Applications of different machine learning methods on nuclear charge radius estimations

In this research, experimental data reported in the literature [4, 5, 7] were used to estimate the nuclear charge radii with different machine learning algorithms. These data include the radius information of 933 nuclei (A ≥ 40 and Z ≥ 20). In this study, 699 randomly selected data were used as training data and the remaining 234 as testing data to evaluate the performance of the models. The predictive variables were taken as in the study of Dong et al [26]. These are mass number (A), proton number (Z), isospin dependence (I²), pairing term (δ), the promiscuity factor (P) related to shell closure effects [28, 29], and a term related to abnormal charge radii behavior in ^181,183,185Hg (LI). The explicit form of the I², δ, P and LI are given by

$$\begin{eqnarray}&&{I}^{2}={\left(\displaystyle \frac{N-Z}{A}\right)}^{2},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\delta =\displaystyle \frac{{\left(-1\right)}^{Z}+{\left(-1\right)}^{N}}{2},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&P=\displaystyle \frac{{\nu }_{p}{\nu }_{n}}{{\nu }_{p}+{\nu }_{n}},\end{eqnarray} \tag{ 3 }$$

and

$$\begin{eqnarray}{LI}=\left\{\begin{array}{ll}1, & {}^{181}\mathrm{Hg}{,}^{183}\mathrm{Hg}{,}^{185}\mathrm{Hg}\\ 0, & \mathrm{else}\end{array}\right.\end{eqnarray} \tag{ 4 }$$

The same input variables were used in all algorithms used in the study to estimate the atomic radius value in the machine-learning training process.

All analyses, calculations, and data visualizations in the research were prepared in the R programming environment [30]. The R library files used in this study are, kernlab for SVR and GPPK algorithms [31], randomForest for RF algorithm [32], quantregForest for QRF algorithms [33, 34], xgboost for XGBoost algorithm [35], RSNNS for Artificial neural network (ANN) algorithm [36], cubist for cubist model [37], earth for multivariate Adaptive Regression Spline (MARS) model [38], caret for data processing, data separation and optimization of machine learning algorithms [39], openair [40] and ggplot2 [41] for cross-validation, scatter diagrams, and data visualization.

In this study, eight algorithms (artificial neural network, Cubist model, Gaussian process with polynomial kernel, multivariate adaptive regression splines, random forest, quantile random forest, support vector regression, and extreme gradient boosting) were used. These algorithms are summarized below.As will be seen in the next sections, the performance metrics of the Cubist model are higher than the others for nuclear charge radii predictions. Therefore more details for the Cubist model algorithm are given.

Artificial Neural Network (ANN): Artificial neural networks (ANNs) [42–44], which mathematically simulate biological nerve cell structure and functionality, have been developing in different structures in recent years to solve different problems [16, 45–49]. Artificial neural networks consist of small processing units called neurons, and each neuron is connected to each other through adaptive synaptic weights [50].

Gaussian Process with Polynomial Kernel (GPPK): Gaussian processes, developed to explain non-parametric relationships on a Bayesian basis, provide point estimates as well as confidence intervals for these estimates [51]. In this approach, the common distribution of training and test data is represented by the multidimensional Gaussian density function obtained on the basis of the Polynomial kernel, so the estimated distribution for each test data is determined by the distribution conditions of the training data [51, 52]. Detailed information for this algorithm can be found in the cited sources [53, 54].

Multivariate Adaptive Regression Splines (MARS): This method, which is a nonparametric modeling technique, was developed by Friedman [55]. In the MARS approach, the entire data set is significantly divided into sub-datasets and a separate linear model is established for each sub-dataset [56]. Thus, with the help of this piecewise linear model, which is automatically adapted to all data, the nonlinear connection between the independent variables and the target variable (splines) can be estimated [57, 58]. MARS is a very useful algorithm for problems where the relationships of the variables in the dataset may be different in each region [59, 60].

Random Forest (RF): This algorithm developed by Breiman [61] has been used in many different fields in recent years due to its high performance in classification and regression problems [62–67]. In the RF approach, the entire dataset is divided into subsets and more than one decision tree is created within each subset. Then Each tree is trained with randomly selected features. As a result, RF, an ensemble learning technique, is affected by specific weights from each tree for the final model result [68]. Also, in the RF model, training each tree with randomly selected features and generating each tree dataset using a subset avoids the overfitting problem [63].

Quantile Random Forest (QRF): Unlike the traditional Random Forest algorithm, QRF uses only values from a certain percentile during splits in each tree [33]. Thus, it allows customizing each tree to include only values in a certain percentile. This method can show better results in some cases than a standard Random Forest algorithm in regression problems by reducing the effect of outliers in the data set and allowing the model to better explain the distribution of the target variable (predicted variable) in certain percentiles [69–71].

Support Vector Regression (SVR): This method is a regression method that utilizes support vectors and the Lagrange multiplier approach for analyzing and predicting data [72]. The SVR algorithm is based on the Support Vector Machine (SVM) algorithm, which provides effective solutions to classification problems [73]. SVR is particularly useful when dealing with outliers and non-linearities in data [74]. The SVR aims to obtain a regression estimate that accurately predicts the response values based on a subset of high-dimensional prediction variables [75]. It utilizes support vectors and the Lagrange multiplier approach to achieve this goal [76, 77].

Extreme Gradient Boosting (XGBoost): This algorithm, developed by Chen and Guestrin [35], has emerged by optimizing the Gradient Boosting Machine (GBM) [78] algorithm and expanding it to a scalable level. XGBoost is particularly successful in large datasets and high-dimensional feature spaces. XGBoost is a machine learning algorithm that models nonlinear relationships using multiple decision trees and is based on the principle of sequential error reduction [79, 80]. In this method, decision trees are created sequentially and each tree is reconstructed according to the predictions of the previous tree with a focus on error reduction. Thus, each tree created learns a new feature that increases the accuracy of the model and reduces the error rate [81]. Furthermore, in the XGBoost model, a weight value is assigned to each tree and the contribution of each tree is determined more clearly in order for the results to have higher performance. In addition, the XGBoost algorithm has many regularization parameters to prevent overfitting [17].

Cubist model: The cubist model is a rule-based model capable of handling both numerical and categorical variables [82, 83]. It uses a 'separate and conquer' methodology, to create a rule-based regression tree that identifies different paths by iteratively splitting predictor variables within the model [84, 85]. The Cubist model results contain several sets of rules that can be broken down into sub-datasets with similar characteristics to represent the entire dataset. A multivariate linear regression model is fitted to the subsets of data generated by these rule sets to reveal the pattern of association between the predictor variables and the target variable [86]. Unlike other rule-based tree models, Cubist uses a combination of models together with a smoothing coefficient to combine the linear models at each node of the tree, as expressed in equation (5) [85, 87].

$$\begin{eqnarray}&&{\hat{y}}_{{par}}=a\times {\hat{y}}_{(k)}+(1-a)\times {\hat{y}}_{(p)}\end{eqnarray} \tag{ 5 }$$

where ${\hat{y}}_{(k)}$ is the prediction from the current model (child model), ${\hat{y}}_{(p)}$ is the prediction from the parent model located above it in the tree, and a is the smoothing coefficient. This coefficient can be determined as expressed in equation (6) [85]:

$$\begin{eqnarray}&&a=\displaystyle \frac{\mathrm{Var}[{e}_{(p)}]-\mathrm{Cov}[{e}_{(k)},{e}_{(p)}]}{\mathrm{Var}[{e}_{(p)}-{e}_{(k)}]}\end{eqnarray} \tag{ 6 }$$

where Var[e_(p)] is the variance of the errors (i.e., $y-{\hat{y}}_{(k)}$) of the parent model, Cov[e_(k), e_(p)] is the covariance of the residuals of the child and parent models, and Var[e_(p) − e_(k)] is the variance of the difference between the residuals. This process is based on the covariance of the residuals of the child and parent models. The covariance indicates that the errors of the two models are linearly related. If the variance of the errors of the parent model is greater than the covariance, the child model is weighted more than the parent model. In the opposite case, Cubist gives more weight to the parent model. Thus, the model with the lowest error value is more dominant in the adjusted model. If the error values of the two models are the same, both models will have equal weight [85]. Cubist combines the linear models at each node according to equation (5), creating a single linear model for each rule. This allows the models to be presented in a more organized and representative way [85]. With new regression models developed at each tree wedding, branches with a high error are pruned, thus preventing overfitting for the model [88, 89].

There are clear specific differences between the Cubist model and other tree-based models, such as committee models (a boosting-like procedure for building iterative model trees), specific techniques used for model smoothing, rule generation, and pruning, and instance-based corrections (using nearby points from the training set data to adjust predictions) [85, 86]. In addition, the Cubist model is highly interpretable and can provide insights into the decision-making process without the need for techniques such as SHAP (SHapley Additive exPlanations) [90] and LIME (Local Interpretable Model-agnostic Explanations) [91]. Moreover, the ability to handle non-linear relationships between dependent and independent variables [17, 92] and to use both numerical and categorical variables as model inputs give the Cubist model significant advantages in terms of high forecasting performance [93].

Machine learning algorithms have their own strengths and weaknesses. The choice of algorithm depends on the specific problem and the characteristics of the dataset. It is important to carefully consider the advantages and disadvantages of each algorithm before selecting the most appropriate one for a given task. The algorithms used in this study have their advantages and disadvantages, which can be summarized as follows.

Artificial neural networks are known for their ability to model complex relationships and handle large amounts of data. They can learn from examples and generalize well to unseen data. However, they can be computationally expensive to train and require a large amount of data to achieve good performance [94]. They can also exhibit illogical behavior when not well trained [95]. Cubist models are rule-based models that can handle both numerical and categorical variables. They are interpretable and can provide insights into the decision-making process. However, they may not perform as well as other algorithms when the relationships between variables are non-linear [93]. Gaussian processes with polynomial kernels are flexible models that can capture complex relationships between variables. They can provide levels of uncertainty for predictions. However, Gaussian processes are computationally intensive and may not scale well to large datasets, and they offer a probabilistic interpretation [96, 97]. Multivariate adaptive regression splines are non-linear models that can capture complex relationships between variables and they are useful when the dataset is large and computation time is not an issue [98]. Random forests are ensemble models that combine multiple decision trees. They are robust to overfitting and can handle high-dimensional data. They can also provide estimates of variable importance. However, they may not perform well when there are strong interactions between variables [99]. Quantile random forests are a variation of random forests that can model the conditional distribution of the response variable. They can provide more information about the uncertainty of predictions. However, they may require more computational resources and may not perform as well as other algorithms when the conditional distribution is highly skewed or heavy-tailed [100]. Support vector regression can handle non-linear relationships between variables and can provide good generalization performance. However, it can be sensitive to the choice of hyperparameters and may not perform well when the dataset is noisy or contains outliers [93]. Extreme gradient boosting (XGBoost) can handle high-dimensional data and capture complex relationships between variables. XGBoost is known for its high performance and scalability, However, XGBoost can be sensitive to outliers and may not perform well on unstructured and sparse data [101]. Additionally, XGBoost may have a slower prediction speed compared to the random forest due to the generation of sequential decision trees [102].

In this study, three basic metrics were used to evaluate the performance of machine learning models. These are the mean absolute error (MAE) and root mean square error (RMSE). The mathematical expressions of these performance measures can be shown as follows.

$$\begin{eqnarray}&&{MAE}=\displaystyle \frac{1}{n}\displaystyle \sum _{i=0}^{n}| {A}_{i}-{P}_{i}| \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{RMSE}=\sqrt{\displaystyle \frac{1}{n}\displaystyle \sum _{i=0}^{n}{\left({A}_{i}-{P}_{i}\right)}^{2}}\end{eqnarray} \tag{ 8 }$$

where n is the total number of data. A_i and P_i indicates actual data and predicted value of ith sample, respectively. The RMSE and MAE metrics should be close to zero for the model with high predictive performance.

In this study, the results of the nuclear charge radius estimations for the atomic nuclei from eight different machine-learning methods were obtained. All of the data (933 nuclei) used in the machine learning process were separated into training (699 nuclei) and test (234 nuclei) data sets. The training process of machine learning algorithms was carried out with 10-fold cross-validation processes and the model hyperparameters that gave the best results for each algorithm were determined. Thus, it aims to increase each model's performance by ensuring that it is optimized within the training process. Table 1 shows the performance metrics and optimized model hyperparameter values determined as a result of the 10-fold cross-validation process. According to table 1, along with the learning performance of all models examined in the study being quite high, it can be determined that the best-performing machine learning algorithm is the Cubist model (RMSE = 0.01199 fm and MAE = 0.0077 fm).

The performances of the models were evaluated by the test data and the results were shown with discrepancy distribution in figure 1. The one with the lowest RMSE value among these methods is the Cubist model, where the deviations of the estimation results from the experimental values are given in figure 1(a). This model's RMSE and MAE values are 0.0102 fm and 0.0075 fm, respectively. As clearly seen from the figure, the deviations from the experimental values are concentrated around the zero line except for a few atomic nuclei. In figure 1(b), the graph of the deviations of the predictions from the XGBoost from the experimental data is presented. The distribution in the curve, where the deviations are clearly seen to be concentrated around the horizontal zero line, appears to be in the range of about −0.05 fm to +0.05 fm. The RMSE and MAE values of the results obtained from the XGBoost model are 0.0125 fm and 0.0125 fm, respectively, which is the method in which the best results are obtained after the Cubist model. After the XGBoost, the results of the RF model, the method with the better predictions, are given in figure 1(c). It is seen that the distribution here is concentrated in the range of −0.03 fm to +0.06 fm, around the horizontal zero line. The RMSE and MAE values of the results obtained from this model were 0.0138 fm and 0.0102 fm, respectively. Next better results are obtained with the QRF model (figure 1(d)). This model's RMSE and MAE values are 0.0140 fm and 0.0104 fm, respectively. As can be seen from the figure, the distribution of deviations from the experimental data of this model is concentrated in the horizontal zero line, in the range of −0.05 fm to +0.06 fm. When the results obtained from the GPPK model given in figure 1(e) are examined, it is seen that the RMSE and MAE values are 0.0346 fm and 0.0273 fm, respectively. The distribution here appears to be in the range of −0.08 fm to +0.11 fm. The distribution of results in this model is concentrated on the positive side of the zero line, indicating that the estimates are generally bigger than the experimental data. The results of the predictions of the MARS method are given in figure 1(f), showing their deviations from the experimental values. The RMSE and MAE values of this model are 0.0346 fm and 0.0277 fm, respectively, and it is seen that the concentration is in the range of −0.06 fm and +0.11 fm around the zero line. In figure 1(g), the distributions of the estimates obtained from the SVR model are given. In the model, in which RMSE and MAE values were obtained as 0.0439 fm and 0.0336 fm, respectively. It is seen that the deviations of the estimations from the experimental data fluctuate around zero. However, the distribution was found to be in the range of −0.11 fm to +0.11 fm. Finally, the results from the ANN model are presented in figure 1(h). In the graph showing the distribution, it is clearly seen that the zero line disappears and the distribution is shifted down. It can also be seen from the figure that the deviations from the experimental values spread around −0.05 fm to +0.12 fm. The RMSE and MAE values of the ANN model were found to be 0.0564 fm and 0.0494 fm, respectively. These results obtained by ANN show close results to the findings of our previous study [21]. It should be noted that the results of Cubist, XGBoost and RF model for RMSE and MAE are better than those of [23, 25, 27]. As seen in this study, the Cubist model gives the most successful results in nuclear charge radius estimations. Cubist model success is based on its algorithm, which correctly partitions the data and generates regression equations that best fit each subset. This is due to its ability to model complex relationships using a combination of both tree structure and regression equations.

Zoom In Zoom Out Reset image size

This study used the bootstrap method to measure the uncertainty of machine learning predictions [103]. In this method, a large number of models (1000 for this study) are trained by resampling from the original training set and their predictions on the test data are recorded. The standard deviation of these predictions represents the uncertainty. For the Cubist model, which gave the best prediction results in our study, the minimum and maximum values of the uncertainty in the test data were determined as 0.000 85 fm (for ¹⁹⁹Hg) and 0.050 21 fm (for ⁷⁷Rb), respectively.

In this subsection, we are searching for the feasibility of considering machine learning methods based on experimental data of Kr and Sr isotopic chains to check their prediction on shell closure at N = 50 because the shell structure of nuclei around N = 50 is clearly visible in the experimental nuclear charge radii data where the lowest values are located at N = 50 for Kr and Sr isotopic chains. (More details for N and Z dependence of nuclear charge radii can be found in [4].) We also compare our results with the prediction of conventional semi-empirical charge radii formulas and mean field models. As it is well known the radius of the nucleus is proportional to its mass number. However, the conventional A-dependent RMS charge radius formulation in equation (9) is not globally valid for all nuclei covered by nuclidic charts because some nuclei have different numbers of neutrons and protons even if they have the same mass numbers. On the other hand, the experimental nuclear charge radii values show that the R/A^1/3 ratio is not constant through the nuclidic chart [104, 105]. Therefore, Z or isospin-dependent formulas have been developed and it has been found that they describe nuclear charge radii of nuclei much better (Details and references can be found in [104, 106]). Some well-known semi-empirical charge radii formulas are given below.

$$\begin{eqnarray}&&{R}_{c}={r}_{A}\,{A}^{1/3}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{R}_{c}={r}_{Z}\,{Z}^{1/3}\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{R}_{c}=\sqrt{\displaystyle \frac{5}{3}}{\left({\left({r}_{p}\,{Z}^{1/3}\right)}^{2}+0.64\right)}^{1/2}\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{R}_{c}={r}_{A}\left(1-b\,\displaystyle \frac{N-Z}{A}\right){A}^{1/3}\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{R}_{c}={r}_{A}\left(1-b\,\displaystyle \frac{N-Z}{A}+c\,\displaystyle \frac{1}{A}\right){A}^{1/3}\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{R}_{c}={r}_{Z}\left(1+b\,\displaystyle \frac{N-N/Z}{Z}\right){Z}^{1/3}\end{eqnarray} \tag{ 14 }$$

In these formulas, R_c , Z, N, and A are charge radii, proton number, neutron number, and mass number of considered nuclei, respectively. r_A , r_Z , r_p , b, and c are fitted parameters by using experimental data. These parameters were fitted by [6] for a better description of nuclear charge radii in the case of A > 40 region. For easy follow, the formulas given in equations (9), (10), (11), (12), (13) and (14) will be named as to be F1, F2, F3, F4, F5 and F6 in the text and graphics, respectively. The prediction of the Cubist model for nuclear charge radii of ⁷²⁻⁹⁶Kr and ⁷⁸⁻¹⁰⁰Sr are shown in figures 2(a) and (b), respectively. The experimental data [5] and the predictions of semi-empirical formulas are shown for comparison. As can be seen in the figures, the F1, F2, F3, and F6 formulas give results far away from experimental data. The F4 and F5 formulas give close results to experimental data up to N = 50 neutron numbers then start to become far. On the right panels of figure 2(a) and (b) kinks around N = 50 are seen which means that shell closure is visible on experimental charge radii of Kr and Sr isotopes. The Cubist model predicts kinks around N = 50 for Kr and Sr isotopic chains very well. It should be noted that the Cubist model predicts the experimental data quite successfully.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Furthermore, phenomenological microscopic nuclear models based on the mean-field approach give close charge radii values to experimental data for nuclei cover nuclidic chart [107, 108]. The kink on nuclear charge radii around N = 50 for isotopic chains is produced very well by the relativistic mean field (RMF) model with NL3^* interaction parameter [109]. Because of this reason, we have compared predictions of charge radius values for ⁷²⁻⁹⁶Kr and ⁷⁸⁻¹⁰⁰Sr isotopes obtained from either Cubist, MARS, SVR, GPPK, XGBoost, RF, QRF, ANN models together with the predictions of microscopic nuclear models and the experimental data. In figures 3 and 4, the predictions of considered machine learning methods for nuclear charge radii of ⁷²⁻⁹⁶Kr and ⁷⁸⁻¹⁰⁰Sr are shown, respectively. The experimental data [5] and calculated results of the HFB (Hartree-Fock-Bogoliubov) method with SLy4 force [108] and RMF model with non-linear NL3^* interaction [109] are shown for comparison. As can be seen from the figures, both microscopic models and four machine learning models (Cubist, XGBoost, RF and QRF) are successful in the prediction of the tendency of nuclear charge radii as a function of mass numbers. However, it should be noted that the kinks on nuclear charge radii around N = 50 in figures 3 and 4 are produced well with the RMF model and our machine learning methods Cubist, XGBoost, RF and QRF.

Zoom In Zoom Out Reset image size