This work focuses on enriching Discontinuous Galerkin (DG) bases to improve the approximation of steady solutions for hyperbolic systems of balance laws. The enrichment is based on a prior, computed using a parametric Physics-Informed Neural Network (PINN), which approximates a family of steady solutions. In practice, we multiply the original elements of the Taylor basis by this prior function, resulting in a new, enriched basis. This amounts to creating a reduced-order model for near-equilibrium simulations. Indeed, using the enhanced basis on a coarse mesh will yield similar errors compared to the classical basis on a fine mesh. This approach has been validated on several linear and nonlinear hyperbolic systems, in one and two space dimensions. Moreover, we are able to prove an error estimate. Under standard assumptions, for any smooth function, the error between the function and its projection onto the enriched basis is shown to be small, with the size of the error depending on how well the prior approximates both the solution and its derivatives. This error estimate demonstrates that the multiplicative correction mainly impacts the error constant of the DG scheme, improving it based on higher-order derivatives of the ratio between the prior and the solution. Therefore, it is essential that the prior function provides a good approximation of the solution and its derivatives, which further validates the effectiveness of using PINNs.