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.

翻译：本文考虑使用神经网络逼近非线性双曲守恒律的熵解。我们通过显式计算揭示了经典PINNs为何无法处理非线性双曲守恒律的间断解，并论证了损失泛函中应使用偏微分方程残差的弱（对偶）范数——该方法近期被称为“弱物理信息神经网络（weak PINNs）”。我们提出若干修改方案以降低弱PINNs的训练难度，数值实验表明这些修改能用更少的训练步骤获得更小的误差。此外，我们将wPINNs拓展至含弱边界条件的标量守恒律以及双曲守恒律方程组，并通过数值实验评估该扩展方法的精度与效率。