In this contribution we propose a data-driven surrogate model for the prediction of magnetic stray fields in two-dimensional random micro-heterogeneous materials. Since data driven models require thousands of training data sets, FEM simulations appear to be too time consuming. Hence, a stochastic model based on Brownian motion, which utilizes an efficient evaluation of stochastic transition matrices, is applied for the training data generation. For the encoding of the microstructure and the optimization of the surrogate model, two architectures are compared, i.e. the so-called UResNet model and the Fourier Convolutional neural network (FCNN). Here we analyze two FCNNs, one based on the discrete cosine transformation and one based on the complex-valued discrete Fourier transformation. Finally, we compare the magnetic stray fields for independent microstructures (not used in the training set) with results from the FE$^2$ method, a numerical homogenization scheme, to demonstrate the efficiency of the proposed surrogate model.

翻译：本文提出了一种基于数据驱动的代理模型，用于预测二维随机微异质材料中的杂散磁场。由于数据驱动模型需要数千组训练数据，有限元模拟往往耗时过长。因此，我们采用基于布朗运动的随机模型，利用随机转移矩阵的高效计算方法来生成训练数据。针对微观结构的编码与代理模型的优化，我们比较了两种架构：UResNet模型与傅里叶卷积神经网络（FCNN）。本文分析了两种FCNN架构，分别基于离散余弦变换与复值离散傅里叶变换。最后，我们将独立微结构（未包含在训练集中）的杂散磁场预测结果与数值均匀化方法FE$^2$的结果进行对比，以验证所提出代理模型的有效性。