang, B. Back, K. Rehm, K. Hagino, G. Montagnoli, A. Stefanini, Heavy-ion fusion reactions at extreme sub-barrier energies, The European Physical Journal A 57 (2021) 1–47.
  * (3) G. Montagnoli, A. M. Stefanini, Recent experimental results in sub-and near-barrier heavy-ion fusion reactions, The European Physical Journal A 53 (8) (2017) 169.
  * (4) G. Montagnoli, A. M. Stefanini, Recent experimental results in sub-and near-barrier heavy ion fusion reactions, The European Physical Journal A 59 (6) (2023) 138.
  * (5) X.-X. Sun, L. Guo, Effects of the tensor force on low-energy heavy-ion fusion reactions: A mini review, Communications in Theoretical Physics.
  * (6) J.-W. Xia, W.-L. Zhan, B.-W. Wei, Y. Yuan, M. Song, W. Zhang, X. Yang, P. Yuan, D. Gao, H. Zhao, et al., The heavy ion cooler-storage-ring project (hirfl-csr) at lanzhou, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 488 (1-2) (2002) 11–25.
  * (7) J. Yang, J. Xia, G. Xiao, H. Xu, H. Zhao, X. Zhou, X. Ma, Y. He, L. Ma, D. Gao, et al., High intensity heavy ion accelerator facility (hiaf) in china, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 317 (2013) 263–265.
  * (8) X. Zhou, J. Yang, H. P. Team, Status of the high-intensity heavy-ion accelerator facility in china, AAPPS Bulletin 32 (1) (2022) 35.
  * (9) B. Hong, Status of the raon project in korea, AAPPS Bulletin 33 (1) (2023) 3.
  * (10) A. Karpov, A. Denikin, M. Naumenko, A. Alekseev, V. Rachkov, V. Samarin, V. Saiko, V. Zagrebaev, Nrv web knowledge base on low-energy nuclear physics, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 859 (2017) 112–124.
  * (11) P. Fröbrich, Fusion and capture of heavy ions above the barrier: analysis of experimental data with the surface friction model, Physics Reports 116 (6) (1984) 337–400.
  * (12) A. Balantekin, N. Takigawa, Quantum tunneling in nuclear fusion, Reviews of Modern Physics 70 (1) (1998) 77.
  * (13) M. Dasgupta, D. Hinde, N. Rowley, A. Stefanini, Measuring barriers to fusion, Annual Review of Nuclear and Particle Science 48 (1) (1998) 401–461.
  * (14) L. Canto, P. Gomes, R. Donangelo, M. S. Hussein, Fusion and breakup of weakly bound nuclei, Physics reports 424 (1-2) (2006) 1–111.
  * (15) A. Diaz-Torres, L. R. Gasques, M. Wiescher, Effects of nuclear molecular configurations on the astrophysical s-factor for 16o+ 16o, Physics Letters B 652 (5-6) (2007) 255–258.
  * (16) H. Zhang, C. Lin, H. Jia, C. Zhang, G. Zhang, F. Yang, Z. Liu, G. An, Z. Wu, X. Xu, et al., Heavy ion reactions around the coulomb barrier, Science China Physics, Mechanics and Astronomy 54 (2011) 6–13.
  * (17) L. Canto, P. Gomes, R. Donangelo, J. Lubian, M. S. Hussein, Recent developments in fusion and direct reactions with weakly bound nuclei, Physics Reports 596 (2015) 1–86.
  * (18) K.-X. Cheng, J. Pu, Y.-T. Wang, Y.-F. Guo, C.-W. Ma, Non-frozen process of heavy-ion fusion reactions at deep sub-barrier energies, Nuclear Science and Techniques 33 (10) (2022) 132.
  * (19) K. Hagino, N. Rowley, A. Kruppa, A program for coupled-channel calculations with all order couplings for heavy-ion fusion reactions, Computer Physics Communications 123 (1-3) (1999) 143–152.
  * (20) P. Wen, O. Chuluunbaatar, A. Gusev, R. Nazmitdinov, A. Nasirov, S. Vinitsky, C. Lin, H. Jia, Near-barrier heavy-ion fusion: Role of boundary conditions in coupling of channels, Physical Review C 101 (1) (2020) 014618.
  * (21) C. Simenel, A. Umar, Heavy-ion collisions and fission dynamics with the time-dependent hartree–fock theory and its extensions, Progress in Particle and Nuclear Physics 103 (2018) 19–66.
  * (22) C. Simenel, Nuclear quantum many-body dynamics: From collective vibrations to heavy-ion collisions, The European Physical Journal A 48 (11) (2012) 152.
  * (23) B. Wang, K. Wen, W.-J. Zhao, E.-G. Zhao, S.-G. Zhou, Systematics of capture and fusion dynamics in heavy-ion collisions, Atomic Data and Nuclear Data Tables 114 (2017) 281–370.
  * (24) M. S. Gautam, R. P. Chahal, S. Duhan, H. Khatri, et al., Comprehensive analysis of heavy-ion fusion reactions around the coulomb barrier, Physica Scripta 97 (4) (2022) 045305.
  * (25) C. Wong, Interaction barrier in charged-particle nuclear reactions, Physical Review Letters 31 (12) (1973) 766.
  * (26) N. Lwin, N. Htike, K. Hagino, Applicability of the wong formula for fusion cross sections from light to heavy systems, Physical Review C 95 (6) (2017) 064601\.
  * (27) Vijay, R. P. Chahal, M. S. Gautam, S. Duhan, H. Khatri, Fusion cross sections and barrier distributions for ${}^{16}\mathrm{O}+^{70,72,73,74,76}\mathrm{Ge}$ and ${}^{18}\mathrm{O}+^{74}\mathrm{Ge}$ reactions at energies near and below the coulomb barrier, Phys. Rev. C 103 (2021) 024607.  doi:10.1103/PhysRevC.103.024607.
URL https://link.aps.org/doi/10.1103/PhysRevC.103.024607

  * (28) D. L. Hill, J. A. Wheeler, Nuclear constitution and the interpretation of fission phenomena, Physical Review 89 (5) (1953) 1102.
  * (29) L. Zhu, Z.-Q. Feng, C. Li, F.-S. Zhang, Orientation effects on evaporation residue cross sections in ca 48-induced hot fusion reactions, Physical Review C 90 (1) (2014) 014612.
  * (30) G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, L. Zdeborová, Machine learning and the physical sciences, Reviews of Modern Physics 91 (4) (2019) 045002.
  * (31) W. He, Q. Li, Y. Ma, Z. Niu, J. Pei, Y. Zhang, Machine learning in nuclear physics at low and intermediate energies, Science China Physics, Mechanics & Astronomy 66 (8) (2023) 282001.
  * (32) W.-B. He, Y.-G. Ma, L.-G. Pang, H.-C. Song, K. Zhou, High-energy nuclear physics meets machine learning, Nuclear Science and Techniques 34 (6) (2023) 88\.
  * (33) Y. Wang, Q. Li, Machine learning transforms the inference of the nuclear equation of state, Frontiers of Physics 18 (6) (2023) 64402.
  * (34) L. Li, Y. Zhang, Y. Cui, J. Liang, Determination of the impact parameter distribution of low-intermediate energy heavy ion collisions using machine learning, Scientia Sinica Physica, Mechanica & Astronomica 52 (5) (2022) 252014\.
  * (35) L.-G. Pang, K. Zhou, N. Su, H. Petersen, H. Stöcker, X.-N. Wang, An equation-of-state-meter of quantum chromodynamics transition from deep learning, Nature communications 9 (1) (2018) 210.
  * (36) G. WEI, Y. WANG, Q. LI, F. LIU, Event-by-event extracting physical information from heavy ion collisions based on machine learning, Atomic Energy Science and Technology 57 (4) (2023) 774.
  * (37) Y. Wang, F. Li, Q. Li, H. Lü, K. Zhou, Finding signatures of the nuclear symmetry energy in heavy-ion collisions with deep learning, Physics Letters B 822 (2021) 136669.
  * (38) K. Zhou, L. Wang, L.-G. Pang, S. Shi, Exploring qcd matter in extreme conditions with machine learning, arXiv preprint arXiv:2303.15136.
  * (39) M. O. Kuttan, K. Zhou, J. Steinheimer, A. Redelbach, H. Stoecker, An equation-of-state-meter for cbm using pointnet, Journal of High Energy Physics 2021 (10) (2021) 1–25.
  * (40) F.-P. Li, H.-L. Lü, L.-G. Pang, G.-Y. Qin, Deep-learning quasi-particle masses from qcd equation of state, Physics Letters B (2023) 138088.
  * (41) Y.-L. Du, D. Pablos, K. Tywoniuk, Jet tomography in heavy-ion collisions with deep learning, Physical Review Letters 128 (1) (2022) 012301.
  * (42) D. Peng, H.-L. Wei, X.-X. Chen, X.-B. Wei, Y.-T. Wang, J. Pu, K.-X. Cheng, C.-W. Ma, Bayesian evaluation of residual production cross sections in proton-induced nuclear spallation reactions, Journal of Physics G: Nuclear and Particle Physics 49 (8) (2022) 085102.
  * (43) C.-W. Ma, X.-B. Wei, X.-X. Chen, D. Peng, Y.-T. Wang, J. Pu, K.-X. Cheng, Y.-F. Guo, H.-L. Wei, Precise machine learning models for fragment production in projectile fragmentation reactions using bayesian neural networks, Chinese Physics C 46 (7) (2022) 074104.
  * (44) Q. Song, L. Zhu, B. Cai, C. Yuan, J. Su, H. Guo, Image processing of isotope yield in neutron-induced fission, Physical Review C 107 (4) (2023) 044609.
  * (45) Z.-A. Wang, J. Pei, Y. Liu, Y. Qiang, Bayesian evaluation of incomplete fission yields, Phys. Rev. Lett. 123 (2019) 122501.  doi:10.1103/PhysRevLett.123.122501.
URL https://link.aps.org/doi/10.1103/PhysRevLett.123.122501

  * (46) Z.-A. Wang, J. Pei, Optimizing multilayer bayesian neural networks for evaluation of fission yields, Physical Review C 104 (6) (2021) 064608.
  * (47) Z. Wang, J. Pei, Y. Chen, C. Qiao, F. Xu, Z. Ge, N. Shu, Bayesian approach to heterogeneous data fusion of imperfect fission yields for augmented evaluations, Physical Review C 106 (2) (2022) L021304.
  * (48) Z. Niu, H. Liang, Nuclear mass predictions based on bayesian neural network approach with pairing and shell effects, Physics Letters B 778 (2018) 48–53.
  * (49) Z.-P. Gao, Y.-J. Wang, H.-L. Lü, Q.-F. Li, C.-W. Shen, L. Liu, Machine learning the nuclear mass, Nuclear Science and Techniques 32 (10) (2021) 109.
  * (50) X.-C. Ming, H.-F. Zhang, R.-R. Xu, X.-D. Sun, Y. Tian, Z.-G. Ge, Nuclear mass based on the multi-task learning neural network method, Nuclear Science and Techniques 33 (4) (2022) 48.
  * (51) X. Wu, L. Guo, P. Zhao, Nuclear masses in extended kernel ridge regression with odd-even effects, Physics Letters B 819 (2021) 136387.
  * (52) T. Zhao, H. Zhang, A new method to improve the generalization ability of neural networks: A case study of nuclear mass training, Nuclear Physics A 1021 (2022) 122420.
  * (53) X.-K. Le, N. Wang, X. Jiang, Nuclear mass predictions with multi-hidden-layer feedforward neural network, Nuclear Physics A (2023) 122707.
  * (54) L. Peng, B. JingHu, N. ZhongMing, N. YiFei, $\backslash$beta-decay half-lives studied using neural network method, Scientia Sinica Physica, Mechanica & Astronomica 52 (5) (2022) 252006.
  * (55) Y.-F. Gao, B.-S. Cai, C.-X. Yuan, Investigation of $\beta$–decay half-life and delayed neutron emission with uncertainty analysis, Nuclear Science and Techniques 34 (1) (2023) 9.
  * (56) J. M. Munoz, S. Akkoyun, Z. P. Reyes, L. A. Pachon, Predicting $\beta$-decay energy with machine learning, Physical Review C 107 (3) (2023) 034308.
  * (57) C.-Q. Li, C.-N. Tong, H.-J. Du, L.-G. Pang, Deep learning approach to nuclear masses and $\alpha$-decay half-lives, Physical Review C 105 (6) (2022) 064306\.
  * (58) X.-X. Dong, R. An, J.-X. Lu, L.-S. Geng, Nuclear charge radii in bayesian neural networks revisited, Physics Letters B 838 (2023) 137726.
  * (59) P. Su, W.-B. He, D.-Q. Fang, Progress of machine learning studies on the nuclear charge radii, Symmetry 15 (5) (2023) 1040.
  * (60) D. Wu, C. Bai, H. Sagawa, H. Zhang, Calculation of nuclear charge radii with a trained feed-forward neural network, Physical Review C 102 (5) (2020) 054323.
  * (61) X.-X. Dong, R. An, J.-X. Lu, L.-S. Geng, Novel bayesian neural network based approach for nuclear charge radii, Physical Review C 105 (1) (2022) 014308.
  * (62) Z.-X. Yang, X.-H. Fan, P. Yin, W. Zuo, Taming nucleon density distributions with deep neural network, Physics Letters B 823 (2021) 136650.
  * (63) T.-S. Shang, J. Li, Z.-M. Niu, Prediction of nuclear charge density distribution with feedback neural network, Nuclear Science and Techniques 33 (12) (2022) 153.
  * (64) T. L. Zhao, X. J. Bao, H. F. Zhang, Improvement of evaporation residual cross sections for superheavy nuclei using a neural network method, Nuclear Physics A 1027 (2022) 122510.
  * (65) D. Dell’Aquila, B. Gnoffo, I. Lombardo, F. Porto, M. Russo, Modeling heavy-ion fusion cross section data via a novel artificial intelligence approach, Journal of Physics G: Nuclear and Particle Physics 50 (1) (2022) 015101\.
  * (66) G. Ke, Q. Meng, T. Finley, T. Wang, W. Chen, W. Ma, Q. Ye, T.-Y. Liu, Lightgbm: A highly efficient gradient boosting decision tree, in: I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, R. Garnett (Eds.), Advances in Neural Information Processing Systems, Vol. 30, Curran Associates, Inc., 2017.
URL
https://proceedings.neurips.cc/paper_files/paper/2017/file/6449f44a102fde848669bdd9eb6b76fa-
Paper.pdf

  * (67) Y. Wang, Z. Gao, H. Lü, Q. Li, Decoding the nuclear symmetry energy event-by-event in heavy-ion collisions with machine learning, Physics Letters B 835 (2022) 137508.  doi:https://doi.org/10.1016/j.physletb.2022.137508.
  * (68) F. Li, Y. Wang, Z. Gao, P. Li, H. Lü, Q. Li, C. Tsang, M. Tsang, Application of machine learning in the determination of impact parameter in the sn 132+ sn 124 system, Physical Review C 104 (3) (2021) 034608.
  * (69) F. Li, Y. Wang, H. Lü, P. Li, Q. Li, F. Liu, Application of artificial intelligence in the determination of impact parameter in heavy-ion collisions at intermediate energies, Journal of Physics G: Nuclear and Particle Physics 47 (11) (2020) 115104.
  * (70) R. N. Sahoo, M. Kaushik, A. Sood, A. Sharma, S. Thakur, P. Kumar, M. M. Shaikh, R. Biswas, A. Yadav, M. K. Sharma, et al., Role of neutron transfer in sub-barrier fusion, Physical Review C 102 (2) (2020) 024615.
  * (71) R. N. Sahoo, M. Kaushik, A. Sood, P. Kumar, A. Sharma, S. Thakur, P. P. Singh, P. Raina, M. M. Shaikh, R. Biswas, et al., Sub-barrier fusion in the cl 37+ te 130 system, Physical Review C 99 (2) (2019) 024607.
  * (72) A. Chauhan, R. Prajapat, G. Sarkar, M. Maiti, R. Kumar, J. Gehlot, S. Nath, A. Parihari, N. Madhavan, et al., Evaporation residue cross section in the cl 37+ zn 68 fusion reaction near the coulomb barrier, Physical Review C 102 (6) (2020) 064606.
  * (73) N. K. Deb, K. Kalita, H. Al Rashid, A. Das, S. Nath, J. Gehlot, N. Madhavan, R. Biswas, R. N. Sahoo, P. K. Giri, et al., Investigation of isotopic dependence on the o+ ni fusion cross section near barrier energies, Physical Review C 105 (5) (2022) 054608.
  * (74) K. Kalita, B. Roy, S. Nath, A. Das, A. Vinayak, J. Gehlot, A. Rani, R. Sahoo, S. Biswas, P. Giri, et al., Role of neutron transfer in the sub-barrier fusion cross section in 18o+ 116sn.
  * (75) R. Prajapat, M. Maiti, R. Kumar, M. Sagwal, C. Kumar, R. Biswas, J. Gehlot, S. Nath, N. Madhavan, et al., Fusion and back-angle quasielastic measurements in si 30+ gd 156 near the coulomb barrier, Physical Review C 105 (6) (2022) 064612\.
  * (76) A. Stefanini, G. Montagnoli, M. D’Andrea, M. Giacomin, C. Dehman, R. Somasundaram, V. Vijayan, L. Zago, G. Colucci, F. Galtarossa, et al., New insights into sub-barrier fusion of 28si+ 100mo, Journal of Physics G: Nuclear and Particle Physics 48 (5) (2021) 055101.
  * (77) G. Colucci, G. Montagnoli, A. Stefanini, K. Hagino, A. Caciolli, P. Čolović, L. Corradi, E. Fioretto, F. Galtarossa, A. Goasduff, et al., Sub-barrier fusion involving odd mass nuclei: The case of 36 s+ 50 ti, 51 v, The European Physical Journal A 55 (2019) 1–6.
  * (78) S. Sanila, A. Vinodkumar, B. Babu, N. Madhavan, S. Nath, J. Gehlot, R. Biswas, C. Kumar, A. Rani, A. Parihari, et al., Fusion studies in c 12+ w 182, 184, 186 reactions at energies below and near the coulomb barrier, Physical Review C 106 (2) (2022) 024614.
  * (79) A. Stefanini, G. Montagnoli, H. Esbensen, P. Čolović, L. Corradi, E. Fioretto, F. Galtarossa, A. Goasduff, J. Grebosz, F. Haas, et al., New results in low-energy fusion of ca 40+ zr 90, 92, Physical Review C 96 (1) (2017) 014603.
  * (80) M. Del Fabbro, A. Stefanini, G. Montagnoli, L. Corradi, E. Fioretto, M. Fornale’, S. Szilner, Influence of the z= 50 shell closure on heavy-ion sub-barrier fusion, The European Physical Journal A 59 (2) (2023) 26.
  * (81) K. Hagino, N. Takigawa, Subbarrier fusion reactions and many-particle quantum tunneling, Progress of theoretical physics 128 (6) (2012) 1061–1106.
  * (82) A. Umar, V. Oberacker, Density-constrained time-dependent hartree-fock calculation of 16 o+ 208 pb fusion cross-sections, The European Physical Journal A 39 (2009) 243–247.
  * (83) X.-X. Sun, L. Guo, A. S. Umar, Microscopic study of the fusion reactions ${}^{40,48}\mathrm{Ca}+^{78}\mathrm{Ni}$ and the effect of the tensor force, Phys. Rev. C 105 (2022) 034601.  doi:10.1103/PhysRevC.105.034601.
URL https://link.aps.org/doi/10.1103/PhysRevC.105.034601

  * (84) D. Bourgin, C. Simenel, S. Courtin, F. Haas, Microscopic study of ca 40+ ni 58, 64 fusion reactions, Physical Review C 93 (3) (2016) 034604.
  * (85) D. Bourgin, S. Courtin, F. Haas, A. Stefanini, G. Montagnoli, A. Goasduff, D. Montanari, L. Corradi, E. Fioretto, J. Huiming, et al., Barrier distributions and signatures of transfer channels in the ca 40+ ni 58, 64 fusion reactions at energies around and below the coulomb barrier, Physical Review C 90 (4) (2014) 044610.
  * (86) S. M. Lundberg, S.-I. Lee, A unified approach to interpreting model predictions, Advances in neural information processing systems 30.