Prediction of magnetization dynamics in a reduced dimensional feature space setting utilizing a low-rank kernel method

• Published in: Journal of computational physics 2021-11, Vol.444, p.110586, Article 110586
• Main Authors: Exl, Lukas, Mauser, Norbert J., Schaffer, Sebastian, Schrefl, Thomas, Suess, Dieter
• Format: Article
• Language: English
• Subjects: Breaking down; Computational physics; Equations of motion; Kernels; Low-rank kernel approximation; Low-rank kernel principal component analysis; Machine learning; Magnetization; Micromagnetics; Model reduction; Nonlinear model order reduction; Nystroem approximation; Partial differential equations; Principal components analysis
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2021.110586

•A machine learning model to predict the dynamics described by the LLG equation with the field as parameter is established.•We introduce low-rank kernel principal component analysis and low-rank kernel ridge regression for larger training sets.•The model is trained entirely in a reduced dimensional feature space obtained from unsupervised learning. We establish a machine learning model for the prediction of the magnetization dynamics as function of the external field described by the Landau-Lifschitz-Gilbert equation, the partial differential equation of motion in micromagnetism. The model allows for fast and accurate determination of the response to an external field which is illustrated by a thin-film standard problem. The data-driven method internally reduces the dimensionality of the problem by means of nonlinear model reduction for unsupervised learning. This not only makes accurate prediction of the time steps possible, but also decisively reduces complexity in the learning process where magnetization states from simulated micromagnetic dynamics associated with different external fields are used as input data. We use a truncated representation of kernel principal components to describe the states between time predictions. The method is capable of handling large training sample sets owing to a low-rank approximation of the kernel matrix and an associated low-rank extension of kernel principal component analysis and kernel ridge regression. The approach entirely shifts computations into a reduced dimensional setting breaking down the problem dimension from the thousands to the tens.