  Along with the practical success of the discovery of dynamics using deep
learning, the theoretical analysis of this approach has attracted increasing
attention. Prior works have established the grid error estimation with
auxiliary conditions for the discovery of dynamics using linear multistep
methods and deep learning. And we extend the existing error analysis in this
work. We first introduce the concept of inverse modified differential equations
(IMDE) for linear multistep methods and show that the learned model returns a
close approximation of the IMDE. Based on the IMDE, we prove that the error
between the discovered system and the target system is bounded by the sum of
the LMM discretization error and the learning loss. Furthermore, the learning
loss is quantified by combining the approximation and generalization theories
of neural networks, and thereby we obtain the priori error estimates for the
discovery of dynamics using linear multistep methods. Several numerical
experiments are performed to verify the theoretical analysis.
