The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimation
The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimation
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DOI：
10.1137/21m140691x
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发表时间：
2021-03
期刊：
影响因子：
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通讯作者：
Q. Du;Yiqi Gu;Haizhao Yang;Chao Zhou
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文献类型：
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作者：
Q. Du;Yiqi Gu;Haizhao Yang;Chao Zhou
从观测数据中识别隐藏的动力学在众多应用中是一项重要且具有挑战性的任务。最近，线性多步法（LMMs）与深度学习的结合已成功用于发现动力学，然而这种方法的完整收敛性分析仍在发展中。在这项工作中，我们考虑基于深度网络的LMMs用于发现动力学。我们利用深度网络的逼近性质为这些方法提出了误差估计。它表明，对于某些LMMs族，$\ell^2$网格误差由$O(h^p)$与网络逼近误差之和所界定，其中$h$是时间步长，$p$是局部截断误差阶。提供了几个与物理相关的实例的数值结果以证明我们的理论。
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.