Extend structural indistinguishability theory to nonlinear systems and incorporate data effects

Develop structural indistinguishability results applicable to nonlinear dynamical systems, such as Hill-function-based ODE gene regulatory models, and incorporate the influence of experimental data on indistinguishability analyses.

Background

Current theoretical results on structural indistinguishability largely pertain to linear systems and do not consider how experimental data quality and sampling affect the ability to distinguish models.

The models studied in this paper are nonlinear and data-limited, underscoring the need for theory that accommodates nonlinearity and data dependence to better interpret practical indistinguishability findings.

References

These theoretical results are largely limited to linear models and cannot currently be applied to non-linear systems, such as Eqs. (\ref{eq1})-(\ref{eq2}) and do not consider the impact of experimental data on indistinguishability.

— Practical indistinguishability in a gene regulatory network inference problem, a case study  (2508.21006 - FitzGerald et al., 28 Aug 2025) in Discussion

A third open problem is to establish what would justify the step from dependence to mechanism. Identifiability theory~\citep{Villaverde2019} offers one route, by characterizing in advance the equivalence classes of models that generate the same observables, so that the limits of what the data could determine are known before fitting. A second is to seek exclusion results rather than confirmations, as in the shared-environment result of Section~\ref{sec:disc:inference}, where a restricted class of indirect mechanisms is shown to produce only a restricted set of signatures. Constructing comparable results for variable elimination and temporal aggregation would turn each indirect route into a hypothesis that observations can reject. Where intervention is possible, perturbing a candidate group remains the most decisive evidence, while expressing reducibility in causal terms would clarify what such an intervention can establish.

— Higher-Order Interactions in Complex Systems: Mechanisms, Behaviour, Representation and Reducibility  (2609.20142 - Pérez-Reche, 17 Sep 2026) in Section 7, subsection “Open problems”