  In this work, we introduce a spatio-temporal kernel for Gaussian process (GP)
regression-based sound field estimation. Notably, GPs have the attractive
property that the sound field is a linear function of the measurements,
allowing the field to be estimated efficiently from distributed microphone
measurements. However, to ensure analytical tractability, most existing kernels
for sound field estimation have been formulated in the frequency domain, formed
independently for each frequency. To address the analytical intractability of
spatio-temporal kernels, we here propose to instead learn the kernel directly
from data by the means of deep kernel learning. Furthermore, to improve the
generalization of the deep kernel, we propose a method for regularizing the
learning process using the wave equation. The representational advantages of
the deep kernel and the improved generalization obtained by using the wave
equation regularization are illustrated using numerical simulations.
