Understanding Charge Radii with Machine Learning: Discovering Physics Expressions

TL;DR

This paper presents a hybrid machine learning framework combining numerical and symbolic regression to accurately predict nuclear charge radii and uncover underlying physical formulas, including for light nuclei often omitted in such studies.

Contribution

It introduces a novel hybrid ML approach that accelerates symbolic regression and derives interpretable physics expressions from complex models, applied to nuclear charge radii.

Findings

Accurate predictions of nuclear charge radii across the nuclear chart.

Derived analytical formulas reveal key physical dependencies.

First-time inclusion of pairing gap in ML-based charge radius modeling.

Abstract

We introduce a robust, interpretable machine learning (ML) framework that combines numerical regression for high-accuracy predictions with symbolic regression to uncover the underlying physics. This hybrid approach effciently derives analytical expressions by leveraging the smoothed predictions of optimized ML models, a significant acceleration over direct symbolic regression on raw experimental data. We apply this framework, as an example, to nuclear charge radii across the nuclear chart, notably including light nuclei that are often excluded from such studies. We employ Light Gradient Boosting Machine (LGBM) and Gaussian Process Regression (GPR) models to map correlations between charge radii and key physical features: mass $A^{1/3}$ and proton number $Z^{1/3}$ dependencies, total binding energy, and for the first time, the pairing gap. Our models are rigorously trained using four-fold cross-validation with automated hyperparameter optimization, ensuring robustness and generalizability, which is critical for the typically small and skewed datasets in nuclear physics. Finally, we distill the knowledge from the initial LGBM and GPR models into simplified, interpretable mathematical expressions via symbolic regression, white-boxing these ML models. The derived formulas provide physical insights comparable to traditional many-body models and demonstrate a powerful pathway for physics expression discovery guided by ML.