An introduction to kernel and operator learning methods for homogenization by self-consistent clustering analysis

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 current deep learning methods is high and unscalable. The article presents a thorough analysis on the mathematical underpinnings of the operator learning paradigm and proposes a kernel learning method that maps between Banach spaces of functions. 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 show 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, at least on the real line. 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.

Automated summary

Recent advances in operator learning theory have led to the development of a kernel learning method that can map between infinite dimensional spaces. However, the high training cost and lack of scalability of current deep learning methods pose challenges for large-scale engineering problems. This article provides a comprehensive analysis of the mathematical foundations of operator learning and proposes an algorithm to analytically approximate piecewise constant functions, suggesting the potential success of neural operators. Additionally, the article discusses the application of a kernel operator learning method for multiscale homogenization and presents preliminary results.