Computing all identifiable functions of parameters for ODE models

Parameter identifiability is a structural property of an ODE model for recovering the values of parameters from the data (i.e., from the input and output variables). This property is a prerequisite for meaningful parameter identification in practice. In the presence of nonidentifiability, it is important to find all functions of the parameters that are identifiable. The existing algorithms check whether a given function of parameters is identifiable or, under the solvability condition, find all identifiable functions. However, this solvability condition is not always satisfied, which presents a challenge. Our first main result is an algorithm that computes all identifiable functions without any additional assumptions, which is the first such algorithm as far as we know. Our second main result concerns the identifiability from multiple experiments (with generically different inputs and initial conditions among the experiments). For this problem, we prove that the set of functions identifiable from multiple experiments is what would actually be computed by input-output equation-based algorithms (whether or not the solvability condition is fulfilled), which was not known before. We give an algorithm that not only finds these functions but also provides an upper bound for the number of experiments to be performed to identify these functions. We provide an implementation of the presented algorithms.

💡 Research Summary

The paper addresses the fundamental problem of structural parameter identifiability in rational ordinary differential equation (ODE) models. Identifiability asks whether the values of model parameters can be uniquely recovered from ideal, noise‑free measurements of inputs and outputs. When a model is not fully identifiable, the practical task becomes to determine all rational functions of the parameters that are nevertheless identifiable. Existing methods either (i) test a given function for identifiability, or (ii) compute the whole set of identifiable functions only under a “solvability condition” (also called the “generic solvability” or “rank‑regularity” condition). This condition fails for many realistic models, causing popular software tools such as DAISY and COMBOS to miss non‑identifiable parameters.

The authors present two major contributions that remove the reliance on the solvability condition.

1. Single‑experiment identifiability (Theorem 11 and Algorithm 1).
For a given ODE system
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