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Nuclear masses are fundamental observables that reflect nuclear structure and stability, playing a key role in nuclear physics and astrophysical processes. Most existing neural network studies focus on predicting either binding energies or neutron/proton separation energies individually, with limited attention to the physical correlations between these observables. Based on the relativistic point-coupling model PCF-PK1, a physics-informed artificial neural network (ANN) was developed to systematically predict nuclear binding energies along with single- and double-neutron/proton separation energies, while preserving the physical self-consistency of the predictions. To assess the impact of incorporating separation-energy constraints, networks were trained with varying loss function weight combinations, enabling a comparison between networks without separation-energy constraints (e.g., ANN1) and those including such constraints (e.g., ANN3). The neural network significantly improves the overall prediction accuracy of binding energies compared with the PCF-PK1 model. Without separation-energy constraints, ANN1 already achieves high precision for binding energies (RMSE $\approx$ 0.147 MeV) and separation energies (RMSE $\approx$ 0.158–0.185 MeV). Incorporating separation-energy constraints in ANN3 results in a slight improvement in overall prediction accuracy. The binding energy predictions improve by approximately 4.6%, while the separation energy predictions increase by 8.9–12.0%. The improvement is particularly noticeable for nuclei where the deviations of ANN1 predictions from experimental values exceed 0.2 MeV. Supporting datasets are publicly accessible at the Science Data Bank ( https://doi.org/10.57760/sciencedb.j00213.00239 ).
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Keywords:
- nuclear mass /
- neural network /
- separation energy
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图 1 引入分离能约束的多层前馈神经网络结构示意图
Figure 1. Diagram of the feedforward neural network with separation energy constraints.
图 2 训练过程中训练集(红色)与验证集(蓝色)均方根偏差随训练步数的变化示意. 当验证集偏差达到最小值时, 模型参数被保存
Figure 2. Root-mean-square deviations of training (red) and validation (blue) sets over epochs; parameters at minimum validation deviation are saved.
图 3 ANN1、ANN2、ANN3、ANN4以及PCF-PK1模型对原子核E、$ S_{\mathrm{n}} $、$ S_{\mathrm{2 n}} $、$ S_{\mathrm{p}} $和$ S_{\mathrm{2 p}} $的预测值与AME2020实验值均方根偏差的对比
Figure 3. Comparison of RMSE between ANN1, ANN2, ANN3, ANN4, and PCF-PK1 predictions and AME2020 experimental values for E, $ S_{\mathrm{n}} $, $ S_{\mathrm{2 n}} $, $ S_{\mathrm{p}} $, and $ S_{\mathrm{2 p}} $.
图 4 ANN1和ANN3预测值相对于实验值的偏差随核质量数的变化, 包括结合能E(a, f); 单、双中子分离能$ S_\mathrm{n} $(b, g)、$ S_\mathrm{2 n} $(c, h) 以及单、双质子分离能$ S_\mathrm{p} $(d, i)、$ S_\mathrm{2 p} $(e, j). 其中(a—e)为ANN1预测结果, (f—j)为ANN3预测结果, 阴影区域对应结合能预测值与实验值偏差的$ \pm $0.1 MeV范围
Figure 4. Deviations of ANN1 and ANN3 predictions from experimental values as a function of nuclear mass number, including binding energy E(a, f), single- and double-neutron separation energies $ S_\mathrm{n} $(b, g) and $ S_\mathrm{2 n} $(c, h), and single- and double-proton separation energies $ S_\mathrm{p} $(d, i) and $ S_\mathrm{2 p} $(e, j). Panels (a–e) show ANN1 predictions, and panels (f–j) show ANN3 predictions. The shaded areas indicate $ \pm $0.1 MeV deviations of binding energies.
图 5 ANN1(红色实线)、ANN2(蓝色虚线)、ANN3(绿色点线)和ANN3_ZNP(橙色点划线)对Ca($ Z=20 $)及Pb($ Z=82 $)同位素链结合能E(a, f)、单中子分离能$ S_{\mathrm{n}} $(b, g)、双中子分离能$ S_{\mathrm{2 n}} $(c, h)、单质子分离能$ S_{\mathrm{p}} $(d, i)以及双质子分离能$ S_{\mathrm{2 p}} $(e, j)的预测值与实验值偏差. 图中深浅不同的阴影分别对应预测值与实验值偏差在$ \pm $0.1 MeV和$ \pm $0.2 MeV的区间
Figure 5. Deviations between the predictions and experimental values of the binding energy E(a, f), single- and double-neutron separation energies $ S_{\mathrm{n}} $, $ S_{\mathrm{2 n}} $(b, g, c, h), single- and double-proton separation energies $ S_{\mathrm{p}} $, $ S_{\mathrm{2 p}} $(d, i, e, j), obtained with ANN1 (red solid line), ANN2 (blue dashed line), ANN3 (green dotted line), and ANN3_ZNP (orange dash-dot line) for the Ca($ Z=20 $) and Pb($ Z=82 $) isotopic chains. Dark and light shaded areas represent deviations from experimental values of $ \pm $0.1 MeV and $ \pm $0.2 MeV, respectively.
图 6 同图4, ANN1(红色实线)、ANN2(蓝色虚线)、ANN3(绿色点线)和ANN3_ZNP(橙色点划线)对$ N=28 $及$ N=126 $同中子素链结合能E(a, f)、单中子分离能$ S_{\mathrm{n}} $(b, g)、双中子分离能$ S_{\mathrm{2 n}} $(c, h)、单质子分离能$ S_{\mathrm{p}} $(d, i)以及双质子分离能$ S_{\mathrm{2 p}} $(e, j)的预测值与实验值偏差. 图中深浅不同的阴影分别对应预测值与实验值偏差在$ \pm $0.1 MeV和$ \pm $0.2 MeV的区间
Figure 6. Similar to Fig. 4. Deviations between the predictions and experimental values of the binding energy E(a, f), single- and double-neutron separation energies $ S_{\mathrm{n}} $, $ S_{\mathrm{2 n}} $(b, g, c, h), single- and double-proton separation energies $ S_{\mathrm{p}} $, $ S_{\mathrm{2 p}} $(d, i, e, j), obtained with ANN1 (red solid line), ANN2 (blue dashed line), ANN3 (green dotted line), and ANN3_ZNP (orange dash-dot line) for the N = 28 and $ N = 126 $ >$ Z=82 $) isotonic chains. Dark and light shaded areas represent deviations from experimental values of $ \pm $0.1 MeV and $ \pm $0.2 MeV, respectively.
图 7 ANN3对整个核素图上原子核结合能E的预测值与实验值的偏差分布
Figure 7. Distribution of prediction residuals of ANN3 with respect to experimental values on the nuclear chart for the binding energy E.
图 8 ANN3对整个核素图上原子核单/双中子分离能$ S_\mathrm{n} $(a), $ S_\mathrm{2 n} $(b)以及单/双质子分离能$ S_\mathrm{p} $(c), $ S_\mathrm{2 p} $(d)的预测值与实验值的偏差分布
Figure 8. Distribution of prediction residuals of ANN3 with respect to experimental values on the nuclear chart for the one- and two-neutron separation energies $ S_\mathrm{n} $ (a) and $ S_\mathrm{2 n} $ (b), as well as one- and two-proton separation energies $ S_\mathrm{p} $ (c) and $ S_\mathrm{2 p} $ (d).
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