论文标题
使用Richards方程的深度学习对数值均质化的预测
Prediction of numerical homogenization using deep learning for the Richards equation
论文作者
论文摘要
对于通过异质培养基的非饱和流作为不饱和流的非线性理查兹方程，我们利用数值均质化构建了一种新的粗尺度近似算法。这种方法遵循深度神经网络（DNN），以快速并经常计算宏观参数。更具体地说，我们训练神经网络，其训练集由随机渗透性实现和相应的计算宏观靶标（有效的渗透率张量，均匀的刚度矩阵和右侧侧向量）。我们提出的深度学习方案在此类渗透性场和宏观特征之间开发了非线性图，而Richards方程非线性的处理中包括在预测的粗尺度同质化刚度矩阵中，这是一种新颖性。通过二维模型问题中的几个数值测试证明了这种策略的良好性能，以预测宏观特性和解决方案。
For the nonlinear Richards equation as an unsaturated flow through heterogeneous media, we build a new coarse-scale approximation algorithm utilizing numerical homogenization. This approach follows deep neural networks (DNNs) to quickly and frequently calculate macroscopic parameters. More specifically, we train neural networks with a training set consisting of stochastic permeability realizations and corresponding computed macroscopic targets (effective permeability tensor, homogenized stiffness matrix, and right-hand side vector). Our proposed deep learning scheme develops nonlinear maps between such permeability fields and macroscopic characteristics, and the treatment for Richards equation's nonlinearity is included in the predicted coarse-scale homogenized stiffness matrix, which is a novelty. This strategy's good performance is demonstrated by several numerical tests in two-dimensional model problems, for predictions of the macroscopic properties and consequently solutions.
