**5. Conclusions**

In this paper, we develop a method using deep learning techniques for fast computation of GMsFEM discretizations. Given a particular permeability field, the main ingredients of GMsFEM, including the multiscale basis functions and coarse-scale matrices, are computed in an offline stage by solving local problems. However, when one is interested in calculating GMsFEM discretizations for multiple choices of permeability fields, repeatedly formulating and solving such local problems could become computationally expensive or even infeasible. Multi-layer networks are used to represent the nonlinear mapping from the fine-scale permeability field coefficients to the multiscale basis functions and the coarse-scale parameters. The networks provide a direct fast approximation of the GMsFEM ingredients in a local neighborhood for any online permeability fields, in contrast to repeatedly formulating and solving local problems. Numerical results are presented to show the performance of our proposed method. We see that, given sufficient samples of GMsFEM discretizations for supervised training, deep neural networks are capable of providing reasonably close approximations of the exact GMsFEM discretization. Moreover, the small consistency error provides good approximations of multiscale solutions.

**Author Contributions:** The authors have contributed equally to the work.

**Funding:** The research of Eric Chung is partially supported by the Hong Kong RGC General Research Fund (Project numbers 14304217 and 14302018) and CUHK Faculty of Science Direct Grant 2017-18. YE would like to acknowledge the support of Mega-grant of the Russian Federation Government (N 14.Y26.31.0013) and the partial support from NSF 1620318.

**Conflicts of Interest:** The authors declare no conflict of interest.
