Physics informed neural networks (PINNs) have emerged as a powerful tool to provide robust and accurate approximations of solutions to partial differential equations (PDEs). However, PINNs face serious difficulties and challenges when trying to approximate PDEs with dominant hyperbolic character. This research focuses on the development of a physics informed deep learning framework to approximate solutions to nonlinear PDEs that can develop shocks or discontinuities without any a-priori knowledge of the solution or the location of the discontinuities. The work takes motivation from finite element method that solves for solution values at nodes in the discretized domain and use these nodal values to obtain a globally defined solution field. Built on the rigorous mathematical foundations of the discontinuous Galerkin method, the framework naturally handles imposition of boundary conditions (Neumann/Dirichlet), entropy conditions, and regularity requirements. Several numerical experiments and validation with analytical solutions demonstrate the accuracy, robustness, and effectiveness of the proposed framework.

翻译：物理信息神经网络已成为求解偏微分方程近似解的强大工具，能够提供稳健且精确的近似结果。然而，在尝试近似具有显著双曲特性的偏微分方程时，物理信息神经网络面临严峻的困难与挑战。本研究致力于开发一个基于物理信息的深度学习框架，用于近似能够产生激波或间断的非线性偏微分方程的解，且无需预先了解解的特征或间断位置。该研究的灵感来源于有限元方法——即在离散化区域的节点上求解解值，并利用这些节点值获得全局定义的解场。该框架建立在间断伽辽金方法的严谨数学基础之上，能够自然处理边界条件（诺伊曼/狄利克雷）、熵条件以及正则性要求。多项数值实验及与解析解的验证结果，证明了所提框架的准确性、鲁棒性与有效性。