This work concerns the enrichment of Discontinuous Galerkin (DG) bases, so that the resulting scheme provides a much better approximation of steady solutions to hyperbolic systems of balance laws. The basis enrichment leverages a prior -- an approximation of the steady solution -- which we propose to compute using a Physics-Informed Neural Network (PINN). To that end, after presenting the classical DG scheme, we show how to enrich its basis with a prior. Convergence results and error estimates follow, in which we prove that the basis with prior does not change the order of convergence, and that the error constant is improved. To construct the prior, we elect to use parametric PINNs, which we introduce, as well as the algorithms to construct a prior from PINNs. We finally perform several validation experiments on four different hyperbolic balance laws to highlight the properties of the scheme. Namely, we show that the DG scheme with prior is much more accurate on steady solutions than the DG scheme without prior, while retaining the same approximation quality on unsteady solutions.

翻译：本文关注间断伽辽金（DG）基底的增强问题，使得改进后的格式能更精确地逼近双曲平衡律系统的稳态解。基底增强利用了先验信息——稳态解的近似——我们提出采用物理信息神经网络（PINN）进行计算。为此，在阐述经典DG格式后，我们展示如何用先验信息增强其基底。随后给出收敛性结果与误差估计，证明带有先验的基底不会改变收敛阶，且误差常数得到改善。为构建先验，我们选择使用参数化PINN，并介绍其定义及从PINN生成先验的算法。最后，我们在四种不同双曲平衡律上开展多项验证实验，以突出格式的特性。具体而言，我们表明：带有先验的DG格式在稳态解上比无先验的DG格式精度显著提升，同时在非稳态解上保持相同的逼近质量。