Operator Learning Meets Numerical Analysis: Improving Neural Networks through Iterative Methods

TL;DR

This paper connects deep neural networks with classical numerical analysis by framing them as iterative operator methods, providing theoretical insights and empirical evidence that iterative approaches enhance performance.

Contribution

It introduces a theoretical framework linking neural networks to iterative operator methods and demonstrates benefits of iteration in architectures like diffusion models and AlphaFold.

Findings

Iterative methods improve neural network performance.

Popular architectures inherently employ iterative operator learning.

Introduction of the PIGN graph neural network with iterative benefits.

Abstract

Deep neural networks, despite their success in numerous applications, often function without established theoretical foundations. In this paper, we bridge this gap by drawing parallels between deep learning and classical numerical analysis. By framing neural networks as operators with fixed points representing desired solutions, we develop a theoretical framework grounded in iterative methods for operator equations. Under defined conditions, we present convergence proofs based on fixed point theory. We demonstrate that popular architectures, such as diffusion models and AlphaFold, inherently employ iterative operator learning. Empirical assessments highlight that performing iterations through network operators improves performance. We also introduce an iterative graph neural network, PIGN, that further demonstrates benefits of iterations. Our work aims to enhance the understanding of deep learning by merging insights from numerical analysis, potentially guiding the design of future networks with clearer theoretical underpinnings and improved performance.