# Identifiability Analysis of Linear Ordinary Differential Equation Systems with a Single Trajectory

arXiv: https://arxiv.org/abs/2103.05660 · HTML: https://mlsanity.com/paper/2103.05660/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Mar 2021.

## Built on by (most cited first, 10 of 10)

- [ODEFormer: Symbolic Regression of Dynamical Systems with Transformers](https://mlsanity.com/paper/2310.05573/) (2023)
- [Marrying Causal Representation Learning with Dynamical Systems for Science](https://mlsanity.com/paper/2405.13888/) (2024)
- [Identifiability and Asymptotics in Learning Homogeneous Linear ODE Systems from Discrete Observations](https://mlsanity.com/paper/2210.05955/) (2022)
- [Generator Identification for Linear SDEs with Additive and Multiplicative Noise](https://mlsanity.com/paper/2310.19491/) (2023)
- [Identifiability Challenges in Sparse Linear Ordinary Differential Equations](https://mlsanity.com/paper/2506.09816/) (2025)
- [When is a System Discoverable from Data? Discovery Requires Chaos](https://mlsanity.com/paper/2511.08860/) (2025)
- [Identifiability Analysis of Linear ODE Systems with Hidden Confounders](https://mlsanity.com/paper/2410.21917/) (2024)
- [Physics from Video: Identifiability of Time-Invariant Second-Order ODEs under Minimal Trajectory Conditions](https://mlsanity.com/paper/2606.00115/) (2026)
- [Limits of Learning Linear Dynamics from Experiments](https://mlsanity.com/paper/2605.12010/) (2026)
- [Identifying ODEs from Unstructured Data with Causal Representation Learning](https://mlsanity.com/paper/2609.37083/) (2026)

## Abstract (arXiv)

Ordinary differential equations (ODEs) are widely used to model dynamical behavior of systems. It is important to perform identifiability analysis prior to estimating unknown parameters in ODEs (a.k.a. inverse problem), because if a system is unidentifiable, the estimation procedure may fail or produce erroneous and misleading results. Although several qualitative identifiability measures have been proposed, much less effort has been given to developing quantitative (continuous) scores that are robust to uncertainties in the data, especially for those cases in which the data are presented as a single trajectory beginning with one initial value. In this paper, we first derived a closed-form representation of linear ODE systems that are not identifiable based on a single trajectory. This representation helps researchers design practical systems and choose the right prior structural information in practice. Next, we proposed several quantitative scores for identifiability analysis in practice. In simulation studies, the proposed measures outperformed the main competing method significantly, especially when noise was presented in the data. We also discussed the asymptotic properties of practical identifiability for high-dimensional ODE systems and conclude that, without additional prior information, many random ODE systems are practically unidentifiable when the dimension approaches infinity.