“Sloppy” nuclear energy density functionals: Effective model reduction

Concepts from information geometry are used to analyze parameter sensitivity for a nuclear energy density functional, representative of a class of semiempirical functionals that start from a microscopically motivated ansatz for the density dependence of the energy of a system of protons and neutrons. It is shown that such functionals are “sloppy,” namely, characterized by an exponential range of sensitivity to parameter variations. Responsive to only a few stiff parameter combinations, sloppy functionals exhibit an exponential decrease of sensitivity to variations of the remaining soft parameters. By interpreting the space of model predictions as a manifold embedded in the data space, with the parameters of the functional as coordinates on the manifold, it is also shown that the exponential distribution of model manifold widths corresponds to the range of parameter sensitivity. Using the manifold boundary approximation method, we illustrate how to systematically construct effective nuclear density functionals of successively lower dimension in parameter space until sloppiness is eventually eliminated and the resulting functional contains only stiff combinations of parameters.

Figure 1

Eigenvectors and eigenvalues of the $7\times 7$ Hessian matrix of second derivatives $\mathcal{M}$ of ${\chi }^2\left(\mathbf{p}\right)$ at the best-fit point in symmetric nuclear matter for the functional defined by the couplings of Eq. (2). The empty and filled bars indicate that the corresponding amplitudes contribute with opposite signs. The last panel displays the logarithmic plot of the widths of the model manifold for each of the seven eigendirections, in comparison to the square roots of the corresponding eigenvalues of the Hessian.

Figure 2

Evolution of the seven parameters of the isoscalar part of the functional defined in Eq. (2), as functions of the affine parametrization, along the geodesic path determined by the eigenvector of the Hessian matrix that corresponds to the smallest eigenvalue (cf. Fig. 1). Parameters of the scalar channel are plotted in the upper panel, and the lower panel displays the three parameters that determine the vector channel of the functional defined by Eq. (2).

Figure 3

The initial (best-fit point) and final (at the boundary of the model manifold) eigenspectrum of the FIM for the functional defined by Eq. (2), with seven parameters in the isoscalar channel (left panel), and the initial and final eigenvectors that correspond to the smallest eigenvalues (panels on the right).

Figure 4

The initial (best-fit point) and final (at the boundary of the model manifold) density-dependent isoscalar coupling functions Eq. (2) (left) and the corresponding initial and final EoS curves (right). Solid curves denote the initial couplings and EoS, while dashed curves refer to the couplings and EoS at the boundary. The pseudodata that represent the microscopic EoS are indicated by (red) circles.

Figure 5

Eigenvectors and eigenvalues of the $6\times 6$ Hessian matrix of second derivatives $\mathcal{M}$ of ${\chi }^2\left(\mathbf{p}\right)$ at the best-fit point in symmetric nuclear matter for the functional defined by Eq. (17). The empty and filled bars indicate that the corresponding amplitudes contribute with opposite signs.

Figure 6

Same as in the caption to Fig. 2 but for the functional defined by the six parameters in Eq. (17), and the eigendirection for the smallest eigenvalue of the Hessian matrix shown in Fig. 5.

Figure 7

Same as in the caption to Fig. 3 but for the functional with six isoscalar parameters defined by Eq. (17).

Figure 8

Same as in the caption to Fig. 4 but for the functional with six isoscalar parameters defined by Eq. (17).

Figure 9

Eigenvectors and eigenvalues of the $4\times 4$ Hessian matrix of second derivatives $\mathcal{M}$ of ${\chi }^2\left(\mathbf{p}\right)$ at the best-fit point in symmetric nuclear matter for the functional defined by Eq. (20). The empty and filled bars indicate that the corresponding amplitudes contribute with opposite signs.

Figure 10

Same as in the caption to Fig. 4 but for the functional with four parameters defined by Eq. (20).

Figure 11

Equation of state of neutron matter, calculated with the isovector channel of the functional DD-PC1 (solid) in comparison to the pseudodata that represent the microscopic EoS and are indicated by (red) circles.