  In this work we introduce a new multiscale artificial neural network based on
the structure of $\mathcal{H}$-matrices. This network generalizes the latter to
the nonlinear case by introducing a local deep neural network at each spatial
scale. Numerical results indicate that the network is able to efficiently
approximate discrete nonlinear maps obtained from discretized nonlinear partial
differential equations, such as those arising from nonlinear Schr\"odinger
equations and the Kohn-Sham density functional theory.
