Meshless computing method for thermal conductivity problem based on physical-informed neural networks
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摘要:
建立了物理神经网络（PINNs）求解导热问题的通用框架，描述了三维非稳态问题、初始条件和三类边界条件及曲面边界的处理方法。使用PINNs求解了一个一维导热问题。求解结果与理论解的最大相对误差为0.0017%，平均相对误差为0.0011%。使用一个简化叶片的导热问题作为案例，将PINNs与传统有限元方法进行对比，探究了PINNs不同的网络架构和超参数对结果的影响。对于简化叶片的导热问题，有限元方法求解时间为11.7 s，PINNs平均求解时间为8.96 s，求解结果的最大误差为1.03%，平均误差为0.139%。微调实心叶片的内冷源强度，在训练收敛的PINNs基础上重新采样计算，新的计算收敛时间为1.41 s，证明了PINNs方法具有设计条件微调时的快速计算能力。
Abstract:A general framework for solving thermal conductivity problems by physics-informed neural networks (PINNs) was developed and the treatment methods of three-dimensional unsteady problems, initial conditions, three types of boundary conditions, and surface boundaries were described. A one-dimensional thermal conductivity problem was solved by PINNs. The maximum relative error and average relative error between the solution and the theoretical solution were 0.0017% and 0.0011%, respectively. Using a simplified blade thermal conductivity problem as a case study, PINNs was compared with conventional finite element methods and the effects of different network architectures and hyperparameters of PINNs on the results were explored. The results showed that for the thermal con-ductivity problem of a simplified blade, the finite element method solved the problem in 11.7 s and PINNs solved in 8.96 s, with a maximum error of 1.03% and an average error of 0.139%. The internal cold source intensity of the solid blade was adjusted slightly, and the new convergence time was 1.41 s based on the PINNs training convergence, proving that PINNs method has the ability of fast calculation in case of minor change of the design condition.
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图 1 针对导热问题的PINNs示意图
Figure 1. Schematic diagram of PINNs for thermal conductivity problems
图 3 PINNs求解结果与理论解结果对比
Figure 3. Comparison of PINNs solution results with theoretical solutions
图 6 叶片表面的采样点及外法线向量
Figure 6. Sampling points and external normal vectors of blade surface
图 7 解决实心叶片导热问题的PINNs结构
Figure 7. PINNs structure to solve the thermal conductivity problem of solid blades
图 8 损失函数和相对误差随迭代步数的变化
Figure 8. Loss function and relative error varying with the iteration steps
图 9 PINNs三维温度场求解结果和误差可视化
Figure 9. PINNs three-dimensional temperature field solution results and error visualization
图 10 内冷源微调后PINNs三维温度场求解结果和误差可视化
Figure 10. PINNs three-dimensional temperature field solution results and error visualization after the minor change of the internal cold sourse
表 1 初始化对于计算时间和计算精度的影响
Table 1. Effect of initialization on computation time and accuracy
实验 计算时间/s 迭代步数 最大误差/% 平均误差/% 第1次 11.63 1053 0.27 0.07 第2次 6.86 553 0.31 0.12 第3次 11.58 948 0.41 0.13 第4次 12.97 1036 0.31 0.09 第5次 8.85 664 0.29 0.09 平均值 10.38 1053 0.32 0.10 
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表 2 不同深度及宽度网络的计算收敛时间
Table 2. Convergence time for different depth and width networks
网络宽度 计算收敛时间/s 10 20 40 80 2层 10.30 10.38 20.90 47.44 4层 21.27 25.37 40.75 93.36 6层 22.85 48.97 55.59 144.07
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表 3 不同深度及宽度网络的计算收敛步数
Table 3. Convergence steps for different depth and width networks
网络宽度 计算收敛参数 10 20 40 80 2层 799 1053 1288 1396 4层 796 1237 1505 1598 6层 766 1355 1539 1694
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表 4 不同深度及宽度网络计算结果的最大相对误差
Table 4. Maximum relative error of the results calculated for different depth and width networks
网络宽度 最大相对误差/% 10 20 40 80 2层 0.30 0.32 0.39 0.36 4层 0.36 0.25 0.38 0.32 6层 0.28 0.28 0.23 0.59
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表 5 不同深度及宽度网络计算结果的平均相对误差
Table 5. Average relative error of the results calculated for different depth and width networks
网络宽度 平均相对误差/% 10 20 40 80 2层 0.088 0.096 0.111 0.104 4层 0.104 0.074 0.102 0.091 6层 0.057 0.055 0.062 0.133
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表 6 不同采样点数量下的网络计算时间和计算精度
Table 6. Computation time and accuracy under different number of sampling points
实验 $ {N}_{0} $ $ {N}_{\mathrm{u}\mathrm{d}} $ $ {N}_{\mathrm{s}} $ 平均计算
时间/s平均计算
步数最大
误差/%平均
误差/%1 100 10 30 8.96 711 1.03 0.139 2 1000 50 300 9.21 708 0.66 0.142 3 10000 500 3000 10.30 799 0.30 0.088 4 50000 2300 15000 26.77 1235 0.29 0.076
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