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Physics-Informed Neural Networks (PINN) are neural networks encoding theproblem governing equations, such as Partial Differential Equations (PDE), as apart of the neural network. PINNs have emerged as a new essential tool to solvevarious challenging problems, including computing linear systems arising fromPDEs, a task for which several traditional methods exist. In this work, wefocus first on evaluating the potential of PINNs as linear solvers in the caseof the Poisson equation, an omnipresent equation in scientific computing. Wecharacterize PINN linear solvers in terms of accuracy and performance underdifferent network configurations (depth, activation functions, input data setdistribution). We highlight the critical role of transfer learning. Our resultsshow that low-frequency components of the solution converge quickly as aneffect of the F-principle. In contrast, an accurate solution of the highfrequencies requires an exceedingly long time. To address this limitation, wepropose integrating PINNs into traditional linear solvers. We show that thisintegration leads to the development of new solvers whose performance is on parwith other high-performance solvers, such as PETSc conjugate gradient linearsolvers, in terms of performance and accuracy. Overall, while the accuracy andcomputational performance are still a limiting factor for the direct use ofPINN linear solvers, hybrid strategies combining old traditional linear solverapproaches with new emerging deep-learning techniques are among the mostpromising methods for developing a new class of linear solvers.

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