Parameter Estimation in ODE Models with Certified Polynomial System Solving

2025-04-24

AI Summary

This work addresses parameter estimation for rational ordinary differential equation (ODE) models. Traditional homotopy continuation methods often fail on ill-conditioned or high-dimensional instances. To overcome this, we propose a novel certified algebraic approach that— for the first time—integrates Rational Univariate Representation (RUR) and real root isolation techniques into ODE parameter estimation, transforming parameter recovery into a rigorously verifiable polynomial system solving problem. Our method combines differential-algebraic elimination, rational interpolation, and RUR construction, and employs HomotopyContinuation.jl for hybrid numerical–symbolic solving. Experiments demonstrate that our RUR-based solver successfully handles multiple challenging benchmarks where homotopy continuation fails, significantly improving estimation accuracy and reliability. Moreover, the results reveal complementary strengths between RUR and homotopy methods, thereby expanding the applicability frontier of certified parameter estimation for rational ODE systems.

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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.

Problem

• Estimating parameters in rational ODE models from data
• Comparing polynomial solvers for parameter estimation
• Certified solver vs homotopy methods for ODE parameters

Innovation

• Uses certified polynomial system solving
• Compares homotopy and RUR solvers
• Applies rational univariate representation