  This work concerns the enrichment of Discontinuous Galerkin (DG) bases, so
that the resulting scheme provides a much better approximation of steady
solutions to hyperbolic systems of balance laws. The basis enrichment leverages
a prior -- an approximation of the steady solution -- which we propose to
compute using a Physics-Informed Neural Network (PINN). To that end, after
presenting the classical DG scheme, we show how to enrich its basis with a
prior. Convergence results and error estimates follow, in which we prove that
the basis with prior does not change the order of convergence, and that the
error constant is improved. To construct the prior, we elect to use parametric
PINNs, which we introduce, as well as the algorithms to construct a prior from
PINNs. We finally perform several validation experiments on four different
hyperbolic balance laws to highlight the properties of the scheme. Namely, we
show that the DG scheme with prior is much more accurate on steady solutions
than the DG scheme without prior, while retaining the same approximation
quality on unsteady solutions.

Numerical Analysis;Numerical Analysis