State and Parameter Estimation of Partially Observed Linear Ordinary Differential Equations with Deterministic Optimal Control

TL;DR

This paper introduces a novel optimal control-based method for estimating parameters and states in linear ODEs from time series data, addressing issues of ill-posedness and improving accuracy over traditional methods.

Contribution

It formulates parameter estimation as an optimal control problem, deriving a new criterion and demonstrating statistical properties like consistency and normality.

Findings

Method outperforms nonlinear least squares in accuracy.

Approach is more reliable even with model misspecification.

Estimates are root-n consistent and asymptotically normal.

Abstract

Ordinary Differential Equations are a simple but powerful framework for modeling complex systems. Parameter estimation from times series can be done by Nonlinear Least Squares (or other classical approaches), but this can give unsatisfactory results because the inverse problem can be ill-posed, even when the differential equation is linear. Following recent approaches that use approximate solutions of the ODE model, we propose a new method that converts parameter estimation into an optimal control problem: our objective is to determine a control and a parameter that are as close as possible to the data. We derive then a criterion that makes a balance between discrepancy with data and with the model, and we minimize it by using optimization in functions spaces: our approach is related to the so-called Deterministic Kalman Filtering, but different from the usual statistical Kalman filtering. e show the root-$n$ consistency and asymptotic normality of the estimators for the parameter and for the states. Experiments in a toy model and in a real case shows that our approach is generally more accurate and more reliable than Nonlinear Least Squares and Generalized Smoothing, even in misspecified cases.