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arxiv: 2506.09816 · v3 · submitted 2025-06-11 · 💻 cs.LG

Identifiability Challenges in Sparse Linear Ordinary Differential Equations

Cecilia Casolo , S\"oren Becker , Niki Kilbertus This is my paper

Pith reviewed 2026-05-19 09:33 UTC · model grok-4.3

classification 💻 cs.LG
keywords identifiabilitysparse linear ODEdynamical systemssingle trajectoryunidentifiability probabilitydata-driven modeling
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The pith

Sparse linear ODEs are unidentifiable from a single trajectory with positive probability in common sparsity regimes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates identifiability of linear ordinary differential equations when the coefficient matrix is sparse. It establishes that unlike dense matrices, which are almost surely identifiable from one observed trajectory, sparse matrices yield unidentifiability with positive probability under sparsity levels typical of real systems. Lower bounds on this probability are provided. Experiments show that modern estimation algorithms inherit the same limitation in practice, even with standard inductive biases. The findings imply that learned sparse models cannot support reliable predictions or control without additional safeguards.

Core claim

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, confirming that sparse systems are also practically unidentifiable and that theoretical limitations are not resolved through inductive biases or optimization dynamics.

What carries the argument

Probabilistic characterization of when a sparse coefficient matrix produces a non-unique linear ODE consistent with a single observed trajectory.

Load-bearing premise

The sparsity pattern of the coefficient matrix follows distributions typical of biological, social, and physical systems.

What would settle it

A concrete sparse matrix drawn from the relevant regime whose observed trajectory admits only one consistent linear ODE would contradict the positive-probability unidentifiability result.

Figures

Figures reproduced from arXiv: 2506.09816 by Cecilia Casolo, Niki Kilbertus, S\"oren Becker.

Figure 1
Figure 1. Figure 1: Identifiability differs between dense and sparse systems. The trajectories in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Proportion of matrices satisfying the conditions i), ii) and iii) at different system dimensions [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Box-plots of smoothed condition numbers (SCN) and distance-to-unidentifiability [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Normalized Hamming distance of reconstructed systems using SINDy (left) and NODE [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Proportion of trajectories that have been well-reconstructed by Sparse Neural ODEs for [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Relative difference in sparsity count (lower the better) between the true and reconstructed [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Box-plots of distance-to-unidentifiability [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Box-plots of smoothed condition numbers (SCN) in log-scale for the least and most [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Box-plots of distance-to-unidentifiability [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Proportion of matrices satisfying the conditions conditions [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Proportion of matrices satisfying the conditions conditions [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Proportion of matrices satisfying the conditions conditions [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Hamming distance for different generating settings for SINDy on trajectories generated [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Hamming distance for different generating settings for NODE on trajectories generated [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper claims that, unlike dense linear ODEs which are almost surely identifiable from a single trajectory, sparse linear ODEs are unidentifiable with positive probability under sparsity regimes typical of biological, social, and physical systems. It derives explicit lower bounds on this unidentifiability probability and provides empirical evidence that state-of-the-art estimation methods do not overcome the theoretical limitations, even with inductive biases.

Significance. If the central claim holds, the work is significant for data-driven dynamical systems modeling. It supplies an independent theoretical characterization of identifiability failure in the sparse regime together with separate empirical validation on practical methods, enabling quantitative assessment of when learned linear ODEs can be trusted for prediction or control in sparse domains.

major comments (1)
  1. [§3] §3 (Theoretical Results), the probability space over sparsity patterns: the lower bounds on P(unidentifiable) are derived under an i.i.d. Bernoulli model for the support of A. This choice is load-bearing for the claim that unidentifiability occurs with positive probability 'in practically relevant sparsity regimes' and 'distributions typical of biological, social, and physical systems,' because configuration, preferential-attachment, or modular models (explicitly named in the abstract) assign probability zero or exponentially small to the disconnected-component and invariant-subspace events that drive the bounds.
minor comments (2)
  1. [Abstract] The abstract and introduction use 'practically relevant sparsity regimes' without an explicit cross-reference to the precise sparsity parameters or random-graph model employed in the theorems.
  2. [§2] Notation for the observed trajectory and the cyclic subspace generated by x0 could be introduced earlier in the preliminaries to improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive and detailed review. We appreciate the recognition of the significance of our results on identifiability challenges for sparse linear ODEs. We address the major comment below and describe the planned revisions.

read point-by-point responses

  1. Referee: [§3] §3 (Theoretical Results), the probability space over sparsity patterns: the lower bounds on P(unidentifiable) are derived under an i.i.d. Bernoulli model for the support of A. This choice is load-bearing for the claim that unidentifiability occurs with positive probability 'in practically relevant sparsity regimes' and 'distributions typical of biological, social, and physical systems,' because configuration, preferential-attachment, or modular models (explicitly named in the abstract) assign probability zero or exponentially small to the disconnected-component and invariant-subspace events that drive the bounds.

    Authors: We thank the referee for this precise observation. Our explicit lower bounds on the unidentifiability probability are derived under the i.i.d. Bernoulli model on the support of A, which permits direct calculation of the probabilities of the relevant events (disconnected components and invariant subspaces). This is a standard modeling choice in the sparse random matrix literature. While the abstract cites configuration, preferential-attachment, and modular models as motivating examples of sparsity in applied domains, the quantitative claims and bounds in §3 are specific to the Bernoulli setting. We agree that, under certain alternative generative models, the probability of the critical events may be zero or exponentially small, which narrows the direct applicability of the stated lower bounds. In the revised manuscript we will add a clarifying paragraph in §3 that (i) states the modeling assumption explicitly, (ii) notes that the positive-probability result is tied to the Bernoulli regime, and (iii) suggests that analogous identifiability questions under other sparsity models constitute a natural direction for future work. This change will make the scope and limitations of the theoretical results more transparent without altering the core contribution. The empirical section, which demonstrates that state-of-the-art estimators fail to recover unique parameters on sparse systems drawn from a variety of practical regimes, remains independent of the specific generative model and continues to support the practical message of the paper. revision: partial

Circularity Check

0 steps flagged

No significant circularity in identifiability analysis for sparse linear ODEs

full rationale

The paper derives lower bounds on the probability of unidentifiability for sparse linear ODEs by considering a probability space over sparsity patterns (such as independent Bernoulli) and analyzing when the cyclic subspace generated by a generic initial condition x0 becomes proper, allowing distinct sparse matrices A' to produce identical trajectories. This follows from standard results in linear algebra and measure theory applied to the support of A, without reducing to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The empirical validation on state-of-the-art estimators is presented separately and does not underpin the theoretical claims. The derivation is self-contained and externally verifiable under the stated assumptions on the sparsity distribution.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, invented entities, or ad-hoc axioms are evident from the abstract; the work builds on standard linear ODE assumptions and known dense-case results.

pith-pipeline@v0.9.0 · 5767 in / 985 out tokens · 30351 ms · 2026-05-19T09:33:54.648894+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

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  1. Limits of Learning Linear Dynamics from Experiments

    cs.LG 2026-05 unverdicted novelty 7.0

    The reachable subspace dynamics of LTI systems remain uniquely identifiable from any experiment, even when the full system is not.

  2. Symbolic recovery of PDEs from measurement data

    cs.LG 2026-02 unverdicted novelty 7.0

    Symbolic rational-function networks recover an admissible PDE from noiseless complete measurements and select the regularization-minimizing parameterization within the architecture.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · cited by 2 Pith papers

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    As expected, the (mean) empirical measure closely matches the theoretical expected distance between a random unit vector x0 and a d0-dimensional subspace of Rn [Vershynin, 2018], given by E[dA(x0) | n, d0] = Γ(n/2)Γ((n − d0 + 1)/2) Γ((n − d0)/2)Γ((n + 1)/2) , hence (in expectation) validating the computation of our distance-to-unidentifiability dA. 0 1 2 ...

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