标题： 在线 GMsFEM 的离散化预测，使用深度学习求解 Richards 方程 显示英文标题

标题： Prediction of discretization of online GMsFEM using deep learning for Richards equation

摘要： 我们开发了一种新的粗尺度近似策略，用于非线性单连续介质Richards方程，作为在异质非周期介质上的非饱和流，使用在线广义多尺度有限元方法（online GMsFEM）和深度学习。 这种方法的一个新颖之处在于，通过利用深度神经网络（DNNs）快速且频繁地计算局部在线多尺度基函数。 更准确地说，我们使用随机渗透率实现的训练集和计算的相关在线多尺度基函数来训练神经网络。 我们提出的深度学习算法建立了这种渗透率场与在线多尺度基函数之间的非线性映射。 也就是说，以一种新的方式，预测的在线多尺度基函数包含了Richards方程的非线性处理，并反映了问题属性的任何时间依赖性变化。 在二维模型问题中的多个数值实验表明了该技术的良好性能，体现在在线多尺度基函数的预测以及因此找到解决方案方面。 显示英文摘要

摘要： We develop a new coarse-scale approximation strategy for the nonlinear single-continuum Richards equation as an unsaturated flow over heterogeneous non-periodic media, using the online generalized multiscale finite element method (online GMsFEM) together with deep learning. A novelty of this approach is that local online multiscale basis functions are computed rapidly and frequently by utilizing deep neural networks (DNNs). More precisely, we employ the training set of stochastic permeability realizations and the computed relating online multiscale basis functions to train neural networks. The nonlinear map between such permeability fields and online multiscale basis functions is developed by our proposed deep learning algorithm. That is, in a new way, the predicted online multiscale basis functions incorporate the nonlinearity treatment of the Richards equation and refect any time-dependent changes in the problem's properties. Multiple numerical experiments in two-dimensional model problems show the good performance of this technique, in terms of predictions of the online multiscale basis functions and thus finding solutions.