  In recent years, deep learning has led to impressive results in many fields.
In this paper, we introduce a multi-scale artificial neural network for
high-dimensional non-linear maps based on the idea of hierarchical nested bases
in the fast multipole method and the $\mathcal{H}^2$-matrices. This approach
allows us to efficiently approximate discretized nonlinear maps arising from
partial differential equations or integral equations. It also naturally extends
our recent work based on the generalization of hierarchical matrices [Fan et
al. arXiv:1807.01883] but with a reduced number of parameters. In particular,
the number of parameters of the neural network grows linearly with the
dimension of the parameter space of the discretized PDE. We demonstrate the
properties of the architecture by approximating the solution maps of non-linear
Schr{\"o}dinger equation, the radiative transfer equation, and the Kohn-Sham
map.
