Classification and determination of sub-grid effects of shallow water flow in porous media using machine learning

Porous shallow water equations have been widely used to model urban inundation since it allows coarse-grid simulation of fluid-structure interaction problems. However, the trade-off between accuracy and efficiency by the coarse-grid model induces loss of sub-grid scale information such as bottom friction, drag force, and turbulence. The present study utilized a machine-learning technique to recover such sub-grid effects. Firstly, reference solutions were obtained by solving nonlinear shallow water equations over the fine resolution meshes and compared with coarse-grid solutions of porous shallow water equations. Secondly, sub-grid effects were classified and modeled from the difference between the reference and coarse-grid solutions using the Gaussian process. Each sub-grid model was formulated as a function of coarse-grid information, in which proper parameterization was considered. Especially for case of isotropic porosity, the trained model was consistent with that obtained by the homogenization method. Extensive numerical experiments were performed to validate the present method and showed good agreements.