Emulation of Proton-Deuteron Scattering via the Reduced Basis Method and Active Learning: Detailed Description

TL;DR

This paper tackles the computational bottleneck of calibrating three-nucleon forces in χEFT by developing model-driven emulators based on the Reduced Basis Method and active learning for proton-deuteron scattering below the deuteron breakup threshold. It presents a comprehensive HH-based high-fidelity framework and implements three emulators: a variational RBM emulator and two ROM-based approaches (G-ROM and LSPG-ROM) that exploit affine parameter dependence in the χEFT Hamiltonian. Greedy snapshot selection driven by a residual error proxy yields extreme efficiency, with relative errors reaching as low as $\lesssim 10^{-7}$ using fewer than 12 training points in a two-dimensional $\boldsymbol{\theta}$-space (spanned by $c_E$ and $c_D$) for both $J^{\pi}=\tfrac{1}{2}^+$ and $\tfrac{1}{2}^-$ channels. The results enable rapid Bayesian calibration of nucleon interactions and can be generalized to other nuclear scattering problems and finite quantum systems, representing a significant step toward practical uncertainty quantification in ab initio nuclear theory.

Abstract

Nucleon-deuteron ($Nd$) scattering can be used to constrain three-nucleon forces in chiral effective field theory ($χ$EFT). However, high-fidelity calculations, such as the Hyperspherical Harmonic (HH) method, are computationally expensive, making it difficult or even prohibitive to explore the vast parameter space of $χ$EFT\xspace. To address this challenge, specifically for proton-deuteron ($pd$) scattering below the deuteron breakup threshold, we developed model-driven emulators based on the Reduced Basis Method (RBM) and active learning techniques, as presented in \href{https://arxiv.org/abs/2511.01844}{arXiv:2511.01844}. The method exploits the similarities between solutions at different parameter points to significantly reduce computational costs. In this companion paper, we provide a comprehensive description of our HH-based high-fidelity calculations and implementation of both variational-method-based and Galerkin-projection-based scattering emulators. We demonstrate the effectiveness of active learning in the form of greedy algorithms for selecting optimal training points in the parameter space, and the high accuracy and speed of the emulators, for two different nucleon forces and two scattering channels (${1/2}^+$ and ${1/2}^-$). For example, in a two-dimensional parameter space, the relative emulation errors can be reduced to $10^{-7}$ with fewer than 10 training points. Our work paves the way for the efficient calibration of $χ$EFT\xspace nucleon interactions using Bayesian statistics, and the methodology can be applied to other nuclear scattering processes (including neutron-deuteron scattering), as well as other finite quantum systems.

Figures (16)

• Figure 1: $R$-matrix elements vs. $c_E$ with fixed $c_D$ values (shown in the legend) and $E = 2$ MeV. The results from the NVIIa and NVIIb forces are presented.
• Figure 2: The progressive reductions (seen in the $(c_E,c_D)$ plane) of the residue $||\vb{r}^{a=1}(\vb*{\theta})||$ (for the $\frac{1}{2}^+$ channel) with increasing number of training points ($N_b$). The residual is computed within the G-ROM emulations; the force is NVIIb. Two different energies are considered, one close to the elastic scattering threshold and the other close to the deuteron breakup threshold.
• Figure 3: The same figure as Fig. \ref{['fig:res_halfplus_GROM_NVIIb']} but now for the NVIIb force with LSPG-ROM emulator.
• Figure 4: Emulation errors at $E=2$ MeV for $\mathcal{R}_{1 \to 1 }$ at its 1st order in the $\frac{1}{2}^+$ channel for the NVIIb force. From left to right, $N_b$ increases, and the errors decrease. Three different emulators are compared from top to bottom: "G" (G-ROM), "LS" (LSPG-ROM), and "Var" (variational).
• Figure 5: The same as Fig. \ref{['fig:emul_error_halfplus_R11_1st_NVIIb_multiple_emulators']} but now for the $\mathcal{R}_{1 \to 2 }$ at its 1st order.