Vanilla Feedforward Neural Networks as a Discretization of Dynamical Systems

TL;DR

This paper demonstrates that vanilla feedforward neural networks can be viewed as numerical discretizations of dynamical systems, offering new insights into their approximation capabilities.

Contribution

It proves that standard feedforward networks, not just residual ones, can be interpreted as discretized dynamical systems using properties of leaky-ReLU and splitting methods.

Findings

Vanilla feedforward networks can be modeled as discretizations of dynamical systems.

The proof utilizes properties of leaky-ReLU functions.

Provides a new perspective on the approximation properties of neural networks.

Abstract

Deep learning has made significant applications in the field of data science and natural science. Some studies have linked deep neural networks to dynamic systems, but the network structure is restricted to the residual network. It is known that residual networks can be regarded as a numerical discretization of dynamic systems. In this paper, we back to the classical network structure and prove that the vanilla feedforward networks could also be a numerical discretization of dynamic systems, where the width of the network is equal to the dimension of the input and output. Our proof is based on the properties of the leaky-ReLU function and the numerical technique of splitting method to solve differential equations. Our results could provide a new perspective for understanding the approximation properties of feedforward neural networks.