Prediction of numerical homogenization using deep learning for the Richards equation

Abstract

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.

32 pages, submitted to Journal of Computational and Applied Mathematics

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• Multicontinuum homogenization and its relation to nonlocal multicontinuum theories
• Hybrid explicit-implicit learning for multiscale problems with time dependent source
• Physics-informed neural networks for learning the homogenized coefficients of multiscale elliptic equations
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Cited by in corpus (2)

• Constraint Energy Minimizing Generalized Multiscale Finite Element Method for multi-continuum Richards equations
• Convolutional Surrogate for 3D Discrete Fracture-Matrix Tensor Upscaling