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.

翻译：我们针对非均质非周期介质中非饱和流动的非线性单连续理查兹方程，提出了一种新的粗尺度近似策略，该策略将在线广义多尺度有限元方法（online GMsFEM）与深度学习相结合。该方法的新颖之处在于，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。更具体地，我们利用随机渗透率实现的数据集及计算得到的相关在线多尺度基函数来训练神经网络。通过我们提出的深度学习算法，建立了渗透率场与在线多尺度基函数之间的非线性映射。即，预测得到的在线多尺度基函数以全新方式纳入了理查兹方程的非线性处理，并反映了问题属性随时间的变化。二维模型问题的多项数值实验表明，该技术在预测在线多尺度基函数及求解方面展现出良好性能。