  We introduce a method based on Gaussian process regression to identify
discrete variational principles from observed solutions of a field theory. The
method is based on the data-based identification of a discrete Lagrangian
density. It is a geometric machine learning technique in the sense that the
variational structure of the true field theory is reflected in the data-driven
model by design. We provide a rigorous convergence statement of the method. The
proof circumvents challenges posed by the ambiguity of discrete Lagrangian
densities in the inverse problem of variational calculus. Moreover, our method
can be used to quantify model uncertainty in the equations of motions and any
linear observable of the discrete field theory. This is illustrated on the
example of the discrete wave equation and Schr\"odinger equation. The article
constitutes an extension of our previous article arXiv:2404.19626 for the
data-driven identification of (discrete) Lagrangians for variational dynamics
from an ode setting to the setting of discrete pdes.
