SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 429-455, January 2021.
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 backward 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 backward 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

SIAM数值分析学报，第59卷，第1期，第429-455页，2021年1月。
线性多步法（LMM）是用于微分方程数值解的流行时间离散化技术。传统上，它们被用于求解给定动力学的状态（正向问题），但在这里我们考虑将其应用于学习给定动力学的状态（反问题）。LMM的这种重用很大程度上是由人们对数据驱动的动力学建模的兴趣日益浓厚推动的，但是事实证明，用于发现的LMM的行为和分析与前瞻性问题的众所周知的现有理论有很大不同。假设高度理想化的环境设置为精确状态且离散动力学的残差为零，我们首次基于一致性和稳定性的精炼概念建立了一个严格的框架，以便使用LMM进行收敛。当将这些概念应用于三个流行的$ M $步长的LMM，Adams-Bashforth，Adams-Moulton和向后微分公式方案时，新理论表明，$ M $的Adams-Bashforth的范围从1到6， $ M = 0 $和$ M = 1 $的Adams-Moulton以及所有正$ M $的后向微分公式都是收敛的，否则，这些方法通常不会收敛。此外，我们提供了数值实验来激发和证实我们的理论分析。这些方法通常不收敛。此外，我们提供了数值实验来激发和证实我们的理论分析。这些方法通常不收敛。此外，我们提供了数值实验来激发和证实我们的理论分析。