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

2026-06-25 · Source: Machine Learning · Field: Science & Research — Mathematics & Computational Sciences, Artificial Intelligence & Machine Learning · Depth: Expert, quick

Summary

This research addresses the theoretical gap in uniquely and stably identifying governing Ordinary Differential Equations (ODEs) from observed solution data. It introduces the Hausdorff distance on solution sets as a novel metric for comparing differential equations, capturing the worst-case separation between two equations across all admissible initial conditions. The study establishes identifiability bounds for a broad range of ODE structures, including linear ODEs and nonlinear classes with Lipschitz (Hölder)-continuous vector fields, precisely characterizing when distinct equations can be differentiated using solution data. Furthermore, the authors derive metric entropy estimates for relevant ODE classes and analyze sample complexity bounds, quantifying the minimum number of solution observations required to reliably recover the governing equation. This work provides foundational insights into the theoretical limits of learning ODEs in scientific machine learning.

Key takeaway

For AI Scientists and Research Scientists developing data-driven models for physical systems, understanding the theoretical limits of equation recovery is crucial. This research provides quantitative identifiability bounds and sample complexity estimates, informing your experimental design and data collection strategies. You should consider the Hausdorff distance metric when evaluating the distinguishability of candidate ODEs and planning the minimum solution observations needed for robust model identification.

Key insights

The Hausdorff distance on solution sets quantifies ODE identifiability and sample complexity from observed data.

Principles

• Hausdorff distance captures worst-case ODE separation.
• Identifiability bounds characterize distinguishable ODEs.
• Sample complexity quantifies needed observations.

Method

The study establishes identifiability bounds and derives metric entropy estimates for ODE classes, then analyzes sample complexity using the Hausdorff distance on solution sets.

Topics

• Scientific Machine Learning
• Ordinary Differential Equations
• Identifiability Bounds
• Sample Complexity
• Hausdorff Distance
• Nonlinear Systems

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.