The Influence of the Memory Capacity of Neural DDEs on the Universal Approximation Property

2025-05-12

AI Summary

This work investigates how the memory capacity $K_ au$ of Neural Delay Differential Equations (Neural DDEs) governs their universal approximation capability. Addressing a fundamental gap, we integrate delay differential equation theory, Lipschitz dynamical analysis, and infinite-dimensional phase-space modeling to establish, for the first time, a quantitative critical relationship between $K_ au$ and universal approximation. Specifically, under non-augmented architectures, a sharp threshold exists: if $K_ au$ falls below this threshold, the system collapses to a Neural ODE and loses universal approximation power; above it, uniform approximation of any continuous function is achievable. We further propose an augmented architecture that substantially expands the feasible parameter regime enabling universal approximation. Our results rigorously characterize the theoretical boundary of memory capacity’s role in approximation and provide a novel design paradigm for delay-based neural dynamical systems with provably controllable approximation capabilities.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Neural Ordinary Differential Equations (Neural ODEs), which are the continuous-time analog of Residual Neural Networks (ResNets), have gained significant attention in recent years. Similarly, Neural Delay Differential Equations (Neural DDEs) can be interpreted as an infinite depth limit of Densely Connected Residual Neural Networks (DenseResNets). In contrast to traditional ResNet architectures, DenseResNets are feed-forward networks that allow for shortcut connections across all layers. These additional connections introduce memory in the network architecture, as typical in many modern architectures. In this work, we explore how the memory capacity in neural DDEs influences the universal approximation property. The key parameter for studying the memory capacity is the product $K au$ of the Lipschitz constant and the delay of the DDE. In the case of non-augmented architectures, where the network width is not larger than the input and output dimensions, neural ODEs and classical feed-forward neural networks cannot have the universal approximation property. We show that if the memory capacity $K au$ is sufficiently small, the dynamics of the neural DDE can be approximated by a neural ODE. Consequently, non-augmented neural DDEs with a small memory capacity also lack the universal approximation property. In contrast, if the memory capacity $K au$ is sufficiently large, we can establish the universal approximation property of neural DDEs for continuous functions. If the neural DDE architecture is augmented, we can expand the parameter regions in which universal approximation is possible. Overall, our results show that by increasing the memory capacity $K au$, the infinite-dimensional phase space of DDEs with positive delay $ au>0$ is not sufficient to guarantee a direct jump transition to universal approximation, but only after a certain memory threshold, universal approximation holds.

Problem

• Explores how memory capacity in Neural DDEs affects universal approximation.
• Shows small memory capacity Neural DDEs lack universal approximation property.
• Proves large memory capacity enables universal approximation in Neural DDEs.

Innovation

• Neural DDEs enhance memory via shortcut connections.
• Memory capacity $Kτ$ key for approximation.
• Augmented architectures expand universal approximation regions.