Analysis of a Skyrme energy density functional with deep learning

Over the past decade, machine learning has been successfully applied in various fields of science. In this study, we employ a deep learning method to analyze a Skyrme energy density functional (Skyrme-EDF), which is a Kohn-Sham type functional commonly used in nuclear physics. Our goal is to construct an orbital-free functional that reproduces the results of the Skyrme-EDF. To this end, we first compute energies and densities of a nucleus with the Skyrme Kohn-Sham $+$ Bardeen-Cooper-Schrieffer method by introducing a set of external fields. Those are then used as training data for deep learning to construct a functional which depends only on the density distribution. Applying this scheme to the ${}^{24}\mathrm{Mg}$ nucleus with two distinct random external fields, we successfully obtain a new functional which reproduces the binding energy of the original Skyrme-EDF with an accuracy of about 0.04 MeV. The rate at which the neural network outputs the energy for a given density is about ${10}^5–{10}^6$ times faster than the Kohn-Sham scheme, demonstrating a promising potential for applications to heavy and superheavy nuclei, including the dynamics of fission.

Figure 1

A neural network employed in this work to learn the Skyrme-EDF, $E\left[\rho \right]$. It consists of 10 hidden layers, all of which are fully connected. Their activation functions are the ReLU, and the sigmoid activation function is employed for the output layer. The number of neurons in each layer is listed below the layers.

Figure 2

Histograms for the results of the Skyrme EDF calculations with 250 000 different external fields based on the SHO (the first row) and the RND (the second row) fields. The left and right panels show the binding energy and the pairing energy in units MeV, respectively. It can be observed that the structure in the shape of the histograms is washed out to a large extent for the RND external fields, which are more random than the SHO cases.

Figure 3

The differences between the Kohn-Sham result and the predicted results from the neural network for $E\left[\rho \right]$ and $E\left[v\right]$ for the RND (the first and the second rows) and the SHO (the third and forth rows) external fields. These are given as a function of the corresponding Kohn-Sham energies, all given in units of MeV. From the left to the right columns, the training results are shown for the binding energy, the pairing energy, and the energy of the external fields. The results for 20 000 test data points are plotted in each figure, in which densely populated (under populated) points are displayed in red (blue).

Figure 4

A neural network with the encoder-decoder structure employed in this work for a mapping from an external $v$ to a particle number density $\rho$. It consists of 10 hidden layers, all of which are fully connected. Their activation functions are ReLU, and the softmax activation function is employed for the output layer.

Figure 5

The absolute error of the density distribution directly generated by a deep learning from a given external field $v$. It is plotted as a function of the binding energy from the corresponding Kohn-Sham calculation. The left and the right panels show the results with the SHO and the RND external fields, respectively. The densely populated points are displayed in red, while the underpopulated points are shown in blue.

Figure 6

Examples of the predicted densities (the panels in the third row) generated directly from the RND external potentials shown in the panels in the first row. For a comparison, the corresponding Kohn-Sham densities are also plotted in the second row. The panels in the fourth row show the difference between the predicted densities and the Kohn-Sham densities, on which the numbers denote the absolute error for each sample. The units of the color coordinate are MeV for the external potentials and ${\mathrm{fm}}^{-3}$ for the densities.

Figure 7

A verification of generalization performance for the present deep learning. The left and right panels show the results with the RND and the SHO external fields, respectively. The horizontal axes denote the energies obtained with the Kohn-Sham calculations. On the other hand, the vertical axes denote the difference between the Kohn-Sham energy of RND $E_{\mathrm{SHO}}\left[{\rho }_{\mathrm{RND}}\right]$ and $E_{\mathrm{RND}}\left[{\rho }_{\mathrm{SHO}}\right]$, that is, the predictions of deep learning trained with the SHO (the left panel) and the RND (the right panel) external fields. Both the training and test data (200 000 data in total) are plotted in each panels because the RND (SHO) dataset are not used in training $E_{\mathrm{SHO}}\left[\rho \right]$ ($E_{\mathrm{RND}}\left[\rho \right])$.

Figure 8

Same as Fig. 2, but for the kinetic energy, the interaction energy, and the energy for the external field.

Figure 9

Same as Fig. 3, but for the kinetic and the interaction energies.