Nuclear mass predictions with machine learning reaching the accuracy required by $r$-process studies

Nuclear masses are predicted with the Bayesian neural networks by learning the mass surface of even-even nuclei and the correlation energies to their neighboring nuclei. By keeping the known physics in various sophisticated mass models and performing the delicate design of neural networks, the proposed Bayesian machine learning mass model achieves an accuracy of $84\mathrm{keV}$, which crosses the accuracy threshold of the $100\mathrm{keV}$ in the experimentally known region. It is also demonstrated the corresponding uncertainties of mass predictions are properly evaluated, while the uncertainties increase by about $50\mathrm{keV}$ each step along the isotopic chains towards the unknown region. The shell structures in the known region are well described and several important features in the unknown region are predicted, such as the new magic numbers around $N=40$, the robustness of $N=82$ shell, the quenching of $N=126$ shell, and the smooth separation energies around $N=104$.

Figure 1

The rms deviations of $M, S_n, S_{2n}, S_p, S_{2p}, S_D$, and $Q_{\beta }$ with respect to the experimental data for the learning set of the BML mass model. The corresponding rms deviations given by the BW2, FRDM12, HFB-31, and DZ28 mass models are shown for comparison.

Figure 2

Microscopic correction energies $E_{\mathrm{mic}}$ of BML. The contours show the boundary of nuclei with known masses in AME2016 and the dotted lines denote the traditional magic numbers.

Figure 3

Same as Fig. 2 but for (a) the two-proton gaps ${\delta }_{2p}$ and (b) the two-neutron gaps ${\delta }_{2n}$.

Figure 4

Mass differences between $M_{\mathrm{th}}$ from the BML, NewBML, HFB-31, FRDM12, and $\mathrm{WS}4+\mathrm{BNN}\text{−}\mathrm{I}4$ mass models and $M_{\exp }$ in AME2016 for the Cr and Nd isotopes. The shaded (white) regions indicate the BML learning (extrapolation) areas, where the accuracy of $M_{\exp }$ in AME2016 is higher (worse) than $100\mathrm{keV}$. The new experimental data [45, 46] after AME2016 are shown with blue solid symbols.

Figure 5

Two-neutron separation energies $S_{2n}$ of $Z=60\text{−}65$ isotopes. The $S_{2n}$ calculated with the experimental data in AME2016 are shown with yellow circles, while the new $S_{2n}$ calculated by including the new experimental data [45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72] are shown with blue circles. The $S_{2n}$ predictions of BML and NewBML are shown with open circles and open squares, respectively. For displaying the data clearly, all $S_{2n}$ of $Z=61\text{−}65$ isotopes are increased by $(Z-60)$ MeV.