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.

我们建立了一个机器学习模型，用于预测作为由 Landau-Lifschitz-Gilbert 方程（微磁运动的偏微分方程）描述的外部场的函数的磁化动力学。该模型允许快速准确地确定对外部场的响应，这由薄膜标准问题说明。数据驱动的方法内部通过无监督学习的非线性模型约简来降低问题的维数。这不仅使对时间步长的准确预测成为可能，而且还决定性地降低了学习过程的复杂性，其中将来自与不同外部场相关联的模拟微磁动力学的磁化状态用作输入数据。我们使用内核主成分的截断表示来描述时间预测之间的状态。由于核矩阵的低秩近似以及核主成分分析和核岭回归的相关低秩扩展，该方法能够处理大型训练样本集。该方法将计算完全转变为降维设置，将问题维度从数千个分解为数十个。