Uncertainty Quantification for Nuclear Density Functional Theory and Information Content of New Measurements

Statistical tools of uncertainty quantification can be used to assess the information content of measured observables with respect to present-day theoretical models, to estimate model errors and thereby improve predictive capability, to extrapolate beyond the regions reached by experiment, and to provide meaningful input to applications and planned measurements. To showcase new opportunities offered by such tools, we make a rigorous analysis of theoretical statistical uncertainties in nuclear density functional theory using Bayesian inference methods. By considering the recent mass measurements from the Canadian Penning Trap at Argonne National Laboratory, we demonstrate how the Bayesian analysis and a direct least-squares optimization, combined with high-performance computing, can be used to assess the information content of the new data with respect to a model based on the Skyrme energy density functional approach. Employing the posterior probability distribution computed with a Gaussian process emulator, we apply the Bayesian framework to propagate theoretical statistical uncertainties in predictions of nuclear masses, two-neutron dripline, and fission barriers. Overall, we find that the new mass measurements do not impose a constraint that is strong enough to lead to significant changes in the model parameters. The example discussed in this study sets the stage for quantifying and maximizing the impact of new measurements with respect to current modeling and guiding future experimental efforts, thus enhancing the experiment-theory cycle in the scientific method.

Figure 1

Univariate and bivariate marginal estimates of the posterior distribution for the 12-dimensional DFT parameter vector of the unedf1 parametrization. The blue lines enclose an estimated 95% region for the posterior distribution found when only the original unedf1 data are accounted for; the green-outlined regions represent the same region for the posterior distribution found when the new CPT mass measurements are included. The ranges of parameter variations are $0.155\le {\rho }_{\mathrm{c}}\le 0.165({\mathrm{fm}}^{-3})$; $-16.0\le E^{\mathrm{NM}}/A\le -15.5(\mathrm{MeV})$; $200\le K^{\mathrm{NM}}\le 240(\mathrm{MeV})$; $28\le a_{\mathrm{sym}}^{\mathrm{NM}}\le 30(\mathrm{MeV})$; $20\le L_{\mathrm{sym}}^{\mathrm{NM}}\le 60(\mathrm{MeV})$; $0.8\le 1/M_s^{*}\le 1.2$; $-60\le C_0^{\rho \mathrm{\Delta }\rho }\le -40(\mathrm{MeV}{\mathrm{fm}}^5)$; $-200\le C_1^{\rho \mathrm{\Delta }\rho }\le -90(\mathrm{MeV}{\mathrm{fm}}^5)$; $-200\le V_0^n\le -150(\mathrm{MeV}{\mathrm{fm}}^3)$; $-220\le V_0^p\le -180(\mathrm{MeV}{\mathrm{fm}}^3)$; $-80\le C_0^{\rho \nabla J}\le -60(\mathrm{MeV}{\mathrm{fm}}^5)$; and $-80\le C_1^{\rho \nabla J}\le 0(\mathrm{MeV}{\mathrm{fm}}^5)$.

Figure 2

Estimated theoretical error bars for the masses of the even-even nuclei measured in Refs. [28, 29, 30], using the posterior distribution for unedf1. Thin green bars show the results of training DFT calculations. Dark blue bands represent the 90% confidence bands due to uncertainty in the EDF parameters; larger, light blue bands also account for model error; black bars show mass residuals.

Figure 3

Comparison between the two-neutron dripline predictions made with unedf1 (solid line) and those made with ${\mathrm{UNEDF}1}_{\text{CPT}}$ (dashed line). The 90% probability spread about the unedf1 predictions is shown in gray.

Figure 4

Comparison between the fission barrier predictions for ${}^{240}\mathrm{Pu}$ made with unedf1 (solid line) and those made with ${\mathrm{UNEDF}1}_{\text{CPT}}$ (dashed line), together with the 90% confidence interval (shaded gray area). The potential energy surface was obtained by following the lowest-energy static fission pathway in a four-dimensional collective space of axial and triaxial quadrupole, axial octupole, and axial hexadecapole mass moments.