Physics‐informed neural network applied to surface‐tension‐driven liquid film flows
Physics‐informed neural network applied to surface‐tension‐driven liquid film flows
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DOI：
10.1002/fld.5093
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发表时间：
2021-09
影响因子：
1.8
通讯作者：
Y. Nakamura;Suguru Shiratori;R. Takagi;Michihiro Sutoh;I. Sugihara;H. Nagano;K. Shimano
中科院分区：
文献类型：
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作者：
Y. Nakamura;Suguru Shiratori;R. Takagi;Michihiro Sutoh;I. Sugihara;H. Nagano;K. Shimano
最近由Raissi等人提出的物理信息神经网络（PINN）[J Comput Phys. 2019;378：686-707]被应用于液膜流动的偏微分方程（PDE）。所考虑的偏微分方程是由于拉普拉斯压力引起的厚度分布h（x，t）$$ h\left（x，t\right）$$的时间演化，其中包含四阶空间导数和四阶非线性项。即使对于这样的偏微分方程，它被证实，PINN可以预测的解决方案具有足够的精度。然而，在训练收敛性和解的准确性方面需要改进。浮点数的精度是当前PDE的关键问题。当使用单精度浮点数执行计算时，由于丢失有效位，优化将终止。自动微分的计算支配训练所需的计算时间，并且随着导数阶数的增加而呈指数增长。通过将原始的四阶单偏微分方程分解为低阶耦合偏微分方程，大大减少了每次训练迭代的计算时间。训练数据的采样密度也显著影响训练收敛性。对于本研究中考虑的问题，通过允许训练数据的采样密度在较早的时间范围内更大来获得改进的收敛性，在较早的时间范围内发生厚度的快速平坦化。
A physics‐informed neural network (PINN), which has been recently proposed by Raissi et al. [J Comput Phys. 2019;378:686–707], is applied to the partial differential equation (PDE) of liquid film flows. The PDE considered is the time evolution of the thickness distribution h(x,t)$$ h\left(x,t\right) $$ owing to the Laplace pressure, which involves fourth‐order spatial derivative and fourth‐order nonlinear term. Even for such a PDE, it is confirmed that the PINN can predict the solutions with sufficient accuracy. Nevertheless, some improvements are needed in training convergence and accuracy of the solutions. The precision of floating‐point numbers is a critical issue for the present PDE. When the calculation is executed with a single precision floating‐point number, the optimization is terminated due to the loss of significant digits. Calculation of the automatic differentiation dominates the computational time required for training and becomes exponentially longer with increasing order of derivatives. By splitting the original fourth‐order single PDE into lower‐order coupled PDEs, the computational time for each training iteration is greatly reduced. The sampling density of training data also significantly affects training convergence. For the problem considered in this study, improved convergence was obtained by allowing the sampling density of training data to be greater in earlier time ranges, where the rapid flattening of the thickness occurs.