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

TL;DR

This paper presents a method to efficiently emulate proton-deuteron scattering calculations using reduced basis and active learning, enabling rapid exploration of parameter space in nuclear physics.

Contribution

It introduces a novel emulator framework combining the Reduced Basis Method and active learning for high-accuracy, low-cost nuclear scattering simulations.

Findings

Emulators achieve errors as low as 10^{-7} with fewer than 10 training points.

Active learning effectively selects optimal training points in parameter space.

The approach significantly reduces computational costs for nuclear scattering calculations.

Abstract

Nucleon-deuteron ($Nd$) scattering can be used to constrain three-nucleon forces in chiral effective field theory ($\chi$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 $\chi$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 $\chi$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.