Kolmogorov-Arnold networks in nuclear binding energy prediction

The gist: Kolmogorov-Arnold networks (KANs) are explored for predicting nuclear binding energies, achieving a root mean square error of 0.26 MeV when incorporating features such as atomic number, neutron number, and shell effects.

Introduction and Motivation

Binding energies (BE), synonymous with nuclear mass, are key characteristics of atomic nuclei and play an indispensable role in elucidating various nuclear phenomena [1]. These phenomena include nuclear shapes, shell effects, pairing effects, and the emergence and disappearance of magic numbers [2–5]. Both theoretical predictions and experimental measurements of nuclear binding energies are essential for advancements in nuclear physics [6–7]. Traditional physical models face challenges when managing complex relationships, while data-driven methods require substantial data and computational resources [13–18]. Kolmogorov-Arnold networks (KANs) offer a promising solution by decomposing complex multi-parameter systems into manageable univariate functions [22]. This approach is based on the Kolmogorov-Arnold representation theorem [23, 24].

Formalism of KANs

The Kolmogorov-Arnold theorem asserts that any continuous multivariate function on a bounded domain can be decomposed into a finite composition of continuous univariate functions and addition [1]. Specifically, for a continuous function f: [0, 1]n → R: f(x) = f(x1, x2,..., xn) = 2Xn+1 q=1 Xn p=1 Φq Xn p=! (1). This decomposition shows that addition is the only truly multivariate operation [2]. KANs improve upon the original framework by allowing networks to have any number of layers and widths, which overcomes the original theorem’s limitations [22]. The structure of a shallow KAN, as illustrated in Figure 1, consists of two input variables, x1 and x2, a hidden layer with five nodes, and a single output node [1]. Each input is connected to the hidden nodes through edges characterized by adaptable functions that are expanded into basis functions like B-spline functions [1].

Data Collection and Feature Selection

The study utilized mass excess values from the Atomic Mass Evaluation (AME2020) [25], focusing on nuclei with atomic number (Z) and neutron number (N) greater than or equal to 8, covering 3456 nuclei [1]. The experimental data were randomly divided into two subsets: 2,856 nuclei for training and 600 nuclei for testing [2]. K-fold cross-validation was applied with k=3, allocating 1/3 of the data for testing and the remaining 2/3 for training in each cycle [2]. Key features included atomic number (Z), neutron number (N), mass number (A), pairing effects captured by ZEO and NEO, isospin asymmetry (N − Z), nuclear magic numbers, and shell structure features like Zshell and Nshell [3]. The network structure was set to a single hidden layer with a uniform width of 12 neurons [4].

Results and Analysis

The KAN-2 model, using only basic properties such as neutron number N and proton number Z, achieved a root mean square error (RMSE) of just 0.87 MeV [3]. When additional features were incorporated, the KAN-11 model reduced the RMSE even further to an impressive 0.26 MeV for the entire data set [4]. The models performed better for medium and heavy nuclei compared to light nuclei, with shell structure features like KAN-9 and KAN-11 achieving lower RMSEs [5]. The analysis of feature importance indicated that the neutron number N exerted the largest influence on the model’s predictions, as evidenced by its highest L1 norm among the variables [7].

Symbolic Regression and Extrapolation

The symbolic regression analysis yielded an analytical expression for nuclear binding energy (6), which aligns well with established models like the Bethe-Weizsäcker formula [9]. This expression achieves an RMSE of 4.2 MeV for these even-even nuclei, suggesting less than 1% error given the range of binding energies [8]. The study explored extrapolation capabilities by comparing KAN predictions with those from the finite-range droplet model (FRDM2012) [36] for nuclei beyond AME2020. The KAN-4 and KAN-11 models demonstrated good agreement with FRDM12 across most regions, while the KAN-2 model exhibited limited extrapolation capability [9].

Conclusion

KANs effectively capture underlying physical relationships in nuclear data, achieving a RMSE as low as 0.26 MeV for the entire dataset when using an expanded set of features [4]. The symbolic regression analysis provided an analytical expression that offers improved interpretability and potential for deeper theoretical understanding [8]. Future work will focus on extending the application of KANs to better model light nuclei and refining network architectures for improved performance [10].

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Kolmogorov-Arnold networks in nuclear binding energy prediction

Hao Liu, Jin Lei, Zhongzhou Ren

School of Physics Science and Engineering, Tongji University, Shanghai 200092, China.

(Dated: September 7, 2026)

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[39] J. Giroux and C. Fanelli “Uncertainty quantification with bayesian higher order relu kans” (2410.01687 [cs.LG]).

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Fig. 9 Mass excess for Zr, Nd, Hg, and Fm isotopic chains compared with KANs predictions, the results from symbolic regression, FRDM12 (black points) [36], and experimental data from AME2020 (cyan stars) [25].

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Fig. 8 Panel (a) shows the original trained network structure, while panel (b) shows the symbolic network structure. Panels (c) and (d) illustrate the absolute value of binding energy differences between predictions and data from AME2020 [25].

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Fig. 7 Visualization of the trained network structure in the panel (a). The transparency of the lines connecting nodes is proportional to tanh(βAl,i,j), where Al,i,j is the mean activation. In panel (b), the L1 norm for each connection is shown on a logarithmic scale, highlighting the significance of each link.

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Fig. 4 Experimental single neutron separation energies for Ca, Zr, Nd, and Hg isotopic chains in comparison with the KAN predictions.

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Fig. 3 The absolute value of binding energy differences between KAN predictions using different features (see Table I) and AME2020 [25].

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TABLE I Feature space and KANs’ structure

Model Feature Structure KAN-2 N, Z [2,12,1]

KAN-4 N, Z, A, N-Z [4,12,1]

KAN-9 N, Z, A 2/3, ZEO NEO [9], [12], [1]

µZ, µN KAN-11 N, Z, A 2/3, ZEO NEO [11], [12], [4]

µZ, µN, Zshell, Nshell

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FIG. 2 The training set (gray circles), test set (red circles) used in the KANs. Both the training and test sets include nuclei from AME2020.

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FIG. 5 The absolute value of mass excess differences between KAN predictions using different features (see Table I) and FRDM12’s prediction [36].

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Fig. 6 Mass excess for Ca, Zr, Nd, Hg, and Hg isotopic chains compared with KANs predictions, FRDM12 (black points) [36], and experimental data from AME2020 (cyan stars) [25].

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I INTRODUCTION The atomic nucleus a quintessential quantum manybody system exhibits remarkable structural complexity [1]. Binding energies (BE), synonymous with nuclear mass, are key characteristics of atomic nuclei and play an indispensable role in elucidating various nuclear phenomena.

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Over the years numerous theories and methods have been developed for predicting nuclear binding energies [1, 2, 8–21]. These include the Bethe-Weizsäcker formula [2, 8], the Thomas-Fermi model [9], the HartreeFock-Bogoliubov mean field model [10], and ab initio methods [1, 11, 12].

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Over the years numerous theories and methods have been developed for predicting nuclear binding energies [2, 8–21]. These include the Bethe-Weizsäcker formula [2, 8], the Thomas-Fermi model [9], the HartreeFock-Bogoliubov mean field model [10], and ab initio methods [1, 11, 12].

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However traditional physical models necessitate a profound understanding of the inherent mechanisms of nuclear physical systems and face challenges when managing complex relationships.

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In this study we aim to explore the potential of KANs in predicting nuclear binding energies.

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In this study, we aim to derive mass formulas using AI, harnessing its power to reveal insights that can enhance our understanding of nuclear binding energies.

1. A KAN-11 architecture incorporating features like isospin asymmetry, and mass number can achieve a significant lower root mean square error (0.26 MeV) for predicting nuclear binding energies across the entire dataset, surpassing traditional models.

2. The symbolic regression analysis yields simplified analytical expressions for binding energies, aligning with classical models like the liquid drop model and the Bethe-Weizsäcker formula, enhancing the interpretability of predictions.

3. The system can perform improved extrapolation by using symbolic regression to replace the polynomial expansion form in KANs, which may possess better extrapolation capabilities due to their flexibility and the ability to capture underlying patterns more effectively.

This study explores the application of Kolmogorov-Arnold networks (KANs) in predicting nuclear binding energies, leveraging their ability to decompose complex multiparameter systems into simpler univariate functions. By utilizing data from the Atomic Mass Evaluation (AME2020) and incorporating features such as atomic number, neutron number, and shell effects, KANs achieved a significant lower root mean square error (0.26 MeV), surpassing traditional models. The symbolic regression analysis yielded simplified analytical expressions for binding energies, aligning with classical models like the liquid drop model and the Bethe-Weizsaecker formula. These results highlight KANs' potential in enhancing the interpretability and understanding of nuclear phenomena, paving the way for future applications in nuclear physics and beyond.