Research on Efficient Solution of Neutron Physics Equations Using NAS-Optimized PINN
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摘要: 为快速且精确地求解堆芯中子扩散和输运这两类方程，可利用物理信息神经网络（PINN）提升偏微分方程求解的速度和效率。然而，由于PINN的预定义结构不够灵活，在一定程度上限制了其在实际应用中的广度和深度。本研究提出了一种寻找最佳PINN结构的创新方法（NAS-PINN），利用神经网络架构搜索（NAS）策略，动态地选择最适合于求解核反应堆中子扩散和输运方程的PINN结构。将搜索得到的PINN模型应用于方程求解中，进行真实值与预测值的实验验证比较。结果表明，NAS-PINN方法在求解不同几何的反应堆方程中具有更高的精度，为复杂的中子方程提供了更加准确、高效的求解方案。
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关键词:
- 中子扩散方程 /
- 中子输运方程 /
- 物理信息神经网络（PINN） /
- 神经网络架构搜索（NAS）
Abstract: To quickly and accurately solve the two kinds of equations of neutron diffusion and transport in the core, Physics-Informed Neural Networks (PINN) can be utilized to enhance the speed and efficiency of solving partial differential equation. However, the predefined structure of PINNs is relatively inflexible, limiting their width and depth in practical applications. This study proposes an innovative approach for determining the optimal PINN structure, (NAS-PINN), which employs a Neural Architecture Search (NAS) strategy to dynamically select the most suitable PINN structure for solving neutron diffusion and transport equations of nuclear reactors. The PINN model identified through this search is applied to equation solving, and the experimental verification comparison is made between the true and predicted values. The results show that the NAS-PINN method has higher accuracy in solving reactor equations with different geometries, and provides a more accurate and efficient solution for complex neutron equations. -
图 4 可视化一维扩散方程下不同方法的预测结果
Figure 4. Visualized Prediction Results of Different Cases for One-Dimensional Diffusion Equations
图 5 可视化二维扩散方程下不同方法的预测值与真实值的绝对误差
Figure 5. Visualized Absolute Error Results of Different Cases for Two-Dimensional Diffusion Equations
表 1 中子扩散方程中的超参数
Table 1. Hyperparameters in Neutron Diffusion Equations
形状 维度 边界点 域点 形状参数 球体 1 50 1950 半径R=1 圆柱体 2 80 8200 半径R=1，高H=1 立方体 3 1200 7800 边长a=1，b=1，c=1 
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表 2 一维扩散方程上最优神经网络架构的实验结果
Table 2. Experimental Results for Optimal Architecture on One-dimensional Diffusion Equation
方法 架构 MSE L2相对误差 Case 1 6.52×10−6 Case 2 [1, 20 × 16, 1] 3.49×10−10 7.59×10−3 Case 3 [1, 8 × 15, 1] 6.26×10−7 1.08×10−2 Case 4 [1, 19 × 15, 1] 3.49×10−6 6.65×10−3 新模型 [1, 12, 8, 9, 9, 12, 14, 16, 19, 12, 18, 19, 19, 17, 14, 8, 1] 1.30×10−10 5.54×10−3
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表 3 二维扩散方程上最优神经网络架构的实验结果
Table 3. Experimental Results for Optimal Architecture on Two-dimensional Diffusion Equation
方法 架构 MSE L2相对误差 Case 1 1.39×10−5 Case 2 [2, 20 × 16, 1] 4.56×10−6 6.77×10−2 Case 3 [2, 8 × 15, 1] 1.82×10−5 1.24×10−1 Case 4 [2, 20 × 15, 1] 8.75×10−5 7.55×10−2 新模型 [2, 17, 12, 10, 11, 12,19, 14, 20, 16, 19, 8, 13, 18, 8, 20, 1] 9.59×10−8 1.14×10−2
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表 4 三维扩散方程上最优神经架构的实验结果
Table 4. Experimental Results for Optimal Architecture on Three-dimensional Diffusion Equation
方法 架构 MSE L2相对误差 Case 1 5.22×10−5 Case 2 [3, 20 × 16, 1] 6.23×10−5 2.30×10−2 Case 3 [3, 16 × 10, 1] 8.80×10−8 3.23×10−2 Case 4 [3, 20 × 10, 1] 6.61×10−8 3.64×10−2 新模型 [3, 16, 19, 18, 18, 19,17, 20, 18, 16, 20, 1] 4.77×10−8 1.51×10−2
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表 5 球形几何临界标注量率的计算值与理论值的比较
Table 5. Comparison between Calculated Result and Theoretical Value of Critical Scalar Flux Density for Spherical Geometry
参数 r/R=0（中心） r/R=0.25 r/R=0.50 r/R=0.75 r/R=1（边界） 数值计算值 0.99879086 0.92413104 0.68932903 0.37161699 0.07440116 归一化计算值 1 0.92524979 0.69016353 0.37206687 0.07449123 归一化理论值 1 0.91612699 0.68954766 0.36621118 −0.00008100 相对误差 0 0.00995801 0.00089315 0.01598993 −0.00008100
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表 6 不同方法的MSE比较结果
Table 6. Comparison Results of MSE for Different Cases
方法 MSE Case 2 5.4885×10−6 Case 3 5.2847×10−6 Case 4 1.4905×10−5 新模型 3.0745×10−6
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