Nuclear binding energies in artificial neural networks

The binding energy or mass is one of the most fundamental properties of an atomic nucleus. Precise binding energies are vital inputs for many nuclear physics and nuclear astrophysics studies. However, due to the complexity of atomic nuclei and the nonperturbative strong interaction, up to now, no conventional physical model can describe nuclear binding energies with a precision below 0.1 MeV, the accuracy needed by nuclear astrophysical studies. In this work, artificial neural networks (ANNs), the so-called “universal approximators”, are used to calculate nuclear binding energies. We show that the ANN can describe all the nuclei in AME2020 with a root-mean-square deviation (RMSD) around 0.2 MeV, better than the best macroscopic-microscopic models, such as FRDM and WS4. The success of the ANN is mainly due to the proper and essential input features we identify, which contain the most relevant physical information, i.e., shell, paring, and isospin-asymmetry effects. We show that the well-trained ANN has excellent extrapolation ability and can predict binding energies for those nuclei inaccessible experimentally. In particular, we highlight the important role of “feature engineering” for physical systems where data are relatively scarce, such as nuclear binding energies.

Figure 1

The architecture of a neural network consists of two input features, two hidden layers of nine and seven nodes, and one output layer.

Figure 2

Absolute deviations of the ANN2 and ANN4 predictions from the experimental binding energies. The gray lines denote the magic numbers.

Figure 3

Absolute deviations of the ANN6 and ANN7 predictions from the experimental binding energies. The gray lines denote the magic numbers.

Figure 4

Distributions of ${\sigma }_{\mathrm{rms}}$ between every 40 mass numbers in test20.

Figure 5

${\sigma }_{\mathrm{rms}}$ as functions of $|N-Z|$, which reflects the ability of ANNs to describe nuclei with large isospin asymmetries. Note that there is no nucleus in test20 with $50<|N-Z|<60$.

Figure 6

Experimental single neutron separation energies in comparison with the ANN predictions.

Figure 7

Experimental two-neutron separation energies in comparison with the ANN predictions.

Figure 8

The distribution of the number of atomic nuclei as functions of deviations between theoretical calculations and experimental data for binding energies. The nuclei for which the BE deviations fall below 0.7 MeV are divided into seven equally spaced groups. The normal distribution curves fit the number of nuclei for which the ${\sigma }_{\mathrm{rms}}$ is lower than 0.7 MeV. We note that 63(51.6%) of the nuclei in ANN2, 32(26.2%) of the nuclei in ANN4, 16(13.1%) of the nuclei in ANN6, and 7(5.7%) of the nuclei in ANN7 have deviations over 0.7 MeV.