An axiomatized PDE model of deep neural networks

TL;DR

This paper establishes a PDE-based framework for deep neural networks, specifically modeling them as convection-diffusion equations, which enhances understanding, robustness, and training methods for ResNets.

Contribution

It introduces a PDE model for DNNs as convection-diffusion equations, providing new theoretical insights and a novel training approach for ResNets.

Findings

The convection-diffusion PDE model explains effective network behaviors.

The model improves neural network robustness.

Experimental results validate the new training method.

Abstract

Inspired by the relation between deep neural network (DNN) and partial differential equations (PDEs), we study the general form of the PDE models of deep neural networks. To achieve this goal, we formulate DNN as an evolution operator from a simple base model. Based on several reasonable assumptions, we prove that the evolution operator is actually determined by convection-diffusion equation. This convection-diffusion equation model gives mathematical explanation for several effective networks. Moreover, we show that the convection-diffusion model improves the robustness and reduces the Rademacher complexity. Based on the convection-diffusion equation, we design a new training method for ResNets. Experiments validate the performance of the proposed method.