Neural network emulation of spontaneous fission

Background: Large-scale computations of fission properties are an important ingredient for nuclear reaction network calculations simulating rapid neutron-capture process (the $r$ process) nucleosynthesis. Due to the large number of fissioning nuclei potentially contributing to the $r$ process, a microscopic description of fission based on nuclear density functional theory (DFT) is computationally challenging.

Purpose: We explore the use of neural networks (NNs) to construct DFT emulators capable of predicting potential energy surfaces and collective inertia tensors across the whole nuclear chart, starting from a minimal set of DFT calculations.

Methods: We use constrained Hartree-Fock-Bogoliubov (HFB) calculations to predict the potential energy and collective inertia tensor in the axial quadrupole and octupole collective coordinates, for a set of nuclei in the $r$-process region. We then employ NNs to emulate the HFB energy and collective inertia tensor across the considered region of the nuclear chart. Least-action pathways characterizing spontaneous fission half-lives and fragment yields are then obtained by means of the nudged elastic band method.

Results: The potential energy predicted by NNs agrees with the DFT value to within a root-mean-square error of 500 keV, and the collective inertia components agree to within an order of magnitude. These results are largely independent of the NN architecture. The exit points on the outer turning line are found to be well emulated. For the spontaneous fission half-lives the NN emulation provides values that are found to agree with the DFT predictions within a factor of ${10}^3$ across more than 70 orders of magnitude.

Conclusions: Neural networks are able to emulate the potential energy and collective inertia well enough to reasonably predict physical observables. Future directions of study, such as the inclusion of additional collective degrees of freedom and active learning, will improve the predictive power of microscopic theory and further enable large-scale fission studies.

Figure 1

$\mathrm{\Delta }V(A,Z)$ (in MeV) for the deepest NN. The different shapes indicate to which data set each nucleus belongs.

Figure 2

(a) The reference PES for ${}^{280}\mathrm{Cm}$ in MeV. (b) The difference between the reference and reconstructed PESs, again for ${}^{280}\mathrm{Cm}$, in MeV. The reference ground state and the reference LAPs are marked with a red X symbol and black lines, respectively, in both panels. The energy range $0.5\le V^{\text{DFT}}\le 12$ MeV is shaded in gray in (b).

Figure 3

The RMSE for a variety of different NN sizes. The dashed line shows the same depth NN, but with input variables normalized to the range [0,1].

Figure 4

The reference components of $\mathcal{M}$, plotted against the NN reconstructions, for all nuclei considered. The black line is the diagonal, ${\mathcal{M}}_{\mu \nu }^{\text{DFT}}={\mathcal{M}}_{\mu \nu }^{\text{NN}}$. The blue squares/green circles/orange triangles correspond to the training/combining/validation data sets. Note that, for use on a log scale, we plot the absolute value of ${\mathcal{M}}_{23}$ (the other components are non-negative). ${\mathcal{M}}_{22}$ is in ${\mathrm{MeV}}^{-1}{\mathrm{b}}^{-2}, {\mathcal{M}}_{23}$ is in ${\mathrm{MeV}}^{-1}{\mathrm{b}}^{-5/2}$, and ${\mathcal{M}}_{33}$ is in ${\mathrm{MeV}}^{-1}{\mathrm{b}}^{-3}$.

Figure 5

The $Q_{30}$ component of the reconstructed lifetime-weighted exit point, minus the reference $Q_{30}$ component, in ${\mathrm{b}}^{3/2}$. These results were computed using configuration (iii), i.e., the NN was used to reconstruct both the PES and the collective inertia.

Figure 6

The half-life predicted using the DFT reference data, $t_{1/2}^{\text{sf-DFT}}$, plotted against the half-life computed using the NN reconstruction, $t_{1/2}^{\text{sf-NN}}$. (a) shows configuration (i), in which only the PES is emulated; (b) shows configuration (iii), in which both the PES and the collective inertia are emulated. The black line marks the diagonal: $t_{1/2}^{\text{sf-DFT}}=t_{1/2}^{\text{sf-NN}}$. Gray bars are drawn at $t_{1/2}^{\text{sf-DFT}}\times {10}^{\pm 3}$, i.e., 3 orders of magnitude above and below the diagonal. Insets show the range ${10}^{-5}–{10}^{10}$ s, to highlight the relevant $r$-process range.

Figure 7

The ratio of the reconstructed half-life, $t_{1/2}^{\text{sf-NN}}$, to the reference half-life, $t_{1/2}^{\text{sf-DFT}}$, plotted against the difference in $Q_{20}$ between the ground state and the exit point, for configuration (iii). Gray bars mark a three orders of magnitude region.