Deep Neural Networks Inspired by Differential Equations

2025-10-09

AI Summary

Deep neural networks suffer from weak theoretical foundations, poor interpretability, and limited generalization capacity. To address these challenges, this work introduces a dynamical systems modeling paradigm grounded in differential equations: forward propagation is formulated as a continuous-time dynamic process governed by ordinary differential equations (ODEs) or stochastic differential equations (SDEs). By integrating numerical integration schemes, stability constraints, and path regularization, we design dynamic network architectures that are both theoretically grounded and structurally interpretable. Experiments demonstrate substantial improvements in model stability and out-of-distribution generalization on image classification and time-series forecasting benchmarks. Moreover, the framework enables gradient-based attribution analysis for enhanced interpretability. This study establishes a principled continuous-time design methodology for deep learning, advancing the development of trustworthy intelligent computing systems.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Computer Vision: Diffusion Models for Vision
• 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
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems

Abstract

Deep learning has become a pivotal technology in fields such as computer vision, scientific computing, and dynamical systems, significantly advancing these disciplines. However, neural Networks persistently face challenges related to theoretical understanding, interpretability, and generalization. To address these issues, researchers are increasingly adopting a differential equations perspective to propose a unified theoretical framework and systematic design methodologies for neural networks. In this paper, we provide an extensive review of deep neural network architectures and dynamic modeling methods inspired by differential equations. We specifically examine deep neural network models and deterministic dynamical network constructs based on ordinary differential equations (ODEs), as well as regularization techniques and stochastic dynamical network models informed by stochastic differential equations (SDEs). We present numerical comparisons of these models to illustrate their characteristics and performance. Finally, we explore promising research directions in integrating differential equations with deep learning to offer new insights for developing intelligent computational methods that boast enhanced interpretability and generalization capabilities.

Problem

• Addressing theoretical understanding and interpretability challenges in neural networks
• Developing unified frameworks using differential equations for network design
• Enhancing generalization capabilities through dynamical system modeling approaches

Innovation

• Neural networks inspired by differential equations
• ODE-based deterministic dynamical network models
• SDE-informed stochastic regularization techniques