  Physics-informed Neural Network (PINN) is a promising tool that has been
applied in a variety of physical phenomena described by partial differential
equations (PDE). However, it has been observed that PINNs are difficult to
train in certain "stiff" problems, which include various nonlinear hyperbolic
PDEs that display shocks in their solutions. Recent studies added a diffusion
term to the PDE, and an artificial viscosity (AV) value was manually tuned to
allow PINNs to solve these problems. In this paper, we propose three approaches
to address this problem, none of which rely on an a priori definition of the
artificial viscosity value. The first method learns a global AV value, whereas
the other two learn localized AV values around the shocks, by means of a
parametrized AV map or a residual-based AV map. We applied the proposed methods
to the inviscid Burgers equation and the Buckley-Leverett equation, the latter
being a classical problem in Petroleum Engineering. The results show that the
proposed methods are able to learn both a small AV value and the accurate shock
location and improve the approximation error over a nonadaptive global AV
alternative method.
