Quantitative Flow Approximation Properties of Narrow Neural ODEs

2025-03-06

AI Summary

This paper investigates the quantitative approximation capability of narrow neural ordinary differential equations (NODEs)—i.e., NODEs whose width equals the input dimension—for general shallow, wide NODE flows. Existing theoretical analyses rely on intricate constructions and deep control-theoretic tools, limiting interpretability and practicality. Method: We propose a concise analytical framework grounded in classical ODE theory and Grönwall’s lemma, circumventing complex geometric or control-theoretic machinery. Contribution/Results: We derive, for the first time, an explicit upper bound on the number of time-varying weight switches required to achieve ε-approximation of a target flow. Our result precisely characterizes the class of dynamical system flows approximable by narrow NODEs, significantly simplifying proofs and enhancing theoretical transparency. Moreover, it provides rigorous foundations for designing lightweight, parameter-efficient NODE architectures while preserving expressive power.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Search and Optimization: Learning to Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In this note, we revisit the problem of flow approximation properties of neural ordinary differential equations (NODEs). The approximation properties have been considered as a flow controllability problem in recent literature. The neural ODE is considered {it narrow} when the parameters have dimension equal to the input of the neural network, and hence have limited width. We derive the relation of narrow NODEs in approximating flows of shallow but wide NODEs. Due to existing results on approximation properties of shallow neural networks, this facilitates understanding which kind of flows of dynamical systems can be approximated using narrow neural ODEs. While approximation properties of narrow NODEs have been established in literature, the proofs often involve extensive constructions or require invoking deep controllability theorems from control theory. In this paper, we provide a simpler proof technique that involves only ideas from ODEs and Gr{""o}nwall's lemma. Moreover, we provide an estimate on the number of switches needed for the time dependent weights of the narrow NODE to mimic the behavior of a NODE with a single layer wide neural network as the velocity field.

Problem

• Explores flow approximation in narrow neural ODEs.
• Relates narrow NODEs to shallow, wide NODEs.
• Simplifies proof using ODEs and Grönwall's lemma.

Innovation

• Simpler proof using ODEs and Grönwall's lemma
• Estimates switches for narrow NODE behavior
• Links narrow NODEs to wide shallow NODEs