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Identifiability Challenges in Sparse Linear Ordinary Differential Equations

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
dynamical systemsidentifiabilitysparsity
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.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
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Overview

Overall Novelty Assessment

The paper characterizes identifiability conditions for sparse linear ordinary differential equations learned from single trajectory observations. It resides in the Single Trajectory Identifiability leaf, which contains only three papers total, indicating a relatively focused research direction within the broader taxonomy of 22 papers. The work challenges the conventional wisdom that linear ODEs are almost surely identifiable from a single trajectory by demonstrating that sparsity introduces fundamental unidentifiability with positive probability in practically relevant regimes. This positions the contribution at the intersection of classical identifiability theory and modern sparse system identification.

The taxonomy reveals that identifiability theory branches into single-trajectory, multi-trajectory, and hidden-confounder scenarios, while neighboring branches address algorithmic sparse system estimation and ODE reconstruction methods. The paper's theoretical focus distinguishes it from sibling categories like Sparse Linear System Estimation, which emphasizes regularized estimators rather than fundamental identifiability limits. The scope notes clarify that identifiability theory excludes estimation algorithms, suggesting this work provides foundational analysis that could inform but does not directly overlap with the algorithmic branches. The sparse regime appears underexplored in the identifiability literature compared to the dense case.

Among ten candidates examined for the empirical demonstration contribution, none provided clear refutation, though all ten were classified as non-refutable or unclear. The theoretical contributions on sparse identifiability characterization and near-unidentifiability analysis had zero candidates examined, suggesting these may represent novel theoretical angles within the limited search scope. The statistics indicate a modest literature search scale focused on semantic similarity rather than exhaustive coverage. The absence of refutable candidates among examined papers suggests the specific combination of sparsity constraints and single-trajectory identifiability analysis may not have direct precedents in the top-ranked semantic matches.

Based on the limited search of ten semantically similar candidates, the work appears to address a gap in sparse linear ODE identifiability theory. The taxonomy structure shows this is a relatively sparse research direction with few directly comparable papers. However, the analysis does not cover the full breadth of dynamical systems literature, and the zero-candidate examination for two contributions limits confidence in assessing their novelty. The empirical validation component shows no overlap within the examined scope, though broader algorithmic literature may contain related experimental studies.

Taxonomy

Core-task Taxonomy Papers
22
3
Claimed Contributions
10
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: identifiability of sparse linear ordinary differential equations from trajectory data. The field centers on determining when and how one can uniquely recover the structure and parameters of dynamical systems from observed time-series measurements. The taxonomy reveals several complementary perspectives: Identifiability Theory and Analysis establishes fundamental conditions under which system parameters can be uniquely determined, often focusing on minimal data requirements such as single-trajectory scenarios explored in works like Single Trajectory Identifiability[1] and Single Trajectory Identifiability[5]. Sparse System Identification Methods develop algorithmic frameworks that exploit sparsity assumptions to reconstruct governing equations, as seen in approaches like Unified Sparse Dynamical Inference[7] and Sparse High-Dimensional Dynamics[4]. ODE Reconstruction from Data emphasizes practical inference techniques that balance model complexity with data fidelity, while Specialized Dynamical System Modeling addresses domain-specific challenges such as hidden confounders or delayed dynamics. Optimization and Regularization Techniques provide the computational backbone, leveraging methods like coordinate descent and sparsity-promoting penalties to handle high-dimensional parameter spaces. A particularly active line of work examines the tension between data efficiency and identifiability guarantees: some studies investigate how much trajectory information is minimally necessary, while others explore sample complexity bounds as in Sample Complexity Sparse Identification[17]. Another contrast emerges between purely theoretical identifiability conditions and practical reconstruction algorithms that must contend with noise and model misspecification. Sparse Linear ODE Identifiability[0] sits within the Single Trajectory Identifiability branch, sharing theoretical concerns with Single Trajectory Identifiability[1] and Single Trajectory Identifiability[5] about minimal observational requirements. However, where earlier works like Single Trajectory Identifiability[5] laid foundational uniqueness results, Sparse Linear ODE Identifiability[0] appears to refine these conditions specifically for sparse linear systems, potentially tightening the interplay between sparsity structure and trajectory-based inference. This positions the work at the intersection of rigorous identifiability theory and the practical demand for data-efficient system discovery.

Claimed Contributions

Characterization of identifiability in sparse linear ODEs

The authors prove that sparse linear ordinary differential equations are unidentifiable with positive probability in realistic sparsity regimes, contrasting with the known result that dense linear ODEs are almost surely identifiable. They provide theoretical lower bounds on the probability of unidentifiability and identify a sharp threshold at sparsity level p = 1 - ln(n)/n.

0 retrieved papers
Theoretical analysis of near-unidentifiability via distance metric

The authors introduce a continuous distance metric dA that measures closeness to unidentifiability when initial conditions lie near proper invariant subspaces. They prove bounds showing how this distance affects the time horizon over which competing systems remain indistinguishable, formalizing the concept of near-unidentifiability.

0 retrieved papers
Empirical demonstration of practical unidentifiability in sparse systems

The authors empirically validate that theoretical unidentifiability manifests in practice by evaluating state-of-the-art methods (SINDy and Neural ODEs) on sparse linear systems. They demonstrate that inductive biases and optimization dynamics do not circumvent the fundamental identifiability challenges predicted by theory.

10 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Characterization of identifiability in sparse linear ODEs

The authors prove that sparse linear ordinary differential equations are unidentifiable with positive probability in realistic sparsity regimes, contrasting with the known result that dense linear ODEs are almost surely identifiable. They provide theoretical lower bounds on the probability of unidentifiability and identify a sharp threshold at sparsity level p = 1 - ln(n)/n.

Contribution

Theoretical analysis of near-unidentifiability via distance metric

The authors introduce a continuous distance metric dA that measures closeness to unidentifiability when initial conditions lie near proper invariant subspaces. They prove bounds showing how this distance affects the time horizon over which competing systems remain indistinguishable, formalizing the concept of near-unidentifiability.

Contribution

Empirical demonstration of practical unidentifiability in sparse systems

The authors empirically validate that theoretical unidentifiability manifests in practice by evaluating state-of-the-art methods (SINDy and Neural ODEs) on sparse linear systems. They demonstrate that inductive biases and optimization dynamics do not circumvent the fundamental identifiability challenges predicted by theory.

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