  We show how to learn discrete field theories from observational data of
fields on a space-time lattice. For this, we train a neural network model of a
discrete Lagrangian density such that the discrete Euler--Lagrange equations
are consistent with the given training data. We, thus, obtain a
structure-preserving machine learning architecture. Lagrangian densities are
not uniquely defined by the solutions of a field theory. We introduce a
technique to derive regularisers for the training process which optimise
numerical regularity of the discrete field theory. Minimisation of the
regularisers guarantees that close to the training data the discrete field
theory behaves robust and efficient when used in numerical simulations.
Further, we show how to identify structurally simple solutions of the
underlying continuous field theory such as travelling waves. This is possible
even when travelling waves are not present in the training data. This is
compared to data-driven model order reduction based approaches, which struggle
to identify suitable latent spaces containing structurally simple solutions
when these are not present in the training data. Ideas are demonstrated on
examples based on the wave equation and the Schr\"odinger equation.
