课程-迁移学习物理信息神经网络用于曲面长时间对流扩散行为模拟
doi: 10.21656/1000-0887.440320
Curriculum-Transfer-Learning-Based Physics-Informed Neural Networks for Simulating Long-Term-Evolution Convection-Diffusion Behaviors on Curved Surfaces
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摘要:
物理信息神经网络(physics-informed neural networks, PINN)将物理先验知识编码到神经网络中，减少了神经网络对于数据量的需求. 但是对于时间相关偏微分方程的长时间问题，传统PINN稳定性差，甚至难以求得有效解. 针对此问题，该文发展了一种基于课程学习和迁移学习的物理信息神经网络(curriculum-transfer-learning-based physics-informed neural networks, CTL-PINN). 该方法的主要思想是：将长时间历程模拟问题转化为该时间域内多个短时间历程模拟问题，引入课程学习的思想，由简到难，通过PINN在小时间段区域内训练，而后逐渐增大所求解的时域范围；进而引入迁移学习方法，在课程学习的基础上进行时域上的迁移，逐步采用PINN进行求解，从而实现曲面上对流扩散行为的长时间模拟. 该文将此CTL-PINN与非本征的曲面算子处理技术相结合，用于复杂曲面上长时间对流扩散行为的模拟，并通过多个数值算例验证了CTL-PINN的有效性和鲁棒性.
Abstract:Physics-informed neural networks (PINNs) encode prior physical knowledge into neural networks, alleviating the need for extensive data volume within the network. However, for long-term problems involving time-dependent partial differential equations, the traditional PINN exhibits poor stability and struggles to obtain effective solutions. To address this challenge, a novel physics-informed neural network based on curriculum learning and transfer learning (CTL-PINN) was introduced. The main idea of this method is to transform the problem of long-term course simulation into multiple short-term course simulation problems within this time domain. Under the concept of curriculum learning, and step by step from simpleness to difficulty, the scope of the time domain to be solved was gradually expanded by training the PINN within small time quanta. Furthermore, the transfer learning method was adopted to transfer across the time domain based on the curriculum learning, and the PINN was gradually employed for solution, thus to achieve long-term simulation of convection-diffusion behaviors on curved surfaces. The CTL-PINN was combined with the extrinsic surface operator processing technology to simulate long-term convection-diffusion behaviors on complex surfaces, and the effectiveness and robustness of the improved physics-informed neural network were verified through multiple numerical examples.
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Key words:
- physics-informed neural network /
- curriculum learning /
- transfer learning /
- convection-diffusion /
- surface /
- long-term evolution
edited-byedited-by1) (我刊青年编委傅卓佳来稿) -
图 4 CTL-PINN与PINN的误差随时域变化
Figure 4. The errors of the CTL-PINN and the PINN changing with the time domain
图 5 CTL-PINN前8步损失函数收敛曲线
Figure 5. Convergence curves of loss functions for the 1st 8 steps of the CTL-PINN
图 7 不同额外监督学习点数量下的误差随时域[0, t]的变化
Figure 7. The errors changing over time domain [0, t] with different numbers of extra supervised learning points
图 8 不同额外监督学习点数量下两点的预测值以及精确解
Figure 8. The exact and predicted values under different numbers of extra supervised learning points
图 9 t=19 s时两个曲面的绝对误差分布
Figure 9. The distributions of absolute errors on the 2 surfaces at t=19 s
表 1 不同神经网络结构下的L2误差
Table 1. Errors L2 under different neural network structures
10 neurons 30 neurons 50 neurons 8 layers 2.76×10-4 2.32×10-4 2.86×10-4 13 layers 7.53×10-4 2.85×10-4 3.58×10-4 15 layers 6.43×10-4 5.15×10-4 3.24×10-4 17 layers 4.33×10-3 3.48×10-4 6.71×10-4 
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表 2 以步长10 s进行迁移学习的L2误差
Table 2. Errors L2 of the transfer learning with a step size of 10 s
time domain T/s [0, 80] [10, 90] [20, 100] [30, 110] [40, 120] L2 7.01×10-3 7.61×10-3 9.67×10-3 1.18×10-2 1.05×10-2
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表 3 复杂曲面上不同Nsp下模型L2误差
Table 3. The L2 errors of the model at different Nsp values on the complex surface
Nsp 0 600 1 200 1 500 1 800 2 400 3 000 bretzel 2 1.04×10-3 1.14×10-3 1.23×10-3 4.96×10-4 2.62×10-3 3.61×10-3 3.48×10-4 CDP 5.27×10-4 8.83×10-4 1.41×10-3 1.68×10-3 3.54×10-3 4.93×10-4 7.46×10-3
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表 4 复杂曲面以步长3 s进行迁移学习的L2误差
Table 4. The L2 errors of the transfer learning on the complex surface with a step size of 3 s
time domain T/s [0, 4] [3, 7] [6, 10] [9, 13] [12, 16] [15, 19] bretzel 2 1.04×10-3 1.14×10-3 1.23×10-3 4.96×10-4 2.62×10-3 3.61×10-3 CDP 5.27×10-4 8.83×10-4 1.41×10-3 1.68×10-3 3.54×10-3 4.93×10-4
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