A Fourier-based machine learning technique with application in engineering

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.

Automated summary

The article introduces a Fourier-based machine learning method that extends a function into a periodic function and uses partial sums of the Fourier series for approximation. This method can serve as an alternative or complement to neural networks and has some attractive features. In addition to examples of high-dimensional analytical functions, the application to a problem in nonlinear conduction is discussed in detail, along with other examples related to global sensitivity analysis, assessing microstructure effective energies, and solving boundary value problems.