  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.
