Predicting nuclear masses with the kernel ridge regression

The kernel ridge regression (KRR) approach is employed to improve the nuclear mass predictions for the first time, and more importantly, a careful study of the expected predictive power in the neutron-rich region of the nuclear chart is presented, which is crucial for astrophysical simulations of nucleosynthesis processes. The ridge penalty influences the optimized kernel function significantly and plays an essential role in reducing the risk of overfitting and improving the accuracy of the predictions. By taking the WS4 mass model as an example, the mass for each nucleus in the nuclear chart is predicted with the KRR network, which is trained with the mass model residuals, i.e., deviations between experimental and calculated masses, of other nuclei with known masses. The resultant root-mean-square mass deviation from the available experimental data for the 2353 nuclei with $Z\ge 8$ and $N\ge 8$ can be reduced to 199 keV. This is at the same level with other machine-learning-rooted approaches, such as the radial basis function and Bayesian neural network approaches. However, with the Gaussian kernel, the present KRR approach provides strikingly different extrapolated masses for nuclei with unknown masses, since it automatically identifies the limit of the extrapolation distance. Therefore, it avoids the risk of worsening the mass description for nuclei at large extrapolation distances.

Figure 1

Statistical distributions of the hyperparameters $\sigma$ (left) and $\lambda$ (right) for the tenfold cross-validations on the 3000 sampled partitions. The mean and standard deviation are also listed.

Figure 2

(a) Mass differences $\mathrm{\Delta }M=M^{\exp }-M^{\mathrm{WS}4}$ between the experimental data [12] and the predictions of the WS4 model [16]. (b) The KRR reconstructed function $S(N,Z)$ for the WS4 mass table. (c) Mass differences $\mathrm{\Delta }{M}^{\prime }=M^{\exp }-(M^{\mathrm{WS}4}+S^{\mathrm{KRR}})$ between the experimental data and the predictions of the WS4 model with the KRR corrections. The magic numbers are indicated by vertical and horizontal dotted lines. The boundary of nuclei with known masses in AME2012 [12] is shown by the black contour lines.

Figure 3

Nuclei in the training set (gray circles) and eight test sets (colored squares with extrapolation distance numbers) for examining the extrapolation power for the neutron-rich nuclei. The inset presents a partial enlargement of the region from $Z=11$ to 19.

Figure 4

The tenfold cross-validation rms deviations as a function of the hyperparameter $\sigma$ in the Gaussian kernel with several selected penalties with different $\lambda$ values. For comparison, the rms deviation for the WS4 mass predictions is also shown with the horizontal dashed line.

Figure 5

Comparison of the extrapolation power between the KRR and RBF approaches for the eight test sets denoted in Fig. 3. (a) The rms deviations ${\mathrm{\Delta }}_{\mathrm{rms}}$ of the calculated masses from the WS4 mass model (squares), the KRR extrapolations (circles), the linear RBF extrapolations (up triangles), and the Gaussian RBF extrapolations (down triangles) with respect to the experimental masses. (b) For a better comparison, the rms deviations ${\mathrm{\Delta }}_{\mathrm{rms}}$ are scaled to the values for the WS4 mass model. (c) The rms averages of the reconstructed functions $\overline{S}(N,Z)$ for each test set from the KRR (circles), linear RBF (up triangles), and Gaussian RBF (down triangles) approaches.