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

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.