Refining mass formulas for astrophysical applications: A Bayesian neural network approach

Background: Exotic nuclei, particularly those near the drip lines, are at the core of one of the fundamental questions driving nuclear structure and astrophysics today: What are the limits of nuclear binding? Exotic nuclei play a critical role in both informing theoretical models as well as in our understanding of the origin of the heavy elements.

Purpose: Our aim is to refine existing mass models through the training of an artificial neural network that will mitigate the large model discrepancies far away from stability.

Methods: The basic paradigm of our two-pronged approach is an existing mass model that captures as much as possible of the underlying physics followed by the implementation of a Bayesian neural network (BNN) refinement to account for the missing physics. Bayesian inference is employed to determine the parameters of the neural network so that model predictions may be accompanied by theoretical uncertainties.

Results: Despite the undeniable quality of the mass models adopted in this work, we observe a significant improvement (of about 40%) after the BNN refinement is implemented. Indeed, in the specific case of the Duflo-Zuker mass formula, we find that the rms deviation relative to experiment is reduced from ${\sigma }_{\mathrm{rms}}=0.503$ MeV to ${\sigma }_{\mathrm{rms}}=0.286$ MeV. These newly refined mass tables are used to map the neutron drip lines (or rather “drip bands”) and to study a few critical $r$-process nuclei.

Conclusions: The BNN approach is highly successful in refining the predictions of existing mass models. In particular, the large discrepancy displayed by the original “bare” models in regions where experimental data are unavailable is considerably quenched after the BNN refinement. This lends credence to our approach and has motivated us to publish refined mass tables that we trust will be helpful for future astrophysical applications.

Figure 1

An example of a feed-forward neural network with a single hidden layer consisting of three nodes. In our case, the two inputs that define the nucleus of interest are $Z$ and $A$, and a single output provides an estimate of $\delta M(Z,A)$; namely, the discrepancy between the bare theoretical prediction and the experimental value.

Figure 2

Correlations coefficients and marginalized probability densities (shown along the diagonal) for the 10-parameter Duflo-Zuker model [22, 50, 53].

Figure 3

Proton and neutron drip lines as predicted by both the bare and BNN-refined 10-parameter Duflo-Zuker (DZ10) model. In the case of the BNN predictions, the neutron drip line evolves into a “drip band” due to statistical uncertainties. Also shown (with green points) is the AME2012 data set [31], with the stable nuclei displayed with black points.

Figure 4

Proton and neutron drip lines and drip bands as predicted by (a) the Duflo-Zuker model [22] and (b) the HFB19 model [23]; in both cases bare and BNN-refined predictions are displayed. The remaining two panels display the same information but now the comparison is between the predictions of both mass models: (c) displays bare-model results whereas (d) shows the corresponding ones after BNN refinement. Also shown (with green points) is the AME2012 data set [31], with the stable nuclei displayed with black points.

Figure 5

Mass predictions for (a) palladium ($Z=46$), (b) cadmium ($Z=48$), (c) indium ($Z=49$), and (d) tin ($Z=50$) for both the Duflo-Zuker [22] and HFB19 [23] mass models. The predictions are relative to a reference mass value taken from either the AME2012 compilation when available [31] or from the bare Duflo-Zuker model when unavailable. Predictions are displayed with statistical error bars for the BNN-improved models.

Figure 6

Two-neutron separation energies for all even-even cadmium isotopes from ${}^{94}\mathrm{Cd}$ to ${}^{164}\mathrm{Cd}$, as predicted by the mic-mac model of Duflo and Zuker [22] and the microscopic HFB19 model [23]. Predictions are displayed without error bars for the bare models and with error bars after the BNN refinement. Experimental data are from Ref. [31].