Study on Neutron Diffusion Calculation Method Based on hp-VPINN
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摘要: 先进的反应堆模拟技术需要基于较少的实际探测数据反演全堆关键参数。针对这一需要，本文基于具有高阶多项式域分解功能的变分残差物理信息神经网络（hp-VPINN）构建计算模型，用于正向、反向求解中子扩散方程。该模型使用神经网络作为试函数，并将其代入中子扩散方程形成变分残差作为损失函数进行梯度下降。为了提高求解精度及效率，本文还根据中子扩散方程的物理特性提出了有效增殖系数智能搜索与反演等创新型关键技术，并基于鲸鱼优化算法（WOA）实现了神经网络超参数自优化。最后通过多个算例进行验证，结果表明该方法在具有较高精度的同时，实现了较低的训练数据依赖，为先进反应堆模拟技术提供了一条少量输入数据且较高精度输出的中子扩散求解途径。
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关键词:
- 变分残差物理信息神经网络（hp-VPINN） /
- 鲸鱼优化算法（WOA） /
- 中子扩散方程 /
- 有效增殖系数 /
- 反应堆模拟技术
Abstract: Advanced reactor simulations require the inversion of key parameters of the whole reactor based on less actual detection data. To meet this need, this paper constructs a computational model based on variational residual physics-informed neural network (hp-VPINN) with high-order polynomial domain decomposition function, which is used to solve the neutron diffusion equation forward and backward. The model uses neural network as trial function, and substitutes it into the neutron diffusion equation to form variational residual as loss function for gradient descent. In order to improve the accuracy and efficiency of the solution, this paper also proposes some innovative key technologies such as effective multiplication factor intelligent search and inversion based on the physical characteristics of neutron diffusion equation, and realizes the self-optimization of neural network hyperparameters based on whale optimization algorithm (WOA). Finally, several examples are verified, and the results show that the method has high accuracy and low dependence on training data, providing a solution with less training data and higher accuracy output for advanced reactor simulations. -
图 1 基于鲸鱼优化算法的神经网络超参数优化流程
Figure 1. Workflow Diagram of Neural Network Hyperparameter Optimization Based on Whale Optimization Algorithm
图 2 相对误差分布以及损失函数下降趋势
Figure 2. Relative Error Distribution and Loss Function Decline Trend
图 3 子域中使用勒让德多项式作为权函数时的求解结果
Figure 3. Solution Results Using Legendre Polynomial as Weight Function in Subdomain
图 4 求解域分区加密前后的相对误差分布
Figure 4. Relative Error Distribution before and after Mesh Refinement in the Solution Domain
图 5 非均匀介质内中子扩散方程的求解结果与相对误差
Figure 5. Solution Results and Relative Error of Neutron Diffusion Equation in Inhomogeneous Media
图 6 一维瞬态中子扩散方程求解结果
Figure 6. Solution Results of 1D Transient Neutron Diffusion Equation
图 10 归一化求解结果与参考解对角线分布
Figure 10. Diagonal Distribution of Normalized Solution Results and Reference Solution
表 1 分区截面参数
Table 1. Zonal Cross-section Parameters
区域 g ${D_g}/{\text{cm}}$ $ \mathrm{\mathit{\Sigma}}_{\mathrm{a},g}/\text{c}\text{m}^{-1} $ $ v\mathit{\mathrm{\mathit{\Sigma}}}_{\mathrm{f},g}/\text{c}\text{m}^{-1} $ ${{{\varSigma}} _{1 \to 2}}/{\text{c}}{{\text{m}}^{ - 1}}$ ①,② 1 1.4 0.01 0.007 0.01 ①,② 2 0.4 0.015 0.2 ③ 1 1.3 0.008 0.003 0.01 ③ 2 0.5 0.05 0.06 
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