  Gaussian processes (GPs) are a popular model for spatially referenced data
and allow descriptive statements, predictions at new locations, and simulation
of new fields. Often a few parameters are sufficient to parameterize the
covariance function, and maximum likelihood (ML) methods can be used to
estimate these parameters from data. ML methods, however, are computationally
demanding. For example, in the case of local likelihood estimation, even
fitting covariance models on modest size windows can overwhelm typical
computational resources for data analysis. This limitation motivates the idea
of using neural network (NN) methods to approximate ML estimates. We train NNs
to take moderate size spatial fields or variograms as input and return the
range and noise-to-signal covariance parameters. Once trained, the NNs provide
estimates with a similar accuracy compared to ML estimation and at a speedup by
a factor of 100 or more. Although we focus on a specific covariance estimation
problem motivated by a climate science application, this work can be easily
extended to other, more complex, spatial problems and provides a
proof-of-concept for this use of machine learning in computational statistics.
