HFBTHO-AD: Differentiation of a nuclear energy density functional code

TL;DR

This work demonstrates how automatic differentiation, via the Tapenade tool, can be integrated with the HFBTHO nuclear energy density functional solver to compute derivatives of outputs with respect to a subset of input parameters. By differentiating the top-level solver and differentiating through BLAS/LAPACK (with DSYEVR treated by a higher-level rule), the authors obtain a Jacobian of size $108\times12$ for $d=108$ residuals across $72$ nucleus configurations, enabling derivative-based optimization and uncertainty quantification in Skyrme-like functionals. Validation across three UNEDF0 parameter points shows strong agreement between AD and finite-difference derivatives, and a detailed performance analysis indicates that AD offers similar walltimes to FD but superior resource efficiency, albeit with higher memory demands. The results pave the way for Levenberg–Marquardt-style optimization and uncertainty quantification in nuclear EDF calibrations, with future work aimed at symmetry restoration and comparisons between Skyrme and Gogny functionals.

Abstract

The HFBTHO code implements a nuclear energy density functional solver to model the structure of atomic nuclei. HFBTHO has previously been used to calibrate energy functionals and perform sensitivity analysis by using derivative-free methods. To enable derivative-based optimization and uncertainty quantification approaches, we must compute the derivatives of HFBTHO outputs with respect to the parameters of the energy functional, which are a subset of all input parameters of the code. We use the algorithmic/automatic differentiation (AD) tool Tapenade to differentiate HFBTHO. We compare the derivatives obtained using AD against finite-difference approximation and examine the performance of the derivative computation.

Figures (5)

• Figure 1: Strong-scaling results for the (top) Intel/LAPACK software configuration and the (bottom) Intel/LAPACK/AD software configuration.
• Figure 2: Noise level determined for the $r_{11}$ residual associated with the ground-state binding energy of $^{248}$Fm at UNEDFpre along ($\mathbf{e}_j$). The noise level is abnormally large, and there is a large deviation between the partial derivative of $r_{11}$ with respect to approximated with AD (78.45) and with forward differences (53.55). (top) ECNoise data collected at UNEDFpre to study the noise level. The two FD evaluations used to approximate the partial derivative of this observable with respect to are shown in red. (bottom) The deviation between the ECNoise data and the predictions made by a linear regression model that was fit to the ECNoise data with least-squares. There is a sharp change in the ECNoise data and the two evaluations used to compute the forward difference lie on either side of this change.
• Figure 3: Absolute deviations between the residual vectors $\mathbf{r}$, obtained at UNEDFpre as computed using (top) Intel/LAPACK/AD and GCC/LAPACK/AD, (middle) Intel/LAPACK/AD and Intel/LAPACK, and (bottom) Intel/LAPACK/AD and Intel/MKL.
• Figure 4: Histograms of relative deviations of all partial derivatives in the Jacobian matrices obtained at the three comparison parameter points between (left) Intel/LAPACK/AD and GCC/LAPACK/AD and (right) between Intel/LAPACK/AD and FD using Intel/LAPACK. All relative deviations were computed as $\left|1.0-{J_{\textrm{other}_{ij}}}/{J_{\textrm{reference}_{ij}}}\right|$ using the Intel/LAPACK/AD as the reference value.
• Figure 5: Comparison of the absolute deviations between the Intel/LAPACK/AD Jacobian and the Intel/LAPACK/FD Jacobian approximation to a theoretical error approximation more2011edn derived from the ECNoise intermediate quantities $\epsilon_{ij}$ and $\mu_{ij}$.