---
title: "从解数据中恢复控制方程：线性和非线性ODE的识别性边界"
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: ["可识别性","微分方程","方程发现","科学机器学习"]
---

# 从解数据中恢复控制方程：线性和非线性ODE的识别性边界

## 速览

该论文研究了从观测解数据中唯一且稳定地识别真实ODE的理论条件。通过引入解集上的Hausdorff距离作为比较微分方程的自然度量，建立了线性及 Lipschitz/Hölder 连续非线性 ODE 类的可识别性边界，并推导了度量熵估计和样本复杂度下界，为科学机器学习中控制方程发现提供了理论保证。

## 关键点

- 提出了解集上的Hausdorff距离作为比较微分方程的自然度量，捕捉最坏情况下的分离程度。
- 建立了线性ODE及非线性（Lipschitz/Hölder连续）ODE的可识别性边界。
- 推导了相关ODE类的度量熵估计和样本复杂度界，量化了可靠恢复控制方程所需解观测数。

## 全文

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.

---

引用前请核对原文;本文由星际流动整理:https://feed.astralor.com/items/itm_e1179e15847a51f4746ae20b