Identifying hidden dynamics from observed data is a significant and challenging task in a wide range of applications. Recently, the combination of linear multistep methods (LMMs) and deep learning has been successfully employed to discover dynamics, whereas a complete convergence analysis of this approach is still under development. In this work, we consider the deep network-based LMMs for the discovery of dynamics. We put forward error estimates for these methods using the approximation property of deep networks. It indicates, for certain families of LMMs, that the $\ell^2$ grid error is bounded by the sum of $O(h^p)$ and the network approximation error, where $h$ is the time step size and $p$ is the local truncation error order. Numerical results of several physically relevant examples are provided to demonstrate our theory.

翻译：在广泛应用中,从观测到的数据中找出隐藏的动态是一项重要而具有挑战性的任务。最近,线性多步方法(LMMs)和深层学习相结合,成功地发现动态,而对这一方法的全面趋同分析仍在开发之中。在这项工作中,我们考虑为发现动态而采用基于网络的深层LMMs。我们用深层网络的近似属性对这些方法提出了错误估计。它表明,对于LMMs的某些家庭来说, $\ell_2美元的网格错误与美元(h ⁇ p)和网络近似错误($h是时间步骤大小,$p$是当地短程误差顺序)。我们提供了几个实际相关实例的数值结果,以证明我们的理论。