Optimization of generator coordinate method with machine-learning techniques for nuclear spectra and neutrinoless double-beta decay: ridge regression for nuclei with axial deformation

Abstract

Background: The generator coordinate method (GCM) is an important tool of choice for modeling large-amplitude collective motion in atomic nuclei. Recently, it has attracted increasing interest as it can be exploited to extend ab initio methods to the description of collective excitations of medium-mass and heavy deformed nuclei, as well as the nuclear matrix elements (NME) of candidates for neutrinoless double-beta (0νββ) decay.Purpose: The computational complexity of the GCM increases rapidly with the number of collective coordinates. It imposes a strong restriction on the applicability of the method. We aim to exploit machine learning (ML) algorithms to speed up GCM calculations and ultimately provide a more efficient description of nuclear energy spectra and other observables such as the NME of 0νββ decay without loss of accuracy.Method: To speed up GCM calculations, we propose a subspace reduction algorithm that employs optimized ML models as surrogates for exact quantum-number projection calculations for norm and Hamiltonian kernels. The model space of the original GCM is reduced to a subspace relevant for nuclear low energy spectra and the NME of ground state to ground state 0νββ decay based on the orthogonality condition (OC) and the energy transition-orthogonality procedure (ENTROP), respectively. Nuclear energy spectra are determined by the GCM through the configuration mixing within this subspace. For simplicity, a polynomial regression algorithm is used to learn the norm and Hamiltonian kernels. The efficiency and accuracy of this algorithm are illustrated for Ge 76 and Se 76 by comparing results obtained using the ML models to direct GCM calculations. The popular non-relativistic and relativistic EDFs, a valence-space shell-model Hamiltonian, and a modern nuclear interaction derived from chiral effective field theory are employed. Results:The low-lying energy spectra of 76 Ge and 76 Se, as well as the 0νββ-decay NME between their ground states, are computed. The results show that the performance of the GCM+OC/ENTROP+ML is more robust than that of the GCM+ML alone, and the former can reproduce the results of the original GCM calculation rather accurately with a significantly reduced computational cost.Conclusions: ML algorithms, when implemented properly, can accelerate GCM calculations without loss of accuracy. In applications with axially deformed configurations, the computation time can be reduced by a factor of three to nine for energy spectra and NMEs, respectively. This factor is expected to increase significantly with the number of employed generator coordinates.