Physics-Informed Neural Networks (PINN) are neural networks encoding the
problem governing equations, such as Partial Differential Equations (PDE), as a
part of the neural network. PINNs have emerged as a new essential tool to solve
various challenging problems, including computing linear systems arising from
PDEs, a task for which several traditional methods exist. In this work, we
focus first on evaluating the potential of PINNs as linear solvers in the case
of the Poisson equation, an omnipresent equation in scientific computing. We
characterize PINN linear solvers in terms of accuracy and performance under
different network configurations (depth, activation functions, input data set
distribution). We highlight the critical role of transfer learning. Our results
show that low-frequency components of the solution converge quickly as an
effect of the F-principle. In contrast, an accurate solution of the high
frequencies requires an exceedingly long time. To address this limitation, we
propose integrating PINNs into traditional linear solvers. We show that this
integration leads to the development of new solvers whose performance is on par
with other high-performance solvers, such as PETSc conjugate gradient linear
solvers, in terms of performance and accuracy. Overall, while the accuracy and
computational performance are still a limiting factor for the direct use of
PINN linear solvers, hybrid strategies combining old traditional linear solver
approaches with new emerging deep-learning techniques are among the most
promising methods for developing a new class of linear solvers.

本篇论文探讨了 PINN 作为线性求解器的潜力，在 Poisson 方程中评估了不同参数下的表现和精度，并且阐述了传递学习的关键作用。同时，论文也提出了将 PINN 与传统线性求解器相结合的方法，以解决高频解的问题，并表明该混合策略在发展具有潜力的新型线性求解器方面具有前景。

传统线性求解器能否被基于物理启发的深度学习所替代？

The Old and the New: Can Physics-Informed Deep-Learning Replace  Traditional Linear Solvers?

MFEM is an open-source, lightweight, flexible and scalable C++ library for
modular finite element methods that features arbitrary high-order finite
element meshes and spaces, support for a wide variety of discretization
approaches and emphasis on usability, portability, and high-performance
computing efficiency. MFEM's goal is to provide application scientists with
access to cutting-edge algorithms for high-order finite element meshing,
discretizations and linear solvers, while enabling researchers to quickly and
easily develop and test new algorithms in very general, fully unstructured,
high-order, parallel and GPU-accelerated settings. In this paper we describe
the underlying algorithms and finite element abstractions provided by MFEM,
discuss the software implementation, and illustrate various applications of the
library.

MFEM 是一款开源的、轻量级的、灵活而可扩展的 C++ 库，提供高阶有限元法网格、离散化和线性求解器算法等。