Parametric Estimation of Ordinary Differential Equations with Orthogonality Conditions

Differential equations are commonly used to model dynamical deterministic systems in applications. When statistical parameter estimation is required to calibrate theoretical models to data, classical statistical estimators are often confronted to complex and potentially ill-posed optimization problem. As a consequence, alternative estimators to classical parametric estimators are needed for obtaining reliable estimates. We propose a gradient matching approach for the estimation of parametric Ordinary Differential Equations observed with noise. Starting from a nonparametric proxy of a true solution of the ODE, we build a parametric estimator based on a variational characterization of the solution. As a Generalized Moment Estimator, our estimator must satisfy a set of orthogonal conditions that are solved in the least squares sense. Despite the use of a nonparametric estimator, we prove the root-$n$ consistency and asymptotic normality of the Orthogonal Conditions estimator. We can derive confidence sets thanks to a closed-form expression for the asymptotic variance, and we give a practical way to optimize the variance by adaptive reweighting. Finally, we compare our estimator in several experiments in order to show its versatility and relevance with respect to classical Gradient Matching and Nonlinear Least Squares estimator