  We consider the approximation of entropy solutions of nonlinear hyperbolic
conservation laws using neural networks. We provide explicit computations that
highlight why classical PINNs will not work for discontinuous solutions to
nonlinear hyperbolic conservation laws and show that weak (dual) norms of the
PDE residual should be used in the loss functional. This approach has been
termed "weak PINNs" recently. We suggest some modifications to weak PINNs that
make their training easier, which leads to smaller errors with less training,
as shown by numerical experiments. Additionally, we extend wPINNs to scalar
conservation laws with weak boundary data and to systems of hyperbolic
conservation laws. We perform numerical experiments in order to assess the
accuracy and efficiency of the extended method.
