Inverse modified differential equations for discovery of dynamics

We introduce \textbf{inverse modified differential equations} (IMDEs) to contribute to the fundamental theory of discovery of dynamics. In particular, we investigate the IMDEs for the neural ordinary differential equations (neural ODEs). Training such a learning model actually returns an approximation of an IMDE, rather than the original system. Thus, the convergence analysis for data-driven discovery is illuminated. The discrepancy of discovery depends on the order of the integrator used. Furthermore, IMDEs make clear the behavior of parameterizing some blocks in neural ODEs. We also perform several experiments to numerically substantiate our theoretical results.