Verifying global identifiability of parametric linear ODE models is NP-hard

2026-05-29

AI Summary

This study addresses the problem of global parameter identifiability for linear ordinary differential equation models in which parameters depend linearly on state variables and rationally on other parameters. By reformulating identifiability as the injectivity of the corresponding input–output map and leveraging tools from algebraic geometry and differential algebra, the authors establish—for the first time—that this identifiability problem is NP-hard. This result resolves a longstanding gap in the computational complexity theory of identifiability for this important class of models, providing a rigorous lower bound on its intrinsic computational difficulty. Consequently, it offers a solid theoretical foundation for the design and performance evaluation of future algorithms aimed at tackling parameter identifiability in such systems.

Abstract

Global parameter identifiability is a property of a parametric ODE model to recover the parameter values uniquely from the input-output data. Not all parametric ODE models have this property, and checking for parameter identifiability is a prerequisite to perform numerical parameter estimation. There are many algorithms and software packages for global parameter identifiability, and frequently the runtime is large. However, the computational complexity for this problem has not been analyzed yet, though there are complexity results for local (finitely many values fit the data) parameter identifiability. In this paper, we estimate the complexity of checking global parameter identifiability over real fields for ODE models that depend linearly on the state variables and rationally on the parameters. In particular, we prove that it is equivalent to the injectivity problem.

Problem

• global identifiability
• parametric ODE models
• computational complexity
• injectivity problem
• parameter estimation

Innovation

• global identifiability
• computational complexity
• NP-hard
• parametric ODE models
• injectivity