Structural Identifiability in Dynamical Models

• Structural identifiability is defined as the property that model parameters can be uniquely recovered from ideal input–output data, ensuring unambiguous model predictions.
• Methodologies such as differential algebra, observability matrices, and symmetry analysis provide frameworks to assess parameter uniqueness in various dynamical systems.
• Applications in biology, engineering, and physics leverage identifiability analysis to improve experimental design, model validation, and parameter estimation strategies.

Structural identifiability is a foundational concept in mathematical modeling of dynamical systems, describing whether the parameters of a model can, in principle, be uniquely determined from ideal (noise-free, continuous, and complete) input–output data. It is critical for reliable parameter estimation, model validation, and experimental design. The notion applies across a wide range of modeling frameworks, including ordinary and partial differential equations, stochastic models, and various classes of network and compartmental representations.

1. Definitions, Concepts, and Theoretical Foundations

Structural identifiability refers to the theoretical property that a model’s parameters (or identifiable functions thereof) can be uniquely (globally or locally) recovered from the model equations and ideal data alone—abstracting away measurement noise and experimental imperfections. If a model is structurally unidentifiable, multiple distinct parameter values can produce identical output trajectories; consequently, even perfect data are insufficient to recover the true parameter values, rendering model predictions and parameter interpretations ambiguous (Villaverde, 2018, Chowell et al., 2022, Meshkat et al., 6 Jul 2025).

Local structural identifiability is defined as a one-to-one mapping from the parameter set to output coefficients in a neighborhood of almost every parameter value but may allow a finite (but greater than one) number of parameter sets globally. Formally, for models mapping parameters $\theta$ to output $y(t; \theta)$, if $y(t; \theta_1) = y(t; \theta_2)$ for all $t$ in some interval implies $\theta_1 = \theta_2$ (always), the model is globally identifiable; if this only holds locally (possibly with finitely many equivalent $\theta$), identifiability is local (Mahdi et al., 2013, Barreiro et al., 26 Jul 2025). Parameters that are only locally but not globally identifiable are sometimes called SLING (Structurally Locally Identifiable but Not Globally identified) (Barreiro et al., 26 Jul 2025).

Structural identifiability is distinct from practical identifiability, which incorporates data limitations, experiment noise, and the sensitivity of outputs to parametric perturbations. Structural identifiability is a necessary, but not sufficient, condition for practical identifiability (Villaverde, 9 Oct 2024, Liyanage et al., 21 Mar 2025).

Mathematically, identifiability can be recast as an injectivity (or finite-to-one-ness) criterion on the map from parameters (including initial conditions if unknown) to a set of data features—typically coefficients in input–output or constitutive equations—or, equivalently, on the set of universal invariants under parameter symmetries (Borgqvist et al., 2 Oct 2024).

2. Methodologies and Algorithmic Approaches

A broad array of methodologies for structural identifiability has been developed and refined to address increasingly complex modeling scenarios.

a. Input–Output (Differential Algebra) Approach

The dominant strategy for ODEs starts by eliminating unobserved state variables to produce an input–output relation: $$p(\partial) y = q_1(\partial) u_1 + \cdots + q_r(\partial) u_r$$ where the observable output $y$, system inputs $u_j$, and their derivatives are linked via polynomials whose coefficients are typically functions of the parameters (Ovchinnikov et al., 2019). Structural identifiability reduces to verifying if the mapping from model parameters to these coefficients is injective (Mahdi et al., 2013, Ovchinnikov et al., 2019, Ovchinnikov et al., 2020).

Rigorous justification of the input–output equation method is provided for linear ODE models and linear compartmental models where every compartment can reach a leak or input; in these settings, the coefficients generate the field of identifiable functions (Ovchinnikov et al., 2019). When models lack rational first integrals, identifiability and input–output identifiability coincide—the field of identifiable functions is precisely generated by the coefficients of a minimal characteristic set of the differential ideal (Ovchinnikov et al., 2020).

b. Differential Geometric and Observability-Based Approaches

By treating parameters as constant state variables, structural identifiability is recast as an observability problem. The method constructs a generalized observability-identifiability matrix using time derivatives of the output (Lie derivatives), augmenting the state with parameters (Villaverde, 2018, Villaverde, 9 Oct 2024): $$\mathcal{O}^{NL}_I(\tilde{x}) = \begin{bmatrix} \frac{\partial g(\tilde{x})}{\partial \tilde{x}} \ \frac{\partial (L_f g(\tilde{x}))}{\partial \tilde{x}} \ \vdots \ \frac{\partial (L_f^{n+q-1} g(\tilde{x}))}{\partial \tilde{x}} \end{bmatrix}$$ Structural identifiability is confirmed if this matrix has full rank ($n + q$, with $n$ state variables, $q$ parameters) (Villaverde, 2018). This methodology is local and can be readily extended to analyze the effect of experimental constraints, input structure, and repeated experiments (Ovchinnikov et al., 2020, Villaverde, 9 Oct 2024).

c. Symmetry-Based and Parameter Symmetry Approaches

Global identifiability can be reframed via parameter symmetries: transformations of the parameter vector that preserve the output (Borgqvist et al., 2 Oct 2024, Barreiro et al., 26 Jul 2025). Using the CaLinInv-recipe, one identifies universal differential invariants under all parameter symmetries, which correspond to the structurally identifiable parameter combinations (Borgqvist et al., 2 Oct 2024). Analysis of discrete symmetries through algebraic or Thomas decomposition characterizes SLING parameters—those locally but not globally identifiable due to, for example, permutation symmetries in mammillary or Goodwin oscillator models (Barreiro et al., 26 Jul 2025).

d. Matrix and Graph Methods for Networks and Compartmental Models

For cyclic or acyclic graphical models (e.g., SEMs), identifiability can be analyzed using Wright’s path coefficient method to generate identifiability matrices encoding the dependency between parameters and observed covariances, with efficient binary matrix reductions revealing the identifiability of each parameter (Wang et al., 2017). For linear compartmental models, identifiability degree can be predicted directly from the associated graph structure, with criteria deduced for tree or cycle graphs and leak interlacing (Meshkat et al., 6 Jul 2025).

e. Extensions to PDEs and Nonlinear/Spatial Models

For linear-in-parameter parabolic PDEs, identifiability can be reframed as an existence and uniqueness problem for an associated elliptic operator, invoking the Fredholm alternative to distinguish “good” and “bad” initial/boundary conditions for identifiability (Salmaniw et al., 26 Nov 2024). For linear reaction-advection-diffusion models, systematic differential-algebraic elimination of unobserved quantities can always be performed in principle, and adding spatial heterogeneity cannot decrease structural identifiability (Browning et al., 2023).

3. Applications Across Domains

Structural identifiability frameworks have been widely applied in diverse domains:

• Biological and Epidemiological Modeling: Analysis of SEIR-type and vector-borne models shows which transmission, recovery, and interaction parameters are reliably estimable given available observations and initial condition assumptions (Chowell et al., 2022, Liyanage et al., 15 May 2025, Meshkat et al., 6 Jul 2025). Identifiability analyses inform reparametrization and guide experimental design (e.g., which compartments to measure) (Chowell et al., 2022). In oncology, identifiability studies of chemotherapy and immunotherapy models clarify the estimability of drug efficacy and resistance rates (Meshkat et al., 6 Jul 2025).
• Engineering and Physical Systems: For viscoelastic networks of springs and dashpots, identifiability of mechanical parameters (spring moduli, dashpot viscosities) is characterized by parameter count versus the number of non-monic coefficients in the constitutive ODE, guided by identifiability tables tracking the aggregation of component types (Mahdi et al., 2013). For LCR circuit networks, identifiability depends on the type and number of coefficients in the composite constitutive equation, and limitations are identified for three-component systems (Bortner et al., 2021).
• Systems Biology and Biochemistry: In kinetic models with nonlinear interaction terms, differential algebra and observability-based criteria have been used to assess the identifiability of rate constants in phosphorylation cascades, metabolic networks, and gene regulatory networks; in many such systems, identifiability hinges on labeling information or experiment multiplicity (Wang et al., 2017).
• Battery Systems: For electrochemical models of Li-ion batteries, identifiability analyses of simplified pseudo-2D or Doyle–Fuller–Newman models clarify the number and combinations of parameters (e.g., conductivity, diffusivity) uniquely recoverable from impedance data, with important consequences for state estimation and battery management (Drummond et al., 2020).
• Partial Differential Equation and Spatial Models: For spatially extended reaction–advection–diffusion systems, differential algebraic methodologies are adapted to characterize parameter identifiability from spatially resolved data, with spatial terms often increasing the richness of identifiable combinations (Browning et al., 2023, Salmaniw et al., 26 Nov 2024).

4. Key Software Tools and Computational Methods

Structural identifiability analysis is enabled by a suite of specialized software tools, with key roles for symbolic computation and algebraic elimination.

• StructuralIdentifiability.jl (Julia): Implements the input–output elimination algorithm; efficiently handles high-dimensional epidemiological and compartmental models, checking internal matrix nonsingularity assumptions and supporting reduction via first integrals (Liyanage et al., 15 May 2025, Liyanage et al., 21 Mar 2025). Demonstrated to outperform DAISY on complex or large-scale models (Liyanage et al., 15 May 2025). Code is available for reproducibility and educational purposes.
• DAISY (Differential Algebra for Identifiability of SYstems): Employs characteristic set computation to uncover parameter identifiability and correlations in differential models; particularly effective when parameter expressions are rational functions (Chowell et al., 2022).
• Web-based Structural Identifiability Analyzer: Integrates differential algebraic (e.g., SIAN, Gröbner basis) and Monte Carlo symbolic algorithms within an accessible cloud interface for parameter-wise identifiability and computation of all identifiable combinations (Ilmer et al., 2021).
• Model Theory and Symbolic Packages (Oscar, Nemo in Julia): Support algebraic manipulations for multi-experiment identifiability calculations, including randomized algorithms for transcendence degree computation (Ovchinnikov et al., 2020).

5. Strategies for Resolving Unidentifiability and Limitations

When a model is structurally unidentifiable, several remedies are available:

• Model Reparametrization: Identify a set of functions in the parameters that are structurally identifiable (possibly via algorithmic computation of invariants), and reformulate the model in these new parameters (Meshkat et al., 6 Jul 2025, Ovchinnikov et al., 2019, Ovchinnikov et al., 2020).
• Adjustment of Experimental Design: Add outputs (measured compartments or observables), fix certain parameters using prior knowledge, or utilize multiple independent experiments (initial condition or input variations) to break ambiguous parameter combinations (Ovchinnikov et al., 2020, Chowell et al., 2022). The number of experiments required for full identifiability can be quantified via defect algorithms (Ovchinnikov et al., 2020).
• Incorporation of Forcing Function Inputs: Introduce known, time-varying inputs correlated with hard-to-identify parameters to “split” intertwined parameter combinations—provably never worsening and potentially improving structural identifiability (Conrad et al., 3 Jul 2024).
• Graph-Theoretic or Symmetry Analysis: Recognize and exploit symmetries, especially discrete ones, that give rise to finite ambiguity; SLING parameters can be isolated, and explicit symmetry groups characterized (Barreiro et al., 26 Jul 2025, Borgqvist et al., 2 Oct 2024).
• Model Simplification: Employ model reduction strategies, such as eliminating state variables using conservation laws (first integrals) where possible, to facilitate the identifiability analysis when direct application to the full model is computationally infeasible (Liyanage et al., 15 May 2025).

Limitations remain in the scalability of symbolic-algebraic approaches to very high-dimensional or nonlinear PDE models, the analysis of systems with non-rational first integrals, and the practical estimation of parameters in the presence of measurement error or under constraints on input/output richness (Browning et al., 2023, Meshkat et al., 6 Jul 2025).

6. Impact, Open Problems, and Future Directions

Structural identifiability has become a mature area, but ongoing research is addressing outstanding questions:

• Global versus Local Identifiability: Many methods yield only local identifiability; rigorous global criteria are being explored through hybrid algebraic and symmetry-based techniques (Borgqvist et al., 2 Oct 2024, Barreiro et al., 26 Jul 2025).
• Extension to PDEs, Stochastic, and Delay Systems: Operator-theoretic and symbolic-algebraic frameworks are being advanced for spatially distributed systems and stochastic settings, with active development on tractable algorithms for PDEs and SDEs (Browning et al., 2023, Salmaniw et al., 26 Nov 2024).
• Automated Experimental Design: Integration of identifiability analysis into optimal experiment design, the selection of informative inputs, or choice of measurement variables is an active area, crucial for practical model calibration (Villaverde, 9 Oct 2024, Ovchinnikov et al., 2020).
• Application to Large-Scale and Biological Networks: Systematic screening of biomolecular reaction networks, epidemiological models, and high-dimensional dynamical systems to establish empirical rules or graph-based predictions for identifiability is ongoing (Meshkat et al., 6 Jul 2025, Wang et al., 2017).
• Novel Computational and Theoretical Methods: Further development of efficient algorithms (e.g., leveraging model theory, numerical algebra, or large-scale symbolic computation) is needed to address computational bottlenecks in non-linear, high-dimensional, or spatial models (Meshkat et al., 6 Jul 2025, Ovchinnikov et al., 2020).

In summary, structural identifiability analysis provides critical theoretical and practical guarantees for parameter identifiability in dynamical systems, with a diverse methodological toolbox and widespread applications. The interplay between algebraic, geometric, and symmetry-based approaches continues to yield new insights, guiding the design, analysis, and interpretation of models in biology, engineering, and the physical sciences.