Nonparametric Estimation under General Nonlinear ODE Constraints: A Comparison with Parametric ODE-Fitting Methods

2026-08-05

AI Summary

This study addresses nonparametric estimation of state variables governed by general first-order nonlinear ordinary differential equations (ODEs). We propose an ODE-constrained local polynomial regression framework (DE-constrained LPR) that integrates symbolic differentiation with nonlinear least squares, requiring only a single local parameter per estimation point. This approach overcomes the limitations of existing methods confined to linear or exponential ODE forms and accommodates any Lipschitz-continuous nonlinear function F. By leveraging the ODE structure to avoid direct estimation of high-order derivatives, we derive an optimal bandwidth selection criterion. In logistic growth simulations, our method consistently outperforms classical local linear regression and demonstrates superior accuracy and robustness over the parameter cascading method (PCODE), particularly in regions of steep curvature, offering an efficient nonparametric alternative when structural parameters are difficult to identify.

Abstract

Many physical, biological, and epidemiological processes are governed by ordinary differential equations (ODEs) that are nonlinear in the state variable, including logistic population growth, chemical reaction kinetics, and epidemiological compartment models. We develop a differential equation-constrained local polynomial regression (DE-constrained LPR) framework for the general first-order ODE constraint g'(x) = F(x, g(x)), where F may be any Lipschitz continuous function, extending prior work restricted to exponential and linear ODE structures. Because F is generally nonlinear in g, the Taylor coefficients of the DE1-k estimator cannot be written in closed form; instead they are obtained by successive symbolic differentiation of F, and the estimator is computed by nonlinear least squares, requiring only a single local parameter at each evaluation point regardless of polynomial degree k. We derive the asymptotic conditional bias and variance of the DE1-k estimator, propose an AIMSE-optimal bandwidth that exploits the ODE structure to avoid direct estimation of high-order derivatives, and evaluate the method in a simulation study based on logistic growth, benchmarking against the parameter cascading method of Ramsay et al. (2007) (PCODE) and classical local linear regression. The DE-constrained estimator consistently outperforms local linear regression and is competitive with PCODE even though it estimates no structural parameter of the ODE; a sensitivity analysis across growth rates shows DE-constrained estimation becomes more accurate and more robust than PCODE as the curve steepens and PCODE's parameter estimation grows less stable. These results position DE-constrained LPR as a practical nonparametric alternative to parametric ODE-fitting methods when structural parameters are difficult to identify reliably.

Problem

• nonparametric estimation
• nonlinear ODE constraints
• structural parameter identification
• ordinary differential equations
• parameter estimation

Innovation

• nonparametric estimation
• ODE-constrained regression
• local polynomial regression
• nonlinear ODE
• symbolic differentiation