Understanding the Theoretical Foundations of Deep Neural Networks through Differential Equations

TL;DR

This paper explores how differential equations can serve as a foundational framework for understanding, analyzing, and enhancing deep neural networks, bridging theory and practical applications.

Contribution

It introduces a differential equations perspective to interpret DNN architectures, analyze their behavior, and guide performance improvements, connecting theoretical insights with real-world applications.

Findings

Differential equations provide a principled understanding of DNN architectures.

Tools from differential equations can improve DNN performance systematically.

The framework connects model design, analysis, and real-world applications.

Abstract

Deep neural networks (DNNs) have achieved remarkable empirical success, yet the absence of a principled theoretical foundation continues to hinder their systematic development. In this survey, we present differential equations as a theoretical foundation for understanding, analyzing, and improving DNNs. We organize the discussion around three guiding questions: i) how differential equations offer a principled understanding of DNN architectures, ii) how tools from differential equations can be used to improve DNN performance in a principled way, and iii) what real-world applications benefit from grounding DNNs in differential equations. We adopt a two-fold perspective spanning the model level, which interprets the whole DNN as a differential equation, and the layer level, which models individual DNN components as differential equations. From these two perspectives, we review how this framework connects model design, theoretical analysis, and performance improvement. We further discuss real-world applications, as well as key challenges and opportunities for future research.