Active learning emulators for nuclear two-body scattering in momentum space

TL;DR

This work develops active-learning reduced-order emulators for momentum-space, coupled-channel nuclear two-body scattering by solving the Lippmann–Schwinger equation with an affine-parameter potential. Using Galerkin and Petrov-Galerkin projections, and a greedy snapshot algorithm, the authors build accurate ROMs with quantified emulator errors and demonstrate substantial computational speed-ups. They validate the approach on Minnesota and GT+ chiral potentials, achieving exponential convergence and sub-10^-7 phase-shift accuracy while enabling a Bayesian calibration framework that incorporates emulator and EFT truncation uncertainties. The framework, implemented in Python/JAX, paves the way for rigorous Bayesian calibrations of NN and 3N interactions against scattering data with fully quantified emulator errors and is poised for extension to additional observables and non-affine parameter regimes.

Abstract

We extend the active learning emulators for two-body scattering in coordinate space with error estimation, recently developed by Maldonado et al. in Phys. Rev. C \textbf{112}, 024002, to coupled-channel scattering in momentum space. Our full-order model (FOM) solver is based on the Lippmann-Schwinger integral equation for the scattering $t$-matrix as opposed to the radial Schrödinger equation. We use (Petrov-)Galerkin projections and high-fidelity calculations at a few snapshots across the parameter space of the interaction to construct efficient reduced-order models (ROMs), trained by a greedy algorithm for locally optimal snapshot selection. Both the FOM solver and the corresponding ROMs are implemented efficiently in Python using Google's JAX library. We present results for emulating scattering phase shifts in coupled and uncoupled channels and cross sections, and assess the accuracy of the developed ROMs and their computational speed-up factors. We also develop emulator error estimation for both the $t$-matrix and the total cross section. The software framework for reproducing and extending our results will be made publicly available. Together with our recent advances in developing active-learning emulators for three-body scattering, these emulator frameworks set the stage for full Bayesian calibrations of chiral nuclear interactions and optical models against scattering data with quantified emulator errors.

Figures (9)

• Figure 1: Convergence of the greedy algorithm for the Minnesota potential \ref{['eq:minn_pot_ms']} at $E_\text{lab} = 20 \, \text{MeV}$ in the ${}^1\text{S}_0$ channel. Each panel shows a color map of the true error \ref{['eq:rom_error']} in the two-dimensional parameter space of the potential (see the legends). The pound symbol "#" specifies the number of greedy iterations passed: Panel (a) depicts the initial configuration, for which we randomly select snapshots in the parameter space using LHS. The remaining panels (b) through (f) show the configuration in the emulator after 2, 4, 6, 8, and 9 iterations, respectively. Note that not all greedy iterations are depicted. As shown, the greedy algorithm places additional snapshots at locations where the (estimated) emulator error is largest, thereby iteratively reducing the emulator error locally and globally. Note that the emulator error vanishes at the origin (which is not a grid point in the panels) because the interaction is zero as there is no parameter-independent term, rendering the ROM equations homogeneous. See the main text for details.
• Figure 2: Emulator convergence and comparison at $E_\text{lab} = 100 \, \text{MeV}$ (left panel) and $200 \, \text{MeV}$ (right panel) for the GT+ potential Gezerlis:2014zia in the ${}^1\text{S}_0$ channel. The greedy algorithm applied to the G-ROM is compared to the POD approach. Both emulators have access to the 150 candidate snapshots in a nine-dimensional parameter space, randomly selected via LHS in the region of $\pm 40\%$ of the best-fit LEC values (obtained in Ref. Gezerlis:2014zia) However, the POD emulator uses all of them, while the greedy emulator selects only a small subset. The $x$-axis shows the size of the emulator basis, corresponding to the snapshot calculations performed (greedy algorithm) or the dominant POD modes used (POD approach). To test their accuracy, we use a validation set of 500, similarly obtained via LHS as the candidate snapshots during training. The boxes indicate the range between the first and third quartiles, and the line within the box represents the median. The whiskers show the range from the 5th to the 95th percentiles. See the main text for details.
• Figure 3: Same as Fig. \ref{['fig:singlet_error']} but for the coupled channel ${}^3\text{S}_1$--${}^3\text{D}_1$.
• Figure 4: High-fidelity (or FOM) and emulated (or ROM) phase shifts and mixing angles as a function of the laboratory energy for the coupled ${}^3\text{S}_1$--${}^3\text{D}_1$ channel based on the GT+ potential. The panels (a) to (c) show the phase shifts in the ${}^3\text{S}_1$ (panel a) and in the ${}^3\text{D}_1$ channel (panel b), as well as the mixing angle $\varepsilon_1$ (panel c). The solid black lines represent the high-fidelity results, while the dots indicate the emulated results at a few selected energies. The panels (d) to (f) show the relative error \ref{['eq:sym_relative_error']} of the G-ROM and LSPG-ROM, respectively, with respect to the high-fidelity calculations. This comparison was made for two requested error tolerances on the $\tilde{t}$-matrix, $\alpha = 10^{-4}$ and $10^{-7}$, as reported in the legend. See the main text for details.
• Figure 5: Total $np$ cross sections obtained with the G-ROM in panel (a) and LSPG-ROM in panel (b) (orange dots) based on the GT+ potential, which depends on nine LECs. The black lines depict the high-fidelity calculations. All partial-wave channels with $j \leqslant 6$ are included in the calculations. Both emulators use the same pool of candidate snapshots for training, comprising 500 random points selected via LHS. The insets show the mean relative error \ref{['eq:sym_relative_error']} of the emulator and the high-fidelity solution based on 1000 random samples in the potential's parameter space (dashed lines) and the best-fit LEC values obtained in Ref. Gezerlis:2014zia (solid lines), for the requested error tolerances of $\alpha = 10^{-4}$ and $10^{-7}$ (see the legends). See the main text for details.