Research on Algorithm of Solving Neutron Equation Based on ResNet-PINN
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摘要: 物理信息神经网络（PINN）作为一种结合物理知识的深度学习方法，其在求解问题的精度方面存在一定的局限性。为进一步提升PINN模型的求解精度，提出了一种基于残差网络（ResNet）结构改进的PINN模型（ResNet-PINN），详细阐述了ResNet-PINN基本原理和数值计算流程，并将其应用于核领域的中子扩散和输运方程的求解。实验验证表明，ResNet-PINN将堆芯中子扩散方程的求解精度提高了2~10倍，输运方程的求解精度提高了3~6倍，有效解决了PINN模型面临的求解精度局限性问题。
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关键词:
- 物理信息神经网络（PINN） /
- 残差网络（ResNet） /
- 中子扩散方程 /
- 中子输运方程
Abstract: As a deep learning method integrating physical knowledge, the Physics-Informed Neural Network (PINN) has certain limitations in terms of the accuracy of problem-solving. To further enhance the solution accuracy of the PINN model, an improved PINN model based on the Residual Network (ResNet) structure (ResNet-PINN) is proposed. The basic principle and numerical calculation process of ResNet-PINN are elaborated in detail, and it is applied to the solution of neutron diffusion and transport equations in the nuclear field. Experimental validation has shown that ResNet-PINN improves the solution accuracy of the reactor core neutron diffusion equation by a factor of 2 to 10 times, and enhances the solution accuracy of the transport equation by a factor of 3 to 6 times., effectively solving the solution accuracy limitations faced by the PINN model. -
图 2 ResNet-PINN求解稳态中子扩散方程的解
Figure 2. Solution of the Steady-State Neutron Diffusion Equation by ResNet-PINN
图 3 ResNet-PINN在不同时间求解方程的预测值与真实值对比图
Figure 3. Comparison of Predicted Values and True Values for Solving Equations at Different Times Using ResNet-PINN
表 1 平板几何机器学习损失函数及样本生成方式
Table 1. Loss Function and Sample Generation Method for Machine Learning in Planar Geometry
损失函数来源类型 约束形式 样本域 样本点个数 生成方式 控制方程 式（5） $ -b\le x\le b,-1\le \mu \le 1 $ 9000 LHS抽样分布 边界条件 $ F(x,\mu {)}^{\prime }\ge 0 $ $ -b\le x\le b,-1\le \mu \le 1 $ 9000 LHS抽样分布 $ x=b,-1\le \mu \le 0,F(x,\mu {)}^{\prime }=0 $ $ x=b,-1\le \mu \le 0 $ 200 等间距分布 $ x=-b,0\le \mu \le 1,F(x,\mu {)}^{\prime }=0 $ $ x=-b,0\le \mu \le 1 $ 200 等间距分布 原函数定解约束 $ \mu =-1,{F}_{0}(x,-1)=0 $ $ -b\le x\le b,\mu =-1 $ 200 等间距分布 特征值约束 $ {F}_{0}(0,1{)}^{\prime }=0.2 $ $ x\in \left\{0,0\right\} $ 1 固定 $ {F}_{0}(0,-1{)}^{\prime }=0.2 $ $ \mu \in \left\{-1,1\right\} $ 1 固定 
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表 2 球几何机器学习损失函数与样本生成方式
Table 2. Loss Function and Sample Generation for Machine Learning in Spherical Geometry
损失函数来源类型 约束形式 样本域 样本点个数 生成方式 控制方程 式（7） $ 0\le h\le R,-1\le \mu \le 1 $ 3000 LHS抽样分布 边界条件 $ F(h,\mu {)}^{\prime }\ge 0 $ $ 0\le h\le R,-1\le \mu \le 1 $ 3000 LHS抽样分布 $ h=R,-1\le \mu \le 0,F(h,\mu {)}^{\prime }=0 $ $ h=R,-1\le \mu \le 0 $ 100 等间距分布 原函数定解约束 $ \mu =-1,{F}_{0}(h,-1)=0 $ $ 0\le h\le R,\mu =-1 $ 100 等间距分布 特征值约束 $ {F}_{0}(0,\mu {)}^{\prime }=0.5 $ $ h=0,-1\le \mu \le 1 $ 100 等间距分布
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