  We introduce the concept of numerical Gaussian processes, which we define as
Gaussian processes with covariance functions resulting from temporal
discretization of time-dependent partial differential equations. Numerical
Gaussian processes, by construction, are designed to deal with cases where: (1)
all we observe are noisy data on black-box initial conditions, and (2) we are
interested in quantifying the uncertainty associated with such noisy data in
our solutions to time-dependent partial differential equations. Our method
circumvents the need for spatial discretization of the differential operators
by proper placement of Gaussian process priors. This is an attempt to construct
structured and data-efficient learning machines, which are explicitly informed
by the underlying physics that possibly generated the observed data. The
effectiveness of the proposed approach is demonstrated through several
benchmark problems involving linear and nonlinear time-dependent operators. In
all examples, we are able to recover accurate approximations of the latent
solutions, and consistently propagate uncertainty, even in cases involving very
long time integration.
