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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1.63MeV, 较之前1.86MeV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 accurately predictions of binding energy are crucial for research on nuclear structure, reactions, and decay. However, traditional mass models exhibit large errors in double magic number regions and heavy nuclei regions. These models struggle to effectively describe shell effects and parity effects in nuclear structures, and also fail to capture the subtle differences observed in experimental results. This study shows the strong modeling capabilities of MLP neural networks, which optimizes the parameters of the nuclear mass model, and reduces prediction errors in key regions and globally. The neural network takes the features as neutron number, proton number, and binding energy, and the labels as mass-model coefficients. The training set is the multiple sets of calculated nuclear mass model coefficients. Through extensive experimentation, the optimal parameters are determined to ensure model convergence speed and stability. The Adam optimizer is employed to adjust the weights and biases of the network, for reducing the mean squared error loss during training. The trained neural network model with minimal loss was used to predict the optimal coefficients of the nuclear mass model based on the AME2020 dataset. The optimized BW2 model significantly reduces root-mean-square errors in double magic number and heavy nuclei regions. Specifically, the optimized model achieved reductions of approximately 28%, 12%, and 18% in root-mean-square errors near Z(N) = 50, Z(N) = 50 (82), and Z(N) = 82 (126), respectively. In heavy nuclei regions, the errors were reduced by 48%. The BW3 model, incorporating higher-order symmetry energy terms, reduced global root-mean-square errors from 1.86 MeV to 1.63 MeV after parameter optimization using the neural network. This work reveals that the model with newly optimized coefficients not only exhibits significant error reductions near double magic numbers but also shows improvements in binding energy predictions for both neutron-rich and neutron-deficient nuclei. Furthermore, the model demonstrates good improvements in describing parity effects, accurately capturing parity-related differences in isotopic chains with varying proton numbers. This study demonstrates the tremendous potential of MLP neural networks in optimizing nuclear mass model parameters and provides a novel method for optimizing parameters in more complex nuclear mass models. Moreover, the proposed method applies to nuclear mass models with implicit or nonlinear relationships, offering new perspectives for further development of 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 初始系数(a)和优化系数(b)下BW2质量公式预测值与实验结合能的偏差对比图
Fig. 3. Comparison plot of the deviation of the predicted value of the BW2 mass formula from the experimental binding energy under the original coefficient (a) and optimization coefficient (b).
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. Different Proton 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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