ReLU Deep Neural Networks from the Hierarchical Basis Perspective

Abstract

We study ReLU deep neural networks (DNNs) by investigating their connections with the hierarchical basis method in finite element methods. First, we show that the approximation schemes of ReLU DNNs for $x^2$ and $xy$ are composition versions of the hierarchical basis approximation for these two functions. Based on this fact, we obtain a geometric interpretation and systematic proof for the approximation result of ReLU DNNs for polynomials, which plays an important role in a series of recent exponential approximation results of ReLU DNNs. Through our investigation of connections between ReLU DNNs and the hierarchical basis approximation for $x^2$ and $xy$, we show that ReLU DNNs with this special structure can be applied only to approximate quadratic functions. Furthermore, we obtain a concise representation to explicitly reproduce any linear finite element function on a two-dimensional uniform mesh by using ReLU DNNs with only two hidden layers.

arXiv comment28 pages

References in corpus (7)

• DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
• DeepXDE: A deep learning library for solving differential equations
• On the Number of Linear Regions of Deep Neural Networks
• Deep Network Approximation for Smooth Functions
• ReLU Deep Neural Networks and Linear Finite Elements
• The Finite Neuron Method and Convergence Analysis
• Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularities

Cited by in corpus (2)

• Approximation Properties of Deep ReLU CNNs
• Power Series Expansion Neural Network