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Along with the practical success of the discovery of dynamics using deeplearning, the theoretical analysis of this approach has attracted increasingattention. Prior works have established the grid error estimation withauxiliary conditions for the discovery of dynamics using linear multistepmethods and deep learning. And we extend the existing error analysis in thiswork. We first introduce the concept of inverse modified differential equations(IMDE) for linear multistep methods and show that the learned model returns aclose approximation of the IMDE. Based on the IMDE, we prove that the errorbetween the discovered system and the target system is bounded by the sum ofthe LMM discretization error and the learning loss. Furthermore, the learningloss is quantified by combining the approximation and generalization theoriesof neural networks, and thereby we obtain the priori error estimates for thediscovery of dynamics using linear multistep methods. Several numericalexperiments are performed to verify the theoretical analysis.

Error analysis based on inverse modified differential equations for discovery of dynamics using linear multistep methods and deep learning