Structural functional identifiability and model discovery in differential equation models

Abstract

Differential equation models are widely used to describe, interpret, and predict dynamical phenomena across science and engineering. In practice, however, the governing dynamics are rarely fully known and must be inferred from observational data. Traditionally, inverse problems in differential equation modelling have focused on estimating unknown parameter values. In this setting, structural identifiability determines whether parameter values can, in principle, be uniquely recovered from ideal observations and is, therefore, a prerequisite for meaningful inference. More recently, the integration of machine learning with mechanistic modelling has enabled the discovery of unknown equations, functions, and constitutive relationships, substantially expanding the space of admissible models. This raises a fundamental question: under what conditions can unknown functional components be uniquely recovered from data? In this paper, we generalise the classical notion of structural parameter identifiability to functional identifiability. We first identify broad classes of models for which unique functional recovery is impossible. We then show how functional identifiability can be assessed for differential equation models using differential algebra-based techniques which are well-established as a means of assessing structural identifiability for ordinary differential equation-based models. Our framework reveals new phenomena that arise in the transition from parametric to functional inference and have no analogue in the classical setting. Finally, we characterise functional identifiability in several common model classes. Taken together, our results demonstrate that functional identifiability provides a theoretical foundation for modern inverse problems in differential equation modelling, particularly those that use machine learning representations of unknown system components.