Along with the practical success of the discovery of dynamics using deep learning, the theoretical analysis of this approach has attracted increasing attention. Prior works have established the grid error estimation with auxiliary conditions for the discovery of dynamics using linear multistep methods and deep learning. And we extend the existing error analysis in this work. We first introduce the concept of inverse modified differential equations (IMDE) for linear multistep methods and show that the learned model returns a close approximation of the IMDE. Based on the IMDE, we prove that the error between the discovered system and the target system is bounded by the sum of the LMM discretization error and the learning loss. Furthermore, the learning loss is quantified by combining the approximation and generalization theories of neural networks, and thereby we obtain the priori error estimates for the discovery of dynamics using linear multistep methods. Several numerical experiments are performed to verify the theoretical analysis.

翻译：随着深度学习在动力学发现中的实际成功，该方法的相关理论分析日益受到关注。已有研究在借助辅助条件使用线性多步法与深度学习进行动力学发现时，建立了网格误差估计。本文进一步拓展了现有的误差分析体系。我们首先引入线性多步法的逆修正微分方程（IMDE）概念，并证明学习模型能够对IMDE进行近似逼近。基于IMDE，我们证明了发现系统与目标系统之间的误差可由LMM离散化误差与学习损失之和界定。进一步，通过结合神经网络的逼近理论与泛化理论对学习损失进行量化，从而获得了使用线性多步法进行动力学发现的先验误差估计。多项数值实验验证了理论分析的正确性。