Wind Field Reconstruction with Adaptive Random Fourier Features

We investigate the use of spatial interpolation methods for reconstructing the horizontal near-surface wind field given a sparse set of measurements. In particular, random Fourier features is compared to a set of benchmark methods including Kriging and Inverse distance weighting. Random Fourier features is a linear model β(pmb x) = sumk=1K βk eiωk pmb x approximating the velocity field, with frequencies ωk randomly sampled and amplitudes βk trained to minimize a loss function. We include a physically motivated divergence penalty term |nabla · β(pmb x)|2, as well as a penalty on the Sobolev norm. We derive a bound on the generalization error and derive a sampling density that minimizes the bound. Following (arXiv:2007.10683 [math.NA]), we devise an adaptive Metropolis-Hastings algorithm for sampling the frequencies of the optimal distribution. In our experiments, our random Fourier features model outperforms the benchmark models.