An Introduction to Kernel and Operator Learning Methods for Homogenization by Self-consistent Clustering Analysis

TL;DR

This paper reviews kernel and operator learning theories, proposes a kernel learning method for function spaces, and demonstrates its application to multiscale homogenization using clustered domain analysis and graph kernel networks.

Contribution

It introduces a novel kernel operator learning approach for homogenization, combining clustering and graph kernels to improve efficiency in multiscale simulations.

Findings

Kernel learning methods can approximate piecewise constant functions.

Clustered domain analysis enhances the feasibility of neural operators.

Graph kernel networks provide a mechanistic reduced order model.

Abstract

Recent advances in operator learning theory have improved our knowledge about learning maps between infinite dimensional spaces. However, for large-scale engineering problems such as concurrent multiscale simulation for mechanical properties, the training cost for the current operator learning methods is very high. The article presents a thorough analysis on the mathematical underpinnings of the operator learning paradigm and proposes a kernel learning method that maps between function spaces. We first provide a survey of modern kernel and operator learning theory, as well as discuss recent results and open problems. From there, the article presents an algorithm to how we can analytically approximate the piecewise constant functions on R for operator learning. This implies the potential feasibility of success of neural operators on clustered functions. Finally, a k-means clustered domain on the basis of a mechanistic response is considered and the Lippmann-Schwinger equation for micro-mechanical homogenization is solved. The article briefly discusses the mathematics of previous kernel learning methods and some preliminary results with those methods. The proposed kernel operator learning method uses graph kernel networks to come up with a mechanistic reduced order method for multiscale homogenization.