Parameter Estimation in ODE Models with Certified Polynomial System Solving

TL;DR

The paper tackles parameter estimation for ODE models with rational dynamics by transforming data into a polynomial system. It compares a certified solver based on Gröbner bases and rational univariate representation (RUR) with real root isolation to the existing HomotopyContinuation.jl approach, across models of varying size. Results show complementary strengths: the RUR-based method solves instances unreachable by HC Julia, while HC Julia handles other cases efficiently; overall, the RUR method is competitive for systems with up to roughly 10 states and 10 parameters. The work demonstrates that certified algebraic methods can provide robust, efficient parameter estimation guidance depending on model identifiability and structure.

Abstract

We consider dynamical models given by rational ODE systems. Parameter estimation is an important and challenging task of recovering parameter values from observed data. Recently, a method based on differential algebra and rational interpolation was proposed to express parameter estimation in terms of polynomial system solving. Typically, polynomial system solving is a bottleneck, hence the choice of the polynomial solver is crucial. In this contribution, we compare two polynomial system solvers applied to parameter estimation: homotopy continuation solver from HomotopyContinuation.jl and our new implementation of a certified solver based on rational univariate representation (RUR) and real root isolation. We show how the new RUR solver can tackle examples that are out of reach for the homotopy methods and vice versa.