A practical identifiability criterion leveraging weak-form parameter estimation

In this work, we define a practical identifiability criterion, (e, q)-identifiability, based on a parameter e, reflecting the noise in observed variables, and a parameter q, reflecting the mean-square error of the parameter estimator. This criterion is better able to encompass changes in the quality of the parameter estimate due to increased noise in the data (compared to existing criteria based solely on average relative errors). Furthermore, we leverage a weak-form equation error-based method of parameter estimation for systems with unobserved variables to assess practical identifiability far more quickly in comparison to output error-based parameter estimation. We do so by generating weak-form input-output equations using differential algebra techniques, as previously proposed by Boulier et al [1], and then applying Weak form Estimation of Nonlinear Dynamics (WENDy) to obtain parameter estimates. This method is computationally efficient and robust to noise, as demonstrated through two classical biological modelling examples.

💡 Research Summary

This paper introduces a novel practical identifiability criterion, termed (e, q)-identifiability, and demonstrates its utility together with a weak‑form parameter estimation framework (WENDy) for dynamical systems that include unobserved states. The authors begin by highlighting the persistent problem in biological modeling where the same model can yield widely divergent parameter estimates due to the interplay of model structure, data quality, and numerical discretization. While structural identifiability (whether a model’s parameters can be uniquely determined from perfect data) is well‑studied using differential algebra, Taylor expansions, or observability analyses, practical identifiability (whether parameters can be reliably estimated from noisy, sparse data) remains computationally demanding. Traditional tools such as the Fisher Information Matrix (FIM), profile likelihood, or simulation‑based average relative error require repeated costly parameter optimizations, especially for nonlinear ODE systems.

To address these challenges, the authors propose two key innovations. First, they define (e, q)-identifiability, where e = σ/RMS(Ω) quantifies the observation noise level relative to the magnitude of the noise‑free output, and q = √(M_i)/|w_i| measures the estimator’s mean‑squared error (MSE) relative to the true parameter magnitude. A parameter w_i is (e, q)-identifiable if, for a given e, its MSE stays below (q · w_i)^2. This criterion simultaneously captures accuracy (low bias) and precision (low variance) and can be applied a priori (to guide experimental design) or a posteriori (to assess a completed study).

Second, the paper leverages weak‑form estimation, specifically the Weak form Estimation of Nonlinear Dynamics (WENDy) algorithm, to evaluate practical identifiability efficiently. The approach proceeds as follows: (1) differential elimination (via Rosenfeld‑Groebner algorithms implemented in the DifferentialAlgebra v4 Python package) is used to eliminate unobserved states and generate an input‑output differential equation that depends only on measured variables and parameters; (2) this equation is cast into its weak form by multiplying with compactly supported test functions φ and integrating, thereby avoiding numerical differentiation of noisy data; (3) the resulting linear system is solved using regularized least squares, yielding parameter estimates. Because the weak form eliminates the need for repeated forward simulations, thousands of Monte‑Carlo repetitions become computationally tractable.

The authors validate the methodology on two canonical biological models—a viral infection dynamics model and an enzyme kinetics model. For each model they generate synthetic data under multiple noise levels (e ranging from 0.01 to 0.1) and sampling frequencies, then perform 100 Monte‑Carlo runs using both WENDy and a conventional output‑error (OE) least‑squares estimator. The results show that (i) WENDy achieves comparable or superior parameter accuracy while requiring roughly an order of magnitude less CPU time than OE; (ii) the (e, q) criterion provides a more nuanced assessment than average relative error, revealing that parameters remain practically identifiable (q ≤ 0.2) even when noise is relatively high (e ≈ 0.05), whereas OE often exceeds acceptable q thresholds; (iii) the weak‑form approach is robust to noise because the integration against smooth test functions effectively filters high‑frequency disturbances.

The paper also discusses limitations. The method relies on the existence of an input‑output equation; for some high‑dimensional or highly coupled systems such equations may be absent or computationally infeasible to derive. Moreover, extending the (e, q) framework to multiple output variables requires a principled scaling strategy, which the authors leave for future work. They suggest possible extensions such as Bayesian integration of (e, q) thresholds, adaptive test‑function design, and coupling with optimal experimental design to pre‑emptively ensure practical identifiability.

In summary, this work contributes a theoretically grounded, easily interpretable practical identifiability metric and demonstrates that weak‑form parameter estimation can dramatically accelerate the assessment of identifiability in partially observed nonlinear dynamical systems. By bridging differential‑algebraic elimination with noise‑robust weak‑form estimation, the authors provide a powerful toolset for modelers in systems biology, epidemiology, and related fields, enabling more reliable inference and better‑informed experimental planning.