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神经网络具有强大的建模能力和对大规模数据的适应性, 在拟合核质量模型参数方面表现出显著效果. 本研究旨在探索神经网络拟合核质量模型参数的问题: 采用多层感知机(multilayer perceptron, MLP)神经网络结构, 评估不同参数下Adam优化器的训练效果, 训练出准确的模型参数. 研究发现, 基于AME2020数据, 更新系数后的BW2核质量模型在双幻数以及重核区域的均方根误差降低明显; BW3模型重新拟合后的全局均方根误差为1.63 MeV, 较之前1.86 MeV有所降低. 结果表明, 该方法能够有效地拟合模型参数, 并具有良好的拟合性能和泛化能力. 这项研究为BW系列核质量模型的系数提供了新的拟合方法, 也为其他核质量寻求最佳拟合参数提供了有益的参考.The nuclear mass model has significant applications in nuclear physics, astrophysics, and nuclear engineering. The accurate prediction of binding energy is crucial for studying nuclear structure, reactions, and decay. However, traditional mass models exhibit significant errors in double magic number region and heavy nuclear region. These models are difficult to effectively describe shell effect and parity effect in the nuclear structure, and also fail to capture the subtle differences observed in experimental results. This study demonstrates the powerful modeling capabilities of MLP neural networks, which optimize the parameters of the nuclear mass model, and reduce prediction errors in key regions and globally. In the neural network, neutron number, proton number, and binding energy are used as training feature values, and the mass-model coefficient is regarded as training label value. The training set is composed of the multiple sets of calculated nuclear mass model coefficients. Through extensive experiments, the optimal parameters are determined to ensure the convergence speed and stability of the model. The Adam optimizer is used to adjust the weight and bias of the network to reduce the mean squared error loss during training. Based on the AME2020 dataset, the trained neural network model with the minimum loss is used to predict the optimal coefficients of the nuclear mass model. The optimized BW2 model significantly reduces root-mean-square errors in double magic number and heavy nuclear regions. Specifically, the optimized model reduces the root-mean-square error by about 28%, 12%, and 18% near Z = 50 and N = 50; Z(N) = 50 and N = 82; Z = 82 and N = 126, respectively. In the heavy nuclear region, the error is reduced by 48%. The BW3 model combines higher-order symmetry energy terms, and after parameter optimization using the neural network, reduces the global root-mean-square error from 1.86 MeV to 1.63 MeV. This work reveals that the model with newly optimized coefficients not only exhibit significant error reduction near double magic numbers, but also shows the improvements in binding energy predictions for both neutron-rich and neutron-deficient nuclei. Furthermore, the model shows good improvements in describing parity effects, accurately capturing the differences related to parity in isotopic chains with different proton numbers. This study demonstrates the tremendous potential of MLP neural networks in optimizing the parameters of nuclear mass model and provides a novel method for optimizing parameters in more complex nuclear mass models. In addition, the proposed method is applicable to the nuclear mass models with implicit or nonlinear relationships, providing a new perspective for further developing the nuclear mass models.
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Keywords:
- nuclear mass model /
- magic numbers /
- multilayer perceptron neural network /
- Adam optimizer
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图 1 神经网络优化核质量模型系数框架图(MLP, 多层感知机神经网络; Exp., 实验结合能)
Fig. 1. Framework diagram of neural network optimizing nuclear mass model coefficients (MLP, multi-layer perceptron neural network; Exp., experimental binding energy).
图 2 Adam优化器不同学习率和权重衰减参数实验对比图, 水平坐标为神经网络训练次数, 垂直坐标为神经网络损失值, 当损失值下降低于0.1%时停止训练($ lr $表示学习率, $ w $表示权重衰减参数)
Fig. 2. Comparison chart of Adam optimizer with different learning rates and weight decay parameters. The horizontal axis represents the number of neural network training iterations, and the vertical axis represents the neural network loss value. Training stops when the loss value drops below 0.1%. ($ lr $ represents the learning rate, $ w $ represents the weight decay parameter)
图 3 BW2质量公式预测值与实验结合能的偏差对比图 (a)初始系数; (b)优化系数
Fig. 3. Comparison plot of the deviation of the predicted value of the BW2 mass formula from the experimental binding energy: (a) Original coefficient; (b) optimization coefficient.
图 4 随机元素结合能的实验值与质量公式初始系数和优化系数计算值之间的差值折线图
Fig. 4. Line plot of the difference between the experimental value of the binding energy of a random element and the calculated value of the original coefficient and optimization coefficient of the mass formula.
图 5 相同质量数下对比不同中子数的结合能偏差图
Fig. 5. Graph of same mass number vs. different neutron numbers.
图 6 中子数相同对比不同质子数的结合能偏差图
Fig. 6. Graph of same neutron number vs. differentproton numbers.
表 1 MLP神经网络寻找的系数组(部分, 单位: MeV)
Table 1. Coefficients identified by the MLP neural network (partial, unit: MeV).
1 2 3 4 5 6 7 8 $ \alpha_{v} $ 16.58 16.22 16.24 16.21 16.22 16.22 16.24 16.05 $ \alpha_{s} $ –26.95 –23.36 –23.42 –23.39 –23.38 –23.36 –23.40 –23.10 $ \alpha_{C} $ –0.77 –0.74 0.74 –0.74 –0.74 –0.75 –0.75 –0.74 $ \alpha_{t} $ –31.51 –31.53 –31.59 –31.54 –31.57 –31.53 –32.60 –31.62 $ \alpha_{xC} $ 2.22 1.39 1.38 1.39 1.40 1.39 1.40 1.59 $ \alpha_{W} $ –43.40 –57.38 –57.40 –57.42 –57.41 –57.40 –57.47 –72.97 $ \alpha_{s t} $ 55.62 54.98 55.02 54.96 55.03 54.99 55.09 64.10 $ \alpha_{p} $ 9.87 10.63 10.61 10.64 10.64 10.63 10.67 10.56 $ \alpha_{R} $ 14.77 9.89 9.94 9.91 9.91 9.89 9.93 9.89 $ \alpha_{m} $ –1.90 –1.89 –1.91 –1.90 –1.89 –1.89 –1.90 –1.88 $ \beta_{m} $ 0.14 0.14 0.13 0.14 0.14 0.15 0.15 0.14 $ b $ — — — — — — — –11.36 $ \sigma $ 1.92 1.90 1.84 1.68 1.76 1.81 1.89 1.63 
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[1] Lunney D, Pearson J M, Thibault C 2003 Rev. Mod. Phys. 75 1021
Google Scholar
[2] 李涛, 黎春青, 周厚兵, 王宁 2021 70 102101
Google Scholar
Li T, Li C Q, Zhou H B, Wang N 2021 Acta Phys. Sin. 70 102101
Google Scholar
[3] Ramirez E M, Ackermann D, Blaum K, Block M, Droese C, Düllmann C E, Dworschak M, Eibach M, Eliseev S, Haettner E, Herfurth F, Heßberger F P, Hofmann S, Ketelaer J, Marx G, Mazzocco M, Nesterenko D, Novikov Y N, Plaß W R, Rodríguez D, Scheidenberger C, Schweikhard L, Thirolf P G, Weber C 2012 Science 337 1207
Google Scholar
[4] Horoi M 2013 International Summer School for Advanced Studies Dynamics of Open Nuclear Systems (Predeal12) Predeal, Romania, July 9–20, 2012 p012020
[5] Wienholtz F, Beck D, Blaum K, Borgmann C, Breitenfeldt M, Cakirli R B, George S, Herfurth F, Holt J D, Kowalska M, Kreim S, Lunney D, Manea V, Menéndez J, Neidherr D, Rosenbusch M, Schweikhard L, Schwenk A, Simonis J, Stanja J, Wolf R N, Zuber K 2013 Nature 498 346
Google Scholar
[6] Burbidge E M, Burbidge G R, Fowler W A, Hoyle F 1957 Rev. Mod. Phys. 29 547
Google Scholar
[7] Ye W, Qian Y, Ren Z 2022 Phys. Rev. C 106 024318
Google Scholar
[8] Bethe H A, Bacher R F 1936 Rev. Mod. Phys. 8 82
Google Scholar
[9] Weizsäcker C F V 1935 Zeitschrift für Physik 96 431
Google Scholar
[10] Kirson M W 2008 Nucl. Phys. A 798 29
Google Scholar
[11] Sorlin O, Porquet M G 2008 Prog. Part. Nucl. Phys. 61 602
Google Scholar
[12] Ozawa A, Kobayashi T, Suzuki T, Yoshida K, Tanihata I 2000 Phys. Rev. Lett. 84 5493
Google Scholar
[13] Gherghescu R A, Poenaru D N 2022 Phys. Rev. C 106 034616
Google Scholar
[14] Björck Å 1990 Handb. Numer. Anal. 1 465
Google Scholar
[15] Jiang B N 1998 Comput. Methods Appl. Mech. Eng. 152 239
Google Scholar
[16] Mohammed-Azizi B, Mouloudj H 2022 Int. J. Mod. Phys. C 33 2250076
Google Scholar
[17] Cao Y, Lu D, Qian Y, Ren Z 2022 Phys. Rev. C 105 034304
Google Scholar
[18] Huang W, Wang M, Kondev F, Audi G, Naimi S 2021 Chin. Phys. C 45 030002
Google Scholar
[19] Wang M, Huang W, Kondev F, Audi G, Naimi S 2021 Chin. Phys. C 45 030003
Google Scholar
[20] Sobiczewski A, Pomorski K 2007 Prog. Part. Nucl. Phys. 58 292
Google Scholar
[21] Yin X, Shou R, Zhao Y M 2022 Phys. Rev. C 105 064304
Google Scholar
[22] Wang N, Liu M, Wu X 2010 Phys. Rev. C 81 044322
Google Scholar
[23] Wu Y C, Feng J W 2018 Wirel. Pers. Commun. 102 1645
Google Scholar
[24] Popescu M C, Balas V E, Perescu-Popescu L, Mastorakis N 2009 WSEAS Trans. Cir. and Sys. 8 579
Google Scholar
[25] Xiang C, Ding S, Lee T H 2005 IEEE Trans. Neural Netw. 16 84
Google Scholar
[26] Pinkus A 1999 Acta Numerica 8 143
Google Scholar
[27] Sharma A, Gandhi A, Kumar A 2022 Phys. Rev. C 105 L031306
Google Scholar
[28] Wu X H, Ren Z X, Zhao P W 2022 Phys. Rev. C 105 L031303
Google Scholar
[29] Gao Z P, Wang Y J, Lü H L, Li Q F, Shen C W, Liu L 2021 Nucl. Sci. Tech. 32 109
Google Scholar
[30] 庞龙刚, 周凯, 王新年 2020 原子核物理评论 37 720
Google Scholar
Pang L G, Zhou K, Wang X N 2020 Nucl. Phys. Rev. 37 720
Google Scholar
[31] Gernoth K A, Clark J W 1995 Neural Networks 8 291
Google Scholar
[32] Yüksel E, Soydaner D, Bahtiyar H 2021 Int. J. Mod. Phys. E 30 2150017
Google Scholar
[33] Liu M, Wang N, Deng Y, Wu X 2011 Phys. Rev. C 84 014333
Google Scholar
[34] Wang N, Liu M 2011 Phys. Rev. C 84 051303
Google Scholar
[35] Utama R, Piekarewicz J, Prosper H B 2016 Phys. Rev. C 93 014311
Google Scholar
[36] Utama R, Piekarewicz J 2018 Phys. Rev. C 97 014306
Google Scholar
[37] Ma C, Zong Y Y, Zhao Y M, Arima A 2020 Phys. Rev. C 102 024330
Google Scholar
[38] Özdoğan H, Üncü Y, Şekerci M, Kaplan A 2022 Appl. Radiat. Isot. 184 110162
Google Scholar
[39] Chen X, Ma Q, Alkharobi T 2009 2nd IEEE International Conference on Computer Science and Information Technology Beijing, China, August 8–11, 2009 p291
[40] Ming X C, Zhang H F, Xu R R, Sun X D, Tian Y, Ge Z G 2022 Nucl. Sci. Tech. 33 48
Google Scholar
[41] Le X K, Wang N, Jiang X 2023 Nucl. Phys. A 1038 122707
Google Scholar
[42] Slowik A, Kwasnicka H 2020 Neural Comput. Appl. 32 12363
Google Scholar
[43] Amine K 2019 Adv. Oper. Res. 2019 8134674
Google Scholar
[44] Wang D, Tan D, Liu L 2018 Soft Computing 22 387
Google Scholar
[45] Chen A, Tan H, Zhu Y 2022 2nd International Conference on Applied Mathematics, Modelling, and Intelligent Computing (CAMMIC 2022) Kunming, China, March 25–27, 2022 p1472
[46] Huang L, Qin J, Zhou Y, Zhu F, Liu L, Shao L 2023 IEEE Trans. Pattern Anal. Mach. Intell. 45 10173
Google Scholar
[47] Xu X Y, Deng L, Chen A X, Yang H, Jalili A, Wang H K 2024 Nucl. Sci. Tech. 35 91
Google Scholar
[48] Möller P, Myers W D, Sagawa H, Yoshida S 2012 Phys. Rev. Lett. 108 052501
Google Scholar
[49] Zhang H F, Wang L H, Yin J P, Chen P H, Zhang H F 2017 J. Phys. G: Nucl. Part. Phys. 44 045110
Google Scholar
[50] Samyn M, Goriely S, Heenen P H, Pearson J, Tondeur F 2002 Nucl. Phys. A 700 142
Google Scholar
[51] Moller P, Nix J, Myers W, Swiatecki W 1995 At. Data Nucl. Data Tables 59 185
Google Scholar
[52] Duflo J, Zuker A 1995 Phys. Rev. C 52 R23
Google Scholar
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