  Covariance estimation is ubiquitous in functional data analysis. Yet, the
case of functional observations over multidimensional domains introduces
computational and statistical challenges, rendering the standard methods
effectively inapplicable. To address this problem, we introduce Covariance
Networks (CovNet) as a modeling and estimation tool. The CovNet model is
universal -- it can be used to approximate any covariance up to desired
precision. Moreover, the model can be fitted efficiently to the data and its
neural network architecture allows us to employ modern computational tools in
the implementation. The CovNet model also admits a closed-form
eigen-decomposition, which can be computed efficiently, without constructing
the covariance itself. This facilitates easy storage and subsequent
manipulation in the context of the CovNet. Moreover, we establish consistency
of the proposed estimator and derive its rate of convergence. The usefulness of
the proposed method is demonstrated by means of an extensive simulation study.
