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 $beta(pmb x) = sum_{k=1}^K beta_k e^{iomega_k pmb x}$ approximating the velocity field, with frequencies $omega_k$ randomly sampled and amplitudes $beta_k$ trained to minimize a loss function. We include a physically motivated divergence penalty term $| abla cdot beta(pmb x)|^2$, 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.