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Bridging Theory and Data: Correcting Nuclear Mass Models with Interpretable Machine Learning
Yanhua Lu, Tianshuai Shang, Pengxiang Du, Jian Li, Haozhao Liang
TL;DR
Nuclear mass prediction must handle a small measured dataset with complex nonlinear dependencies, while existing models leave residual structure insufficiently described. The paper applies an interpretable KAN to learn these residuals, improving WS4 predictions and extending the correction across five additional mass models. Feature analysis identifies proton number as the most important residual-related input.
Problem
Nuclear mass prediction is constrained by about 2500 measured masses and complex nonlinear dependencies, while existing theoretical models leave residual components poorly described.
Method
The study uses a simple, interpretable KAN to learn residuals from theoretical nuclear mass models, with network visualization and feature-importance analysis.
Results
The KAN-WS4 model reduced RMS error from approximately 0.3 MeV for WS4 to 0.167 MeV overall, with 0.157 MeV on training data and 0.205 MeV on test data.
Takeaways & Limitations
Proton number Z was the most significant residual-related feature, and KAN correction significantly improved prediction accuracy for five other nuclear mass models.
Abstract
from arXiv · showhide
Nuclear mass prediction is one of the core issues in nuclear physics research, yet it faces the challenge of small-sample datasets with high complexity. This study introduces the Kolmogorov-Arnold Network (KAN) into the refinement of nuclear mass models, proposing an efficient and interpretable solution. By constructing the KAN-WS4 hybrid model, the prediction accuracy is significantly improved (the root mean square error is reduced from 0.3 MeV to 0.16 MeV). Furthermore, leveraging the intrinsic interpretability of KAN, feature importance analysis reveals that the proton number is the most critical factor influencing residuals, indicating potential systematic biases in proton-related terms within existing theoretical models. The method's generality is demonstrated across five mass models. This study shows that KAN provides a novel approach to small-sample, high-complexity scientific problems. Its interpretability facilitates the data-driven discovery of physical laws, promising broad applicability to key nuclear physics issues.
I. INTRODUCTION
Nuclear mass models have progressed from macroscopic formulas to more accurate global approaches, but existing methods remain insufficient for demanding applications and residual structure. This study uses an interpretable KAN correction to address small, complex nuclear-mass data.
- Nuclear masses inform nuclear structure, effective interactions, stellar energy generation, and the origin of elements.
- Approximately 0.3 MeV accuracy was achieved by macro–micro models such as FRDM and WS, while microscopic density-functional models reached approximately 0.5 MeV.
- Approximately 0.1 MeV accuracy is required for r-process studies, motivating further improvement in nuclear mass predictions.
- Only about 2500 nuclear masses have been measured, and their complex nonlinear dependencies make machine-learning prediction difficult.
- The study constructs a simple KAN model to correct poorly described mass residuals and uses visualization and feature importance to improve interpretability.
A. Kolmogorov-Arnold network
KANs replace fixed node activations and edge weights with learnable univariate functions on edges, offering a data-efficient representation of complex mappings. Their construction is motivated by the Kolmogorov-Arnold theorem and supports greater interpretability than conventional MLPs.
- MLPs model nonlinear mappings by stacking linear transformations with nonlinear activation functions, but their parameter inefficiency can require substantial training data.
- The Kolmogorov-Arnold theorem represents continuous multivariate functions through compositions of continuous univariate functions and addition.
- Equation (2) makes addition the only truly multivariate operation, reducing high-dimensional learning to learning multiple one-dimensional functions with polynomial scaling.
- KANs place learnable activation functions on edges and perform simple summation at nodes, unlike MLPs, whose learnable weights lie on edges and fixed activations on nodes.
B. Small-sample datasets with high complexity
The method models nuclear-mass residuals as small, complex deviations left after theoretical models capture the smooth mass component. It uses physically motivated nuclear variables as KAN inputs to learn nonlinear residual patterns.
- Nuclear mass is decomposed into a smooth component Mpre described by theory and a small, high-frequency residual Mδ that is difficult to predict.
- The study uses 2340 nuclei with Z, N ≥8 from AME2020, retaining only measurements with experimental uncertainties no greater than 100 keV.
- The KAN inputs are proton number Z, neutron number N, pairing-related quantity P, and shell-effect-related quantity S.
- P and S are derived from Z and N to represent pairing and shell effects, respectively.
- The input design directs KAN toward nonlinear residual patterns that traditional theoretical models do not capture.
III. RESULTS AND DISCUSSION
The KAN-WS4 model substantially improves nuclear-mass prediction accuracy across the nuclide chart and selected chains while providing interpretable feature rankings. The analysis identifies proton number as the dominant residual-related feature and shows that KAN correction generalizes across multiple theoretical mass models.
- Global prediction accuracy: 0.167 MeV RMS error is achieved across the entire nuclide chart, compared with approximately 0.3 MeV for the classical WS4 model.The training and test RMS errors are 0.157 MeV and 0.205 MeV, respectively.
- Local chain analysis: KAN-WS4 prediction errors are significantly smaller than WS4 errors across the Sn, Pb, N = 82, and N = 126 chains.The figure compares KAN-WS4 and WS4 deviations, separating training and randomly selected test regions.
- Interpretable architecture: KAN uses learnable activation functions on edges and summation at nodes, enabling nonlinear transformations of each input for feature extraction.This architecture is contrasted with MLPs, which place learnable weights on edges and fixed activation functions at nodes.
- Feature importance: Proton number Z is the most important residual-related feature, followed by shell-effect quantity S and pairing-related quantity P, which both outrank neutron number N.Feature importance is quantified using the L1 norms of activation functions connected to each input variable.
- Physical interpretation: The prominence of Z likely reflects systematic deviations in WS4 Coulomb-energy terms, while S and P capture additional shell and pairing corrections.The authors interpret the lower independent importance of N as possible evidence that its contributions are already captured or partly embedded in other features.
- Cross-model generality: KAN correction significantly improves five additional mass models, including Bhagwat, DZ28, FRDM, HFB31, and KTUY.The results suggest that KAN can learn residual effects across models with both small and large initial errors.
IV. SUMMARY AND PROSPECTS
The KAN-WS4 model improves nuclear-mass prediction while providing interpretable residual analysis. Proton number emerges as the most critical residual feature, and the framework improves five other mass models, although symbolic-law discovery remains limited to simple physical laws.
- 0.3 MeV to about 0.16 MeV: KAN-WS4 improves WS4 nuclear-mass prediction by learning residuals between theoretical and experimental values.The approach decomposes nuclear mass into a smooth global component and complex fluctuating residuals.
- Proton number is the most critical feature determining mass residuals, suggesting potential biases in proton-number-dependent terms of global models.The cited terms include Coulomb energy and proton shell effects.
- Significant accuracy improvements across five other mass models demonstrate the KAN framework’s broad applicability for correcting nuclear mass models.
- KAN’s current symbolic-regression utility is largely restricted to rediscovering simple physical laws, limiting direct law discovery in complex systems.The passage identifies small-sample, high-complexity datasets as especially challenging because minor deviations can be obscured by error propagation.