A Unified Approach to Integration and Optimization of Parametric Ordinary Differential Equations

TL;DR

This paper presents a unified method that reformulates parameter estimation in ordinary differential equations as a boundary value problem, improving convergence to the global optimum especially in complex systems.

Contribution

It introduces a novel boundary value problem formulation for parameter estimation in ODEs, enabling simultaneous solution and better global convergence.

Findings

Effective in oscillatory systems with multiple local optima

Demonstrated on Lotka-Volterra systems with full and partial observations

Enhances convergence to the global maximum likelihood estimate

Abstract

Parameter estimation in ordinary differential equations, although applied and refined in various fields of the quantitative sciences, is still confronted with a variety of difficulties. One major challenge is finding the global optimum of a log-likelihood function that has several local optima, e.g. in oscillatory systems. In this publication, we introduce a formulation based on continuation of the log-likelihood function that allows to restate the parameter estimation problem as a boundary value problem. By construction, the ordinary differential equations are solved and the parameters are estimated both in one step. The formulation as a boundary value problem enables an optimal transfer of information given by the measurement time courses to the solution of the estimation problem, thus favoring convergence to the global optimum. This is demonstrated explicitly for the fully as well as the partially observed Lotka-Volterra system.