Parameters of differential equations are essential to characterize intrinsic behaviors of dynamic systems. Numerous methods for estimating parameters in dynamic systems are computationally and/or statistically inadequate, especially for complex systems with general-order differential operators, such as motion dynamics. This article presents Green's matching, a computationally tractable and statistically efficient two-step method, which only needs to approximate trajectories in dynamic systems but not their derivatives due to the inverse of differential operators by Green's function. This yields a statistically optimal guarantee for parameter estimation in general-order equations, a feature not shared by existing methods, and provides an efficient framework for broad statistical inferences in complex dynamic systems.

翻译：微分方程中的参数对于刻画动态系统的内在行为至关重要。现有众多动态系统参数估计方法在计算效率或统计性能上存在不足，尤其针对包含一般阶微分算子（如运动动力学）的复杂系统。本文提出Green's匹配方法——一种计算可处理且统计高效的两步法。该方法利用格林函数实现微分算子的逆变换，仅需近似动态系统的轨迹而无需估计其导数。这为一般阶方程的参数估计提供了统计最优性保证（现有方法不具备此特性），并为复杂动态系统的广泛统计推断建立了高效框架。