Green's matching: an efficient approach to parameter estimation in complex dynamic systems

TL;DR

The paper addresses parameter estimation for dynamic systems described by general-order differential operators by introducing Green's matching, a two-step method that uses local smoothing to obtain trajectories and Green's functions to invert the differential operator, thereby avoiding derivative estimation. It proves root-$n$ consistency and asymptotic normality for Green's matching and shows it outperforms gradient and integral matching, especially as the differential order grows, with favorable computational properties. Through extensive simulations on gene regulatory networks, spring-mass systems, and oscillatory models, plus an equation-discovery example for a nonlinear pendulum, Green's matching demonstrates superior accuracy and efficient computation, accompanied by well-behaved confidence intervals. The work also discusses limitations (dense-trajectory data requirements) and outlines extensions to learn the differential order $K$, enable sparsity-driven equation discovery, and tackle causal-network tasks.

Abstract

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.

Figures (4)

• Figure 1: The estimated derivatives for a curve (in a spring-mass system) by local polynomial regressions, where the estimation is performed based on the discrete noisy data (blue circles) obtained from different sample sizes $n$ (subtitles). The red, green, and purple regions represent the weights assigned to the observations, controlled by the bandwidths $h_0$, $h_1$, and $h_2$, respectively. For more information about the spring-mass system and the bandwidth selection scheme, please refer to Section \ref{['sim_1']} and Part 1.1 in the Supplementary Materials, respectively.
• Figure 2: RBIAS and RSD of three methods with different sample sizes $n$ and $\gamma$s (sub-title), where RSDs are colored in gray, and RBIASs of order 2 / 1 gradient matching and Green's matching are colored in blue, orange, and red, respectively.
• Figure 3: A. Reconstructed curves for $X_1(\cdot)$, $DX_1(\cdot)$, and $D^2X_1(\cdot)$ (in the oscillatory dynamic directional model) by three methods from 100 simulations, where the solid black lines indicate the true curves. B. Confidence intervals of $a_{1}$, $b_{1}$, $c_{1}$, and $d_{1}$ (in the oscillatory dynamic directional model) estimated by three methods from 100 simulations, where the solid black lines indicate the true values of parameters.
• Figure 4: The true vector field and the average vector fields from 100 simulations for the nonlinear pendulum, where the color represents the magnitude of the Euclidean norm of $(DX(t),D^2X(t))^T$.

Theorems & Definitions (3)

• Theorem 1: Consistency
• Theorem 2: Asymptotic normality
• Theorem 3: Estimation biases and rates of convergence