The generic problem in supervised machine learning is to learn a function f from a collection of samples, with the objective of predicting the value taken by f for any given input. In effect, the learning procedure consists in constructing an explicit function that approximates f in some sense. In this article is introduced a Fourier‐based machine learning method which could be an alternative or a complement to neural networks for applications in engineering. The basic idea is to extend f into a periodic function so as to use partial sums of the Fourier series as approximations. For this approach to be effective in high dimension, it proved necessary to use several ideas and concepts such as regularization, Sobol sequences and hyperbolic crosses. An attractive feature of the proposed method is that the training stage reduces to a quadratic programming problem. The presented method is first applied to some examples of high‐dimensional analytical functions, which allows some comparisons with neural networks to be made. An application to a homogenization problem in nonlinear conduction is discussed in detail. Various examples related to global sensitivity analysis, assessing effective energies of microstructures, and solving boundary value problems are presented.

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基于傅立叶的机器学习技术及其在工程中的应用

监督机器学习中的通用问题是从样本集合中学习函数f，目的是预测f对于任何给定输入所取的值。实际上，学习过程在于构造一个在某种意义上近似于f的显式函数。本文介绍了一种基于傅立叶的机器学习方法，该方法可以替代或补充神经网络以用于工程应用。基本思想是扩展f转换为周期函数，以便使用傅立叶级数的部分和作为近似值。为了使这种方法在高维度上有效，事实证明有必要使用一些思想和概念，例如正则化，Sobol序列和双曲线叉。所提出的方法的一个吸引人的特征是训练阶段减少了二次规划问题。所提出的方法首先应用于高维分析函数的一些示例，从而可以与神经网络进行一些比较。详细讨论了非线性传导中均质化问题的一个应用。给出了与全局灵敏度分析，评估微结构的有效能量以及解决边界值问题有关的各种示例。