Nuclear mass predictions based on deep neural network and finite-range
  droplet model (2012)
A neural network with two hidden layers is developed for nuclear mass
prediction, based on the finite-range droplet model (FRDM12). Different
hyperparameters, including the number of hidden units, the choice of activation
functions, the initializers, and the learning rates, are adjusted explicitly
and systematically. The resulting mass predictions are achieved by averaging
the predictions given by several different sets of hyperparameters with
different regularizers and seed numbers. It can provide us not only the average
values of mass predictions but also reliable estimations in the mass prediction
uncertainties. The overall root-mean-square deviations of nuclear mass have
been reduced from $0.603$ MeV for the FRDM12 model to $0.200$ MeV and $0.232$
MeV for the training set and validation set, respectively.
To Chung Yiu, Haozhao Liang, Jenny Lee
# Nuclear mass predictions based on deep neural network and finite-range droplet model (2012)

###### Abstract

A neural network with two hidden layers is developed for nuclear mass prediction, based on the finite-range droplet model (FRDM12). Different hyperparameters, including the number of hidden units, the choice of activation functions, the initializers, and the learning rates, are adjusted explicitly and systematically. The resulting mass predictions are achieved by averaging the predictions given by several different sets of hyperparameters with different regularizers and seed numbers. It can provide us not only the average values of mass predictions but also reliable estimations in the mass prediction uncertainties. The overall root-mean-square deviations of nuclear mass have been reduced from 0.603 MeV for the FRDM12 model to 0.200 MeV and 0.232 MeV for the training set and validation set, respectively.

nuclear mass, machine learning, deep neural network

## I Introduction

Nuclear mass is one of the most fundamental nuclear properties [1; 2]. It not only represents the static properties of nuclei, but also determines the reaction energies in different nuclear processes, such as \(\beta\) decay, neutron capture, and fission [3]. All these processes play important roles in the origin of elements and the abundance of elements in the universe [3; 4; 5].

Nowadays, there are around 2500 nuclei with experimental masses. These nuclei are estimated to be only 27.8% of around 9000 bounded nuclei estimated theoretically [2; 6]. In order to have a better understanding of nuclear mass for all nuclei, theoretical mass models are required. There are several different theoretical models for mass prediction. Some of them are the microscopic models such as Hartree-Fock-Bogoliubov (HFB) mass models [7; 8] and relativistic mean-field (RMF) mass models [9], while the others are the macroscopic-microscopic models such as finite-range droplet model [6] and Weizsacker-Skyrme (WS) model [10]. These theoretical models give a root-mean-square (RMS) deviation between the theoretical and experimental masses from 0.3 to 2.3 MeV [11]. To reduce RMS deviation, which is critical for a better description of final element abundance [3; 4], machine learning model is developed.

Neural network, one of the algorithms in machine learning, has been widely used in different research fields [12; 13; 14]. From several recent studies, it has been proven that neural network is able to improve the accuracy of the models for several different nuclear properties, such as the \(\beta\)-decay half-lives [15; 16], neutron capture rate [17], nuclear charge radii [18], and ground-state and excited energies [19]. Neural network has also been used in nuclear mass prediction in several previous studies [11; 20; 21; 22]. In particular, a significant improvement in mass predictions has been achieved by using several neural networks [22].

Meanwhile, each neural network in these studies includes several kinds of hyperparameters, e.g., the number of hidden units, the choice of activation functions, the initializers, and the learning rates [16; 23]. These hyperparameters play very important roles in both the training process and the final performance of the neural network. In other words, it is essential to investigate such hyperparameters in an explicit and systematic way.

In this paper, a deep neural network will be used to build a mass model improving the current finite-range droplet model [6]. Different hyperparameters will be in particular investigated in this study to achieve a better neural network model. Also, several different sets of hyperparameters will be used together to predict the nuclear masses and provide the uncertainty of the result.

The details about the neural network used in this study will be given in Sec. II. The results of the nuclear mass neural network model and its performance will be discussed in Sec. III. Finally, a summary will be presented in Sec. IV. For all the hyperparameters adjusted in this work, they will be given in Appendices A, B, C, and D.

## II Neural network model

In this study, a deep neural network (DNN) consisting of an input layer, two hidden layers, and an output layer is used, as shown in Fig. 1. The relation between input,