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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

闵建 傅卓佳 郭远

袁镒吾. 关于平面应变和反平面应变复合型裂纹尖端的理想塑性应力场[J]. 应用数学和力学, 1987, 8(11): 1007-1014.
引用本文: 闵建, 傅卓佳, 郭远. 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[J]. 应用数学和力学, 2024, 45(9): 1212-1223. doi: 10.21656/1000-0887.440320
Yuan Yi-wu. On Perfectly Stress Field at a Mixed-Mode Crack Tip under Plane and Anti-Plane Strain[J]. Applied Mathematics and Mechanics, 1987, 8(11): 1007-1014.
Citation: MIN Jian, FU Zhuojia, GUO Yuan. Curriculum-Transfer-Learning-Based Physics-Informed Neural Networks for Simulating Long-Term-Evolution Convection-Diffusion Behaviors on Curved Surfaces[J]. Applied Mathematics and Mechanics, 2024, 45(9): 1212-1223. doi: 10.21656/1000-0887.440320

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
(我刊青年编委傅卓佳来稿)
基金项目: 

国家自然科学基金 12122205

国家自然科学基金 12372196

详细信息
    作者简介:

    闵建(2001—),男,硕士生(E-mail: 231308010033@hhu.edu.cn)

    通讯作者:

    傅卓佳(1985—),男,教授,博士(通讯作者. E-mail: paul212063@hhu.edu.cn)

  • 中图分类号: TP183;O34

Curriculum-Transfer-Learning-Based Physics-Informed Neural Networks for Simulating Long-Term-Evolution Convection-Diffusion Behaviors on Curved Surfaces

(Contributed by FU Zhoujia, M.AMM Youth Editorial Board)
  • 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.

    1)  (我刊青年编委傅卓佳来稿)
  • 图  1  CTL-PINN求解步骤示意图

    Figure  1.  Steps of the CTL-PINN

    图  2  CTL-PINN结构示意图

       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  2.  Structure of the CTL-PINN

    图  3  不同时间步长的训练结果

    Figure  3.  Training results with different time steps

    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  6  不同时刻的绝对误差分布

    Figure  6.  The distributions of absolute errors at different moments

    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
    下载: 导出CSV

    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
    下载: 导出CSV

    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
    下载: 导出CSV

    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
    下载: 导出CSV
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出版历程
  • 收稿日期:  2023-10-25
  • 修回日期:  2024-02-22
  • 刊出日期:  2024-09-01

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