Building Surrogate Models of Nuclear Density Functional Theory with Gaussian Processesand Autoencoders

TL;DR

The paper tackles the computational bottleneck of predictive nuclear density functional theory by developing two surrogate modeling approaches for HFB solutions: Gaussian processes to interpolate mean-field and pairing potentials across a two-dimensional PES, and deep autoencoders to learn compact latent representations of canonical wavefunctions. The GP emulators achieve sub-MeV accuracy in total energy and percent-level accuracy in collective inertia on smooth PES regions, whereas the autoencoder learns a low-dimensional latent space (around $D=10$–$20$) that preserves essential physics and enables post-hoc physics validation. Together, these surrogates enable efficient uncertainty quantification and PES densification, offering a path toward rapid, large-scale nuclear-structure and fission simulations with controlled errors. The study demonstrates practical feasibility and highlights limitations, such as GP sensitivity to discontinuities and the need for physics-aware loss and latent representations in deep models.

Abstract

From the lightest Hydrogen isotopes up to the recently synthesized Oganesson (Z=118), it is estimated that as many as about 3000 atomic nuclei could exist in nature. Most of these nuclei are too short-lived to be occurring on Earth, but they play an essential role in astrophysical events such as supernova explosions or neutron star mergers that are presumed to be at the origin of most heavy elements in the Universe. Understanding the structure, reactions, and decays of nuclei across the entire chart of nuclides is an enormous challenge because of the experimental difficulties in measuring properties of interest in such fleeting objects and the theoretical and computational issues of simulating strongly-interacting quantum many-body systems. Nuclear density functional theory (DFT) is a fully microscopic theoretical framework which has the potential of providing such a quantitatively accurate description of nuclear properties for every nucleus in the chart of nuclides. Thanks to high-performance computing facilities, it has already been successfully applied to predict nuclear masses, global patterns of radioactive decay like $β$ or $γ$ decay, and several aspects of the nuclear fission process such as, e.g., spontaneous fission half-lives. Yet, predictive simulations of nuclear spectroscopy or of nuclear fission, or the quantification of theoretical uncertainties and their propagation to applications, would require several orders of magnitude more calculations than currently possible. However, most of this computational effort would be spent into generating a suitable basis of DFT wavefunctions. Such a task could potentially be considerably accelerated by borrowing tools from the field of machine learning and artificial intelligence. In this paper, we review different approaches to applying supervised and unsupervised learning techniques to nuclear DFT.

Figures (13)

• Figure 1: Potential energy surface of $^{240}$Pu with the SkM* EDF for the grid $(q_{20},q_{30}) \in [ 0\, {\rm b}, 300\, {\rm b}] \times [0\, {\rm b}^{3/2}, 51\, {\rm b}^{3/2}]$ with steps $\delta q_{20}=6\, {\rm b}$ and $\delta q_{30}= 3\, {\rm b}^{3/2}$. The black crosses are the training points, the white circles the validation points. Energies indicated by the color bar are in MeV relatively to -1820 MeV.
• Figure 2: Left: Histogram of the error on the GP-predicted total HFB energy and zero-point energy correction across the validation points. Bin size is 100 keV. Right: Size of the error on the GP-predicted total HFB energy across the validation set. Gray circles have an error lower than 500 keV and the size of the markers correspond to energy bins of 100 keV. Black circles have an error greater than 500 keV and are binned by 400 keV units. Energies indicated by the color bar are in MeV relatively to -1820 MeV.
• Figure 3: Left: Histogram of the error on the GP-predicted values of the multipole moments. The bin size is 0.2 $b^{\lambda/2}$ with $\lambda=2$ (quadrupole moment) or $\lambda=3$ (octupole moment). Right: Histogram of the relative error, in percents, on the GP-predicted values of the components of the collective inertia tensor. The bin size is 1, corresponding to 1% relative errors.
• Figure 4: Upper left: Central part of the mean-field potential for protons, $U_p(r,z)$ for the configuration $(q_{20},q_{30}) = (198\, \mathrm{b}, 30\, \mathrm{b}^{3/2})$; bottom left: Error in the GP fit for that same configuration. Upper right: Central part of the mean-field potential for protons, $U_p(r,z)$ for the configuration $(q_{20},q_{30}) = (138\, \mathrm{b}, 51\, \mathrm{b}^{3/2})$; bottom right: Error in the GP fit for that same configuration. For all figures, $i_{\rm GH}$ and $j_{\rm GL}$ refer to the index $i$ and $j$ on the Gauss-Hermite and Gauss-Laguerre quadrature grid, and the energy given by the error bar is in MeV.
• Figure 5: An autoencoder is the association of two blocks. The first one, on the left, compresses the input data into a lower-dimensional representation, or code, in the latent space. The second one, on the right, decompresses the code back into the original input.