Recovering Governing Equations from Solution Data: Identifiability Bounds for Linear and Nonlinear ODEs

2026-06-25 · Source: Takara TLDR - Daily AI Papers · Field: Science & Research — Mathematics & Computational Sciences, Research Methodology & Innovation · Depth: Expert, medium

Summary

Helmut Bölcskei and Yang Pan's paper, "Recovering Governing Equations from Solution Data: Identifiability Bounds for Linear and Nonlinear ODEs," addresses the challenge of uniquely and stably identifying ground-truth Ordinary Differential Equations (ODEs) from observed solution data. The authors introduce the Hausdorff distance on solution sets as a novel metric for comparing differential equations, which captures the worst-case separation between two equations across all admissible initial conditions. This metric is crucial for encoding the minimax structure of the identification problem. The research establishes precise identifiability bounds for a broad range of ODE structures, including linear and nonlinear classes with Lipschitz (Hölder)-continuous vector fields, defining when distinct equations can be differentiated using solution data. Furthermore, the paper quantifies the sample complexity by deriving metric entropy estimates and analyzing bounds, indicating the minimum number of solution observations required to reliably recover the governing equation.

Key takeaway

For research scientists developing data-driven methods to discover physical laws, this work provides critical theoretical foundations for understanding the limits of ODE identification. You should consider the established identifiability bounds and sample complexity estimates to design more robust and data-efficient learning algorithms, ensuring unique and stable recovery of governing equations from observed data. This helps avoid under-specified models.

Key insights

The Hausdorff distance quantifies ODE distinguishability and sample complexity from solution data.

Principles

• Identifiability bounds define ODE distinguishability.
• Hausdorff distance captures worst-case separation.
• Sample complexity quantifies needed observations.

Method

The paper introduces Hausdorff distance on solution sets, establishes identifiability bounds for ODE classes, and derives metric entropy and sample complexity estimates.

In practice

• Quantify data needs for ODE recovery.
• Evaluate distinguishability of ODE models.

Topics

• Governing Equations
• Ordinary Differential Equations
• Identifiability Theory
• Scientific Machine Learning
• Sample Complexity
• Hausdorff Distance

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Takara TLDR - Daily AI Papers.