Identifiability Challenges in Sparse Linear Ordinary Differential Equations

2025-06-11

AI Summary

This work investigates the identifiability of sparse linear ordinary differential equations (ODEs) from a single trajectory. Unlike dense linear ODEs—which are almost surely identifiable—sparse structures, prevalent in biological and social systems, induce non-negligible probability of unidentifiability, a phenomenon previously lacking systematic characterization. We establish the first theoretical identifiability boundary for sparse linear ODEs, deriving a rigorous lower bound on the unidentifiability probability and proving that neither inductive bias nor optimization dynamics can overcome this fundamental limitation. Our methodology integrates random matrix theory, directed graph modeling, and structural identifiability analysis. We empirically evaluate mainstream sparse learning algorithms—including SINDy—under realistic sparsity regimes. Results demonstrate severe estimation inaccuracies of existing methods at typical sparsity levels. Furthermore, we introduce the first quantitative framework for assessing model credibility in sparse ODE identification.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Identifiability challenges in sparse linear ODEs
• Sparse systems are unidentifiable in relevant regimes
• Theoretical unidentifiability affects practical estimation methods

Innovation

• Characterizing identifiability in sparse linear ODEs
• Providing lower bounds for unidentifiability probability
• Empirical study on state-of-the-art estimation methods