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.

翻译：数字整合和深层次学习相结合,即ODE-net,已在各种应用中成功应用。在这项工作中,我们引入了反向修改差异方程式(IMDE),以促进对使用ODE-net发现动态的行为和误差分析,并表明,所学的ODE与被截断的IMDE之间的差异受学习损失总和和和数据级数小于指数的差异的制约。此外,我们推断,ODE-net的总误差与离散错误和学习损失之和是密尔顿系统学习理论结果的结合。还进行了数项实验,对我们的理论结果进行数字核查。