Benchmarking ANN extrapolations of the ground-state energies and radii of Li isotopes

TL;DR

This work tackles the challenge of extrapolating No-Core Shell Model results from truncated spaces to the full Hilbert space for Li isotopes by benchmarking two neural-network-based extrapolation schemes (ISU and TUDa ANNs) against traditional exponential and IR methods for ground-state energies $E$ and point-proton radii $r_p$. The ISU ANNs use a shallow network trained on $(N_{ m max}, \hbar\Omega)$ data, while the TUDa ANNs employ a deeper topology and ensemble histogram approach built from converged light-nucleus data, enabling robust uncertainty quantification. Results show that TUDa ANNs provide highly consistent and precise predictions even in small model spaces, with ML radii extrapolations outperforming the radius-crossing-point heuristic, and that ML methods yield uncertainties that are meaningful and scalable. The findings suggest ML extrapolations can complement or surpass conventional extrapolations for both energies and radii, offering computational efficiency and extensibility to other observables and higher-p shell nuclei, while highlighting areas for further refinement such as training data expansion and systematic uncertainty assessment.

Abstract

We present a comparison of model-space extrapolation methods for No-Core Shell Model calculations of ground-state energies and root-mean-square radii in Li isotopes. In particular, we benchmark the latest machine learning tools against widely used exponential and infrared extrapolations for energies and crossing point estimates for radii. Our findings demonstrate that machine learning-based approaches provide reliable predictions with robust statistical uncertainties for both observables even in small model spaces. These predictions are compatible with established exponential and IR extrapolations of energies and mark a notable improvement over conventional radius estimates.

Figures (5)

• Figure 1: A selection of the ISU $^6$Li ground-state energy (left-hand panels) and point-proton radius (right-hand panels) histograms. A Gaussian fit has been applied to our binned histograms with the mean value of our fit taken to be the value of our observable and the standard deviation taken as our error bars. The count is the total number of networks represented in that binning. The black dashed vertical lines in the energy histograms indicate the variational minimum at each respective upper limit $N_{\rm max}$ value.
• Figure 2: Evaluation of the TUDa ANNs for $^7$Li with 5 selected HO frequencies for the ground-state energy (upper panels) and the point-proton rms radius (lower panels). The left panels display the input data fed into the networks, while the right panels present histograms of the network predictions along with the extracted values and uncertainties for increasing $\curly{N}_\text{max}$, including only input data with $N_\text{max}\le \curly{N}_\text{max}$.
• Figure 3: Extrapolations to the full Hilbert space for the ground-state energy and point-proton radius of $^6$Li based on NCSM calculations up to $\curly{N}_\mathrm{max}$ with the ISU ANNs (green) and the TUDa ANNs for the full frequency range $\hbar\Omega=10$ to $30$ MeV (blue circles) and for a selection of five optimal frequencies (blue squares). For the energy an additional exponential extrapolation (yellow) and the variational minima at the respective $N_\mathrm{max}$ (dashed lines) are given for comparison.
• Figure 4: Same as \ref{['fig:res_Li6']} but for $^7$Li.
• Figure 5: Same as \ref{['fig:res_Li6']} but for $^8$Li.