Symmetry-Based Structural Identifiability and Observability Analysis of PDE and ODE Models
DOI:
https://doi.org/10.17979/ja-cea.2026.47.13688Keywords:
Structural Identifiability, Observability, Scaling, Symmetry, ODEs (Ordinary Differential Equations), PDEs (Partial Differential Equations)Abstract
Structural identifiability and observability are properties that describe the ability to determine the parameters and states of a model from knowledge of its equations, inputs, and outputs. A model is structurally unidentifiable if different parameter sets produce identical outputs. This issue may arise in partially observed systems. In this work, we follow an approach, to which we refer as the SIM algorithm, to analyse identifiability and observability in dynamical systems based on finding its scaling symmetries. The proposed method relies on analysing the invariance of the system under parameter and state scaling transformations and can be systematically applied regardless of the model’s dimensionality. We report the implementation of the SIM method for ODE models in the STRIKE-GOLDD MATLAB toolbox, and extend it to partial differential equation (PDE) models through the newly introduced SIM-PDE algorithm. The approach is straightforward to implement and computationally efficient, making it suitable even for highly nonlinear and large-scale models.
References
Browning, A. P., Tască, M., Falcó, C., Baker, R. E., 2024. Structural identifiability analysis of linear reaction–advection–diffusion processes in mathematical biology. Proceedings of the Royal Society A 480 (2286), 20230911. URL: https://arxiv.org/abs/2309.15326
Byrne, H., Harrington, H., Ovchinnikov, A., Pogudin, G., Rahkooy, H., Soto, P., 01 2025. Algebraic identifiability of partial differential equation models. Nonlinearity 38. DOI: 10.1088/1361-6544/ada510
Castro, M., de Boer, R. J., 11 2020. Testing structural identifiability by a simple scaling method. PLOS Computational Biology 16 (11), 1–15. DOI: 10.1371/journal.pcbi.1008248
Díaz-Seoane, S., Rey Barreiro, X., Villaverde, A., 11 2022. Strike-goldd 4.0: user-friendly, efficient analysis of structural identifiability and observability. Bioinformatics 39. DOI: 10.1093/bioinformatics/btac748
Díaz-Seoane, S., Sellán, E., Villaverde, A. F., 2023. Structural identifiability and observability of microbial community models. Bioengineering 10 (4). DOI: 10.3390/bioengineering10040483
DiStefano III, J., 2015. Dynamic systems biology modeling and simulation. Academic Press.
Holmberg, A., 1982. On the practical identifiability of microbial growth models incorporating michaelis-menten type nonlinearities. Mathematical Biosciences 62 (1), 23–43.
Ljung, L., Glad, T., 1994. On global identifiability for arbitrary model parametrizations. Automatica 30 (2), 265–276. DOI: 10.1016/0005-1098(94)90029-9
Merkt, B., Timmer, J., Kaschek, D., 07 2015. Higher-order lie symmetries in identifiability and predictability analysis of dynamic models. Physical review. E, Statistical, nonlinear, and soft matter physics 92, 012920. DOI: 10.1103/PhysRevE.92.012920
Raue, A., Kreutz, C., Maiwald, T., Bachmann, J., Schilling, M., Klingm¨uller, U., Timmer, J., 2009. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25 (15), 1923–1929.
Renardy, M., Kirschner, D., Eisenberg, M., 01 2022. Structural identifiability analysis of age-structured pde epidemic models. Journal of Mathematical Biology 84. DOI: 10.1007/s00285-021-01711-1
Stigter, J. D., Molenaar, J., 2015. A fast algorithm to assess local structural identifiability. Automatica 58, 118–124.
Villaverde, A. F., Massonis, G., 10 2021. On testing structural identifiability by a simple scaling method: Relying on scaling symmetries can be misleading. PLOS Computational Biology 17 (10), 1–8. DOI: 10.1371/journal.pcbi.1009032
Walter, E., Pronzato, L., 1997. Identification of parametric models: from experimental data. Springer.
Wieland, F.-G., Hauber, A. L., Rosenblatt, M., T¨onsing, C., Timmer, J., 2021. On structural and practical identifiability. Current Opinion in Systems Biology 25, 60–69. DOI: 10.1016/j.coisb.2021.03.005
Wu, H., Zhao, Y., Xu, X., Lou, J., 2025. Global stability and identifiability of the immuno-epidemiological model. Advances in Continuous and Discrete Models 2025 (1), 131. DOI: 10.1186/s13662-025-03983-z
Yates, J. W., Evans, N. D., Chappell, M. J., 2009. Structural identifiability analysis via symmetries of differential equations. Automatica 45 (11), 2585–2591. DOI: 10.1016/j.automatica.2009.07.009
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Mahmoud Shams Falavarjani, Paula Sánchez Calvo, Alejandro F. Villaverde
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
