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Physics-Informed Neural Network Method for Forward and Backward Advection-Dispersion Equations
Water Resources Research ( IF 5.4 ) Pub Date : 2021-06-24 , DOI: 10.1029/2020wr029479
QiZhi He 1 , Alexandre M. Tartakovsky 1, 2
Affiliation  

We propose a discretization-free approach based on the physics-informed neural network (PINN) method for solving the coupled advection-dispersion equation (ADE) and Darcy flow equation with space-dependent hydraulic conductivity urn:x-wiley:00431397:media:wrcr25398:wrcr25398-math-0001. In this approach, urn:x-wiley:00431397:media:wrcr25398:wrcr25398-math-0002, hydraulic head, and concentration fields are approximated with deep neural networks (DNNs). We assume that K(x) is given by its values on a grid, and we use these values to train the K DNN. The head and concentration DNNs are trained by minimizing the residuals of the flow equation and ADE and using the initial and boundary conditions as additional constraints. The PINN method is applied to one- and two-dimensional forward ADEs, where its performance for various Péclet numbers (Pe) is compared with the analytical and numerical solutions. We find that the PINN method is accurate with errors of less than 1% and outperforms some conventional discretization-based methods for large Pe. Next, we demonstrate that the PINN method remains accurate for the backward ADEs, with the relative errors in most cases staying under 5% compared to the reference concentration field. Finally, we show that when available, the concentration measurements can be easily incorporated in the PINN method and significantly improve (by more than 50% in the considered cases) the accuracy of the PINN solution of the backward ADE.

中文翻译:

前向和后向对流-色散方程的物理信息神经网络方法

我们提出了一种基于物理信息神经网络 (PINN) 方法的无离散化方法,用于求解耦合对流-弥散方程 (ADE) 和具有空间相关水力传导率的达西流动方程骨灰盒:x-wiley:00431397:媒体:wrcr25398:wrcr25398-math-0001。在这种方法中,骨灰盒:x-wiley:00431397:媒体:wrcr25398:wrcr25398-math-0002使用深度神经网络 (DNN) 来近似 、水头和浓度场。我们假设K ( x ) 由它在网格上的值给出,我们使用这些值来训练K DNN。通过最小化流动方程和 ADE 的残差并使用初始和边界条件作为附加约束来训练头部和浓度 DNN。PINN 方法应用于一维和二维前向 ADE,其中它对各种 Péclet 数的性能(Pe ) 与解析解和数值解进行比较。我们发现 PINN 方法是准确的,误差小于 1%,并且在处理大Pe 时优于一些传统的基于离散化的方法。接下来,我们证明 PINN 方法对于反向 ADE 仍然准确,与参考浓度场相比,大多数情况下的相对误差保持在 5% 以下。最后,我们表明,当可用时,浓度测量可以很容易地合并到 PINN 方法中,并显着提高(在所考虑的情况下超过 50%)后向 ADE 的 PINN 解决方案的准确性。
更新日期:2021-07-22
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