Surrogate Models for Linear Response

TL;DR

The paper tackles the computational bottleneck of quasiparticle RPA (QRPA) for nuclear linear response by introducing two surrogate models: EM1, a physics-informed reduced-order emulator that reconstructs the strength function $S(oldsymbol{eta};oldsymbol{ u})$ with a small set of Lorentzians and a parametric matrix, and EM2, a data-driven Parametric Matrix Model (PMM) that maps EDF parameters directly to observables. Both emulators are trained on full QRPA data and achieve sub-percent accuracy for key observables, including the electric dipole polarizability $\alpha_D$ of $^{180}$Yb and the beta-decay half-life $T_{1/2}$ of $^{80}$Ni, while offering speedups of 6–7 orders of magnitude over state-of-the-art solvers. EM1 tends to generalize better to unseen parameter regions due to its physics-informed latent space, whereas EM2 demonstrates strong performance with relatively few training points by exploiting the PMM framework. Together, they enable Bayesian calibration and large-scale uncertainty quantification for EDF-based nuclear models, and the approach is adaptable to other linear-response problems in physics and chemistry.

Abstract

Linear response theory is a well-established method in physics and chemistry for exploring excitations of many-body systems. In particular, the quasiparticle random-phase approximation (QRPA) provides a powerful microscopic framework by building excitations on top of the mean-field vacuum; however, its high computational cost limits model calibration and uncertainty quantification studies. Here, we present two complementary QRPA surrogate models and apply them to study response functions of finite nuclei. One is a reduced-order model that exploits the underlying QRPA structure, while the other utilizes the recently developed parametric matrix model algorithm to construct a map between the system's Hamiltonian and observables. Our benchmark applications, the calculation of the electric dipole polarizability of ${}^{180}$Yb and the $β$-decay half-life of ${}^{80}$Ni, show that both emulators can achieve 0.1\%--1\% accuracy while offering a six to seven orders of magnitude speedup compared to state-of-the-art QRPA solvers. These results demonstrate that the developed QRPA emulators are well-positioned to enable Bayesian calibration and large-scale studies of computationally expensive physics models describing the properties of many-body systems.

Figures (8)

• Figure 1: Flowchart illustrating the computational algorithms for the full order model described in Sec. \ref{['subsec: FOM']} and the two emulators described in Secs. \ref{['subsec:emulator 1']} and \ref{['subsec:emulator 2']}. The three approaches aim to connect the controlling EDF parameters $\boldsymbol{\alpha}$ (purple box at the top) with the two observables $\alpha_D$ and $T_{1/2}$ (green boxes at the bottom). A detailed description of the diagram is given in the text.
• Figure 2: The total isovector electric dipole strength function in ${}^{180}$Yb calculated with the FAM QRPA. Starting from the DD-PC1 EDF parameterization, each line corresponds to an individual QRPA calculation at fixed parameters $b_{TV}$ and $d_{TV}$; each curve is colored according to the value of $b_{TV}$. A total of 225 calculations were performed on a regular grid in the parameter space spanning $b_{TV} \in [1.0,4.5]$ fm$^2$ and $d_{TV} \in [0.1,2.0]$. (See Fig. \ref{['fig:YbNiPerformance']}(a) for a graphical representation of the parameter distribution, and Fig. \ref{['fig:ManyCurves']}(a) for the response of $\alpha_D$ to changes in $b_{TV}$ and $d_{TV}$.)
• Figure 3: The Gamow-Teller strength function in ${}^{80}$Ni, within the $Q_\beta$-window, calculated with the linear response QRPA using DD-PC1 EDF parameterization at fixed parameters $g_0$ and $V_0^{is}$; each curve is colored according to its value of $V_0^{is}$. The red solid line indicates the the phase-space factor $f(E_0,Z,A)$ at $E_0 = \Delta_{np} - \omega$. A total of 225 calculations were performed on a regular grid in the parameter space spanning $V_0^{is} \in [0.0,2.0]$ and $g_0 \in [0.0,1.0]$. (See Fig. \ref{['fig:YbNiPerformance']}(c) for a graphical representation of the parameter distribution, and Fig. \ref{['fig:ManyCurves']}(b) for the response of $T_{1/2}$ to changes in $g_0$ and $V_0^{is}$.)
• Figure 4: Evolution of the Gamow-Teller strength function in $^{80}$Ni as a function of $V_0^{is}$ at $g_0=0.5$. The color coding follows the pattern of Fig. \ref{['fig:gamow_teller_example']}. The FOM strength functions $S(\omega)$ at five values of $V_0^{is}=[0.0,0.5,1.0,1.5,2.0]$ are shown. The tracks in the horizontal plane represent the subset of $n_0$ eigenvalues of matrix $\hat{\mathcal{M}}$. These eigenvalues represent the locations of the approximate energies $\hat{E}_i$ that are used to construct the emulated strength function in Eq. \ref{['eq: approx S']}. The values of $S(\hat{E}_i)$ are shown as vertical curves. The dotted line shows the lowest eigenvalue that is not retained in emulation; its behavior influences the overall dynamics. The vertical green plane shows the cutoff at $\Delta_{nH} = 0.782$ MeV, representing the limit of the $Q_\beta$-window. For an interactive 3D visualization of the figure see Ref. smlrWebsite.
• Figure 5: Performance of EM1 in reproducing isovector dipole strength in $^{180}$Yb (left panels), and the Gamow-Teller strength in ${}^{80}$Ni (right panels). The emulator dimension was chosen as $n_1=13\,(n_0 = 8)$ for the isovector dipole strength, and $n_1=13\,(n_0=12)$ for the Gamow-Teller strength. The top panels show the relative error $\chi_{S(i)}^2$(\ref{['eq:relative_error_strength_individual']}) across the explored parameter range shown in Table \ref{['tab:case_ranges']}. In both cases, 121 points inside the black rectangle were used for training, while external 104 points form the test set. Bottom panels show three strength functions as calculated in QRPA (solid lines) and EM1 (dashed lines) for the selected values of parameters (shown in panels (a) and (c) as thick circles matching the colors of $S(\omega)$).