  Finite element-based high-order solvers of conservation laws offer large
accuracy but face challenges near discontinuities due to the Gibbs phenomenon.
Artificial viscosity is a popular and effective solution to this problem based
on physical insight. In this work, we present a physics-informed machine
learning algorithm to automate the discovery of artificial viscosity models in
a non-supervised paradigm. The algorithm is inspired by reinforcement learning
and trains a neural network acting cell-by-cell (the viscosity model) by
minimizing a loss defined as the difference with respect to a reference
solution thanks to automatic differentiation. This enables a dataset-free
training procedure. We prove that the algorithm is effective by integrating it
into a state-of-the-art Runge-Kutta discontinuous Galerkin solver. We showcase
several numerical tests on scalar and vectorial problems, such as Burgers' and
Euler's equations in one and two dimensions. Results demonstrate that the
proposed approach trains a model that is able to outperform classical viscosity
models. Moreover, we show that the learnt artificial viscosity model is able to
generalize across different problems and parameters.
