  While deep learning algorithms demonstrate a great potential in scientific
computing, its application to multi-scale problems remains to be a big
challenge. This is manifested by the "frequency principle" that neural networks
tend to learn low frequency components first. Novel architectures such as
multi-scale deep neural network (MscaleDNN) were proposed to alleviate this
problem to some extent. In this paper, we construct a subspace decomposition
based DNN (dubbed SD$^2$NN) architecture for a class of multi-scale problems by
combining traditional numerical analysis ideas and MscaleDNN algorithms. The
proposed architecture includes one low frequency normal DNN submodule, and one
(or a few) high frequency MscaleDNN submodule(s), which are designed to capture
the smooth part and the oscillatory part of the multi-scale solutions,
respectively. In addition, a novel trigonometric activation function is
incorporated in the SD$^2$NN model. We demonstrate the performance of the
SD$^2$NN architecture through several benchmark multi-scale problems in regular
or irregular geometric domains. Numerical results show that the SD$^2$NN model
is superior to existing models such as MscaleDNN.
