⚛️ nuclear theory

Kolmogorov-Arnold networks in nuclear binding energy prediction

This study demonstrates that Kolmogorov-Arnold networks (KANs) significantly outperform traditional models in predicting nuclear binding energies with a low error of 0.26 MeV while simultaneously providing interpretable analytical expressions that align with classical nuclear physics theories.

Original authors: Hao Liu, Jin Lei, Zhongzhou Ren

Published 2026-09-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hao Liu, Jin Lei, Zhongzhou Ren

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

At the heart of every atom lies a tiny, dense core called the nucleus, a crowded room where protons and neutrons huddle together in a delicate balance. The force that holds this room together is immense, and the energy required to keep it intact is known as the binding energy. This energy is not just a number; it is the key to understanding why some atoms are stable while others fall apart, how stars forge heavy elements, and how scientists might one day create new, super-heavy materials. For decades, physicists have tried to write a single, perfect equation that predicts this energy for every known atom, much like a master recipe for the universe's building blocks. Traditional formulas work well for many cases, but they often struggle with the messy, complex details that arise when the number of particles changes, leaving gaps in our understanding of the atomic world.

A team of researchers at Tongji University in Shanghai has taken a fresh approach to this old problem by using a new type of artificial intelligence called a Kolmogorov-Arnold network, or KAN. Unlike standard computer programs that learn by adjusting fixed internal knobs, this new system is designed to break down a complicated problem into a series of simpler, single-variable steps. Imagine trying to understand a complex machine by taking it apart and studying each gear individually rather than trying to guess how the whole thing works at once. By applying this method to the vast database of known atomic masses, the researchers were able to predict the binding energy of thousands of nuclei with remarkable precision. Their best model achieved an error rate of just 0.26 million electron volts, a level of accuracy that surpasses many of the long-standing physical theories used in the field today.

The researchers started by feeding the network data on the number of protons and neutrons in each nucleus, along with other known factors like how the particles pair up and how they arrange themselves in shells. They tested the system with different combinations of these details, finding that while the simplest version using only the basic counts of protons and neutrons was already quite good, adding information about the nuclear structure made the predictions significantly sharper. The most advanced version of their model, which included details about shell closures and particle pairing, reduced the prediction error to a level that rivals the most sophisticated physics-based simulations. This success suggests that the new network is not just memorizing numbers but is actually learning the underlying rules that govern how atomic nuclei hold together.

One of the most exciting aspects of this work is that the new system does more than just predict numbers; it can also reveal the mathematical formulas hidden inside the data. In many artificial intelligence models, the internal logic is a "black box" that is impossible for humans to read. However, because the KAN system is built to decompose problems into simple functions, the researchers were able to extract a clean, readable equation from its results. This derived formula looks very much like the classic equations physicists have used for years, containing terms for volume, symmetry, and surface effects, but with specific adjustments that the computer found on its own. This ability to translate a complex computer model back into a human-readable law of nature offers a rare glimpse into the mathematical structure of the atomic nucleus.

The study also tested how well these models could guess the properties of atoms that have never been seen or measured, a task known as extrapolation. When the researchers asked the network to predict the mass of nuclei far outside the range of its training data, the results were generally reliable for heavy atoms, staying close to the predictions of established physical models. However, the system struggled more with very light atoms, where the interactions between just a few particles create unique and difficult patterns. This finding highlights that while the new method is powerful, the behavior of the smallest nuclei still presents a unique challenge that requires further refinement. The researchers suggest that future versions of this technology could help uncover even deeper insights, potentially leading to new theories about how matter is constructed and how it behaves under extreme conditions.

By combining the pattern-recognition power of modern machine learning with the need for clear, understandable scientific laws, this work bridges the gap between raw data and theoretical insight. The researchers have shown that it is possible to use artificial intelligence not just as a calculator, but as a tool for discovery that can help rewrite the rules of nuclear physics. As these models continue to evolve, they promise to offer a clearer view of the atomic world, helping scientists navigate the unknown territories of the periodic table and understand the fundamental forces that shape our universe.

Technical Summary: Kolmogorov-Arnold Networks in Nuclear Binding Energy Prediction

Problem Statement
The prediction of nuclear binding energies (BE) is fundamental to understanding nuclear structure, shell effects, pairing, and astrophysical processes like the r-process. While traditional physics-based models (e.g., Bethe-Weizsäcker, Hartree-Fock-Bogoliubov) offer interpretability, they struggle with complex non-linear relationships. Conversely, data-driven machine learning methods (e.g., multilayer perceptrons, Gaussian processes) handle non-linearity well but often act as "black boxes," lacking interpretability and requiring substantial computational resources. The challenge lies in developing a model that balances high predictive accuracy with the ability to extract meaningful, interpretable physical laws from nuclear data.

Methodology
The study employs Kolmogorov-Arnold Networks (KANs), a neural network architecture based on the Kolmogorov-Arnold representation theorem. Unlike traditional Multilayer Perceptrons (MLPs) that use fixed activation functions on nodes, KANs place learnable, adaptable activation functions (specifically B-splines) on the edges. This allows the network to decompose complex multivariate functions into sums of univariate functions.

Data: The model utilizes mass excess values from the Atomic Mass Evaluation (AME2020), focusing on 3,456 nuclei with atomic number ( $Z$ ) and neutron number ( $N$ ) $\ge 8$ . The dataset is split into training (2,856 nuclei) and testing (600 nuclei) sets, with $k$ -fold cross-validation ( $k=3$ ) employed to ensure robustness.
Feature Selection: The study evaluates various feature sets ranging from basic nucleon numbers to complex physical descriptors. Key features include:
- Fundamental: $N$ , $Z$ , Mass number $A$ .
- Derived: Isospin asymmetry ( $N-Z$ ), surface term ( $A^{2/3}$ ).
- Structural: Pairing indicators ( $ZEO$ , $NEO$ ), shell closure excesses ( $\mu_Z$ , $\mu_N$ ), and shell orbital configurations ( $Z_{shell}$ , $N_{shell}$ ).
Training & Regularization: The network uses a shallow architecture (e.g., [Input, 12, 1]). To prevent overfitting and enhance interpretability, the loss function combines Mean Squared Error with L1 regularization and cross-entropy loss, encouraging sparsity in the network connections.
Symbolic Regression: A core component of the methodology involves extracting analytical expressions from the trained KAN. After pruning the network to a sparse structure, the learned B-spline activation functions are approximated using a library of elementary functions (polynomials, exponentials, logarithms) via least-squares fitting to derive a closed-form formula.

Key Results

Predictive Accuracy: The KAN models significantly outperform many traditional microscopic models.
- A minimal model (KAN-2) using only $N$ and $Z$ achieved a Root Mean Square Error (RMSE) of 0.87 MeV.
- The full-featured model (KAN-11), incorporating shell and pairing effects, achieved an RMSE of 0.26 MeV across the entire dataset. This surpasses the performance of models like WS4 (0.285 MeV), FRDM2012 (0.559 MeV), and HFB-31 (0.576 MeV).
Feature Importance: Analysis of the L1 norms of connections revealed that $N$ , $Z$ , and $A$ are the dominant drivers of binding energy. While shell and pairing terms had smaller direct weights, their inclusion was crucial for reducing errors, particularly near closed shells and for light nuclei.
Extrapolation: When tested on nuclei far from the training set (up to $A=339$ ), KANs showed reasonable agreement with the Finite-Range Droplet Model (FRDM2012) for regions not too distant from known data. However, significant deviations occurred for nuclei far beyond the training boundaries, attributed to the polynomial behavior of B-spline edges at boundaries.
Symbolic Derivation: The symbolic regression process yielded an analytical expression for binding energy that aligns with the liquid drop model structure (Volume, Symmetry, Coulomb, and Surface terms). While this symbolic formula had a higher RMSE (4.2 MeV for even-even nuclei) compared to the raw KAN, it successfully captured the underlying physical dependencies.

Significance and Claims
The paper claims that KANs offer a unique bridge between data-driven prediction and theoretical understanding in nuclear physics.

Interpretability: Unlike standard deep learning models, KANs facilitate symbolic regression, allowing researchers to derive simplified analytical expressions that reveal the governing laws of nuclear binding directly from data.
Performance: The study demonstrates that KANs can achieve state-of-the-art accuracy (0.26 MeV RMSE) with relatively simple architectures, outperforming many established physics-based models.
Physical Insight: The derived symbolic formula confirms the relevance of classical terms (volume, symmetry, surface) while highlighting the network's ability to implicitly learn complex structural effects (like shell closures) even when not explicitly provided as inputs.
Future Potential: The authors posit that KANs represent a promising tool for exploring complex multivariate systems in physics, offering a path to refine theoretical models and improve predictions in uncharted nuclear regions, provided that extrapolation limits are carefully managed.

The study concludes that while challenges remain in modeling light nuclei and extreme extrapolation, the combination of high accuracy and interpretability makes KANs a valuable addition to the nuclear physics toolkit.

Technical Summary: Kolmogorov-Arnold Networks in Nuclear Binding Energy Prediction

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