Flow field prediction method integrating residual learning and physics-informed neural network
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摘要:
研究了基于残差神经网络（ResNet） 与物理信息神经网络（PINN）融合的高精度流场预测方法。基于数值模拟数据构建训练集，建立融合残差神经网络的PINN模型（Res-PINN）用于求解圆柱绕流问题。该模型利用流场采样点的速度和压强信息进行训练，以预测不同区域内的流场和压强分布。同时探究了网络结构类型、激活函数、训练集规模、网络层数、神经元数量及损失函数权重对预测结果的影响，寻求网络的优化方法。结果表明：Res-PINN模型能有效重构圆柱绕流场的流速和压强分布，预测精度与直接数值模拟结果高度吻合，物理量相对误差均低于5%；对比分析显示，Res-PINN相较于传统PINN，预测误差均下降超过45%，显著提升了预测精度与稳定性；多种网络超参数对预测效果均有明显影响，优化时需综合考虑并平衡这些参数。
Abstract:A high-precision prediction method for flow field was developed by integrating residual neural network (ResNet) with physics-informed neural network (PINN). Based on numerical simulation data, a PINN model integrating residual neural network (Res-PINN) solved the cylinder flow problem. The model used velocity and pressure data from sampled flow-field points to predict flow-field and pressure distributions in different regions. The influences of network type, activation function, dataset size, network layers, number of neurons and loss function weight on the prediction results were also explored. Results demonstrated that the Res-PINN effectively reconstructed flow-field velocity and pressure distributions. Its prediction accuracy was highly consistent with the direct numerical simulation results. The relative error of physical quantities was less than 5%. Comparative analysis showed that compared with traditional PINN, Res-PINN reduced prediction errors by over 45%, significantly improving the prediction accuracy and stability. Furthermore, multiple network hyperparameters had a significant impact on prediction performance, and optimization should be made to comprehensively consider and balance these parameters.
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图 4 不同时刻x=π/2 m处计算结果对比
Figure 4. Comparison of calculation results at x=π/2 m at different times
图 5 不同时刻x=π/2 m处计算结果相对误差
Figure 5. Relative error of calculation results at x=π/2 m at different times
图 10 Res-PINN与对照数据的计算结果对比（t=0 s）
Figure 10. Comparison of calculation results between Res-PINN and reference data （t=0 s）
图 11 Res-PINN重构结果的相对误差（t=0 s，y/H=0.2）
Figure 11. Relative error of Res-PINN reconstruction results（t=0 s，y/H=0.2）
图 13 不同初始值方程参数随训练步数的变化
Figure 13. Variation of equation parameters of different initial values with training steps
图 14 两类PINN的流场计算结果对比（t=0.2 s）
Figure 14. Comparison of flow field calculation results of two kinds of PINN （t=0.2 s）
图 15 y/H=0.2处计算结果对比（t=0.2 s）
Figure 15. Comparison of calculation results at y/H=0.2 （t=0.2 s）
图 16 y/H=0.5处计算结果对比（t=0.2 s）
Figure 16. Comparison of calculation results at y/H=0.5 （t=0.2 s）
图 17 y/H=0.2处计算结果相对误差对比（t=0.2 s）
Figure 17. Comparison of relative error of calculation results at y/H=0.2 （t=0.2 s）
图 18 y/H=0.5处计算结果相对误差对比（t=0.2 s）
Figure 18. Comparison of relative error of calculation results at y/H=0.5 （t=0.2 s）
图 19 y/H=0.5处不同激活函数计算结果对比（t=0.2 s）
Figure 19. Comparison of calculation results for different activation functions at y/H=0.5 （t=0.2 s）
图 20 y/H=0.5处不同激活函数计算结果相对误差对比（t=0.2 s）
Figure 20. Comparison of relative errors in calculation results for different activation functions at y/H=0.5 （t=0.2 s）
图 21 y/H=0.5处不同训练集计算结果对比（t=0.2 s）
Figure 21. Comparison of calculation results of different training sets at y/H=0.5 （t=0.2 s）
图 22 y/H=0.5处不同训练集计算结果相对误差对比（t=0.2 s）
Figure 22. Comparison of relative error of calculation results of different training sets at y/H=0.5 （t=0.2 s）
图 23 y/H=0.5处不同网络计算结果对比（t=0.2 s）
Figure 23. Comparison of calculation results of different networks at y/H=0.5 （t=0.2 s）
图 24 y/H=0.5处不同网络计算结果相对误差对比（t=0.2 s）
Figure 24. Comparison of relative error of calculation results of different networks at y/H=0.5 （t=0.2 s）
图 25 y/H=0.5处不同网络计算结果对比（t=0.2 s）
Figure 25. Comparison of calculation results of different networks at y/H=0.5 （t=0.2 s）
图 26 y/H=0.5处不同网络计算结果相对误差对比（t=0.2 s）
Figure 26. Comparison of relative error of calculation results of different networks at y/H=0.5 （t=0.2 s）
图 27 y/H=0.5处不同权重计算结果对比（t=0.2 s）
Figure 27. Comparison of calculation results of different weights at y/H=0.5 （t=0.2 s）
图 28 y/H=0.5处不同权重计算结果相对误差对比（t=0.2 s）
Figure 28. Comparison of relative error of calculation results of different weights at y/H=0.5 （t=0.2 s）
表 1 不同初始值方程参数最终收敛值及其误差
Table 1. Final convergence value and error of equation parameters with different initial values
参数 初始值 收敛值 误差/% $ {\lambda }_{1} $ 2 1.0124 1.24 3 1.0166 1.66 3 1.0297 2.97 5 1.5359 53.59 $ {\lambda }_{2} $ 0 0.0102 2.00 −1 0.0098 2.00 2 0.0124 24.00 1 0.0107 7.00 
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表 2 两类PINN计算结果全场平均相对误差
Table 2. Average relative error of two kinds of PINN calculation results in the whole field
模型种类 平均相对误差/% u v p FCNN 1.22256 5.42128 4.91965 Res-PINN 0.53589 3.03291 2.59881
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表 3 不同激活函数的网络计算结果全场平均相对误差
Table 3. Average relative error of the network calculation results in the whole field with different activation functions
激活函数 平均相对误差/% u v p Tanh 0.53589 3.03291 2.59881 ReLU 1.27666 4.63103 5.81088 Sigmoid 1.31649 4.59453 4.44788
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表 4 不同训练集的网络计算结果全场平均相对误差
Table 4. Average relative error of the network calculation results in the whole field with different training sets
训练集 平均相对误差/% u v p 70%数据集 0.52276 2.49547 2.68607 50%数据集 0.53549 3.03786 2.58273 20%数据集 1.02041 5.90285 3.11327 5%数据集 1.38852 12.06042 23.31184
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表 5 不同结构的网络计算结果全场平均相对误差
Table 5. Average relative error of the calculation results of networks with different structures in the whole field
网络结构 平均相对误差/% u v p Net_4 0.81508 3.32581 3.41453 Net_8 0.53549 3.03786 2.58273 Net_16 0.58297 3.01136 3.14837
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表 6 不同神经元数量的网络计算结果全场平均相对误差
Table 6. Average relative error of the networks calculation results in the whole field with different numbers of neurons
每层神经元数 平均相对误差/% u v p 32 0.87378 3.78247 4.46573 64 0.53589 3.03291 2.59881 128 0.44402 2.68027 2.02615 256 0.30140 2.63678 2.24459
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表 7 不同损失函数权重的网络计算结果全场平均相对误差
Table 7. Average relative error of the networks calculation results in the whole field with different the weights of loss functions
$ {\omega }_{{\mathrm{pde}}} $ 平均相对误差/% u v p 0.1 0.42687 2.41145 2.23912 0.5 0.36408 2.77276 2.15901 1 0.53549 3.03786 2.58273 2 0.61921 3.23586 3.26256 5 0.99728 4.17862 3.48449
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