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

arXiv: https://arxiv.org/abs/2212.00802 · HTML: https://mlsanity.com/paper/2212.00802/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 74 references that are in the map.
- First posted on arXiv 1 Dec 2022.

## Builds on (most cited first, 15 of 15)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [The Principles of Deep Learning Theory](https://mlsanity.com/paper/2106.10165/) (2021)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [Recurrent Localization Networks applied to the Lippmann-Schwinger Equation](https://mlsanity.com/paper/2102.00063/) (2021)
- [Adaptivity for clustering-based reduced-order modeling of localized history-dependent phenomena](https://mlsanity.com/paper/2109.11897/) (2021)
- [Stochastic additive manufacturing simulation: from experiment to surface roughness and porosity prediction](https://mlsanity.com/paper/2208.02907/) (2022)

## Built on by (most cited first, 6 of 6)

- [Fourier Neural Operator Surrogate Model to Predict 3D Seismic Waves Propagation](https://mlsanity.com/paper/2304.10242/) (2023)
- [Enhancing Fourier Neural Operators with Local Spatial Features](https://mlsanity.com/paper/2503.17797/) (2025)
- [Interpreting and generalizing deep learning in physics-based problems with functional linear models](https://mlsanity.com/paper/2307.04569/) (2023)
- [A Pretraining-Finetuning Computational Framework for Material Homogenization](https://mlsanity.com/paper/2404.07943/) (2024)
- [A Unified Generative-Predictive Framework for Deterministic Inverse Design](https://mlsanity.com/paper/2512.15746/) (2025)
- [Separable Neural Architectures as Physical World Models: from Mathematical Theory to Applications](https://mlsanity.com/paper/2606.14934/) (2026)

## Abstract (arXiv)

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.