Reduced Order Modeling for Neutron Transport Equation Based on Operator Inference
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摘要: 为建立瞬态中子输运方程的快速预测模型，本文采用仿射参数化算子推断构建中子输运方程的降阶模型。算子推断通过奇异值分解与构造最优化问题，非侵入式地拟合降阶空间中的动力学方程，同时能够保留原控制方程所描述的物理规律。而仿射参数化结构有效地处理瞬态中子输运方程中常见的时变参数问题，在无需参数空间插值的情况下，实现时变堆芯参数到物理量的快速求解。研究结果表明，基于高保真数据与仿射参数化算子推断构建的降阶模型具有较好的泛化能力，能够对不同时变参数下的瞬态问题进行准确求解。因此，本文构建的降阶模型能够用于高保真中子输运方程的快速预测。Abstract: To establish a real-time prediction model for the time-dependent neutron transport equation, the affine-parametric operator inference is employed to train a reduced-order model of the neutron transport equation. Operator inference, through singular value decomposition and solving optimization problem, non-intrusively fits the operators of the reduced dynamic equations in the subspace while preserving the physical structure described by the original governing equations. The affine-parametric structure effectively addresses the issue of time-varying parameters commonly encountered in time-dependent neutron transport equations, enabling rapid computation from time-varying core parameters to physical quantities without the need for parameter space interpolation. The numerical results show that the reduced-order model based on high-fidelity data and affine-parametric operator inference has good generalization ability, accurately solving transient problems under different time-varying parameters. Therefore, the reduced-order model proposed in this study can be used for real-time prediction of high-fidelity neutron transport equations.
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图 2 计算问题中的时变堆芯插入深度相对值${\mu _1}$与${\mu _2}$
Figure 2. Time-varying Parameters ${\mu _1}$ and ${\mu _2}$ in the Problems
图 5 训练集中降阶状态向量$ \hat{\text{φ}} $中的各阶分量$\hat{\text{φ}}[r_i] $随时间的变化
$\hat{\text{φ}}[r_i] $—从向量${\text{φ}} $取出的第$r_i $个元素
Figure 5. Time-Variant Coefficients of the Reduced State Vector in the Training Set
图 6 不同阶数与正则化系数对模型预测性能的影响
Figure 6. Impact of Different Orders and Regularization Coefficients on Model Prediction Performance
表 1 模型训练与在线预测的计算耗时
Table 1. Computational Cost for Model Training and Online Predictions
计算时间 降阶模型r=11 全阶模型 在线阶段：每步平均求解时间/s 3.3×10−5 0.17 core-hours 在线阶段：每步平均重构时间/s 6.4×10−3 离线阶段：训练时间/s 13.3 core-hours—使用一个核心执行计算工作1个小时 
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表 2 预测功率分布误差
Table 2. Error of Power Distributions
问题 功率分布L2误差/% 功率分布最大误差/% 1 0.02 0.14 2 0.65 3.05 3 0.84 3.19
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