  Recent studies revealed the mathematical connection of deep neural network
(DNN) and dynamic system. However, the fundamental principle of DNN has not
been fully characterized with dynamic system in terms of optimization and
generalization. To this end, we build the connection of DNN and continuity
equation where the measure is conserved to model the forward propagation
process of DNN which has not been addressed before. DNN learns the
transformation of the input distribution to the output one. However, in the
measure space, there are infinite curves connecting two distributions. Which
one can lead to good optimization and generaliztion for DNN? By diving the
optimal transport theory, we find DNN with weight decay attempts to learn the
geodesic curve in the Wasserstein space, which is induced by the optimal
transport map. Compared with plain network, ResNet is a better approximation to
the geodesic curve, which explains why ResNet can be optimized and generalize
better. Numerical experiments show that the data tracks of both plain network
and ResNet tend to be line-shape in term of line-shape score (LSS), and the map
learned by ResNet is closer to the optimal transport map in term of optimal
transport score (OTS). In a word, we conclude a mathematical principle of deep
learning is to learn the geodesic curve in the Wasserstein space; and deep
learning is a great engineering realization of continuous transformation in
high-dimensional space.
