Deep learning approach to nuclear masses and $\alpha$-decay half-lives

Ab initio calculations of nuclear masses, binding energy, and $\alpha$-decay half-lives are intractable for heavy nucleus because of the curse of dimensionality in many-body quantum simulations as proton number ($N$) and neutron number ($Z$) grow. We take advantage of the powerful nonlinear transformation and feature representation ability of deep neural network (DNN) to predict the nuclear masses and $\alpha$-decay half-lives. For nuclear binding energy prediction problem we achieve standard deviation $\sigma =0.263$ MeV on 10-fold cross validation on 2149 nuclei. Word-vectors which are high-dimensional representation of nuclei from the hidden layers of mass-regression DNN help us to calculate $\alpha$-decay half-lives. For this task, we get $\sigma =0.797$ on 100 times 10-fold cross validation on 350 nuclei on ${\text{log}}_{10}{T}_{1/2}$ and $\sigma =0.731$ on 486 nuclei. DNN is also used to reduce the residual of three-parameter Gamow formula on 159 even-even nuclei, from 0.3627 to 0.2297 on ${\text{log}}_{10}{T}_{1/2}$, using 100 times 10-fold cross validation. We find physical a priori such as shell structure, magic numbers and augmented inputs inspired by finite-range droplet model are important for this small data regression task.

Figure 1

The adjustable neural network structure with the number of hidden layers changed by $n$ and the width of hidden layers changed by $m$. The output layer is the residual defined in Eq. (2).

Figure 2

The performance of the atomic mass prediction using 10-fold cross validation. The bands represent the range of root-mean-square error using different numbers of hidden layers and different numbers of neurons per layer in the architecture of the deep neural network. The optimal RMSE is around 0.22 MeV using 10 hidden layers with 1,024 neurons in the first and the last hidden layer and 256 neurons for eight other hidden layers, which correspond to width = 256 in the plot.

Figure 3

The optimal performance of the atomic mass prediction using 10-fold cross validation. The RMS error of the 10-fold cross validation for (A) 26 features and (B) 3 features as the input of the optimal network $(n=8,m=256)$.

Figure 4

The prediction error for all the nuclei as compared with LDM. Both training and testing data are used with (A) 26 features and (B) 3 features as the input of the optimal network $(n=8,m=256)$.

Figure 5

The mass-residual prediction error for 322 new elements in AME2020 as compared with LDM.

Figure 6

The $Q$-value prediction error for 486 nuclei.

Figure 7

The $Q$ value from deep neural network (solid circles) as compared with experimental measurements (square) for nuclei with proton number = 86, 87, 88, 89, 92, 93.

Figure 8

Prediction for $\alpha$-decay half-lives using native inputs as compared with word-vector representations on 350 nuclei. Word vector 1 means it is obtained from the first hidden layer of the deep neural network we used to predict nuclear mass; for the details of those native features, see the Appendix.

Figure 9

The performance of word-vector from different hidden layers compared with native features as inputs.

Figure 10

Result of $\alpha$-decay half-lives prediction on 486 nuclei using 64 native features as input.

Figure 11

Best network model prediction on 486 $\alpha$-decay half-lives data.

Figure 12

The distribution and the mean residual of the Gamow formula as compared with the DNN improvement, on the $\alpha$-decay half-lives of even-even nuclei.

Figure 13

The distribution and the mean residual of modified Gamow formula on the $\alpha$-decay half-lives of even-even nuclei.

Figure 14

The binding energy per nucleon as a function of proton number $Z$ and neutron number $N$.

Figure 15

The residual of binding energy per nucleon as a function of number of protons (left) and number of neutrons (right) on valence shells. Different shells contribute to different bands in the figure.

Figure 16

The binding energy per nucleon as a function of number of protons (left) and number of neutrons (right) on valence shells 3 and 4.

Figure 17

The binding energy per nucleon as a function of number of protons (left) and number of neutrons (right) on valence shells of heavy nucleus.