# Learning time-stepping by nonlinear dimensionality reduction to predict magnetization dynamics

## TL;DR

This paper introduces a data-driven, nonlinear dimensionality reduction method using kernel techniques to efficiently predict magnetization dynamics in micromagnetism, reducing computational complexity without needing field evaluations after training.

## Contribution

It presents a novel time-stepping learning algorithm based on kernel methods that accurately predicts magnetization dynamics with minimal degrees of freedom in feature space.

## Key findings

- Only two degrees of freedom sufficed to describe the dynamics.
- The method requires no field evaluations after training.
- It is applicable regardless of spatial discretization.

## Abstract

We establish a time-stepping learning algorithm and apply it to predict the solution of the partial differential equation of motion in micromagnetism as a dynamical system depending on the external field as parameter. The data-driven approach is based on nonlinear model order reduction by use of kernel methods for unsupervised learning, yielding a predictor for the magnetization dynamics without any need for field evaluations after a data generation and training phase as precomputation. Magnetization states from simulated micromagnetic dynamics associated with different external fields are used as training data to learn a low-dimensional representation in so-called feature space and a map that predicts the time-evolution in reduced space. Remarkably, only two degrees of freedom in feature space were enough to describe the nonlinear dynamics of a thin-film element. The approach has no restrictions on the spatial discretization and might be useful for fast determination of the response to an external field.