Machine learning methods for the prediction of micromagnetic magnetization dynamics

TL;DR

This paper explores machine learning techniques, including kernel methods and neural network autoencoders, to efficiently predict micromagnetic magnetization dynamics, aiming to automate and accelerate solutions of complex parameter-dependent PDEs.

Contribution

It introduces novel kernel and neural network-based dimensionality reduction methods for micromagnetic simulations, with theoretical insights and computational validation.

Findings

Kernel ridge regression is fast and accurate.

Neural network autoencoders can outperform kernel methods in accuracy.

Methods are validated on a standard micromagnetic problem.

Abstract

Machine learning (ML) entered the field of computational micromagnetics only recently. The main objective of these new approaches is the automatization of solutions of parameter-dependent problems in micromagnetism such as fast response curve estimation modeled by the Landau-Lifschitz-Gilbert (LLG) equation. Data-driven models for the solution of time- and parameter-dependent partial differential equations require high dimensional training data-structures. ML in this case is by no means a straight-forward trivial task, it needs algorithmic and mathematical innovation. Our work introduces theoretical and computational conceptions of certain kernel and neural network based dimensionality reduction approaches for efficient prediction of solutions via the notion of low-dimensional feature space integration. We introduce efficient treatment of kernel ridge regression and kernel principal component analysis via low-rank approximation. A second line follows neural network (NN) autoencoders as nonlinear data-dependent dimensional reduction for the training data with focus on accurate latent space variable description suitable for a feature space integration scheme. We verify and compare numerically by means of a NIST standard problem. The low-rank kernel method approach is fast and surprisingly accurate, while the NN scheme can even exceed this level of accuracy at the expense of significantly higher costs.