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.

翻译：我们认为使用神经网络的非线性双曲 PDE 的微弱解决方案近似于传统 PINN 方法,但使用残余物的弱(双)规范。 这是最近所谓的“弱 PINN ” 的变体。 我们提供了一些明确的计算方法,强调传统的 PINN 方法对非线性双曲保护法的不连续解决方案效果不佳,我们建议对弱的 PINN 方法进行一些修改,从而导致效率更高的计算和较小的错误。