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：本研究关注间断伽辽金（DG）基函数的增强方法，旨在使改进后的格式能更精确地逼近双曲型守恒律方程组的稳态解。基函数增强过程利用了先验知识——即稳态解的近似值，我们提出采用物理信息神经网络（PINN）计算该先验解。为此，在介绍经典DG格式后，我们展示了如何利用先验知识增强其基函数。随后给出收敛性结果与误差估计，证明了带先验的基函数不会改变收敛阶数，且能改善误差常数。在先验解的构建中，我们选用参数化PINN方法（本文亦将介绍该方法）以及从PINN生成先验解的算法。最后，我们在四个不同的双曲型守恒律方程上进行了多组验证实验，以凸显该格式的特性。实验表明：在保持非稳态解近似质量不变的前提下，带先验的DG格式对稳态解的逼近精度显著优于无先验的经典DG格式。