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Deep Neural Networks Inspired by Differential Equations

Yongshuai Liu, Lianfang Wang, Kuilin Qin, Qinghua Zhang, Faqiang Wang, Li Cui, Jun Liu, Yuping Duan, Tieyong Zeng

TL;DR

This survey surveys how deep learning can be viewed through the lens of differential equations, unifying deterministic (ODE/PDE) and stochastic (SDE) dynamics to guide architecture and training. It categorizes approaches into DDE-driven architectures (e.g., ResNets, LM-ResNets, RevNets, PDE-CNNs) and deterministic dynamics (Neural ODEs, flow-based models), as well as stochastic methods (stochastic regularization and Neural SDEs) that inform robust learning and generative modeling. The paper provides a structured synthesis of theoretical foundations, representative models, and empirical evidence (e.g., CIFAR-10/100 performance) and discusses future directions toward higher-order, delayed, and fractional DEs, non-Gaussian noise, and domain-specific constraints. By linking continuous-time dynamics, regularization, and diffusion-based generative processes, it underscores the potential for more interpretable, stable, and data-efficient deep learning systems with rigorous mathematical underpinnings.

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.

Deep Neural Networks Inspired by Differential Equations

TL;DR

This survey surveys how deep learning can be viewed through the lens of differential equations, unifying deterministic (ODE/PDE) and stochastic (SDE) dynamics to guide architecture and training. It categorizes approaches into DDE-driven architectures (e.g., ResNets, LM-ResNets, RevNets, PDE-CNNs) and deterministic dynamics (Neural ODEs, flow-based models), as well as stochastic methods (stochastic regularization and Neural SDEs) that inform robust learning and generative modeling. The paper provides a structured synthesis of theoretical foundations, representative models, and empirical evidence (e.g., CIFAR-10/100 performance) and discusses future directions toward higher-order, delayed, and fractional DEs, non-Gaussian noise, and domain-specific constraints. By linking continuous-time dynamics, regularization, and diffusion-based generative processes, it underscores the potential for more interpretable, stable, and data-efficient deep learning systems with rigorous mathematical underpinnings.

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.

Paper Structure

This paper contains 35 sections, 49 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fundamentals of Deep Learning
  4. Stochastic Differential Equations
  5. First-Order ODEs
  6. Higher-Order ODEs
  7. Systems of Ordinary Differential Equations (ODESs)
  8. Partial Differential Equations
  9. Stochastic Differential Equations
  10. From DDEs to Deep Neural Networks
  11. DDE-Driven Architecture
  12. ResNets he2016deep
  13. Linear Multistep ResNets lu2018beyond
  14. Reversible Residual Networks (RevNets) gomez2017reversible
  15. PDE-CNNs ruthotto2020deep
  16. ...and 20 more sections

Figures (3)

  • Figure 1: Overview of DDE-driven neural network architectures (left, Sec.\ref{['sec:DDEs_NN']}) and stochastic regularization methods inspired by SDEs (right, Sec.\ref{['Sec:Network_regularition']}).
  • Figure 2: Overview of ODE-guided deterministic dynamics (left, Sec. \ref{['Sec:modeling_DD']}) and SDE-guided stochastic dynamics in deep generative modeling (right, Sec. \ref{['sec:dynamic_SDEs']}).
  • Figure 3: Schematics of representative network architectures inspired by discretizations of DEs. The diagrams illustrate the computational pathways and connections characteristic of each architecture.