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.

开发了一种基于物理信息的深度学习框架，用于近似解非线性偏微分方程，可以在不预先了解解的性质或不连续点的位置的情况下发展出震荡或不连续的解。

解决双曲型偏微分方程的深度学习框架：第一部分

A Deep Learning Framework for Solving Hyperbolic Partial Differential  Equations: Part I

We introduce a new relativistic astrophysics code, SpECTRE, that combines a
discontinuous Galerkin method with a task-based parallelism model. SpECTRE's
goal is to achieve more accurate solutions for challenging relativistic
astrophysics problems such as core-collapse supernovae and binary neutron star
mergers. The robustness of the discontinuous Galerkin method allows for the use
of high-resolution shock capturing methods in regions where (relativistic)
shocks are found, while exploiting high-order accuracy in smooth regions. A
task-based parallelism model allows efficient use of the largest supercomputers
for problems with a heterogeneous workload over disparate spatial and temporal
scales. We argue that the locality and algorithmic structure of discontinuous
Galerkin methods will exhibit good scalability within a task-based parallelism
framework. We demonstrate the code on a wide variety of challenging benchmark
problems in (non)-relativistic (magneto)-hydrodynamics. We demonstrate the
code's scalability including its strong scaling on the NCSA Blue Waters
supercomputer up to the machine's full capacity of 22,380 nodes using 671,400
threads.

该研究介绍了一种新的相对论天体物理代码，使用间续 Galerkin 方法和基于任务的并行模型来实现更准确的解决方案，可用于解决波动方程、核心坍缩超新星和双中子星并合等相对论天体物理问题，并展示了代码在各种具有挑战性的基准问题上的可扩展性和可靠性。