Linear multistep methods (LMMs) are popular time discretization techniques for the numerical solution of differential equations. Traditionally they are applied to solve for the state given the dynamics (the forward problem), but here we consider their application for learning the dynamics given the state (the inverse problem). This repurposing of LMMs is largely motivated by growing interest in data-driven modeling of dynamics, but the behavior and analysis of LMMs for discovery turn out to be significantly different from the well-known, existing theory for the forward problem. Assuming a highly idealized setting of being given the exact state with a zero residual of the discrete dynamics, we establish for the first time a rigorous framework based on refined notions of consistency and stability to yield convergence using LMMs for discovery. When applying these concepts to three popular $M-$step LMMs, the Adams-Bashforth, Adams-Moulton, and Backwards Differentiation Formula schemes, the new theory suggests that Adams-Bashforth for $M$ ranging from $1$ and $6$, Adams-Moulton for $M=0$ and $M=1$, and Backwards Differentiation Formula for all positive $M$ are convergent, and, otherwise, the methods are not convergent in general. In addition, we provide numerical experiments to both motivate and substantiate our theoretical analysis.

中文翻译：

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使用线性多步法发现动力学

线性多步法 (LMM) 是用于微分方程数值求解的流行时间离散化技术。传统上，它们用于求解给定动态的状态（前向问题），但在这里我们考虑它们用于学习给定状态的动态（逆问题）。LMM 的这种再利用很大程度上是由于人们对动态数据驱动建模的兴趣日益浓厚，但 LMM 的行为和分析发现与众所周知的前向问题的现有理论有很大不同。假设一个高度理想化的设置被赋予离散动力学的零残差的精确状态，我们首次建立了一个基于一致性和稳定性的精炼概念的严格框架，以使用 LMM 进行发现以产生收敛。当将这些概念应用于三个流行的 $M-$step LMM，Adams-Bashforth、Adams-Moulton 和 Backwards Differentiation Formula 方案时，新理论表明 Adams-Bashforth 的 $M$ 范围从 $1$ 到 $6$，Adams -Moulton 对于 $M=0$ 和 $M=1$，以及所有正 $M$ 的向后微分公式是收敛的，否则，这些方法一般不收敛。此外，我们提供数值实验来激励和证实我们的理论分析。这些方法一般不收敛。此外，我们提供数值实验来激励和证实我们的理论分析。这些方法一般不收敛。此外，我们提供数值实验来激励和证实我们的理论分析。