Inverse modified differential equations for discovery of dynamics

The combination of numerical integration and deep learning, i.e., ODE-net, has been successfully employed in a variety of applications. In this work, we introduce inverse modified differential equations (IMDE) to contribute to the behaviour and error analysis of discovery of dynamics using ODE-net. It is shown that the difference between the learned ODE and the truncated IMDE is bounded by the sum of learning loss and a discrepancy which can be made sub exponentially small in the data step size. In addition, we deduce that the total error of ODE-net is bounded by the sum of discrete error and learning loss. Furthermore, with the help of IMDE, theoretical results on learning Hamiltonian system are derived. Several experiments are performed to numerically verify our theoretical results.