Convergence Is Not Accuracy: Transmission Conditions and Neural Training in FEM–PINN Coupling

Hybrid solvers can compute one region of a domain with the finite-element method (FEM) and a neighboring region with a physics-informed neural network (PINN). The two solvers exchange boundary values and fluxes across their shared interface until the exchange settles. We show that when the PINN is inexact, the exchange can settle reliably on the wrong answer. For the classical Robin–Neumann exchange we prove that the Robin weight decides whether and how fast the exchange settles, while the answer it settles on is fixed by the PINN's interface response: for a fixed learned solver, every convergent Robin weight produces the same biased solution, with a closed-form error. Experiments on two-dimensional diffusion problems confirm this prediction to within $3\times 10^{-10}$. Couplings that pass the same stopping test differ by more than an order of magnitude in error, and enlarging the learned solver's feature space removes most of the gap. Classical solves stopped at a loose tolerance show the same premature stopping. Frequency probes trace the PINN's response error to both its features and its finite training budget, and the budget also changes how well the training loss tracks the coupled error. Hybrid couplings should therefore report the exchanged response and the physical error, not only the iteration progress.