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Algorithms based on deep neural networks (DNNs) have attracted increasingattention from the scientific computing community. DNN based algorithms areeasy to implement, natural for nonlinear problems, and have shown greatpotential to overcome the curse of dimensionality. In this work, we utilize themulti-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 andlocalized activation function. Several numerical examples of multi-scaleelliptic problems with separable or non-separable scales in low-dimensional andhigh-dimensional Euclidean spaces are used to demonstrate the effectiveness andaccuracy of the MscaleDNN numerical scheme.

A multi-scale DNN algorithm for nonlinear elliptic equations with multiple scales