ROSE: A reduced-order scattering emulator for optical models

A new generation of phenomenological optical potentials requires robust calibration and uncertainty quantification, motivating the use of Bayesian statistical methods. These Bayesian methods usually require calculating observables for thousands or even millions of parameter sets, making fast and accurate emulators highly desirable or even essential. Emulating scattering across different energies or with interactions such as optical potentials is challenging because of the nonaffine parameter dependence, meaning the parameters do not all factorize from individual operators. Here we introduce and demonstrate the reduced-order scattering emulator (rose) framework, a reduced basis emulator that can handle nonaffine problems. rose is fully extensible and works within the publicly available band framework software suite for calibration, model mixing, and experimental design. As a demonstration problem, we use rose to calibrate a realistic nucleon-target scattering model through the calculation of elastic cross sections. This problem shows the practical value of the rose framework for Bayesian uncertainty quantification with controlled trade-offs between emulator speed and accuracy as compared to high-fidelity solvers. Planned extensions of rose are discussed.

Figure 1

(a) High-fidelity, $S$-wave solutions (solid, colored lines) alongside emulated solutions (dashed, black lines) at $E_{\mathrm{c}.\mathrm{m}.}=14$ MeV ${}^{40}\mathrm{Ca}(n,n)$ and four different test parameter points. (b) Basis states for the expansion of $\widehat{\varphi }$ as defined in Eq. (14). The ${\varphi }_0$ term (in black) represents the solution in the absence of an interaction $U(s;\mathit{\alpha })$, and it is used to enforce the boundary conditions at $s\sim 0$. The other three functions ${\varphi }_k$ with $k=[1,2,3]$ (colored solid lines) are the first three principal components of the differences of snapshots with ${\varphi }_0$ as defined in Eq. (15). Only the real components are displayed.

Figure 2

(a) High-fidelity potentials (solid, colored lines) alongside EIM-emulated potential (dashed, black lines) at four different test parameter points. Pink crosses indicate the points $s_j$ at which agreement is enforced. (b) $\text{PCA}$ basis components of the emulated potential in Eq. (21). Only the real parts of the potential are shown.

Figure 3

Flowchart illustrating the offline-online separation in rose. In the offline stage, for a fixed channel $(\ell ,j)$, we create reduced bases by using principal component analysis over the wave functions and potentials associated with sets of training parameters $\mathit{\alpha }$. These reduced bases are used to create the building blocks of the approximate equations: the reduced operators ${\widehat{\mathit{M}}}^{\left(i\right)}$, the reduced boundary terms ${\widehat{\mathit{c}}}^{\left(i\right)}$, a set of interpolation points $s_j$, and the inverse of the empirical interpolation matrix, ${\mathit{U}}^{\text{EIM}}$. In the online stage, the emulator avoids performing operations that scale with the high-fidelity dimension $\mathcal{N}$: the query parameters $\mathit{\alpha }$ are used to interpolate the potential at the selected locations $s_j$ to assemble the approximate reduced equations $\widehat{\mathit{M}}\mathit{a}=\widehat{\mathit{c}}$. Solving this linear system for each channel $(\ell ,j)$ provides the coefficients $\mathit{a}\left(\mathit{\alpha }\right)$ to be used for calculating observables for different query parameters.

Figure 4

Calculated differential cross section for (a) ${}^{40}\mathrm{Ca}(n,n)$ and (b) ${}^{40}\mathrm{Ca}(p,p)$ at 14 MeV for three parameter configurations from the respective test set selected to reflect the range of outcomes. The high-fidelity calculations obtained through the Runge-Kutta method (colored solid lines) are well reproduced by the emulator with configuration $(n_{\varphi },n_U)=(15,15)$ (black dashed lines).

Figure 5

Computational accuracy vs time plot showing the trade-off between accuracy and speed of both rose and the high-fidelity solver when calculating the differential cross section of ${}^{40}\mathrm{Ca}(n,n)$ at 14 MeV for 100 test parameter values. The $x$ axis represents the time taken for each evaluation, while the $y$ axis represents the accuracy defined in Eq. (30) as the maximum relative error in the entire differential cross section when compared to a high-fidelity solver set to very high precision. For rose, we varied the number of bases of the wave functions, $n_{\varphi }$ [Eq. (14)], and the number of bases in the potential, $n_U$ [Eq. (20)], between 7 and 15. The red crosses, “(15,15) - [En],” represent an emulator trained across the energy window [10–30] MeV, while the other three configurations were trained at the same 14 MeV energy. For the high-fidelity solver, which uses the Runge-Kutta method, we varied the relative and absolute tolerance between ${10}^{-2}$ and ${10}^{-5}$, numbers that are shown next to the blue hexagons representing their performance. The very high precision high-fidelity configuration was ${10}^{-9}$. The box delimited by dashed lines represents the target zone consisting of, as explained in the text, more than one million samples per hour with less than $10\%$ relative error in the differential cross section. rose's performance enters the box and is clearly superior to the traditional high-fidelity method, surpassing its speed by more than two orders of magnitude for a comparable accuracy.

Figure 6

Corner plot [82] for the calibration of the optical model through the ${}^{40}\mathrm{Ca}(n,n)$ reaction at 14 MeV. This plot shows all the one- and two-dimensional projections of the approximated posterior probability distributions [Eq. (13)]. The black histograms represent the posterior [Eq. (10)], approximated by 1 million samples visited by 20 MCMC walkers, while the blue solid contours represent the Gaussian prior [Eq. (31)]. The red lines show the values of the true parameters ${\mathbf{\omega }}^{\mathit{t}}$ obtained from Ref. [83] and used to generate the data while the blue lines show the prior means ${\mathbf{\omega }}^{\mathrm{pr}}$. Both sets of parameters are defined in Table 1.

Figure 7

Predictive posterior distribution for the differential cross section in (a) ${}^{40}\mathrm{Ca}(n,n)$ and (b) ${}^{40}\mathrm{Ca}(p,p)$ at 14 MeV. The differential cross sections were calculated using 50000 random parameters obtained during the MCMC sampling in Figs. 6 and 8. The $95\%$ credible interval is calculated from Eq. (12) by taking into account the error structure of the data. The synthetic data with a $95\%$ error bar (in red) was created from Eq. (9) by using the true parameters (black curve) defined in Tables 1 and 2.

Figure 8

Corner plot [82] for the calibration of the optical model through the ${}^{40}\mathrm{Ca}(p,p)$ reaction at 14 MeV. This plot shows all the one- and two-dimensional projections of the approximated posterior probability distributions [Eq. (13)]. The black histograms represent the posterior [Eq. (10)], approximated by 1 million samples visited by 20 MCMC walkers, while the blue solid contours represent the Gaussian prior [Eq. (31)]. The red lines show the values of the true parameters ${\mathbf{\omega }}^{\mathit{t}}$ obtained from Ref. [83] and used to generate the data while the blue lines show the prior means ${\mathbf{\omega }}^{\mathrm{pr}}$. Both sets of parameters are defined in Table 2.

Figure 9

Residuals in the phase shift ${\delta }_{\ell }$ when comparing rose emulation vs the high-fidelity solver across energies for ${}^{40}\mathrm{Ca}(n,n)$ for $\ell \in [0,10]$. No evidence of Kohn anomalies (see Ref. [16]) was found.