论文标题
改善双曲线保护法的弱PINN：双重规范计算，边界条件和系统
Improving Weak PINNs for Hyperbolic Conservation Laws: Dual Norm Computation, Boundary Conditions and Systems
论文作者
论文摘要
我们考虑使用神经网络的非线性双曲保护法的熵溶液的近似。我们提供了明确的计算，这些计算强调了为什么经典的PINN对非线性双曲线保护定律不连续的解决方案不起作用，并表明PDE残留物的弱（双重）规范应在损失功能中使用。该方法最近被称为“弱Pinns”。我们建议对弱的PINN进行一些修改，使他们的训练更容易，从而导致较小的训练会导致较小的训练，如数值实验所示。此外，我们将WPINN扩展到具有弱边界数据和双曲线保护法的标量保护定律。我们执行数值实验，以评估扩展方法的准确性和效率。
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.
