论文标题
基于逆修改微分方程的错误分析，用于使用线性多步法和深度学习发现动力学
Error analysis based on inverse modified differential equations for discovery of dynamics using linear multistep methods and deep learning
论文作者
论文摘要
除了使用深度学习发现动力学的实际成功之外，对这种方法的理论分析引起了人们越来越多的关注。先前的工作已经建立了使用线性多步法和深度学习发现动力学的辅助条件的网格误差估计。我们扩展了这项工作中现有的错误分析。我们首先介绍了线性多步法方法的反修改微分方程（IMDE）的概念，并表明学习模型返回IMDE的紧密近似。基于IMDE，我们证明发现的系统和目标系统之间的误差是由LMM离散误差和学习损失的总和界定的。此外，通过结合神经网络的近似和概括理论来量化学习损失，从而获得了使用线性多步法方法发现动力学的先验误差估计。进行了几个数值实验以验证理论分析。
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.
