Machine learning the nuclear mass

The mass of nuclei, which is of fundamental importance to explore nuclear landscape and properties of nuclear force, plays a crucial role in understanding many issues in both the fields of nuclear physics and astrophysics 1 ; 2 ; 3 ; 4 ; Niu:2018olp ; 44 . It is known that more than 7000 nuclei in the nuclear landscape from H (Z=1) to Og (Z=118) are predicted to be existed according to various theoretical models, while about 3000 nuclei have been found or synthesized in experimental and the masses of about 2500 nuclei have been measured accurately 5 ; 6 . Exploring the masses of the remains is of particular interest for both nuclear experimental and theoretical community. On the experimental side, facilities, such as HIRFL-CSR in China, RIBF at RIKEN in Japan, cooler-storage ring ESR and SHIPTRAP at GSI in Germany, CPT at Argonne and LEBIT at MSU in US, ISOLTRAP at CERN, JYFLTRAP at Jyväskylä in Finland, TITAN at TRIUMPH in Canada, are partly dedicated to measuring the nuclear mass, especially for nuclei round the driplines. On the theoretical side, various models have been developed to study nuclear mass by considering different physics, such as finite-range droplet model (FRDM) 7 ; 8 , the Weizsäcker-Skyrme (WS) model 9 , Hartree-Fock-Bogoliubov (HFB) mass models 10 ; 11 ; 12 , the relativistic mean-field (RMF) model Geng:2005yu , relativistic continuum Hartree-Bogoliubov (RCHB) theory Xia:2017zka . Though tremendous progress has been made in both experimental and theoretical sides, exploring mass of nuclei around dirplines is still a great challenge for both sides.

Machine learning which is the subset of artificial intelligence has been widely applied for analyzing data in many branches of science, such as in physics, e.g., Refs. ROMP ; Nature1 . In nuclear physics, a Bayesian neural network (BNN) has been applied to reduce the mass residuals between theory and experiment, and a significant improvement in the mass predictions of several theoretical models was obtained after BNN refinement Utama:2015hva ; Utama:2017wqe ; Niu:2018csp , e.g., the root mean square deviation (RMSD) of the liquid-drop model (LDM) was reduced from about 3 MeV to 0.8 MeV. Later on, BNN approach is also applied to study nuclear charge radii Utama:2016tcl , $\beta$-decay half-lives Niu:2018trk , fission product yield PJC , fragment Production in spallation reaction Ma:2020bic ; Ma:2020mbd . Besides BNN, other machine learning or deep learning algorithms also have been employed in studying of nuclear reactions, e.g., Refs. PLG ; Du:2019civ ; Steinheimer:2019iso ; Song:2021rmm ; Wang:2020tgb ; Li:2020qqn . Focusing on nuclear mass, besides BNN in Refs. Utama:2015hva ; Utama:2017wqe ; Niu:2018csp , the Levenberg-Marquardt neural network approach Zhang:2017zvb , Gaussian processes shelley ; Neufcourt:2018syo , decision tree algorithm Carnini:2020lvr , the Multilayer Perceptron (MLP) algorithm MLP also have been applied to refine nuclear mass models.

Indeed, studying nuclear mass with machine learning algorithms is not a new topic and it can be traced back to at least 1993, see e.g., Refs. Gernoth:1993dqa ; Athanassopoulos:2003qe ; Clark:2006ua and reference therein. In Ref. Gernoth:1993dqa , the capability of multilayer feedforward neural networks for learning the systematics of atomic masses and nuclear spins and parities with high accuracy has been found. This topic is flourishing again because of the rapid development of computer science and artificial intelligence. In 2016, Light Gradient Boosting Machine (LightGBM) which is a tree based learning algorithm was developed by Microsoft LightGBM . It is a state-of-the-art machine learning algorithm which has achieved better performances in many machine learning tasks. Therefore, it would be interesting to explore whether LightGBM algorithm can achieve better accuracy than BNN on the task of predicting nuclear mass.

The paper is organized as follows: In Sec.II, we will introduce the LightGBM model and ten input features. The predicted binding energy and neutron separation energy obtained with LightGBM are discussed in detail in Sec.III. The conclusion and outlook are given in Sec.IV.

LightGBM refers to a recent improvement of gradient boosting decision tree (GBDT) that provides efficient implementation of gradient boosting algorithms. It is becoming more popular by the day due to its efficiency and capability of handling large amounts of data. LightGBM has leaf-wise growth of trees, rather than a level-wise growth. After the first partition, the next split is performed only on the leaf node that adds more to the information gain.

The primary advantage of LightGBM is the change in training algorithm that speeds up the optimization process dramatically and results in a more effective model in many cases. More concretely, to speed up the training process, LightGBM uses a histogram-based methodology to select the best segmentation. For any continuous variable, instead of using individual values, these are divided into bins or buckets, which can accelerate the training process and reduce memory usage. In addition, LightGBM contains two novel techniques: Gradient bases One-Side Sampling (GOSS), which keeps all the instances of large gradient and performs random sampling on the instances with small gradient, and Exclusive Feature Bundling (EFB), which helps to bundle multiple features into a single feature without losing any information. Furthermore, as a decision tree-based algorithm, LightGBM also has high level of interpretability, allowing the results obtained in machine learning model to be checked against previous knowledge regarding nuclear mass. For example, one can find which feature is more important for predicting nuclear mass, this would be helpful to further improve nuclear mass model.

In this work, the binding energies of 2408 nuclei between ¹⁶O and ²⁷⁰Ds from the atomic mass evaluation (AME2016) 6 are employed as the training and testing dataset. LightGBM is trained to learn the residual between the theoretical prediction and the experimental binding energy, $\delta(Z,A){\rm{}}={\rm{}}{B_{{\rm{th}}}}(Z,A)-{B_{{\rm{exp}}}}(Z,A)$. Four theoretical mass models are adopted in this work to obtain $B_{th}$, including the LDM Zhang:2017zvb , DZ Duflo:1995ep , FRDM 7 ; 8 , and WS4 9 . After LightGBM learns the behaviour of the residual $\delta(Z,A)$, the binding energy of an unknown mass nucleus can be obtained via $B_{\rm{LightGBM}}(Z,A)$=$B_{\rm{th}}(Z,A)$+$\delta(Z,A)$. It is found that the RMSD of these four theoretical mass models can be significantly improved after LightGBM refinement.

For the LDM model, nucleus is regarded as a non-compressible droplet, which contains the volume energy, surface energy, Coulomb energy of proton repulsion, the symmetry energy related to the ratio of neutrons to protons, and the pairing energy of the neutron-proton pairing effect. It can be described as follows:

Where $E_{p}$ is the pairing energy given by the following expression:

In the above formula, A, Z, N and $T_{z}$ are the mass number, proton number, neutron number and the third component of isospin $(T_{z}=\frac{1}{2}(Z-N))$, $a_{v}$, $k_{v}$, $a_{s}$, $k_{s}$, $a_{c}$, $f_{p}$, $d_{n}$, $d_{p}$, $d_{np}$ are adjustable parameters with the values given in Table 1. Based on these parameters the binding energy of 2408 nuclei theoretical and experimental values as an RMSD is $2.463$ MeV.

This work mainly aims to find the relationship between the feature quantity of each nucleus and $\delta(Z,A)$ with the LightGBM model. For each nucleus, we selected 10 physical quantities (cf. Table 2) as the input features. It is known that nuclear binding energy and nuclear structure are linked, therefore, we selected four physical quantities related to the shell structure, among which $Z_{m}$ and $N_{m}$ are the shells where the last proton and neutron are located, and the level of the shell is given by the magic numbers. The number of protons between 8, 20, 50, 82 and 126 corresponds to $Z_{m}$ of 1, 2, 3, 4, and the number of neutrons between 8, 20, 50, 82, 126 and 184 corresponds to $N_{m}$ of 1, 2, 3, 4, 5. In addition, $|Z-m|$ and $|N-m|$ are the absolute values of the difference between the number of protons, the number of neutrons, and the nearest magic number, respectively, which represent the distance between the number of protons, the number of neutrons, and the nearest magic number. N pair is an index that considers the proton-neutron pairing effect, the odd-odd nucleus is 0, the odd-even nucleus or even-odd nucleus is 1, and the even-even nucleus is 2.

In this work, the value of num_boost_round (maximum number of decision trees allowed) is 50000, num_leaves (maximum number of leaves allowed per tree) is 10, max_depth (maximum depth allowed per tree) is -1 and other parameters are basically set as their default values of the LightGBM model. Varying these parameters would not alter the results significantly. During the training process, LightGBM will generate a decision tree based on the relevant information between the features of the training set and $\delta(Z,A)$. 10-fold cross-validation, which is a technique to evaluate models by partitioning the original dataset into 10 equal size subsamples., is also applied to prevent overfitting and selection bias. After training, the model will make predictions on the testing set. Each nucleus in the testing set traverses the decision tree grown during model training. Each decision tree will give its contribution to the predicted value according to the feature quantity of each nucleus. The sum of the contributions of all decision trees is the predicted value given by the final model.

To summarize, several features are fed into the LightGBM algorithm to study the residual $\delta(Z,A)$ between the theoretical and experimental binding energies, it turns out that LightGBM algorithm can mimic the patterns of $\delta(Z,A)$ with high accuracy, so as to refine theoretical mass models. In this work, significant reductions on the RMSD of about 90%, 65%, 40%, and 60% after the LightGBM refinement on the LDM, DZ, WS4, and FRDM are obtained, indicating the strong capability of LightGBM to improve theoretical nuclear mass models. In addition, the RMSD for various mass models with respect to the 66 newly measured nuclei that appeared in AME2020 (compared with AME2016) is reduced on the same level as well. Furthermore, it is found that single and two-neutron separation energies obtained with the LightGBM-refined mass models are in good agreement with the newly appeared experimental data. By using the SHAP package, the most relevant input features to predict the residual $\delta(Z,A)$ for each mass model are found out, which may provide guidance for the further developments of nuclear mass models.

The good performance of machine learning method on refining the nuclear mass model gives us a new tool to further investigate other properties of nuclei that we are interested in, such as, superheavy nuclei, halo nuclei, and nuclei around drip-line. In addition, with the development of the interpretable machine learning methods, more physical hints can be obtained thereby improving our understanding of present nuclear models.