A Neural-network-based multiscale Hybridizable Discontinuous Galerkin method for solving PDEs in porous media

Abstract: We develop a neural-network-accelerated multiscale hybridizable discontinuous Galerkin method for elliptic problems with heterogeneous coefficients. The method preserves the standard MsHDG local-to-global structure: fine-scale HDG problems on coarse blocks define discrete Dirichlet-to-Neumann operators, which are assembled through the standard MsHDG global skeleton equations. To reduce the cost of constructing these local operators, we train a neural network on coefficient fields defined on a reference block and use the predicted operators in place of repeated fine-scale local solves. The numerical experiments assess both the accuracy and online efficiency of the resulting NN-MsHDG method. For two-dimensional binary permeability fields with moderate contrast, the neural method reproduces the standard MsHDG solution with relative modeling errors of a few percent while reducing the total online computational cost by factors of approximately 5 to 16, depending on the coarse trace dimension. In the high-contrast regime, the chosen polynomial coarse trace spaces already yield substantial discretization errors in the standard MsHDG method, indicating the need for more effective coarse spaces, such as coefficient-adapted spectral trace spaces. In addition, the learned local operators introduce modeling errors that become severe as the trace space is enriched. These results demonstrate the potential of neural surrogates for accelerating multiscale HDG computations while also highlighting the need for improved operator representations and a better understanding of error amplification in high-contrast problems.