Wavefunction-Based Emulation of Coupled-Channels Scattering with Non-Affinely Parametrized Interactions

TL;DR

This work extends reduced-basis emulation from single-channel scattering to a coupled-channel framework (CC-RBM) by developing a physics-informed, projection-based emulator that handles non-affine radial dependencies via Empirical Interpolation. The CC-RBM constructs channel-specific reduced bases from high-fidelity training data using PCA, and employs a Petrov–Galerkin projection to form a small linear system for the coefficients that describe the radial wavefunctions across all CC channels. By applying the Empirical Interpolation Method to the radial form factors, the method achieves affine-like online efficiency, enabling rapid evaluation of cross sections for elastic and inelastic neutron–nucleus scattering within deformed rotor models. Benchmarking on 48Ca and 208Pb demonstrates accurate reproduction of cross sections with median errors well below the target percent level and speedups of roughly one to two orders of magnitude, validating the approach for uncertainty quantification and large-scale reaction studies. The framework is generic, scalable to moderate channel counts, and ripe for extensions to broader couplings and GPU-accelerated implementations.

Abstract

Physics based emulators offer a fast and reliable replacement for an exact solution of the scattering problem in nuclear physics. Previous work developed a reduced-basis emulator for single-channel elastic scattering using an optical potential. Since many reactions of interest can be cast as a coupled-channel problem, the purpose of this work is to extend the RBM to a coupled-channel framework (CC-RBM). Although the framework derived is general, in this work we apply it to reactions where the Hamiltonian coupling term comes from assuming a rotational structure model for the target. From a set of training coupled-channel wavefunctions, we perform a singular value decomposition to obtain a reduced set of basis wavefunctions, and then solve the extended (Petrov-)Galerkin equations. In addition, the empirical interpolation method is used to expand the potentials. We apply the CC-RBM method to elastic and inelastic scattering of neutrons on 48Ca including a quadrupole coupling to populate the first 2+ state, and neutrons on 208Pb, including an octupole coupling to populate its first 3- state. We demonstrate that the CC-RBM calculated cross sections match those obtained using traditional finite-difference methods. We show that the CC-RBM results can reliably reproduce the nuclear scattering cross sections at different energy regimes. The computational accuracy versus time plots demonstrate that the CC-RBM method efficiently increases precision with increasing basis size. Most importantly, for the precisions required in reaction calculations (a percent on the cross section), we find the CC-RBM method offers roughly one and a half orders of magnitude gain in computational speed compared to the traditional coupled-channels solver. However, we also discuss how this scaling becomes less favorable, the larger the number of channels included in the coupled-channel set.

Figures (4)

• Figure 1: Real parts of the first four PCA components for the two coupled channels in the $^{48}$Ca$(n,n')$$^{48}$Ca$(2^+)$ system at $E_{\mathrm{lab}} = 12$ MeV and $J^\pi = 0^+$. Panel (a) shows the $l = 0$, $I_t = 0^+$ channel, and panel (b) the $l = 2$, $I_t = 2^+$ channel.
• Figure 2: Calculated differential cross sections for $^{48}$Ca$(n,n')^{48}$Ca inelastic scattering at two incident energies. Panels (a)–(b) show the elastic and inelastic cross sections at $E_\text{lab}=12$ MeV, while panels (c)–(d) show the corresponding results at $E_\text{lab}=26$ MeV. Solid colored lines denote the fresco calculations; dashed black lines denote the emulator results with $N_\psi=12$ and $N_U=12$. Each curve corresponds to a different choice of interaction parameters $\boldsymbol{\alpha}$.
• Figure 3: Computational Accuracy vs. Time (CAT) plot illustrating the trade-off between accuracy and computational speed for both the CC emulator and the high-fidelity fresco solver in the calculation of the differential cross section for the elastic (open markers) and inelastic (filled markers) channels of the $^{48}$Ca$(n,n')$$^{48}$Ca$(2^+)$ reaction at $E_{\text{lab}} = 12$ MeV [panel (a)] and $E_{\text{lab}} = 26$ MeV [panel (b)]. The horizontal axis represents the evaluation time per calculation, while the vertical axis shows the accuracy, defined as the median relative error in the differential cross section with respect to the fresco result, computed across 50 test parameter sets centered on the corresponding KD values. For the CC emulator, the number of basis functions in the wavefunction expansion ($N_{\psi}$) and the EIM representation ($N_{\text{U}}$) were varied between 8 and 16. The vertical red dashed line indicates the average fresco solver evaluation time of 60.9 ms. The emulator achieves relative errors below $10^{-1}$ while providing speedups approaching one and a half orders of magnitude, depending on the basis size employed.
• Figure 4: Computational Accuracy vs. Time (CAT) plot illustrating the trade-off between accuracy and computational speed for both the coupled-channel (CC) emulator and the high-fidelity fresco solver in the calculation of the differential cross section for the elastic (open markers) and inelastic (filled markers) channels of the $^{208}$Pb$(n,n')$$^{208}$Pb$(3^-)$ reaction at $E_{\text{lab}} = 12$ MeV [panel (a)] and $E_{\text{lab}} = 26$ MeV [panel (b)]. The horizontal axis represents the evaluation time per calculation, while the vertical axis shows the accuracy, defined as the median relative error in the differential cross section with respect to the fresco result, computed across 50 test parameter sets centered on the corresponding KD values. For the CC emulator, the number of basis functions in the wavefunction expansion ($N_{\psi}$) and the EIM representation ($N_{\text{U}}$) were varied between 8 and 16. The vertical red dashed line indicates the fresco solver evaluation time of 75.1 ms. The emulator achieves relative errors below $10^{-1}$ while providing speedups around one and a half orders of magnitude, depending on the basis size employed.