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Accurate and Efficient Emulation of Proton-Deuteron Scattering via the Reduced Basis Method and Active Learning

Alex Gnech, Xilin Zhang, Christian Drischler, R. J. Furnstahl, Alessandro Grassi, Alejandro Kievsky, Laura E. Marcucci, Michele Viviani

Abstract

We introduce highly accurate and efficient emulators for proton-deuteron scattering below the deuteron breakup threshold. We explore two different reduced-basis method strategies: one based on the Kohn variational principle and another on Galerkin projections of the underlying system of linear equations. We use the adaptive greedy algorithm previously developed for two-body scattering for optimal selection of high-fidelity training points in the input parameter space. We demonstrate that these emulators reproduce ab initio hyperspherical harmonics calculations of $R$-matrix elements with remarkable precision, achieving relative errors as low as $10^{-7}$ with a small number of training points, even in regions of strong nonlinear parameter dependence. They also dramatically accelerate the exploration of the scattering predictions in the parameter space, a capability highly desired for calibrating (chiral) three-nucleon forces against scattering measurements. Our formalism can be further generalized to handle nucleon-deuteron scattering above the breakup threshold. These emulator developments will provide valuable tools to accelerate uncertainty quantification and rigorous parameter inference in the study of nuclear forces.

Accurate and Efficient Emulation of Proton-Deuteron Scattering via the Reduced Basis Method and Active Learning

Abstract

We introduce highly accurate and efficient emulators for proton-deuteron scattering below the deuteron breakup threshold. We explore two different reduced-basis method strategies: one based on the Kohn variational principle and another on Galerkin projections of the underlying system of linear equations. We use the adaptive greedy algorithm previously developed for two-body scattering for optimal selection of high-fidelity training points in the input parameter space. We demonstrate that these emulators reproduce ab initio hyperspherical harmonics calculations of -matrix elements with remarkable precision, achieving relative errors as low as with a small number of training points, even in regions of strong nonlinear parameter dependence. They also dramatically accelerate the exploration of the scattering predictions in the parameter space, a capability highly desired for calibrating (chiral) three-nucleon forces against scattering measurements. Our formalism can be further generalized to handle nucleon-deuteron scattering above the breakup threshold. These emulator developments will provide valuable tools to accelerate uncertainty quantification and rigorous parameter inference in the study of nuclear forces.

Paper Structure

This paper contains 8 equations, 2 figures.

Figures (2)

  • Figure 1: The residuals defined as $||\vb{r}^{a=1}(\vb*{\theta})||$ for the $\mathcal{R}_{1 \to 1 }$ estimation for different $(c_E, c_D)$ and scattering energy $E = 1$ MeV, when increasing the number of the training points $N_b$ sampled using greedy algorithm, for $pd$ scattering ($\frac{1}{2}^+$). The G-ROM emulation and NVIIb force are used. The training points are effectively marked by the darkest regions.
  • Figure 2: Relative emulation errors for different $R$-matrix elements, averaged over the parameter space as a function of the scattering energy $E$, for the $pd$ scattering ($\frac{1}{2}^+$), using the NVIIb force. "G", "LS", and "Var" in the legend stand for G-ROM, LSPG-ROM, and variational emulation, respectively.