Bayesian model mixing with multi-reference energy density functional

TL;DR

The paper tackles robust prediction of nuclear observables across the chart by marrying multi-reference energy density functional theory with Bayesian model mixing. It builds eight MR-EDF models from particle-number and angular-momentum projection for four Skyrme functionals, then combines them via hierarchical stacking using Dirichlet priors or additive log-ratio transformations to model locally varying, potentially positively correlated weights. Results show substantial improvements in predicting two-neutron separation energies $S_{2n}$, with RMS reductions of roughly 40–60% and GP-based weight models achieving sub-0.5 MeV accuracy in many regions, though extrapolations remain challenging. The work demonstrates a robust, uncertainty-aware framework to fuse beyond-mean-field theories and suggests extending it by co-fitting energy functionals with the BMM to further improve predictive reliability.

Abstract

Reliably predicting nuclear properties across the entire chart of isotopes is important for applications ranging from nuclear astrophysics to superheavy science to nuclear technology. To this day, however, all the theoretical models that can scale at the level of the chart of isotopes remain semi phenomenological. Because they are fitted locally, their predictive power can vary significantly; different versions of the same theory provide different predictions. Bayesian model mixing takes advantage of such imperfect models to build a local mixture of a set of models to make improved predictions. Earlier attempts to use Bayesian model mixing for mass table calculations relied on models treated at single-reference energy density functional level, which fail to capture some of the correlations caused by configuration mixing or the restoration of broken symmetries. In this study we have applied Bayesian model mixing techniques within a multi-reference energy density functional (MR-EDF) framework. We considered predictions of two-particle separation energies from particle number projection or angular momentum projection with four different energy density functionals - a total of eight different MR-EDF models. We used a hierarchical Bayesian stacking framework with a Dirichlet prior distribution over weights together with an inverse log-ratio transform to enable positive correlations between different models. We found that Bayesian model mixing provide significantly improved predictions over results from single MR-EDF calculations.

Figures (8)

• Figure 1: Location of nuclei with available experimental data included in this study with the training set marked with black squares and the validation set by red squares.
• Figure 2: Histograms of the prediction error for each of the eight initial models and the BMM predictions, for each of the four types of BMM considered in this work: a) Generalized Linear Dirichlet, GLD; b) Generalized Linear Log-Ratio, GLLR; c) Gaussian Process Dirichlet, GPD; d) Gaussian Process Log-Ratio, GPLR.
• Figure 3: Prediction error of the two-neutron separation energy in Lead isotopes from the proton to the neutron dripline for each of the four BMM models: a) Generalized Linear Dirichlet, GLD; b) Generalized Linear Log-Ratio, GLLR; c) Gaussian Process Dirichlet, GPD; d) Gaussian Process Log-Ratio, GPLR.
• Figure 4: Posterior weights of each model in the BMM along the Lead isotopic line from the proton to the neutron dripline. a) Generalized Linear Dirichlet, GLD; b) Generalized Linear Log-Ratio, GLLR; c) Gaussian Process Dirichlet, GPD; d) Gaussian Process Log-Ratio, GPLR.
• Figure 5: Correlation matrix of the eight model weights in $^{208}$Pb. Dirichlet models are shown in panels (a) (Generalized Linear, GLD) and (c) (Gausian Process, GPD); models based on the inverse log-ratio transformation are shown in panels (b) (Generalized Linear, GLLR) and (d) (Gaussian Process, GPLR).