Discovering Nuclear Models from Symbolic Machine Learning

TL;DR

The paper tackles the challenge of unifying nuclear models by applying symbolic machine learning to rediscover traditional relations and propose simpler, more predictive forms. It introduces MISR, a Multi-objective Iterated Symbolic Regression framework that fits multiple observables with uncertainty-aware, analytic expressions and iteratively refines models by residual boosting. The authors demonstrate MISR on nuclear binding energies and charge radii, yielding simple, interpretable formulas whose predictions compete with state-of-the-art models, and they further combine MISR with the Duflo–Zuker model via Bayesian ARD to estimate limits of nuclear stability. This hybrid, uncertainty-aware approach highlights the potential of physics-informed symbolic ML to enhance extrapolations in the nuclear chart and guide future exploration of complex many-body systems.

Abstract

Numerous phenomenological nuclear models have been proposed to describe specific observables within different regions of the nuclear chart. However, developing a unified model that describes the complex behavior of all nuclei remains an open challenge. Here, we explore whether novel symbolic Machine Learning (ML) can rediscover traditional nuclear physics models or identify alternatives with improved simplicity, fidelity, and predictive power. To address this challenge, we developed a Multi-objective Iterated Symbolic Regression approach that handles symbolic regressions over multiple target observables, accounts for experimental uncertainties and is robust against high-dimensional problems. As a proof of principle, we applied this method to describe the nuclear binding energies and charge radii of light and medium mass nuclei. Our approach identified simple analytical relationships based on the number of protons and neutrons, providing interpretable models with precision comparable to state-of-the-art nuclear models. Additionally, we integrated this ML-discovered model with an existing complementary model to estimate the limits of nuclear stability. These results highlight the potential of symbolic ML to develop accurate nuclear models and guide our description of complex many-body problems.

Figures (11)

• Figure 1: Diagrammatic representation of Multi-objective Iterated Symbolic Regression (MISR) inner pipeline. The process iteratively refines the symbolic regressed models. See the text for more details.
• Figure 2: Convergence of different observables fitting the MISR on the nuclear binding energies. The colored area illustrates the standard deviation of the residuals among the training set and the dashed line shows the mean over the test nuclei. The LDM results are illustrated as horizontal lines for reference.
• Figure 3: a) Predicted binding energy per nucleon $(BE/A)$ as a function of neutron number with associated model uncertainties denoted by the color bar. b) Absolute error on $\|BE/A\|$ predictions (Experiment - $BE_{MISR}$), showcasing the distribution of discrepancies for different neutron numbers. c) Residuals of the binding energy (Experiment - Model) obtained for our MISR10 results and the Duflo-Zucker models. Vertical dashed lines show the traditional nuclear magic numbers.
• Figure 4: Charge radii differences between the experimental value and prediction obtained by the MISR10 model. The magnitude of these differences is shown with different colors as a function of the neutron and proton numbers.
• Figure 5: Top: Distribution of the residual on the charge radii as a function of $Z$ for the first term in MISR (Experiment - $r_{MISR1}$). Bottom: Same but for binding energy (Experiment - $BE_{MISR1}$). MISR was trained only using the blue points.