  Physics-Informed Neural Networks (PINN) are neural networks that encode the
problem governing equations, such as Partial Differential Equations (PDE), as a
part of the neural network training. PINNs have emerged as an essential tool to
solve various challenging problems, such as computing linear and non-linear
PDEs, completing data assimilation and uncertainty quantification tasks. In
this work, we focus on evaluating the PINN potential to replace or accelerate
traditional approaches for solving linear systems. We solve the Poisson
equation, one of the most critical and computational-intensive tasks in
scientific computing, with different source terms. We test and evaluate PINN
performance under different configurations (depth, activation functions, input
data set distribution, and transfer learning impact). We show how to integrate
PINN with traditional scientific computing approaches, such as multigrid and
Gauss-Seidel methods. While the accuracy and computational performance is still
a limiting factor for the direct use of PINN for solving, hybrid strategies are
a viable option for the development of a new class of linear solvers combining
emerging deep-learning and traditional scientific computing approaches.
