Research on Lattice Boltzmann Solution of Generalized Convection Diffusion Equation Based on Physical Fusion Neural Network
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摘要: 为提高深度学习方法的网络复用性，构建一种适应于不同控制方程和不同物性参数条件的深度网络模型，本研究提出了基于物理融合神经网络的格子Boltzmann方法（PFNN-LBM）。在格子Boltzmann框架下建立了不同特征控制方程的统一格式离散速度Boltzmann方程，并使用单一网络的参数化物理信息约束神经网络求解，可以在一次训练后同时求解不同形式和不同物理参数的控制方程。为测试PFNN-LBM的准确性和适应性，选取了四种类型的宏观方程开展预测分析，包括扩散方程、非线性导热方程、Sine-Gordon方程和Burgers-Fisher方程，同时测试了不同物理参数条件的预测性能并对双群中子扩散问题进行了测试。计算结果表明，所提出的PFNN-LBM可以在一次训练后高精度地求解不同形式和不同物理参数的控制方程。这项工作可以为高效灵活地求解不同类型的方程提供一个新的框架，对于工程应用，这项工作在多物理场耦合计算方面可能具有突出优势。
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关键词:
- 物理信息约束神经网络 /
- 格子Boltzmann /
- 深度学习 /
- 非线性对流扩散方程
Abstract: This paper proposes the physics fusion neural network based lattice Boltzmann method (PFNN-LBM) for solving the nonlinear convection-diffusion equations. The unified discrete-velocity Boltzmann equation for equations with different characteristics is established under lattice Boltzmann method, and solved using the parameterize physics–informed neural network with a single network. The PFNN-LBM can simultaneously solve governing equations with different forms and different physical parameters within one single training. Four types of equations are considered to verify the accuracy and adaptability of proposed PFNN-LBM, including diffusion equation, nonlinear heat conduct equation, Sine–Gordon equation, and Burgers–Fisher equation, with different physical parameters. The two-group neutron diffusion equations are also tested. The calculation results show that the proposed PFNN-LBM can solve equations for different forms and physical parameters with high accuracy, and only one training is required. This work can provide a novel framework for solving different types of equations efficiently and flexibly, and for engineering application, this work may have outstanding advantages in multi-physics coupling calculations. -
图 2 PFNN-LBM对于扩散问题在t=0.25、0.50 s和0.75 s的全场中子注量率分布及误差分布
Figure 2. Neutron Flux and Error Distributions of PFNN-LBM for Diffusion Problems at t=0.25 s, 0.50 s, and 0.75 s
图 3 PFNN-LBM对于扩散问题在t=0.25、0.50 s和0.75 s沿对角线的中子注量率分布及误差预测结果
Figure 3. Neutron Flux and Error Distributions along Diagonal Lines of PFNN-LBM for Diffusion Problems at t=0.25 s, 0.50 s, and 0.75 s
图 4 PFNN-LBM对于非线性导热问题在t=0.25、0.50 s和0.75 s的全场中子注量率分布及误差分布
Figure 4. Neutron Flux and Error Distributions of PFNN-LBM for Nonlinear Heat Conduction Problems at t=0.25 s, 0.50 s, and 0.75 s
图 5 PFNN-LBM对于非线性导热问题在t=0.25、0.50 s和0.75 s沿对角线的中子注量率分布及误差预测结果
Figure 5. Neutron Flux and Error Distributions along Diagonal Lines of PFNN-LBM for Nonlinear Heat Conduction Problems at t=0.25 s, 0.50 s, and 0.75 s
图 6 PFNN-LBM对于Sine-Gordon问题在t=0.25、0.50 s和0.75 s的全场中子注量率分布及误差分布预测结果
Figure 6. Neutron Flux and Error Distributions of PFNN-LBM for Sine-Gordon Problems at t=0.25 s, 0.50 s, and 0.75 s
图 7 PFNN-LBM对于Sine-Gordon问题在t=0.25、0.50 s和0.75 s沿对角线的中子注量率分布及误差预测结果
Figure 7. Neutron Flux and Error Distributions along Diagonal Lines of PFNN-LBM for Sine-Gordon Problems at t=0.25 s, 0.50 s, and 0.75 s
图 8 PFNN-LBM对于Burgers-Fisher问题在t = 0.25、0.50 s和0.75 s的全场中子注量率分布及误差分布预测结果
Figure 8. Neutron Flux and Error Distributions of PFNN-LBM for Burgers-Fisher Problems at t = 0.25 s, 0.50 s, and 0.75 s
图 9 PFNN-LBM对于Burgers-Fisher问题在t=0.25、0.50 s和0.75 s沿对角线的中子注量率分布及误差预测结果
Figure 9. Neutron Flux and Error Distributions along Diagonal Lines of PFNN-LBM for Burgers-Fisher Problems at t=0.25 s, 0.50 s, and 0.75 s
图 10 PFNN-LBM对于不同扩散系数下扩散问题沿对角线的中子注量率分布及误差预测结果
Figure 10. Neutron Flux and Error Distributions along Diagonal Lines of PFNN-LBM for Diffusion Problems under Different Diffusion Coefficients
图 11 快群中子注量率全局预测结果对比
Figure 11. Fast Group Neutron Flux Global Prediction Results Comparison
图 12 热群中子注量率全局预测结果对比
Figure 12. Thermal Group Neutron Flux Global Prediction Results Comparison
图 13 PINN计算不同方程结果分别及对比
Figure 13. Different Equations' Results, Calculate using PINNs and Compare
表 1 不同方程离散速度Boltzmann参数
Table 1. Discrete Velocity Boltzmann Parameters of Different Equations
参数 扩散
方程非线性
导热方程Sine-Gordon
方程Burgers-Fisher
方程B0 0 0 0 2 C11 0 0 0 36/5 C0 0 0 1 0 D11 1 0 1 1 D0 0 1 0 0 F0 0 0 −1 0 F1 1 1 0 1 F2 0 −1 0 0 F3 0 0 0 −1 g0 1 1 0 1 g1 0 0 1 0 
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表 2 PFNN-LBM在不同问题预测中的L2误差 cm–2·s–1
Table 2. L2 Error of PFNN-LBM in Different Problems
方程类型 t=0.25 s t=0.50 s t=0.75 s 扩散方程 1.7×10−3 1.9×10−3 2.0×10−3 非线性导热方程 1.3×10−3 1.3×10−3 1.3×10−3 Sine-Gordon方程 3.6×10−3 7.7×10−3 1.1×10−3 Burgers-Fisher方程 1.8×10−3 9.8×10−3 6.6×10−3
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表 3 沿对角线的中子注量率绝对误差对比
Table 3. Comparison of the Absolute Error of the Fluxes along Diagonal Lines
方程类型 PFNN-LBM/(cm–2·s–1) PINN/(cm–2·s–1) 扩散方程 4.35×10−3 9.990×10−4 非线性导热方程 2.96×10−3 4.796×10−3 Sine-Gordon方程 4.46×10−3 9.918×10−4 Burgers-Fisher方程 1.83×10−3 2.452×10−3
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