---
title: "Recovering Governing Equations from Solution Data: Identifiability Boundaries for Linear and Nonlinear ODEs"
source: "arXiv AI/CL/LG（论文）"
source_tier: "t2"
url: "https://arxiv.org/abs/2606.27285v1"
permalink: "https://feed.astralor.com/items/itm_e1179e15847a51f4746ae20b"
published: "2026-06-25T17:02:08.000Z"
score: 54
channel: "ai"
tags: ["Identifiability","Differential Equations","Equation Discovery","Scientific machine learning"]
---

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

## At a glance

This paper investigates the theoretical conditions for uniquely and stably identifying a ground-truth ODE from multiple solution observations. By introducing the Hausdorff distance on solution sets as a natural metric for comparing differential equations, it establishes identifiability bounds for linear and Lipschitz/Hölder continuous nonlinear ODE classes, and derives metric entropy estimates and sample complexity lower bounds, providing theoretical guarantees for governing equation discovery in scientific machine learning.

## Key points

- Proposes the Hausdorff distance on solution sets as a natural metric for comparing differential equations, capturing worst-case separation.
- Establishes identifiability bounds for linear ODEs and nonlinear (Lipschitz/Hölder continuous) ODE classes.
- Derives metric entropy estimates and sample complexity bounds for relevant ODE classes, quantifying the number of solution observations needed for reliable recovery.

## Full text

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.

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