A regularization method for the parameter estimation problem in ordinary differential equations via discrete optimal control theory

Abstract: We present a parameter estimation method in Ordinary Differential Equation (ODE) models. Due to complex relationships between parameters and states the use of standard techniques such as nonlinear least squares can lead to the presence of poorly identifiable parameters. Moreover, ODEs are generally approximations of the true process and the influence of misspecification on inference is often neglected. Methods based on control theory have emerged to regularize the ill posed problem of parameter estimation in this context. However, they are computationally intensive and rely on a nonparametric state estimator known to be biased in the sparse sample case. In this paper, we construct criteria based on discrete control theory which are computationally efficient and bypass the presmoothing step of signal estimation while retaining the benefits of control theory for estimation. We describe how the estimation problem can be turned into a control one and present the numerical methods used to solve it. We show convergence of our estimator in the parametric and well-specified case. For small sample sizes, numerical experiments with models containing poorly identifiable parameters and with various sources of model misspecification demonstrate the acurracy of our method. We finally test our approach on a real data example.