Efficient wPINN-Approximations to Entropy Solutions of Hyperbolic Conservation Laws

We consider the approximation of weak solutions of nonlinear hyperbolic PDEs using neural networks, similar to the classical PINNs approach, but using a weak (dual) norm of the residual. This is a variant of what was termed "weak PINNs" recently. We provide some explicit computations that highlight why classical PINNs will not work well for discontinuous solutions to nonlinear hyperbolic conservation laws and we suggest some modifications to the weak PINN methodology that lead to more efficient computations and smaller errors.