Solving Equations With Discontinuous Solutions Based on the AGA-PINNs Method
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摘要: 物理信息神经网络(physics-informed neural networks, PINNs)是求解偏微分方程的重要工具，在偏微分方程的数值求解中，具有间断解的方程是目前的研究难题，PINNs及其现有的改进算法通常无法捕捉到间断的特性，然而在流体力学领域需要考虑许多间断问题. 针对PINNs处理具有间断解的方程的不足，本文提出了自适应梯度消灭的物理信息神经网络(adaptive gradient-annihilated PINNs, AGA-PINNs)方法来求解具有间断解的Burgers方程与Allen-Cahn方程. 该方法利用与梯度相关的权重函数来优化损失函数，并在训练过程中根据当前的残差分布动态调整训练点，以逐步强化模型对间断区域的学习能力. 算例结果表明，AGA-PINNs方法比传统的PINNs方法、gPINNs方法、GA-PINNs方法、高阶DG方法等对于物理量的预测精度有显著提升，在求解Burgers方程与Allen-Cahn方程时均方误差降低了大约一个数量级，准确地再现了Burgers方程中冲击波的特征与Allen-Cahn方程中相分离现象.
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关键词:
- AGA-PINNs /
- Burgers方程 /
- Allen-Cahn方程 /
- 自适应方法
Abstract: The physical information neural networks (PINNs) are an important tool for solving partial differential equations. In the numerical solution of partial differential equations, equations with discontinuous solutions are currently a research challenge. The PINNs and their existing improved algorithms often cannot capture the characteristics of discontinuities. However, many discontinuous problems need to be considered in the field of fluid mechanics. In response to the shortcomings of the PINNs in handling equations with discontinuous solutions, an adaptive gradient-annihilated PINNs (AGA-PINNs) method was proposed to solve the Burgers equations and the Allen-Cahn equations with discontinuous solutions. The gradient related weight functions were utilized to optimize the loss function, and the training points were dynamically adjusted based on the current residual distribution during the training process to gradually enhance the model's learning ability in discontinuous regions. The experimental results show that, the AGA-PINNs method significantly improves the accuracy of predicted physical quantities compared to the traditional PINNs, the gradient-enhanced physics-informed neural networks (gPINNs), the gradient-annihilated physics-informed neural networks (GA-PINNs) method, and the high-order deep Galerkin (DG)method. In solving the Burgers equation and the Allen-Cahn equation, the mean square errors decrease by approximately 1 order of magnitude, accurately reproducing the characteristics of shock waves in the Burgers equation and the phase separation phenomena in the Allen-Cahn equation.-
Key words:
- AGA-PINNs /
- Burgers equation /
- Allen-Cahn equation /
- adaptive method
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图 2 不同采样方法生成256个训练点的可视化结果
Figure 2. Visualization results of 256 training points generated with different sampling methods
图 3 不同βu值时AGA-PINNs预测物理量的均方误差
Figure 3. δMSE of predicting physical quantities using AGA-PINNs with different βu values
图 4 Burgers方程精确解
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.
Figure 4. The exact solution of the Burgers equation
图 5 求解Burgers方程初始训练后PDE残差
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.
Figure 5. PDE residuals after initial training in solving the Burgers equation
图 6 不同m值AGA-PINNs求解Burgers方程均方误差
Figure 6. δMSE in solving the Burgers equation using AGA PINNs with different m values
图 7 m=30时AGA-PINNs求解Burgers方程预测结果
Figure 7. Predicting results of AGA-PINNs with m=30 for solving Burgers equation
图 8 不同采样方法AGA-PINNs求解Burgers方程预测结果
Figure 8. Predicted results of the Burgers equation using the AGA-PINNs with different sampling methods
图 10 求解Allen-Cahn方程初始训练后PDE残差
Figure 10. PDE residuals after initial training in solving the Allen-Cahn equation
图 11 不同m值AGA-PINNs求解Allen-Cahn方程均方误差
Figure 11. δMSE in solving the Allen-Cahn equation using the AGA-PINNs with different m values
图 12 m=30时AGA-PINNs求解Allen-Cahn方程预测结果
Figure 12. Predicted results with the AGA-PINNs for m=30 in solving the Allen-Cahn equation
图 13 不同方法求解Burgers方程的Riemann问题的预测结果
Figure 13. Prediction results of the Burgers equation for the Riemann problem solved with different methods
图 14 m=30时AGA-PINNs求解Bugers方程的Riemann问题的预测结果
Figure 14. Predicting results of AGA-PINNs for m=30 in solving the Burgers equation of the Riemann problem
表 1 不同方法求解Burgers方程损失及δMSE
Table 1. Losses and δMSE in solving the Burgers equation with different methods
method loss δMSE GA-PINNs 1.06×10-6 6.23×10-5 PINNs 1.49×10-5 8.10×10-4 gPINNs 7.48×10-5 6.01×10-3 GAN 4.82×10-6 9.17×10-5 PINNsFormer 9.58×10-6 2.62×10-4 high-order DG - 7.99×10-5 
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表 2 不同m值AGA-PINNs求解Burgers方程损失及δMSE
Table 2. Losses and δMSE in solving the Burgers equation with the AGA-PINNs for different m values
number of adaptive points loss δMSE 10 1.08×10-6 7.13×10-5 20 6.06×10-7 3.19×10-6 30 3.71×10-7 1.31×10-6 40 8.98×10-7 8.02×10-6 50 1.03×10-6 5.55×10-5
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表 3 不同方法求解Allen-Cahn方程损失及δMSE
Table 3. Losses and δMSE in solving the Allen-Cahn equation with different methods
method loss δMSE GA-PINNs 9.18×10-6 7.88×10-4 PINNs 8.23×10-5 2.37×10-3 gPINNs 1.75×10-5 8.05×10-4 GAN 1.92×10-5 9.39×10-4 PINNsFormer 6.46×10-5 1.69×10-3 high-order DG - 3.22×10-4
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表 4 不同m值AGA-PINNs求解Allen-Cahn方程损失及δMSE
Table 4. Losses and δMSE in solving the Allen-Cahn equation with the AGA-PINNs for different m values
number of adaptive points loss δMSE 10 1.88×10-6 3.19×10-4 20 8.96×10-7 2.57×10-4 30 2.02×10-7 3.52×10-5 40 6.47×10-7 1.01×10-4 50 9.25×10-7 2.41×10-4
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表 5 不同m值AGA-PINNs求解Burgers方程的Riemann问题的损失及δMSE
Table 5. Losses and δMSE in solving the Burgers equation of the Riemann problem with the AGA-PINNs for different m values
number of adaptive points loss δMSE 10 7.49×10-6 8.12×10-4 20 1.18×10-6 6.44×10-4 30 4.26×10-7 7.19×10-5 40 6.58×10-7 9.83×10-5 50 9.32×10-7 3.57×10-4
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