  Algorithms based on deep neural networks (DNNs) have attracted increasing
attention from the scientific computing community. DNN based algorithms are
easy to implement, natural for nonlinear problems, and have shown great
potential to overcome the curse of dimensionality. In this work, we utilize the
multi-scale DNN-based algorithm (MscaleDNN) proposed by Liu, Cai and Xu (2020)
to solve multi-scale elliptic problems with possible nonlinearity, for example,
the p-Laplacian problem. We improve the MscaleDNN algorithm by a smooth and
localized activation function. Several numerical examples of multi-scale
elliptic problems with separable or non-separable scales in low-dimensional and
high-dimensional Euclidean spaces are used to demonstrate the effectiveness and
accuracy of the MscaleDNN numerical scheme.
