用微分方程解析深度神经网络，打通理论与实践的桥梁
Understanding the Theoretical Foundations of Deep Neural Networks through Differential Equations
- 将整个网络或单层看作微分方程，建立统一理论框架
- 提供可解释的模型设计与性能优化路径
- 适合对理论深度和模型可解释性感兴趣的读者
深度神经网络（DNN）在实践中取得了显著成功，但缺乏系统的理论基础仍限制其发展。本文综述了微分方程作为理解、分析和改进DNN的理论基础。围绕三个核心问题展开：一、微分方程如何为DNN架构提供原理性解释；二、如何利用微分方程工具以原则性方式提升DNN性能；三、哪些实际应用受益于基于微分方程的DNN建模。从模型层面（将整个DNN视为微分方程）与层层面（将各组件建模为微分方程）双重视角，梳理该框架如何连接模型设计、理论分析与性能改进。进一步讨论实际应用场景，以及未来研究的关键挑战与机遇。
原文摘要 · Abstract (English)
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.
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