  The recently developed physics-informed machine learning has made great
progress for solving nonlinear partial differential equations (PDEs), however,
it may fail to provide reasonable approximations to the PDEs with discontinuous
solutions. In this paper, we focus on the discrete time physics-informed neural
network (PINN), and propose a hybrid PINN scheme for the nonlinear PDEs. In
this approach, the local solution structures are classified as smooth and
nonsmooth scales by introducing a discontinuity indicator, and then the
automatic differentiation technique is employed for resolving smooth scales,
while an improved weighted essentially non-oscillatory (WENO) scheme is adopted
to capture discontinuities. We then test the present approach by considering
the viscous and inviscid Burgers equations , and it is shown that compared with
original discrete time PINN, the present hybrid approach has a better
performance in approximating the discontinuous solution even at a relatively
larger time step.
