Wind Field Reconstruction with Adaptive Random Fourier Features

TL;DR

This paper introduces an adaptive random Fourier features approach for reconstructing near-surface wind fields from sparse data, outperforming traditional methods like Kriging and inverse distance weighting.

Contribution

It develops a novel adaptive Fourier features model with divergence and Sobolev penalties, including a derived sampling density and an adaptive sampling algorithm for improved wind field reconstruction.

Findings

Random Fourier features outperform benchmarks in wind field reconstruction.

The method incorporates physically motivated divergence penalties.

An adaptive sampling algorithm improves frequency selection.

Abstract

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 $\beta(\pmb x) = \sum_{k=1}^K \beta_k e^{i\omega_k \pmb x}$ 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 $|\nabla \cdot \beta(\pmb x)|^2$, 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.