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Identifiability Challenges in Sparse Linear Ordinary Differential Equations
Cecilia Caso · 2026-05-11 · via cs.LG updates on arXiv.org

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Abstract:Dynamical systems modeling is a core pillar of scientific inquiry across natural and life sciences. Increasingly, dynamical system models are learned from data, rendering identifiability a paramount concept. For systems that are not identifiable from data, no guarantees can be given about their behavior under new conditions and inputs, or about possible control mechanisms to steer the system. It is known in the community that "linear ordinary differential equations (ODE) are almost surely identifiable from a single trajectory." However, this only holds for dense matrices. The sparse regime remains underexplored, despite its practical relevance with sparsity arising naturally in many biological, social, and physical systems. In this work, we address this gap by characterizing the identifiability of sparse linear ODEs. Contrary to the dense case, we show that sparse systems are unidentifiable with a positive probability in practically relevant sparsity regimes and provide lower bounds for this probability. We further study empirically how this theoretical unidentifiability manifests in state-of-the-art methods to estimate linear ODEs from data. Our results corroborate that sparse systems are also practically unidentifiable. Theoretical limitations are not resolved through inductive biases or optimization dynamics. Our findings call for rethinking what can be expected from data-driven dynamical system modeling and allows for quantitative assessments of how much to trust a learned linear ODE.
Comments: 9 pages, 4 figures
Subjects: Machine Learning (cs.LG)
Cite as: arXiv:2506.09816 [cs.LG]
  (or arXiv:2506.09816v3 [cs.LG] for this version)
  https://doi.org/10.48550/arXiv.2506.09816

arXiv-issued DOI via DataCite
Journal reference: The Fourteenth International Conference on Learning Representations, ICLR 2026

Submission history

From: Cecilia Casolo [view email]
[v1] Wed, 11 Jun 2025 14:55:36 UTC (4,142 KB)
[v2] Thu, 12 Jun 2025 11:03:30 UTC (4,142 KB)
[v3] Fri, 8 May 2026 09:32:31 UTC (4,962 KB)

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