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

2026-06-25

AI Summary

Current theory lacks a quantitative characterization of the conditions and sample complexity required for uniquely and stably identifying the true ordinary differential equation (ODE) from observed solution data. This work proposes the Hausdorff distance between solution sets as a natural metric for comparing ODEs and establishes, for the first time, an identifiability framework based on this metric, encompassing both linear and nonlinear systems with Lipschitz or Hölder continuous vector fields. By leveraging metric entropy estimates, minimax analysis, and function space theory, the study derives precise learnability conditions, sharp distinguishability criteria, and upper bounds on sample complexity for classes of ODEs. These results provide rigorous theoretical guarantees for equation discovery tasks and fill a critical gap in the theoretical understanding of ODE identifiability within scientific machine learning.

Abstract

Learning governing equations from observed solution data is a fundamental challenge in scientific machine learning \cite{bruntonDiscoveringGoverningEquations2016,kovachkiNeuralOperatorLearning2023,longPDENetLearningPDEs2018,rudyDatadrivenDiscoveryPartial2017,raonicConvolutionalNeuralOperators2023}, yet the theoretical conditions under which a ground-truth ODE can be uniquely and stably identified from multiple solution observations remain largely undeveloped, and no quantitative analysis of the sample complexity of such learning tasks exists in the literature. To address this gap, we introduce the Hausdorff distance on solution sets as the natural metric for comparing differential equations, since it captures the worst-case separation between two equations over all admissible initial conditions and thus encodes the minimax structure of the identification problem. We establish identifiability bounds for governing ODEs across a wide class of structure equations--ranging from linear ODEs to nonlinear classes with Lipschitz (Hölder)-continuous vector fields--characterizing precisely when two distinct equations can be distinguished from solution data. Using this metric, we derive metric entropy estimates for the relevant ODE classes and analyze sample complexity bounds, quantifying how many solution observations are needed to reliably recover the governing equation.

Problem

• governing equations
• identifiability
• ordinary differential equations
• sample complexity
• solution data

Innovation

• Hausdorff distance
• identifiability bounds
• sample complexity
• governing equations
• metric entropy