  Identifying hidden dynamics from observed data is a significant and
challenging task in a wide range of applications. Recently, the combination of
linear multistep methods (LMMs) and deep learning has been successfully
employed to discover dynamics, whereas a complete convergence analysis of this
approach is still under development. In this work, we consider the deep
network-based LMMs for the discovery of dynamics. We put forward error
estimates for these methods using the approximation property of deep networks.
It indicates, for certain families of LMMs, that the $\ell^2$ grid error is
bounded by the sum of $O(h^p)$ and the network approximation error, where $h$
is the time step size and $p$ is the local truncation error order. Numerical
results of several physically relevant examples are provided to demonstrate our
theory.
