  One of the central problems in the study of deep learning theory is to
understand how the structure properties, such as depth, width and the number of
nodes, affect the expressivity of deep neural networks. In this work, we show a
new connection between the expressivity of deep neural networks and topological
entropy from dynamical system, which can be used to characterize depth-width
trade-offs of neural networks. We provide an upper bound on the topological
entropy of neural networks with continuous semi-algebraic units by the
structure parameters. Specifically, the topological entropy of ReLU network
with $l$ layers and $m$ nodes per layer is upper bounded by $O(l\log m)$.
Besides, if the neural network is a good approximation of some function $f$,
then the size of the neural network has an exponential lower bound with respect
to the topological entropy of $f$. Moreover, we discuss the relationship
between topological entropy, the number of oscillations, periods and Lipschitz
constant.
